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frameworks struggle to explain all the dynamics observed. ### Dropouts -![Time profiles of lowenergy He ion intensities recorded by the Wind/LEMT sensor showing a gradual SEP event beginning on 1997 November 6. A dropout in ion intensity lasting about 2 hr can be seen during the decay phase of the gradual event. [@tanTurbulentOriginsParticle2023]](./figures/tanTurbulentOriginsParticle2023-fig1b.png) +![Time profiles of lowenergy He ion intensities recorded by the Wind/LEMT sensor showing a gradual SEP event beginning on 1997 November 6. A dropout in ion intensity lasting about 2 hr can be seen during the decay phase of the gradual event. [@tanTurbulentOriginsParticle2023]](./figures/ref/tanTurbulentOriginsParticle2023-fig1b.png) -![dropout period having $θ_{BV} \sim 0°$](./figures/tanTurbulentOriginsParticle2023-fig4.png) +![dropout period having $θ_{BV} \sim 0°$](./figures/ref/tanTurbulentOriginsParticle2023-fig4.png) ### Reservoir A region where the intensities and energy spectra throughout much of the inner heliosphere (see Fig. 52: top right panel) at different azimuthal, radial, and latitudinal locations are nearly identical -![Intensity-time profiles for protons in the 1979 March 1 event at 3 spacecraft are shown in the upper left panel with times of shock passage indicated by S for each. Energy spectra in the “reservoir” at time R are shown in the upper right panel while the paths of the spacecraft through a sketch of the CME are shown below [@reamesTwoSourcesSolar2013]](figures/reamesTwoSourcesSolar2013-fig6.png) +![Intensity-time profiles for protons in the 1979 March 1 event at 3 spacecraft are shown in the upper left panel with times of shock passage indicated by S for each. Energy spectra in the “reservoir” at time R are shown in the upper right panel while the paths of the spacecraft through a sketch of the CME are shown below [@reamesTwoSourcesSolar2013]](figures/ref/reamesTwoSourcesSolar2013-fig6.png) Effective cross-field and non-diffusive transport [@larioHeliosphericEnergeticParticle2010] @@ -67,7 +67,7 @@ $$ ### Turbulent Magnetic Fluctuations -![PDF of the out-of-plane electric current density $J_z$ from a 2D MHD simulation, compared to a reference Gaussian (standard deviation $σ$). For each region I, II, and III, magnetic field lines (contours of constant magnetic potential $A_z$: > 0 solid, < 0 dashed) are shown; the colored (red) regions are places where the selected band (I, II, or III) contributes. [@grecoPartialVarianceIncrements2017]](./figures/grecoPartialVarianceIncrements2017-fig1.png) +![PDF of the out-of-plane electric current density $J_z$ from a 2D MHD simulation, compared to a reference Gaussian (standard deviation $σ$). For each region I, II, and III, magnetic field lines (contours of constant magnetic potential $A_z$: > 0 solid, < 0 dashed) are shown; the colored (red) regions are places where the selected band (I, II, or III) contributes. [@grecoPartialVarianceIncrements2017]](./figures/ref/grecoPartialVarianceIncrements2017-fig1.png) ::: {.notes} A physically appealing interpretation emerges: region I consists of very low values of fluctuations that lie mainly in the lanes between magnetic islands. Region II consists of sub-Gaussian current cores that populate the central regions of the magnetic islands (or flux tubes). Region III is composed of the coherent small-scale current sheet-like structures that form the sharp boundaries between the magnetic flux tubes. This classification provides a real-space picture of the nature of intermittent MHD turbulence. diff --git a/docs/others/phd/2025_atc/thesis_outline.qmd b/docs/others/phd/2025_atc/thesis_outline.qmd index 24d7d62..7236098 100644 --- a/docs/others/phd/2025_atc/thesis_outline.qmd +++ b/docs/others/phd/2025_atc/thesis_outline.qmd @@ -46,7 +46,7 @@ SEPs are primarily accelerated through two distinct mechanisms [@reamesTwoSource Gradual SEP events typically last for several days and are predominantly proton-rich, often associated with fast CMEs driving shocks in the solar corona and interplanetary space. These shocks accelerate particles over extended regions, producing widespread and intense radiation storms. In contrast, impulsive SEP events are related to short duration (less than 1 h) solar flares. These events typically have shorter durations, lasting from minutes to a few hours, and feature characteristically higher electron-to-proton ratios and enrichments of heavy ions (${ }^3 \mathrm{He} /{ }^4 \mathrm{He}$ and $\mathrm{Fe} / \mathrm{O}$ ratios). -![The two-class picture for SEP events. @desaiLargeGradualSolar2016](figures/desaiLargeGradualSolar2016-fig3.png) +![The two-class picture for SEP events. @desaiLargeGradualSolar2016](figures/ref/desaiLargeGradualSolar2016-fig3.png) In the decay phase of large gradual SEP events, a characteristic phenomenon known as the **reservoir effect** frequently occurs, where particle intensities measured by widely separated spacecraft become nearly uniform across large regions and exhibit similar temporal evolutions. One traditional explanation for reservoir formation suggests that particles become trapped behind a CME-driven magnetic structure, resulting in spatially uniform spectra that adiabatically decrease in intensity as the confining magnetic bottle expands. However, high heliolatitude observations from the Ulysses mission revealed the three-dimensional character of the reservoir effect and favor the cross-field diffusion explanation [@larioHeliosphericEnergeticParticle2010; @dallaPropertiesHighHeliolatitude2003]. diff --git a/docs/others/phd/2026_grad/.gitignore b/docs/others/phd/2026_grad/.gitignore new file mode 100644 index 0000000..ea16841 --- /dev/null +++ b/docs/others/phd/2026_grad/.gitignore @@ -0,0 +1,2 @@ +typstapp +uclathesis \ No newline at end of file diff --git a/docs/others/phd/2026_grad/_acknowledgments.qmd b/docs/others/phd/2026_grad/_acknowledgments.qmd new file mode 100644 index 0000000..4372bb6 --- /dev/null +++ b/docs/others/phd/2026_grad/_acknowledgments.qmd @@ -0,0 +1 @@ +# Acknowledgments diff --git a/docs/others/phd/2026_grad/_intro.qmd b/docs/others/phd/2026_grad/_intro.qmd new file mode 100644 index 0000000..0dda8fa --- /dev/null +++ b/docs/others/phd/2026_grad/_intro.qmd @@ -0,0 +1,35 @@ +# Introduction + +The solar wind is a hot, magnetized plasma that continuously expands from the solar corona into interplanetary space, driven by the pressure gradient between the dense coronal plasma and the surrounding interstellar medium [@parkerDynamicsInterplanetaryGas1958; @parkerDynamicalTheorySolar1965; @velliSupersonicWindsAccretion1994; @velliHydrodynamicsSolarWind2001]. As it propagates outward, the solar wind accelerates to super-Alfvénic speeds, transporting mass, momentum, and energy throughout the heliosphere. + +Because the solar wind is highly electrically conductive, magnetic field lines are frozen into the plasma flow to a good approximation. The resulting large-scale interplanetary magnetic field (IMF) is shaped by the combination of radial expansion and solar rotation, producing the Archimedean spiral geometry described by the Parker model—a configuration well supported by observations when averaged over sufficiently long intervals [@zhaoStatisticsInterplanetaryMagnetic2025; @zhaoStatisticsInterplanetaryMagnetic2025a]. + +Superposed on this ordered background, however, are substantial fluctuations spanning a broad range of scales. These fluctuations are a manifestation of MHD turbulence [@schekochihinMHDTurbulenceBiased2022; @matthaeusTurbulenceSpacePlasmas2021; @brunoTurbulenceSolarWind2016; @brunoSolarWindTurbulence2013; @tuMHDStructuresWaves1995; @goldsteinMagnetohydrodynamicTurbulenceSolar1995], which is multiscale, intermittent, and strongly non-Gaussian [@chandranIntermittentReflectiondrivenStrong2025; @brunoIntermittencySolarWind2019; @verscharenMultiscaleNatureSolar2019; @matthaeusIntermittencyNonlinearDynamics2015]. A key feature of this turbulence is that energy does not distribute uniformly across space as it cascades from large to small scales. Instead, it concentrates into localized, coherent structures—magnetic flux ropes, magnetic holes, plasma waves, and, most prominently, current sheets [@perroneCoherentEventsIon2020; @khabarovaCurrentSheetsPlasmoids2021; @pezziCurrentSheetsPlasmoids2021]. These structures are not passive byproducts of the cascade; they feed back into the ambient plasma and magnetic environment, contributing significantly to plasma heating, particle acceleration, and departures from classical MHD behavior [@degiorgioCoherentStructureFormation2017; @borovskyContributionStrongDiscontinuities2010; @liEffectCurrentSheets2011; @grecoPartialVarianceIncrements2017]. + +The heliospheric magnetic field thus exhibits a dual character: while the Parker spiral describes the mean magnetic topology of the heliosphere, a complete physical picture of the solar wind requires explicit consideration of turbulence and embedded coherent structures. This multiscale structure has important consequences for a range of heliophysical processes, particularly the transport and acceleration of energetic particles. + +Solar energetic particles (SEPs) are high-energy ions and electrons episodically accelerated in the solar and interplanetary environment [@anastasiadisSolarEnergeticParticles2019; @kleinAccelerationPropagationSolar2017; @desaiLargeGradualSolar2016; @reamesTwoSourcesSolar2013]. They originate either in the low solar corona during eruptive events, or at interplanetary shock fronts driven by fast coronal mass ejections, where diffusive shock acceleration can energize particles over extended spatial and temporal scales. SEP events pose radiation hazards to spacecraft electronics, astronauts, and high-altitude aviation, and they provide a natural laboratory for studying particle acceleration and transport in magnetized plasmas—with implications extending to astrophysical systems more broadly. + +Early theoretical descriptions of SEP propagation treated the interplanetary medium as a smooth background threaded by broadband turbulent waves, within which particles scatter quasi-linearly. However, growing observational and numerical evidence demonstrates that the fine-scale magnetic structure of the solar wind can exert a decisive influence on particle dynamics [@malaraEnergeticParticleDynamics2023; @malaraChargedparticleChaoticDynamics2021; @artemyevSuperfastIonScattering2020]. When a particle's gyroradius becomes comparable to the thickness of a current sheet, classical guiding-center theory breaks down: the particle undergoes a non-adiabatic interaction that can produce large, abrupt changes in pitch angle. Because current sheets are abundant throughout the heliosphere and their thickness overlaps with the gyroradii of suprathermal and energetic ions, such interactions are not rare events but a systematic feature of SEP propagation. They can enhance pitch-angle scattering, induce trapping or reflection, enable localized acceleration, and produce non-diffusive transport that cannot be captured by models based solely on homogeneous wave turbulence. + +Despite their importance, a quantitative understanding of how current sheets influence SEP transport has remained incomplete, for two reasons. First, the statistical properties of kinetic-scale current sheets—their occurrence rate, thickness distribution, and internal magnetic configuration—have not been systematically characterized across different regions of the heliosphere. Second, the connection between these microphysical structural properties and macroscopic transport coefficients has not been rigorously established. This dissertation addresses both gaps. + +Specifically, this dissertation (1) develops a coherent observational framework to characterize the properties and occurrence rates of current sheets from kinetic to large scales across different heliospheric regions, and (2) constructs a quantitative, statistics-based model of SEP interactions with current sheets that describes pitch-angle scattering induced by these structures and estimates the resulting spatial transport coefficients. Together, these contributions provide a bridge between the microphysics of individual current sheet encounters and the macroscopic diffusion of energetic particles through the heliosphere, offering new insight into heliospheric particle dynamics and a foundation for improved space weather prediction. + +## Thesis Organization + + + +This thesis is organized into four parts, progressing from background reviews through observational characterization of current sheets, to their impact on energetic particle transport, and finally to the methodological developments that support and extend the primary scientific results. + +**Part I** (Chapters 2–3) provides the observational and theoretical background. *Chapter 2* reviews the observational properties of solar wind current sheets, including the identification and characterization methods, their kinetic nature, and the statistical characteristics reported in the literature. *Chapter 3* introduces the solar energetic particle context: the sources and classification of SEP events, the transport frameworks within which particle scattering is parameterized, and the motivation—rooted in solar wind intermittency—for going beyond classical quasilinear theory. This chapter develops the quasi-adiabatic theory of particle motion near magnetic field reversals and the mechanisms by which the quasi-adiabatic invariant is destroyed at separatrix crossings. + +**Part II** (Chapters 4–5) presents the observational characterization of kinetic-scale current sheets across the inner heliosphere from a coordinated set of spacecraft: Parker Solar Probe (PSP) at distances as close as 0.1 AU, Wind, ARTEMIS, and STEREO near 1 AU, and Juno during its cruise phase out to 5 AU. These chapters systematically quantify the statistical distributions of current sheet thickness and current density—two parameters that directly determine the magnetic field configurations. Because the magnetic field geometry directly influences particle motion, these structural properties serve as essential inputs for understanding particle scattering and transport processes. The observational results therefore provide the empirical foundation for the particle-transport studies that follow. + +**Part III** (Chapters 6–7) investigates how kinetic-scale current sheets scatter and transport energetic particles. *Chapter 6* quantifies pitch-angle scattering through test-particle simulations and statistical analysis, establishing how specific current sheet configurations produce non-adiabatic behavior and how an ensemble of current sheets contributes to long-term pitch-angle diffusion. *Chapter 7* extends this analysis to spatial transport, deriving parallel and perpendicular diffusion coefficients from ensembles of particles interacting with statistically representative current sheet populations. Together, these chapters bridge microscopic scattering physics between local structure physics and macroscopic heliospheric transport. + +**Part IV** (Chapters 8–9) presents two complementary developments. *Chapter 8* introduces a multi-fluid current sheet model developed to interpret plasma velocity structure and magnetic field configuration within a current sheet. It is constructed to allow non-zero normal magnetic field components while maintaining constant magnetic field magnitude, thereby extending beyond simplified one-dimensional MHD configurations. *Chapter 9* describes the Julia-based software ecosystem designed to support the large-scale statistical and computational analyses performed throughout this work. This computational framework allows interactive scientific exploration while maintaining near-compiled performance, making large ensemble studies computationally feasible. + +*Chapter 10* summarizes the principal findings and discusses open questions and future directions, with emphasis on the origins of solar wind current sheets and on strategies for incorporating intermittency effects into operational SEP transport models. \ No newline at end of file diff --git a/docs/others/phd/2026_grad/_juno.qmd b/docs/others/phd/2026_grad/_juno.qmd new file mode 100644 index 0000000..1aba4e1 --- /dev/null +++ b/docs/others/phd/2026_grad/_juno.qmd @@ -0,0 +1,210 @@ +# Solar wind discontinuities in the outer heliosphere: spatial distribution between 1 and 5 AU + +## Motivation + +Solar wind discontinuities (SWDs), commonly defined as localized and abrupt changes in the magnetic field [@colburnDiscontinuitiesSolarWind1966], are mesoscale plasma structures with typical thicknesses on the order of ion kinetic scales and horizontal extensions that often exceed magnetohydrodynamic (MHD) scales. They are observed throughout the heliosphere, from the inner heliosphere [@liuCategorizingMHDDiscontinuities2022; @lotekarKineticscaleCurrentSheets2022] to the heliosheath [@burlagaCurrentSheetsHeliosheath2011]. Despite being discovered over half a century ago, their specific origins and the circumstances of their formation remain subjects of ongoing debate [see discussion in @vaskoKineticscaleCurrentSheets2022; @vaskoKineticScaleCurrentSheets2024, and references therein]. +Historically, SWDs were thought to originate inside 1 AU, probably near or at the Sun, and are then passively convected outward by the solar wind [@tsurutaniInterplanetaryDiscontinuitiesTemporal1979; @sodingRadialLatitudinalDependencies2001; @borovskyFluxTubeTexture2008]. However, the good agreement observed in the waiting-time analysis between solar wind data and MHD turbulence simulations [@grecoStatisticalAnalysisDiscontinuities2009] indicates that a significant portion of these current sheet-like structures constitute the sharp boundaries between the magnetic flux tubes, intermittent structures arising spontaneously and forming rapidly and locally within MHD turbulence. +The annihilation mechanisms for these discontinuities are also uncertain. They may be unstable and some could rapidly wane or decay [@neugebauerTangentialDiscontinuitiesSolar1986]. Magnetic reconnection may be a a natural process that destroys them [e.g., @goslingMagneticReconnectionSolar2012; @wilsonParticleincellSimulationsCollisionless2016; @phanParkerSolarProbe2020]. +Further complicating this picture is the presence of different types of discontinuities in interplanetary space - specifically tangential and rotational discontinuities [see discussion in @neugebauerCommentAbundancesRotational2006; @artemyevKineticPropertiesSolar2019; @wangSolarWindCurrent2024], which are classically regarded as static solutions to the MHD equations [@hudsonDiscontinuitiesAnisotropicPlasma1970]. These distinct types may arise from different sources and may undergo unique evolutionary paths. Investigating discontinuities across large radial distances has the potential to address these fundamental questions about their origin and evolution. + +Discontinuities have long been recognized as key signatures of intermittency in solar wind turbulence, where they contribute to the non-Gaussian character of magnetic field fluctuations [@borovskyContributionStrongDiscontinuities2010; @grecoPartialVarianceIncrements2017]. Often localized at kinetic scales, these discontinuities are central to understanding solar wind heating and how energy cascades from larger MHD scales down to ion and electron scales [@osmanIntermittencyLocalHeating2012; @tesseinAssociationSuprathermalParticles2013]. In particular, @tesseinAssociationSuprathermalParticles2013 used ACE data to show that intense magnetic discontinuities are more strongly correlated with suprathermal particles than interplanetary shocks, indicating that multiple mechanisms, including localized coherent structures, can accelerate these suprathermals. Likewise, @osmanIntermittencyLocalHeating2012 found that although coherent structures constitute only 19\% of the data, they contribute around 50\% of the total plasma internal energy, underscoring the importance of intermittent heating in current sheets. Numerical simulations further reinforce these observations: @dmitrukTestParticleEnergization2004 demonstrated that electrons tend to be energized along discontinuities by parallel electric fields, whereas protons gain perpendicular energy from large-scale velocity shears, offering a multiscale picture of how discontinuities drive local heating. Observations also suggest that magnetic reconnection in discontinuities (effectively 1D current sheets) is associated with localized plasma heating [@goslingMagneticReconnectionSolar2012]. From a theoretical standpoint, numerous studies and MHD simulations have linked the formation and destruction of discontinuities to the nonlinear dynamics of Alfv\'en waves and Alfv\'enic turbulence [@lerchePropagationMagneticDisturbances1975; @medvedevDissipativeDynamicsCollisionless1997; @grecoIntermittentMHDStructures2008; @yangFormationRotationalDiscontinuities2015], producing strong departures from the otherwise adiabatic evolution of solar wind flow [@matteiniIonKineticsSolar2012; @tsurutaniReviewAlfvenicTurbulence2018]. Indeed, spacecraft observations reveal that the solar wind's magnetic field follows the Parker model only on average [@svirzhevskyHeliosphericMagneticField2021], whereas localized current sheets, often far more intense than Parker theory predicts, are ubiquitous [@colburnDiscontinuitiesSolarWind1966; @burlagaMicroscaleStructuresInterplanetary1968; @turnerOrientationsRotationalTangential1971]. Consequently, understanding how the thickness, current density, and other properties of SWDs evolve with radial distance from the Sun is crucial for revealing their role in solar wind thermodynamics and turbulence. Addressing whether these discontinuities maintain their kinetic-scale character (e.g., remain a few ion inertial lengths wide) and whether their current density weakens in tandem with the radial drop in magnetic field can help constrain theories of their local generation, annihilation, and their overall impact on energy dissipation throughout the heliosphere. + +Investigation of the evolution of SWD properties with radial distance (or alternatively, investigation of properties of SWD generation at different radial distances) has mostly been limited to comparison of SWD statistics obtained from different missions during varying phases of solar activity [e.g., @marianiVariationsOccurrenceRate1973; @tsurutaniInterplanetaryDiscontinuitiesTemporal1979; @sodingRadialLatitudinalDependencies2001]. However, the data used in these studies were rarely compared with measurements at fixed radial distances, and the methods employed to identify discontinuities were not necessarily optimized for studying discontinuities with weak magnetic fields and long durations, such as those encountered at large radial distances. Furthermore, some measurements at different radial distances were significantly separated in time, with limited time ranges covered [@sodingRadialLatitudinalDependencies2001]. Consequently, there are still some ambiguities in interpreting the results of these studies. For example, it is unclear how much of the observed variation in SWD properties is caused by radial distance itself versus changing solar wind conditions. +This study presents a statistical analysis of discontinuities observed from Earth's orbit ($\sim 1$ AU) to Jupiter's orbit ($\sim 5$ AU). We utilize data collected by the Juno spacecraft [@boltonJunoMission2017] during its cruise phases (2011-2016), with its orbits shown in Panel (a) of @fig-overview. Our primary goal is to investigate the radial dependence of the SWD occurrence rate and properties to gain insights into their evolution and origin. Continuous measurements of SWDs at 1 AU from missions such as Wind [@acunaGlobalGeospaceScience1995], ARTEMIS [@angelopoulosARTEMISMission2011], and STEREO [@kaiserSTEREOMissionIntroduction2008] have been utilized to distinguish temporal effects from spatial variations by examining the characteristics of discontinuities at two radial distances ($1$ AU and at Juno location) simultaneously. This distinction is crucial because temporal variations could easily be misinterpreted as spatial variations [see discussion in @tsurutaniInterplanetaryDiscontinuitiesTemporal1979]. Due to the potential misidentification of the SWD normal direction with single spacecraft [see discussion in @hausmanDeterminingNatureOrientation2004; @liuFailuresMinimumVariance2023; @wangSolarWindCurrent2024], we do not differentiate between shocks and different types of discontinuities, whether they are tangential, rotational, or either [@neugebauerReexaminationRotationalTangential1984]. + +The paper is structured as follows. First, we outline the missions, instruments, and data utilized in this study (Section @sec-dataset). Next, we briefly describe the method used to identify discontinuities and the model employed to estimate the plasma state at the Juno location, necessitated by the absence of plasma data during its cruise phase (Section @sec-methods). In Section @sec-results, we present the results concerning the spatial variation of discontinuities, focusing on their occurrence rate and characteristics, such as their spatial scales and current density. Finally in Section @sec-discussion, we discuss the implications of our findings in the context of the origin and evolution of these discontinuities in the solar wind. + + +## Dataset, models, and methods +\label{dataset-models-and-methods} + +@fig-overview provides an overview of Juno's cruise phase (2011{\textendash}2016) and the corresponding solar wind conditions. The solar wind plasma state, as evidenced by the sunspot number (shown in Panel (c)), plays a crucial role in understanding the dynamics of discontinuities. Early in the Juno mission [@boltonJunoMission2017], the sunspot numbers reached a peak, indicating a period of increased solar activity. By 2016, however, they had declined significantly. This variation underscores the need to account for temporal, solar-cycle-related variability when calibrating and interpreting discontinuity properties. + +::: {#fig-overview} +![](figures/juno/fig_overview.pdf) + +Overview. **(a)** Juno's orbit during its cruise phase (2011-2016). **(b)** Absolute difference in heliographic longitude between Juno and Earth (blue) and between Juno and Stereo-A (STA, green). **(c)** Monthly and smoothed sunspot numbers. **(d-g)** Solar wind plasma density and speed from Near-Earth (OMNI) and STEREO-A missions. +::: + + +@fig-overview (b) shows the heliographic longitudinal difference between Juno and Earth, and between Juno and STEREO-A. Notably, when there is a substantial longitudinal difference between Juno and Earth, the difference between Juno and STEREO-A tends to be minimal, and vice versa. This complementarity allows us to compare SWD behavior at $1$ AU and further outward at Juno with minimal absolute longitudinal separations $<120 \degree$, alternating between Earth and STEREO-A as the SWD observational platform at $1$ AU. + + + +### Dataset +\label{sec-dataset} + +We analyze Juno's observations during its cruise phase from August 25, 2011, to July 30, 2016, spanning approximately 1,800 days. Data from missions such as STEREO, ARTEMIS (THEMIS-B/C), and Wind complement Juno's observations, providing a broader perspective on heliospheric conditions and enabling multi-point analysis of discontinuity properties [@velliUnderstandingOriginsHeliosphere2020]. +Specifically, we utilize magnetic field measurements from Juno's Fluxgate Magnetometer instrument [@boltonJunoMission2017; @connerneyJunoMagneticField2017] and solar wind parameters from the ion sensor of the Jovian Auroral Distributions Experiment (JADE) [@mccomasJovianAuroralDistributions2017]. For ARTEMIS [@angelopoulosARTEMISMission2011], we use magnetic field data from its Fluxgate Magnetometer instrument [@austerTHEMISFluxgateMagnetometer2008] and ion bulk velocity and plasma density from the Electrostatic Analyzer instrument [@mcfaddenTHEMISESAPlasma2009]. For STEREO [@kaiserSTEREOMissionIntroduction2008], we incorporate magnetic fields from the magnetic field experiment on IMPACT [@acunaSTEREOIMPACTMagnetic2008; @luhmannSTEREOIMPACTInvestigation2008] and ion bulk velocity and plasma density from PLASTIC [@galvinPlasmaSuprathermalIon2008]. For Wind [@acunaGlobalGeospaceScience1995], we include magnetic fields from the Magnetic Field Investigation instrument [@leppingWINDMagneticField1995], ion bulk velocity, and plasma density from the Solar Wind Experiment instrument [@ogilvieSWEComprehensivePlasma1995]. +The time resolution of magnetic field and plasma data varies across the missions analyzed in this study. We use magnetic field data from Juno at 1 Sample/S (S/s), ARTEMIS at 5 S/s, Wind at 11 S/s, and STEREO at 8 S/s. We use plasma data from ARTEMIS at 0.25 S/s, Wind at 0.33 S/s, and STEREO at 1-minute intervals. To ensure the robustness of our results, we employ methods (to be introduced in the next section) that are specifically designed to minimize the influence of time resolution on the identification of current sheets and the calculation of their properties. These synergistic multi-satellite observations provide a comprehensive perspective on solar wind discontinuities, enabling the analysis of their radial distribution and temporal evolution. + +Since in-situ measurements from JADE are only available for the final 40 days of its cruise phase prior to Jupiter arrival [@wilsonSolarWindProperties2018], we rely on solar wind propagation models to estimate the thickness and current density of discontinuities for the remainder of the cruise. A direct comparison between the modeled solar wind properties and JADE observations during the overlapping interval is presented in Panels (a-d) of @fig-juno_sw_comparison in the Appendix. Specifically, we employ the Two-Dimensional Outer Heliosphere Solar Wind Modeling (MSWIM2D) [@keeblerMSWIM2DTwodimensionalOuter2022] to determine the ion bulk velocity ($v$) and plasma density ($n$) at Juno's location. This model, which utilizes the BATSRUS MHD solver[@tothAdaptiveNumericalAlgorithms2012], simulates the propagation of the solar wind from 1 to 75 astronomical units (AU) in the ecliptic plane, effectively covering the region pertinent to our study. The MSWIM2D model provides output data with an hourly time resolution, as shown in @fig-model. After averaging Juno data to the same time resolution, comparing magnetic field magnitudes from MSWIM2D with those of Juno reveals a strong correlation, confirming the model's applicability to our study. + +::: {#fig-model} +![](figures/juno/juno_model_validation_full.pdf) + +**(a)** Magnetic field magnitude from MSWIM2D and Juno. **(b-c)** Plasma speed and density from MSWIM2D model. **(d)** Juno radial distance from the Sun. +::: + + + +### Methods +\label{sec-methods} + +We employ the method of @liuMagneticDiscontinuitiesSolar2022 to identify discontinuities in the solar wind. This method is better suited for discontinuities with small magnetic field changes and is more robust for conditions in the outer heliosphere compared to traditional methods, namely the $B$-criterion [@burlagaDirectionalDiscontinuitiesInterplanetary1969] and the $TS$-criterion [@tsurutaniInterplanetaryDiscontinuitiesTemporal1979]. For each sampling time $t$, we define three intervals: the pre-interval $[-1,-1/2]\cdot T+t$, the middle interval $[-1/2,1/2]\cdot T+t$, and the post-interval $[1/2,1]\cdot T+t$, where $T$ represents time lags. The magnetic field time series in these three intervals are labeled as ${\mathbf B}_-$, ${\mathbf B}_0$, and ${\mathbf B}_+$, respectively. We first apply the following three detection criteria to establish the time interval that encompasses each discontinuity candidate: +$$ +\sigma({\mathbf B}_0) &> 2\max\left(\sigma({\mathbf B}_-), \sigma({\mathbf B}_+)\right) \\ +\sigma\left({\mathbf B}_-+{\mathbf B}_+\right) &>\sigma({\mathbf B}_-)+\sigma({\mathbf B}_+) \\ +|\Delta {\mathbf B}| &>|{\mathbf B}_{bg}|/10 +$$ + +Here, $\sigma$ denotes the standard deviation, ${\mathbf B}_{bg}$ represents the background magnetic field magnitude, and $\Delta {\mathbf B}={\mathbf B}(t+T/2)-{\mathbf B}(t-T/2)$. The first two conditions ensure that the field changes are distinguishable from stochastic fluctuations, while the third condition serves as a supplementary measure to reduce recognition uncertainty [see discussion in @liuMagneticDiscontinuitiesSolar2022]. + +Notably, this method shares certain similarities with the Partial Variance of Increments (PVI) approach [@grecoPartialVarianceIncrements2017] for identifying coherent structures, since both calculate indices based on standard deviations (with this method focusing on the standard deviation of the magnetic field while PVI focuses on magnetic field increments). However, the method of @liuMagneticDiscontinuitiesSolar2022 is better suited for discontinuity identification by comparing data within neighboring time windows rather than across a correlation scale (a quantity used in the PVI method that may vary with radial distance and solar wind conditions). As a result, this approach effectively excludes wave-like coherent structures, eliminating the need for manual inspection. Additionally, PVI generates discrete data points at each time step, which subsequently require an additional processing step{\textemdash}grouping consecutive points exceeding a threshold into continuous event clusters, thereby converting individual timestamps into contiguous time ranges. In contrast, our method inherently outputs the time ranges of interest without this additional step. + +Because discontinuities may exhibit different spatial scales at varying radial distances from the Sun, we conducted the identification procedure using multiple time lags ($T = 20, 30, 40, 50, 60$ s). The resulting events from each search were then combined into a comprehensive final list, removing any duplicates. The choice of a 20-second lower limit was driven by the typical sampling rate ($\sim$1 sample/s) of the Juno spacecraft. However, it should be noted that shorter-duration discontinuities may be underrepresented, particularly near 1 AU. This limitation becomes less significant as Juno travels to greater radial distances, where discontinuities generally exhibit longer durations. Discontinuities with spatial scales exceeding 60 seconds were not considered in this analysis. Nonetheless, such large-scale discontinuities are expected to be relatively rare in the outer heliosphere, as previous studies indicate a decreasing frequency of SWDs with increasing spatial scale [@vaskoKineticscaleCurrentSheets2022; @vaskoKineticScaleCurrentSheets2024]. + +More than 130,000 SWD intervals have been identified in the solar wind data from Juno, with approximately 17,000 intervals detected within [4, 5] AU. After identifying each discontinuity candidate, we construct a distance matrix using the magnetic field sequence [@dokmanicEuclideanDistanceMatrices2015] to determine the leading edge and trailing edge of the discontinuity within this preliminary interval. Specifically, we define the distance between two magnetic field vectors $\mathbf B(t_a)$ and $\mathbf B(t_b)$ as the Euclidean norm of their difference: $d(a,b) = \|\mathbf{B}(t_a) - \mathbf{B}(t_b)\|.$ We locate the leading edge at $t_a$ and the trailing edge at $t_b$ such that $t_a < t_b$ and $d(a,b)$ attains its maximum value within the time interval under consideration. After that, we use the minimum variance analysis (MVA) [@sonnerupMinimumMaximumVariance1998; @sonnerupMagnetopauseStructureAttitude1967] to transform the magnetic field into the boundary normal ($LMN$) coordinate system. Then, the maximum variance (main) component, $B_l$, is fitted with a hyperbolic tangent to extract parameters characterizing the discontinuity, inspired by the standard Harris current sheet model [@harrisPlasmaSheathSeparating1962]: +\begin{equation} +\label{eq-fit} +B_l(t) = B_{l,i} \tanh\left(\frac{t - t_i}{\ensuremath{\Delta} t_i}\right) + c_i +\end{equation} +where $B_{l,i}$ is the amplitude of the magnetic field change in the maximum variance direction, $t_i$ is the detection time, $\ensuremath{\Delta} t_i$ is the transit time, and $c_i$ is the offset of the magnetic field change for the $i$-th discontinuity event. While $c_i$ should theoretically be zero for an ideal rotational discontinuity, it may not necessarily be zero for tangential discontinuities or shocks. Nonetheless, including $c_i$ in data processing enhances the fitting accuracy. Finally, we combine the magnetic field data and plasma data (from the model) to obtain the thickness and current density of the discontinuity. + +Assuming that the SWD structures are planar, the normal direction can be obtained by minimizing the variance of the magnetic field{\textquoteright}s normal component in MVA. However, previous studies have indicated that the normal direction obtained through MVA is frequently unreliable [see discussion in @knetterFourpointDiscontinuityObservations2004; @wangSolarWindCurrent2024]. Improved accuracy can be achieved by either using the cross product of the magnetic field measured at the boundaries as the normal direction [@knetterFourpointDiscontinuityObservations2004] or by enforcing conditions such as $\ensuremath{\Delta}|{\bf B}|/|{\bf B}| > 0.05$ or $\ensuremath{\omega} > 60{\degree}$ [@liuFailuresMinimumVariance2023], where $\ensuremath{\Delta}|{\bf B}|$ is the change in the magnitude of the magnetic field and $\ensuremath{\omega}$ is the field rotation angle across the discontinuity. The former approach requires a small magnetic field component ($B_n$) along the discontinuity normal, while the latter approach limits the dataset to a subset that satisfies these conditions. These two methods yield closely aligned normal directions when $B_n$ is small and either $\ensuremath{\Delta}|{\bf B}|/|{\bf B}| > 0.05$ or $\ensuremath{\omega} > 60{\degree}$ is satisfied. In our analysis, we applied both techniques and found that they yielded statistically consistent results. A direct comparison of the discontinuity properties obtained from these two methods is presented in panels (e-j) of @fig-juno_sw_comparison in the Appendix, using intervals from Juno's approach to Jupiter when solar wind parameters were available. + +@fig-examples shows several examples of solar wind discontinuities detected by different spacecraft. Panels (a-c) show three examples of SWDs observed by Juno at 1, 3, and 5 AU distances, whereas Panels (d-f) show three examples of SWDs observed by Wind, ARTEMIS, and STEREO-A (for these three, we also show solar wind flow speed). All discontinuities share the same magnetic field configuration (in local $LMN$ coordinates): the main magnetic field component, $B_l$, reverses sign across the discontinuity, whereas the decrease of the magnetic pressure, $\sim B_l^2/2\mu_0$, is compensated by local enhancement of $B_m$ component. Such magnetic field rotation across the discontinuity keeps $|{\bf B}|$ almost constant [typical magnetic field configuration for SWDs, see @artemyevKineticNatureSolar2019; @vaskoKineticscaleCurrentSheets2022; @lotekarKineticscaleCurrentSheets2022; @vaskoKineticScaleCurrentSheets2024]. The solar wind flow speed (three panels for Wind, THEMIS-B, and STEREO-A) shows a typical jump across the discontinuities [see discussion in @artemyevIonNongyrotropySolar2020]. + +::: {#fig-examples} +![](figures/juno/fig_examples.pdf) + +Examples of solar wind discontinuities observed by various spacecraft at different heliocentric distances and times. Panels (a)–(c) display data from Juno at 1 AU (a), 3 AU (b), and 5 AU (c), showing magnetic field components $B_l$ (blue), $B_n$ (red), $B_m$ (green), and the magnetic field magnitude $B_{\text{total}}$ (black). Panels (d)–(f) show magnetic field and ion speed observations at 1 AU from other spacecraft: THEMIS-B (ARTEMIS) on March 27, 2012 (d), STEREO-A on June 28, 2016 (e), and Wind on August 26, 2011 (f). +::: + + + +## Results +\label{sec-results} + + +### Waiting time +\label{waiting-time} + +To fully understand the radial evolution of the SWD occurrence rate, particularly the average occurrence rate, it is essential to examine the statistical distributions underlying the observed SWD samples. In this regard, we derive the probability distribution of the waiting time $\ensuremath{\tau} = t_{i} - t_{i-1}$, where $t_i$ is the detection time for the $i$-th event from Equation @eq-fit, sorted in ascending order ($\ensuremath{\tau}$ is also referred to as the interarrival interval, e.g., by @tsurutaniInterplanetaryDiscontinuitiesTemporal1979). We employ the Weibull distribution [@weibullStatisticalDistributionFunction1951] for its flexibility in capturing varying occurrence rates. This choice allows for a more accurate representation and improved fit compared to the exponential distribution previously used by @tsurutaniInterplanetaryDiscontinuitiesTemporal1979. The probability density function of the Weibull distribution is given as follows: +\[ +f(\ensuremath{\tau}; \ensuremath{\alpha}, \ensuremath{\theta}) = \frac{\ensuremath{\alpha}}{\ensuremath{\theta}} \left( \frac{\ensuremath{\tau}}{\ensuremath{\theta}} \right)^{\ensuremath{\alpha}-1} e^{-(\ensuremath{\tau}/\ensuremath{\theta})^\ensuremath{\alpha}}, \quad \ensuremath{\tau} \ge 0 +\] +where $\ensuremath{\alpha}$ and $\ensuremath{\theta}$ represent the shape and scale parameters, respectively. The complementary cumulative distribution function of the Weibull distribution $F(\ensuremath{\tau}; \ensuremath{\alpha}, \ensuremath{\theta}) = 1 - \mathrm{e}^{-(\ensuremath{\tau}/\ensuremath{\theta})^\ensuremath{\alpha}}$ is a stretched exponential function (the Weibull distribution reduces to the exponential distribution when $\ensuremath{\alpha} = 1$). + +::: {#fig-waitingTime} +![](figures/juno/fig_wt_dist_no_duplicates.pdf) + +Waiting time probability density functions $p(\tau)$ for Juno at 1 AU in 2011 (top) and 5 AU in 2016 (bottom). Observed data (black) are fitted with Weibull (blue) and exponential (orange) distributions. Vertical dashed lines denote the mean waiting times for each fitted distribution. %In 2011, the Weibull distribution is characterized by parameters $\alpha = 0.99, \theta = 30.29$, and a mean waiting time of 30.41 minutes, while the exponential distribution has a mean of 30.42 minutes. In 2016, the Weibull distribution has parameters $\alpha = 0.65, \theta = 110.50$, and a mean of 150.96 minutes, whereas the exponential distribution has a mean of 165.02 minutes. +::: + + + +The observed waiting time distribution, along with the fitted distribution, is shown in @fig-waitingTime. For $\alpha < 1$, the probability of observing longer waiting times is higher than would be expected from a purely exponential distribution. This indicates that the occurrence rate of discontinuities is not constant, but rather varies with time{\textemdash}likely reflecting the evolving state of the solar wind (see the discussion in @tsurutaniInterplanetaryDiscontinuitiesTemporal1979). Across the largest range of waiting times, the observed probabilities exceed those modeled by the fitted distribution. This may be due to data gaps that artificially extend the observed waiting times between discontinuities. + +The mean waiting time, $\bar{\ensuremath{\tau}}$, can be determined by integrating the probability density function $\bar{\ensuremath{\tau}} = \int_0^\infty \ensuremath{\tau} f(\ensuremath{\tau}) d\ensuremath{\tau}$. The mean occurrence rate, in turn, is simply the reciprocal of the mean waiting time. At 1 AU, the occurrence rate is approximately 150 events per day, while at 5 AU, Juno observes around 30 events per day. These are consistent with previous results reported by @vaskoKineticscaleCurrentSheets2021; @vaskoKineticScaleCurrentSheets2024. Importantly, the mean value derived from the fitted Weibull distribution is more reliable, as it reduces the impact of unusually long waiting times often caused by data gaps. + + +### Occurrence rate +\label{occurrence-rate} + +The occurrence rate of SWDs (number of events per effective observation time) is investigated for Juno, ARTEMIS, STEREO, and Wind. @fig-rate shows the number of discontinuities detected per effective day for each mission, using the same identification criteria ($T = 60 s$). Because data are not continuously available on all satellites, the occurrence rate is corrected to account for data gaps. For ARTEMIS, an additional correction is needed because it only observes the pristine solar wind when the Moon is outside Earth's magnetosphere{\textemdash}roughly 30\% of its orbit is spent within the magnetosphere, and additional time can be spent in the lunar wake if apoapsis aligns with that region. After these corrections, ARTEMIS shows an occurrence rate comparable to Wind and STEREO for similar solar wind conditions. + +The SWD occurrence rate observed by Juno is consistent with the occurrence rate observed by 1-AU missions when Juno is around $1$ AU. This number decreases with distance as Juno moves from $1$ AU to $5$ AU (corresponding to time after $\sim 2014$; the radial distance of Juno for 2011-2016 is shown in @fig-model). + +The occurrence rate depends linearly on the solar wind velocity [@sodingRadialLatitudinalDependencies2001]; however, plasma velocity data from Juno are not available during its cruise phase. Therefore, we do not normalize the occurrence rate with respect to the solar wind speed. This influence of the solar wind speed on the occurrence rate might be mitigated by the fact that all spacecraft are in the ecliptic plane, and the solar wind speed is not expected to change significantly with radial distance from the Sun beyond 1 AU. We normalize the occurrence rate from Juno to the occurrence rate from 1-AU missions to account for the effect of solar wind structures on the occurrence rate. + +The normalized occurrence rate is shown in Panel (b) of @fig-rate. The occurrence rate of SWDs drops with the radial distance from the Sun, following a $1/r$ law. Spacecraft at $\sim$1 AU record a steady rate over the same interval, indicating that solar-wind temporal variability cannot explain this trend. Instead, the observed decrease in the occurrence rate is a result of the increasing radial distance from the Sun (likely due to the geometric spreading of discontinuities across expanding spherical shells, see discussion below) and confirms previous results [@sodingRadialLatitudinalDependencies2001]. + +::: {#fig-rate} +![](figures/juno/fig_occurence_rate.pdf) + +Left: the occurrence rate of discontinuities measured by Juno, STEREO-A, THEMIS-B, and Wind. Right: the normalized occurrence rate as a function of radial distance, where the radial distance of Juno for 2011-2016 is shown in @fig-model(d). +::: + + + +### Current density and thickness +\label{current-density-and-thickness} + +The current density $J_m$ associated with the main magnetic field reversal across a discontinuity is determined by +\begin{equation} +\label{eq-j}{ +J_m = - \frac{1}{\mu_0 V_n} \frac{d B_l}{d t}, +}\end{equation} +where $\mu_0$ is the vacuum permeability and $V_n$ is the normal component of the proton flow velocity. In the context of our fitting method, the derivative of the magnetic field can be estimated as $\max(d B_l / d t) = B_{l,i} / \Delta t$. This estimate is more robust than directly differentiating the magnetic field, because the fitting method is less sensitive to the noise in the data and the time resolution of the data. Since the normal direction from the MVA method can be unreliable, we also compute an alternative velocity, $V_{nm} = \sqrt{V^2 - V_l^2}$, as an upper limit for $V_n$. We check that using either the MVA-derived $V_n$ or this upper bound $V_{nm}$ to estimate $J_m$ and subsequent $d$ leads to very similar results. + +The thickness $d$ of a discontinuity is determined by $d = V_n \Delta t$, where $\Delta t$ is the transit time derived from the fitting method. + +To understand the relationship between the discontinuity and the local plasma, we normalize the discontinuity thickness by the ion (proton) inertial length, $d_i=c/\omega_{pi}$, and the current density by the Alfv\'en current density, $J_A = e N V_A = B / \mu_0 d_i$, both of which represent natural plasma scales. +The ion inertial length characterizes the scale at which ions decouple from the magnetic field, while the Alfv\'en current density corresponds to the characteristic current generated by the relative drift between electrons and ions moving at the local Alfv\'en speed $V_A$, particularly relevant in the context of MHD turbulence and reconnection studies [@zhdankinStatisticalAnalysisCurrent2013; @franciMagneticReconnectionDriver2017]. @fig-junoDistribution presents these quantities for the SWDs measured by Juno and grouped by their radial distance from the Sun. +Panels (a,c) show that the thickness increases with radial distance. However, after normalization, the thickness slightly decreases, with the most probable value around 2-4 $d_i$. With increasing radial distance, a larger fraction of SWDs exhibit thicknesses less than 1 $d_i$, transitioning into sub-proton scale structures. Panels (b,d) show that the current density decreases with increasing radial distance. However, after normalization, it slightly increases with radial distance, with the most probable value around 0.05-0.15 $J_A$. It should be noted that the variation of normalized thickness and current density with radial distance is not significant compared to the overall spread of their distributions [see also discussion in @vaskoKineticScaleCurrentSheets2024]. + +@fig-windDistribution presents the thickness and current density distributions of the SWDs observed by 1-AU satellites (Wind, ARTEMIS, and STEREO-A) during the five-year Juno cruise phase. They are grouped by year of observation. Panels (a,c) show that the distributions of the thickness and normalized thickness of SWDs remain almost constant with the year of observation. Most discontinuities have thicknesses around 350-3500 km at $1$ AU, with the most probable value around 1000 km. Panel (b) shows that the current density increases slightly with the year of observation, with the most probable value remaining around $5\;\text{nA}/\text{m}^{2}$, consistent with previous studies [@vaskoKineticscaleCurrentSheets2022]. However, the year-to-year variation is weak and lies well within the overall variance of the distribution. This modest trend may be associated with gradual changes in solar wind conditions over the solar cycle, as illustrated in @fig-swParameters. Toward solar minimum, the average magnetic field strength and the fitted magnetic field associated with discontinuities tend to increase slightly, but these changes are not statistically significant. Consequently, a weak upward trend in current density is observed. This pattern suggests that the apparent increase in current density and Alfv\'enic current density may reflect broader solar cycle variations while the normalized current density remains statistically stable, as shown in @fig-windDistribution(d). + + +::: {#fig-junoDistribution} +![](figures/juno/juno_distribution_r_sw.pdf) + +Distribution of various properties of SWDs observed by Juno, grouped by radial distance from the Sun (with color coding shown at the top). The label data indicates distributions calculated using solar wind properties from JADE observations rather than model predictions. Panels show: (a) discontinuity thickness, (b) current density, (c) normalized thickness, and (d) normalized current density. +::: + + +::: {#fig-windDistribution} +![](figures/juno/wind_distribution_time.pdf) + +Distribution of various properties of SWDs observed by 1-AU satellites (Wind, ARTEMIS and STEREO-A), grouped by the year of observation (with colors shown at the top). Panel (a) thickness, (b) current density, (c) normalized thickness, (d) normalized current density. +::: + + +::: {#fig-swParameters} +![](figures/juno/wind_sw_paramters.pdf) + +Solar wind parameters associated with the SWDs observed by 1-AU satellites (Wind, ARTEMIS and STEREO-A) grouped by the year of observation. Panel (a) solar wind density, (b), plasma beta, (c) magnetic field, (d) fitted magnetic field $B_{l,i}$ in Equation @eq-fit. +::: + + + +## Discussion +\label{sec-discussion} + + +### Spatial variations of discontinuity properties +\label{spatial-variations-of-discontinuity-properties} + +Continuous Juno observations of SWDs between 1 and 5 AU demonstrate a clear variation of their properties (thickness and current density magnitude) with radial distance. These results agree with and extend the previous statistics collected by different missions at different radial distances [@sodingRadialLatitudinalDependencies2001; @lotekarKineticscaleCurrentSheets2022; @liuCharacteristicsInterplanetaryDiscontinuities2021; @vaskoKineticscaleCurrentSheets2022; @vaskoKineticScaleCurrentSheets2024]. +The current sheet spatial scale (thickness) increases with radial distance $d\propto r$, and scales with the local ion inertial length $d_i \propto 1/\sqrt{N_p} \propto r$. Meanwhile, the current density decreases with radial distance, scaling with the local Alfv\'en current density $J_A=eN_pV_A\propto B\sqrt{N_p} \propto B/r$. Here, the magnitude of the local magnetic field $B$ evolves with radial distance in a manner similar to that of the interplanetary magnetic field, which generally follows $B\propto 1/r$. Consequently, the current density decays as $J \propto 1/r^2$, and this rate is consistent with the proportionality $J \propto B/d\propto 1/r^2$. The observed scaling relationship, $d\propto d_i$, can be interpreted as evidence supporting the local generation of SWDs. However, the same scaling is also expected for solutions of the kinetic nonlinear Schr{\"o}dinger equation describing SWD formation as a result of nonlinear dynamics of Alfv\'en waves [see @medvedevFluidModelsKinetic1996; @medvedevDissipativeDynamicsCollisionless1997]. Thus, if the adiabatic evolution of SWDs follows this equation, it is expected that the scaling $d\propto d_i$ would be observed for SWDs propagating with the solar wind. + + + +### Radial dependence of occurrence rate + +The observed increase in the average waiting time (time between two discontinuities) with radial distance prompts us to explore different hypotheses to explain this phenomenon. At any given radial distance, discontinuities may originate from two primary sources: those generated near the Sun and convected outward by the solar wind and those formed locally. + +First, we consider the limitations of our method for SWD detection. An increase in the duration (thickness) of discontinuities may render some unidentifiable if their duration becomes comparable to the time interval used in our analysis, $T$. Although we observe a slight increase in the average duration of discontinuities with radial distance, this increase is relatively small. The overall population of discontinuities is still predominantly characterized by those with short durations, typically less than $20$ seconds{\textemdash}which are effectively captured with $T = 20$ seconds (see @fig-TEffect in the Appendix). Although increasing $T$ beyond $60$ seconds may capture a few additional events, their number remains small. Importantly, our conclusions are not sensitive to a single $T$ value; instead, the critical factor is the broad coverage provided by the range of $T$. In addition, since our method normalizes the change in the magnetic field relative to the surrounding field, any overall drop in the magnetic field magnitude with radial distance does not itself remove potential discontinuities from detection. Thus, it is unlikely that our methodology causes the observed decrease in occurrence rate at larger radial distances. + +%Notably, our results differ from previous studies [@tsurutaniInterplanetaryDiscontinuitiesTemporal1979], which reported typical durations of 2 to 3 seconds at 1 AU and 10 to 30 seconds at 5 AU. In our analysis, however, the observed durations remain below 10 seconds at 5 AU, suggesting that our method may be more effective at identifying shorter-duration events than those reported before. + +If most discontinuities are generated locally, their generation mechanism should depend on plasma parameters, mainly plasma $\ensuremath{\beta}$ [see discussion in @chenIonscaleSpectralBreak2014; @franciPlasmaBetaDependence2016]. This dependence might potentially explain the variation of the discontinuity occurrence rate with radial distance. To test this hypothesis, we examine the relationship between waiting time $\ensuremath{\tau}$ and $\ensuremath{\beta}$. We use the Wind spacecraft at 1 AU and calculate the average plasma beta $\bar{\ensuremath{\beta}} = (\ensuremath{\beta}_{i} + \ensuremath{\beta}_{i-1})/2$, where $\ensuremath{\beta}_i$ is the plasma beta of the $i$-th discontinuity event. However, even at a 99\% statistical significance level, the results of our analysis do not provide sufficient evidence to reject the null hypothesis that these two variables $\ensuremath{\tau}, \bar{\ensuremath{\beta}}$ are independent. Therefore, if discontinuities are generated primarily locally, factors other than plasma beta must influence their generation rates. + +Alternatively, we consider the hypothesis that the majority of discontinuities are generated close to the Sun and then transported outward by the solar wind (since the solar wind speed is significantly higher than the Alfv\'en speed—which characterizes the propagation speed of rotational discontinuities—the fraction of inward-propagating to outward-propagating discontinuities is expected to be negligible). To estimate the total number of discontinuities in a spherical shell around the Sun at radius $r$, we account for the fact that a spacecraft observes only a subset of all existing structures{\textemdash}specifically, those whose projected surface area intersects its trajectory. Assuming that discontinuities are planar structures oriented at an angle $\ensuremath{\theta}_{n,v}$ with respect to the solar wind flow (an orientation shaped by the Archimedean Parker spiral), the effective surface area they present to the spacecraft is given by $A_{\rm eff} \sim w l \sin\ensuremath{\theta}_{n,v}$, where $w$ and $l$ are the width and length of the discontinuity. Under the assumption that both width and length scale with thickness $d$ [@zhdankinStatisticalAnalysisCurrent2013], this simplifies to $A_{\rm eff} \sim d^2 \sin\theta_{n,v}$. Normalizing this by the area of the spherical shell, $4\pi r^2$, gives the fractional coverage. Consequently, the global occurrence rate $f_{\rm global}$ is proportional to the local (spacecraft-measured) rate $f_{\rm local}$, scaled by the inverse of this fractional area: $f_{\rm global} \propto f_{\rm local}/(\bar{d}^2 \sin\bar{\ensuremath{\theta}}_{n,v} / r^2)$ where $\bar{d}$ and $\bar{\ensuremath{\theta}}_{n,v}$ are the average values of $d$ and $\ensuremath{\theta}_{n,v}$ at that radial distance. Accounting for this geometric effect significantly reduces the apparent discrepancy in global occurrence rates between 1 AU (e.g., Wind) and 5 AU (e.g., Juno){\textemdash}from $5:1$ to $1.3:1$. The residual difference likely results from additional physical processes, such as discontinuity annihilation through mechanisms like magnetic reconnection during solar wind propagation. Future modeling efforts of SWD generation and evolution should therefore incorporate both geometric and physical factors to fully capture the observed trends. + + +It is worth noting that our findings on the radial variation of the occurrence rate are qualitatively consistent with the trends observed in the inner heliosphere, for example $f_{\rm local} \propto 1/ r^2$ in @liuCharacteristicsInterplanetaryDiscontinuities2021 and $f_{\rm local} \propto 1/ r^{1.3}$ in @sodingRadialLatitudinalDependencies2001. While these scalings are steeper than that derived here, the difference may arise from several factors. First, the identification criteria and detection thresholds differ across studies, potentially influencing the inferred occurrence rates. Second, the dominant magnetic field component transitions from $B_r$ in the inner heliosphere (where $B_r \propto 1/r^2$) to $B_\phi$ in the outer heliosphere (where $B_\phi \propto 1/r$), leading to different geometric projection effects through $\sin \ensuremath{\theta}_{n,v}$. In addition, the different scaling with radial distance for rotational and tangential discontinuities reported in @liuCharacteristicsInterplanetaryDiscontinuities2021 may reflect differences in their orientation and its evolution with radial distance, which in turn affects the geometric factor and the resulting observed scaling. + +## Conclusion +\label{conclusion} + +We have collected 5 years of solar wind discontinuity data from Juno, Wind, THEMIS-B, and STEREO-A. The results of our analysis can be summarized as follows: + +- The occurrence rate (normalized to that at 1 AU) of SWDs decreases with radial distance from the Sun, following a $1/r$ law within $[1,5]$ AU. +- The thickness of SWDs increases with radial distance from the Sun; however, after normalization to ion inertial length, the thickness slightly decreases or remains almost constant within the 1-sigma spread of the distribution. The thickness distribution remains almost constant over a one-year observation period. +- The current density of SWDs decreases with radial distance from the Sun, but after normalization to the Alfv\'en current, it slightly increases or remains almost constant well within the 1-sigma spread of the distribution. The current density distribution slightly increases from 2011 to 2016, but after normalization, it remains identical across the five years, indicating that there is no effect of long-term (i.e., solar cycle) temporal variations on the normalized current density. + +These findings provide key observational constraints for future models of SWD generation, evolution, and dissipation. Our study advances previous research by systematically characterizing SWD properties across a broad range of radial distances, using a single spacecraft (Juno) and validating the results through consistent comparisons with multi-spacecraft observations at $1$ AU. By revealing consistent normalization behaviors (such as thickness and current density relative to ion inertial length and Alfv\'en current, respectively), we clarify the evolutionary trends of SWDs in the heliosphere. We conclude that the majority of SWDs likely originate close to the Sun and subsequently evolve dynamically as they propagate outward, with their persistence and structure strongly influenced by both geometric spreading and physical dissipation processes. These insights improve our understanding of the origins of SWD and their influence on the dynamics of solar wind, offering a robust framework for future theoretical and observational studies. + +![Comparison of solar wind properties (top) and discontinuity properties (bottom) between / using model (x-axis) and JADE observations (y-axis). (a-d) Solar wind velocity, density, temperature, and plasma beta. (e-h) Discontinuity thickness, current density, normalized thickness, and normalized current density. Blue dots indicate values derived using the cross-product normal method, while yellow dots correspond to values obtained using minimum variance analysis.](figures/fig_juno_sw_comparision.png){#fig-juno-sw-comparision} \ No newline at end of file diff --git a/docs/others/phd/2026_grad/_psp.qmd b/docs/others/phd/2026_grad/_psp.qmd new file mode 100644 index 0000000..a600d74 --- /dev/null +++ b/docs/others/phd/2026_grad/_psp.qmd @@ -0,0 +1,187 @@ +# Comparison of Solar Wind Current Sheets in the Inner Heliosphere + +## Introduction + +Current sheets are ubiquitous mesoscale structures in the solar wind, where the magnetic field changes direction abruptly over ion-scale thicknesses; whereas their lateral extents can exceed typical magnetohydrodynamic (MHD) scales [@zhdankinStatisticalAnalysisCurrent2013]. They are most readily identified by sharp magnetic-field rotations, but can also be evident in simultaneous changes of plasma density, bulk velocity, and temperature [@colburnDiscontinuitiesSolarWind1966; @sodingRadialLatitudinalDependencies2001; @shenComparingPlasmaAnisotropy2024a]. + +In turbulent solar wind plasmas, current sheets are widely recognized as signatures of intermittency, contributing to the non-Gaussian nature of magnetic field fluctuations and localized energy dissipation [@borovskyContributionStrongDiscontinuities2010; @grecoPartialVarianceIncrements2017]. They play a crucial role in mediating the transfer of energy across scales, from large-scale MHD fluctuations down to ion and electron kinetic scales, and are believed to be important sites of plasma heating and particle acceleration [@osmanIntermittencyLocalHeating2012; @tesseinAssociationSuprathermalParticles2013]. + +Understanding the formation, structure, and evolution of current sheets at different heliocentric distances is crucial for elucidating the mechanisms that govern solar wind turbulence and heating. Several statistical studies have investigated the properties of solar wind current sheets over a wide range of radial distances [@marianiStatisticalStudyMagnetohydrodynamic1983; @artemyevDynamicsIntenseCurrents2018; @perroneCoherentEventsIon2020; @liuCharacteristicsInterplanetaryDiscontinuities2021; @lotekarKineticscaleCurrentSheets2022; @vaskoKineticscaleCurrentSheets2022; @vaskoKineticScaleCurrentSheets2024]. Despite this progress, several important gaps remain. +First, many earlier studies rely on datasets obtained at different times and different distances with varying temporal resolution of magnetic field measurements, which complicates consistent comparisons across heliocentric distances. Few investigations have examined simultaneous observations from multiple radial positions [@velliUnderstandingOriginsHeliosphere2020; @telloniSpacecraftRadialAlignments2023], limiting our understanding of how the properties of current sheets evolve as the solar wind expands. Second, existing current sheet identification methods often depend on fixed thresholds—such as absolute magnetic-field jumps—or are highly sensitive to data resolution, as in the case with partial variance increment techniques [@grecoPartialVarianceIncrements2017]. To enable meaningful cross-mission comparisons, more robust approaches based on normalized, resolution-independent criteria are needed. Finally, most prior work has classified discontinuities within the ideal MHD framework—for instance, distinguishing between tangential and rotational discontinuities [@neugebauerCommentAbundancesRotational2006; @madarDirectionalDiscontinuitiesInner2024]. While useful in some contexts and extensively studied, these categories become inadequate at ion scales, where kinetic and multi-fluid effects play a central role. In this study, we move beyond strict, ideal, and single-fluid MHD classifications and explore the Alfvénic character of current sheets—that is, the extent to which plasma and magnetic-field variations follow Alfvénic correlations [@shenComparingPlasmaAnisotropy2024]. Although this perspective is fundamental for understanding the origin and evolution of current sheets, it has received relatively limited attention in prior studies. + +The Parker Solar Probe (PSP) mission [@foxSolarProbeMission2016] offers an unprecedented opportunity to probe current sheets in the near-Sun environment with high cadence and to compare these observations with near-Earth measurements. In this study, we focus on three PSP perihelion intervals (Encounters 7–9) selected for their differing geometric alignments with the near-Earth spacecraft ARTEMIS and WIND. Encounter 7 features strong radial alignment that enables sampling of similar solar wind streams, while Encounter 8 exhibits weaker alignment, and Encounter 9 places PSP roughly 180$^\circ$ away in longitude. These contrasting configurations enable a controlled examination of current sheet properties and, more broadly, provide insight into how those properties depend on the solar wind's source region. + +Using a robust automated detection algorithm, we identify and analyze more than 45,000 current sheets across the three selected intervals (approximately 17,000 from PSP). We systematically characterize their rotation profiles, thicknesses, current densities, magnetic field jumps, and Alfvénic properties. The remainder of this paper is organized as follows: Section 2 describes the datasets and analysis methods; Section 3 presents the results and discusses their physical implications; and Section 4 summarizes the main conclusions. + +## Dataset and method + +In this study, we utilize magnetic field and plasma measurements from PSP [@foxSolarProbeMission2016], ARTEMIS [@angelopoulosARTEMISMission2011], and WIND [@acunaGlobalGeospaceScience1995]. For PSP, high-resolution magnetic field data are obtained from the FIELDS instrument [@baleFIELDSInstrumentSuite2016], with a time cadence of up to 3.4 ms. Plasma measurements are provided by the Solar Wind Electrons, Alphas, and Protons (SWEAP) suite [@kasperSolarWindElectrons2016], with a typical cadence of 1.7 s for ion moments from the Solar Probe ANalyzer for Ions (SPAN-I). Proton and electron temperature anisotropies are derived from measurements by the Solar Probe ANalyzer for Ions (SPAN-I) and Electrons (SPAN-E) electrostatic analyzers [@liviSolarProbeANalyzer2022; @whittleseySolarProbeANalyzers2020]. +For ARTEMIS, we use magnetic-field measurements from the Fluxgate Magnetometer sampled at approximately 16 samples per second [@austerTHEMISFluxgateMagnetometer2008], along with plasma velocity and density measurements from the Electrostatic Analyzer (ESA) instrument, which provides moments at a cadence of 4 s [@mcfaddenTHEMISESAPlasma2009]. +For WIND, we use magnetic-field data from the Magnetic Field Investigation (MFI) instrument sampled at 11 samples per second [@leppingWINDMagneticField1995], together with ion (proton and alpha-particle) moments from the ion electrostatic analyzers (PESA) of the Plasma Analyzer (3DP), measured at a cadence of approximately 3.1 seconds [@linThreedimensionalPlasmaEnergetic1995]. + +It is worth noting that for PSP, although ion data is also available from SPC, the SPAN-i data was chosen due to its higher temporal resolution. Plasma density estimates delivered by the quasi-thermal noise spectroscopy (QTN) are also available [@moncuquetFirstSituMeasurements2020]. We generally find that the results using QTN data do not differ significantly from those derived from SPAN-I data. Due to the limitations of SPAN-I data during the early mission phases, when a significant portion of the solar wind distributions was out of the instrument's view, our study concentrates on data beginning with Encounter 7, where SPAN-I provides more accurate measurements. + +Our study investigates the evolution of current sheet properties during the radial expansion of the solar wind. To distinguish between spatial and temporal variations, we analyze three distinct time intervals corresponding to PSP Encounters 7 through 9, each characterized by different PSP–Earth alignments [@telloniSpacecraftRadialAlignments2023; @velliUnderstandingOriginsHeliosphere2020]. +Encounter 7 features a favorable alignment between PSP and Earth, enabling nearly simultaneous multi-point observations (PSP approached perihelion on 2021-01-17 17:40; data intervals are 2021-01-15 – 2021-01-20 for PSP and 2021-01-17 – 2021-01-25 for ARTEMIS and WIND). Encounter 8 exhibits moderate alignment (PSP perihelion at 2021-04-29 08:48; data intervals 2021-04-27 – 2021-05-02 for PSP and 2021-04-29 – 2021-05-04 for ARTEMIS and WIND). In Encounter 9, PSP and Earth are located at substantially different heliolongitudes, with most PSP footpoints lying on the far side of the Sun (PSP perihelion on 2021-08-09 at 19:11; data intervals 2021-08-07 – 2021-08-12 for PSP and 2021-08-09 – 2021-08-14 for ARTEMIS and WIND). By comparing these intervals, we assess how the derived current sheet properties depend on the relative geometry of the heliospheric magnetic field and the temporal context of the observations. Our results show that although specific plasma parameters vary slightly among intervals, the overall statistical characteristics of the current sheets remain largely consistent. This suggests that the observed statistics are not strongly sensitive to either the relative geometry or the temporal context of the measurements. + +@fig-psp-overview provides an overview of the selected time interval from Encounter 7, comparing measurements from the PSP (left panels) and WIND (right panels) missions. From top to bottom, the panels display 8-minute averages of the magnetic field magnitude, proton number density, and proton bulk flow speed, followed by ion and electron temperatures resolved in the parallel and perpendicular directions. To facilitate comparison, WIND measurements are scaled to $20 R_\odot$ using the empirical relations for the magnetic-field magnitude $B$ and plasma number density $n$: $B_{\text{scaled}}/B = ( 20 R_\odot / 1\ \text{AU} )^{-1.59}$, $n_{\text{scaled}}/n = ( 20 R_\odot / 1\ \text{AU} )^{-1.96}$ [@perroneRadialEvolutionSolar2019]. +The fifth and sixth panels present, respectively, the median Alfvén ratio and the total number of current sheets identified within each four-hour interval. The Alfvén ratio is defined as $R_{VB} = \lvert \Delta \mathbf{V} \rvert / \lvert \Delta \mathbf{V}_A \rvert$, +where $\mathbf{V}$ is the plasma velocity and $\mathbf{V}_A = \mathbf{B}/\sqrt{\mu_0 \rho}$ is the Alfvén velocity, with $\rho$ the mass density and $\mu_0$ the vacuum permeability. +Including the anisotropy factor $\Lambda = \mu_0 (p_{\parallel,e} - p_{\perp,e} + p_{\parallel,i} - p_{\perp,i})/B^2$ does not significantly affect the Alfvén ratio $R_{VB}^* = R_{VB} / \sqrt{1 - \Lambda}$, where $p_{\parallel}$ and $p_{\perp}$ denote the pressures parallel and perpendicular to the magnetic field for electrons $(e)$ and ions $(i)$; its average value is approximately 0.5 for PSP and 0.4 for WIND over the entire time interval. The number of current sheets varies from roughly 100 to 300 per four-hour window, corresponding to an average of about 1,000 and 800 current sheets per day for PSP and WIND, respectively. The PSP occurrence rates are broadly consistent with those of previous studies at similar heliocentric distances [@lotekarKineticscaleCurrentSheets2022; @liuCharacteristicsInterplanetaryDiscontinuities2021]. +Compared with the occurrence rate of about 150 events per day identified at 1 AU by @vaskoKineticscaleCurrentSheets2022; @zhangSolarWindDiscontinuities2025b, our analysis detects substantially more current sheets, mainly because we do not limit our survey to only the most intense kinetic-scale current sheets (see Methodology below). + +Panels (a.7) and (b.7) present the time series of cross helicity and residual energy, calculated every 8 minutes for PSP and WIND, respectively. These quantities characterize the Alfvenic nature of and energy balance between the ambient plasma flow and magnetic fluctuations, and are computed in the standard form [@wicksCorrelationsLargeScales2013; @huangTemperatureAnisotropyHelium2025]: $\sigma_c = \frac{2 \langle \delta\mathbf{V} \cdot \delta\mathbf{V}_A \rangle}{ \langle \delta\mathbf{V}^2\rangle + \langle \delta\mathbf{V}_A^2\rangle }, \quad \sigma_r = \frac{\langle \delta \mathbf{V}^2\rangle-\langle \delta\mathbf{V}_A^2\rangle}{\langle \delta\mathbf{V}^2\rangle+\langle \delta\mathbf{V}_A^2\rangle},$ where $\delta\mathbf{V}$ and $\delta\mathbf{V}_A$ are the fluctuating velocity and Alfvén velocity, respectively. +Alfvénic fluctuations are typically characterized by $|\sigma_C| \approx 1$ and $|\sigma_R| \approx 0$. A large magnitude of the normalized cross-helicity, $\sigma_C$, indicates that the fluctuations are dominated by either outward- or inward-propagating Alfvénic modes, while a small magnitude of the normalized residual energy, $\sigma_R$, implies near-equipartition between kinetic and magnetic fluctuation energies. +For PSP, $\sigma_C$ displays extended intervals with values close to $\pm 1$, signifying prolonged periods of strongly Alfvénic fluctuations. Notably, a sharp transition in $\sigma_C$ occurs around 12:00 on 2021-01-17, coincident with a heliospheric current sheet crossing. The residual energy $\sigma_r$ remains mostly negative throughout, consistent with magnetic-energy–dominated turbulence [@sioulasMagneticFieldSpectral2023; @mcintyrePropertiesUnderlyingVariation2023]. In contrast, both $\sigma_c$ and $\sigma_r$ measured by WIND fluctuate more irregularly and lack the long, coherent intervals of high Alfvénicity seen by PSP. The cross helicity varies broadly between $-0.5$ and $+0.5$, reflecting the more evolved and mixed solar wind at 1 AU where Alfvénic correlations weaken due to turbulence evolution and stream interactions. +The final two panels display 8-minute averages of helium abundance and plasma beta, providing complementary information on the compositional and thermal conditions of the solar wind. The helium abundance is sensitive to the coronal heat flux into the chromosphere and transition region, with local variations in magnetic topology modulating how this heat flux is transported and ultimately imprinting source-region signatures in the solar wind [@altermanCrossHelicityHelium2025]. Plasma beta, which measures the ratio of thermal to magnetic pressure, is typically enhanced in slow solar wind originating from the streamer belt, forming a high-$\beta$ equatorial plasma sheet in the vicinity of heliospheric current sheet crossings [@huangParkerSolarProbe2023]. Together, these parameters provide critical diagnostics for identifying the source regions of different solar wind streams. + +::: {#fig-psp-overview} + +![](figures/psp/overview-7-all.pdf) + +Overview of Encounter 7 comparing PSP (left) and WIND (right) observations. Panels show 8-minute averages of (a.1, b.1) magnetic field magnitude, (a.2, b.2) proton density, (a.3, b.3) flow speed, (a.4, b.4) ion and electron temperatures, (a.5, b.5) Alfvén ratio, (a.6, b.6) current-sheet counts, (a.7, b.7) cross helicity and residual energy, (a.8, b.8) helium abundance, and (a.9, b.9) plasma beta. WIND measurements are radially scaled to $20 R_\odot$ for comparison. +::: + + + +### Methodology + +To identify current sheets, we employ an improved automated detection algorithm adapted from previous studies [@zhangSolarWindDiscontinuities2025b; @liuMagneticDiscontinuitiesSolar2022]. The magnetic field data are divided into consecutive time intervals, each lagged by a time $T$. For each interval, we compute the standard deviation of the vector magnetic field, $\sigma(\mathbf{B})$, and compare it with those of the adjacent intervals, $\sigma(\mathbf{B}_{-})$ and $\sigma(\mathbf{B}_{+})$. An interval is flagged as a potential current sheet when its standard deviation significantly exceeds those of its neighbors, indicating a sharp magnetic field variation: $\sigma(\mathbf{B}) > 2 \max(\sigma(\mathbf{B}_{-}), \sigma(\mathbf{B}_{+}))$. +For each candidate interval, we refine the boundaries of the potential current sheet by using the distance between two magnetic field vectors, $d(t_a, t_b) = \|\mathbf{B}(t_a) - \mathbf{B}(t_b)\|$. The leading and trailing edges, $t_a$ and $t_b$ ($t_a < t_b$), are defined such that $d(t_a, t_b)$ attains its maximum value within the interval. To minimize false detections—particularly those caused by wave-like or hole-like structures [@karlssonMagneticHolesSolar2021] that may mimic current sheet signatures—we impose the additional requirement $|\Delta\mathbf{B}| / \langle |\mathbf{B}| \rangle \ge 1/10$, where $\langle |\mathbf{B}| \rangle$ is the mean magnetic-field magnitude averaged over the duration of the current sheet crossing. We also evaluate the magnetic field orientation across each candidate structure by computing the angles between magnetic field vectors at the leading and trailing edges and comparing them with the angles between each edge and the midpoint of the interval. Only intervals where the edge-to-edge angle exceeds all interior angles are retained as valid current sheet candidates. Because current sheets span a range of spatial and temporal scales, we performed the identification procedure using multiple time lags ($T = 2, 4, 8, 16, 32, 64$ s). The resulting event lists from each search were merged into a comprehensive catalog after duplicates were removed. A subset of automatically detected intervals was visually inspected. All visually inspected events show clear current sheet signatures, such as (1) well-bounded regions with localized rotations of the magnetic field vector and (2) magnetic reversals in the maximum-variance direction. These features confirm the robustness and reliability of the automated detection procedure. + +We also compared our approach to the Partial Variance of Increments (PVI) method for current sheet identification [@grecoPartialVarianceIncrements2017]. Although the PVI technique is effective for detecting short-duration, kinetic-scale current sheets characterized by sharp gradients, it often fails to capture larger-scale structures that exhibit smoother yet coherent magnetic field variations. Moreover, the PVI method is highly sensitive to data resolution, complicating comparisons across missions with instruments of different temporal cadences. When the analysis is restricted to the smallest lag available in all datasets (e.g., $T=2$ s), the PVI method identifies a comparable number of current sheets and produces similar statistical properties to our standard detection approach. However, because PVI does not reliably capture the full range of larger-scale events and is difficult to apply uniformly across missions, we do not proceed to utilize PVI-derived detections. + +Once the current sheets are identified, we apply a minimum variance analysis (MVA) [@sonnerupMinimumMaximumVariance1998] to transform the magnetic field and plasma data into the local LMN coordinate system, where $L$, $M$, and $N$ correspond to the directions of maximum, intermediate, and minimum variance, respectively. The maximum variance component, $B_L$, is then fitted with a hyperbolic tangent profile to extract parameters characterizing each current sheet, following the standard Harris current sheet model [@harrisPlasmaSheathSeparating1962]: $B_L(t) = B_{L,i} \tanh \left(\frac{t-t_i}{\Delta t_i/2} \right) + c_i$ where $B_{L,i}$ is the magnitude of the magnetic field change in the maximum variance direction, $t_i$ is the detection time, $\Delta t_i$ is the temporal duration, and $c_i$ is a magnetic field offset. + +Assuming that the current sheets are one-dimensional planar structures, their spatial scale (thickness) can be estimated once the normal direction is determined. The normal vector can be obtained either from the minimum variance direction given by MVA or from the cross product of the magnetic field vectors at the current sheet boundaries. However, previous studies have shown that the MVA-derived normal direction can often be unreliable [@knetterFourpointDiscontinuityObservations2004; @wangSolarWindCurrent2024]. The accuracy can be improved by imposing additional constraints such as $\Delta|B|/ |B| > 0.05$ or $\omega > 60°$ [@liuFailuresMinimumVariance2023], where $\Delta|B|$ denotes the change in magnetic field magnitude and $\omega$ is the field rotation angle across the discontinuity. The cross-product method, on the other hand, requires that the magnetic field component along the normal direction ($B_n$) be small, making it unsuitable when $B_n$ is large. In this study, we apply both methods and report results based on the subset of events that satisfy $\Delta|B|/ |B| > 0.05$ or $\omega > 60°$. The results derived using the cross-product method are presented in the Appendix. + +The spatial thickness of each current sheet is estimated as $\delta = V_n \Delta t$, where $V_n$ is the plasma velocity projected along the normal direction. The current density associated with the main magnetic field reversal is calculated from $J_m = - \frac{1}{\mu_0 V_n} \frac{dB_l}{dt}$. Within our fitting framework, the magnetic field derivative can be approximated as $\max(dB_L/ dt) = 2 B_{L,i}/ \Delta t_i$. This approach provides a more robust estimate than direct differentiation, as it is less sensitive to measurement noise and limited temporal resolution. The mean values of proton density, temperature, and alpha-particle density of each interval are used to compute the Alfvén velocity, pressure anisotropy factor, plasma beta, and alpha-particle abundance, thereby providing a comprehensive characterization of the plasma environment associated with each current sheet. + +The upstream and downstream conditions are defined using timestamps immediately before and after the current sheet boundaries, respectively, such that $\Delta\mathbf{V} = \mathbf{V}_{\text{downstream}} - \mathbf{V}_{\text{upstream}}$. +The cosine of the angle between the velocity jump, $\Delta\mathbf{V}$, and the Alfvén velocity jump, $\Delta\mathbf{V}_A$, is then calculated as $\cos \theta = \frac{\Delta\mathbf{V} \cdot \Delta\mathbf{V}_A}{\|\Delta\mathbf{V}\| \|\Delta\mathbf{V}_A\|}$, serving as a measure of Alfvénic alignment. +Finally, the cross helicity, $\sigma_c$, and residual energy, $\sigma_r$, are computed within intervals extending from each current sheet edge to eight times the current sheet duration (or at least 30 s on either side). These parameters quantify the degree of Alfvénicity and the relative contributions of kinetic and magnetic energy fluctuations in the ambient solar wind. + +@fig-event-example presents two representative current sheet examples observed by PSP (left panels) and ARTEMIS (right panels). From top to bottom, the panels display, in LMN coordinates, the magnetic field, proton velocity, and the shifted Alfvén velocity $\mathbf{V}_A = \frac{\mathbf{B}}{\sqrt{\mu_0 \rho}} - \mathbf{V}_{A,0}$, where $\mathbf{V}_{A,0}$ is the Alfvén velocity at the center of the current sheet. The cosine of the angle between the velocity jump, $\Delta\mathbf{V}$, and the Alfvén velocity jump, $\Delta\mathbf{V}_A$, is approximately -0.82 for PSP and -0.95 for ARTEMIS, indicating strong anti-correlation. The corresponding Alfvén ratio, $R_{VB}$, is approximately 0.32 for both PSP and ARTEMIS. + +::: {#fig-event-example} + +![](figures/psp/event-example.pdf) + +Representative examples of current sheets observed by the Parker Solar Probe (PSP; panels a.1–a.3) and ARTEMIS (panels b.1–b.3) in the local LMN coordinate system. Panels (a.1) and (b.1) show the magnetic field components $B_L$, $B_M$, and $B_N$, together with the magnetic field magnitude $B$. The red dashed lines indicate the hyperbolic tangent fits used to derive current sheet parameters. Panels (a.2) and (b.2) display the proton bulk velocity components $V_L$, $V_M$, and $V_N$. Panels (a.3) and (b.3) show the shifted Alfvén velocity components $V_{A,L}$, $V_{A,M}$, and $V_{A,N}$. The vertical gray dashed lines mark the leading and trailing edges of each identified current sheet. + +::: + + +## Results and Discussion + +@fig-properties-hist presents the probability density functions (PDFs) of current sheet thickness, $\delta$, current density, $J$, and magnetic field jump magnitude, $|\Delta\mathbf{B}|$, derived from different missions and time intervals. +For thickness and current density, we display here a subset of the dataset where MVA normals are trustworthy. The results using the cross product method for the current sheet normals are displayed in Appendix @fig-properties-hist-cross and do not show any significant difference. Note, we do not restrict the dataset for the statistics of magnetic field jump magnitude since the jump magnitude does not depend on the normal direction. +To better understand how these properties relate to the local plasma environment [@vaskoKineticscaleCurrentSheets2022; @vaskoKineticScaleCurrentSheets2024; @zhangSolarWindDiscontinuities2025b], we normalize the current sheet thickness by the ion (proton) inertial length, $d_i=c/\omega_{pi}$, and the current density by the Alfvén current density, $J_A = e N V_A = B / \mu_0 d_i$. Both parameters represent natural plasma scales: the ion inertial length characterizes the spatial scale where ions decouple from the magnetic field, while the Alfvén current density corresponds to the characteristic current generated by the drift of electrons and ions relative to each other at the local Alfvén speed, $V_A$. These quantities are particularly relevant in studies of MHD turbulence and magnetic reconnection [@zhdankinStatisticalAnalysisCurrent2013; @franciMagneticReconnectionDriver2017]. + +The absolute thickness, $\delta$, increases from PSP to ARTEMIS and WIND, indicating a broadening of current sheets with solar wind expansion. After normalization by the ion inertial length, however, the characteristic scale $\delta / d_i$ decreases slightly with increasing distance. Similar behavior has been reported by @madarDirectionalDiscontinuitiesInner2024 using Parker Solar Probe and Solar Orbiter data, where the thickness of rotational discontinuities in ion inertial length units decreases rapidly until approximately 0.3 AU. We additionally examined the scaling proposed by @shaikhBetadependentPropertiesSolar2026, which accounts for the dependence of current sheet thickness on plasma beta. Applying the proton-beta-dependent normalization, $\delta / d_i \beta^{0.23}$, does not appreciably modify the distributions of normalized thicknesses shown in @fig-properties-hist, indicating that proton beta effects do not play a dominant role in shaping the observed radial trends in our dataset. + +The normalized current density, $J_m / J_A$, shows remarkably similar distributions across the PSP, WIND, and ARTEMIS datasets. This suggests that the relative strength of current sheets, when expressed in local plasma units, is largely independent of radial distance from the Sun. This trend is consistent with previous studies based on joint WIND–Ulysses measurements between 1 and 5 AU [@vaskoKineticscaleCurrentSheets2022; @vaskoKineticScaleCurrentSheets2024] and Juno observations from 1–5 AU [@zhangSolarWindDiscontinuities2025b]. All of which similarly found that the normalized current density remains nearly constant while the normalized thickness gradually decreases with increasing heliocentric distance. + +The distributions of magnetic field jumps ($|\Delta\mathbf{B}|$ and $|\Delta\mathbf{B}|/\langle|\mathbf{B}|\rangle$) exhibit pronounced differences across the missions. In particular, PSP observations display a markedly distinct distribution compared with WIND and ARTEMIS, characterized by substantially larger magnetic field jumps, even after normalization. These enhanced jumps indicate stronger magnetic field reversals in the near-Sun environment and are likely associated with the frequent occurrence of switchbacks [@baleHighlyStructuredSlow2019; @witSwitchbacksNearsunMagnetic2020]. Previous studies have shown that switchbacks progressively dissipate with increasing heliocentric distance, as their amplitudes decrease more rapidly than the background turbulent fluctuations [@teneraniEvolutionSwitchbacksInner2021; @soniSwitchbackPatchesEvolve2024; @shiPatchesMagneticSwitchbacks2022]. The discrepancy between ARTEMIS and WIND is more plausibly related to the ARTEMIS spacecraft's proximity to the bow shock. The foreshock region hosts strong compressional fluctuations and reflected hot ions, favoring a higher occurrence of compressional current sheets compared to the pristine solar wind. This environmental effect likely contributes to the larger (normalized) magnetic-field jumps observed by ARTEMIS [@kropotinaSolarWindDiscontinuity2021]. + +::: {#fig-properties-hist} +![](figures/psp/properties_hist-mva.pdf) + +Probability density functions of current sheet properties obtained from different missions (color-coded) and encounters (distinguished by line styles). The left column shows unnormalized quantities, while the right column presents values normalized by characteristic plasma scales. From top to bottom, panels display the distributions of (a-b) current sheet thickness $\delta$ and its normalized form $\delta / d_i$; (c-d) current density $J_m$ and normalized current density $J_m / J_A$; and (e-f) magnetic field jump magnitude $|\Delta\mathbf{B}|$ and its normalized value $|\Delta\mathbf{B}| / \langle |\mathbf{B}| \rangle$. The ion inertial length is $d_i = c / \omega_{pi}$, and the Alfvén current density is + +::: + +@fig-joint-properties presents the joint distributions between (a) current density and current sheet thickness, (b) normalized current density and normalized thickness, (c) normalized thickness and normalized magnetic field jump, and (d) normalized current density and normalized magnetic field jump. A clear inverse correlation is evident between the (normalized) current density and (normalized) current sheet thickness, indicating that stronger currents are generally associated with thinner sheets—that is, smaller-scale current sheets tend to be more intense. +After normalization, the scatter distributions of current density versus thickness are highly similar across different missions (different radial distances), both in overall shape and in the range of values. This similarity suggests that the statistical relationship between normalized current density and scale is largely independent of heliocentric distance. + +In contrast, the relationship between the magnetic field jump and current sheet thickness, shown in Panel (c), is relatively weak. For a given range of magnetic field jump amplitudes, the corresponding spatial scales exhibit a broad distribution, spanning up to two orders of magnitude. Moreover, Panel (d) showcases there is no clear one-to-one correspondence between the magnetic field jump amplitude and the current density. +Scale-dependent properties of current sheets have been reported by several previous studies using PSP, WIND, and Ulysses observations [@lotekarKineticscaleCurrentSheets2022; @vaskoKineticscaleCurrentSheets2022; @vaskoKineticScaleCurrentSheets2024]. However, whereas earlier studies primarily focused on kinetic-scale current sheets, our analysis extends this investigation to a broader range of scales—covering nearly two additional orders of magnitude in normalized current density and normalized current sheet thickness. + +Results shown in @fig-joint-properties remain largely unchanged when the current sheet normal is determined using the magnetic field cross-product method rather than the minimum variance analysis. This consistency across methods confirms that the observed correlations are not strongly affected by the specific approach used to estimate the current sheet geometry. + +::: {#fig-joint-properties} +![](figures/psp/joint_properties-mva.pdf) + +Scatter plots of (a) current density versus current sheet thickness, (b) normalized current density versus normalized current sheet thickness, (c) magnetic field jump magnitude versus normalized current sheet thickness, and (d) normalized magnetic field jump magnitude versus normalized current density. +::: + +For an ideal MHD rotational discontinuity (RD), variations in the plasma velocity and magnetic field satisfy the Walén relation, $\Delta \mathbf{V} = \pm \Delta \mathbf{V}_A$, and the the magnetic field is approximately incompressible across the current sheet, such that $\Delta B \approx 0$ [@hudsonDiscontinuitiesAnisotropicPlasma1970]. The Walén relation requires agreement in both magnitude and direction between the plasma velocity change and the corresponding Alfvén velocity change. However, most observed current sheets deviate from these ideal conditions. + +To quantify the degree of Alfvénicity and the deviations from ideal RD behavior, we examine both the magnitude and angular differences between $\Delta \mathbf{V}$ and $\Delta \mathbf{V}_A$, as well as the variation in magnetic field strength across each current sheet. +@fig-Alfvenicities summarizes the statistical results. Panel (a) shows the ratio of velocity-jump magnitude to Alfvén velocity-jump magnitude. Across all missions, the ratio is generally less than unity, with mean values of approximately 0.6 for PSP and 0.5 for the combined WIND and ARTEMIS datasets. +Panel (b) displays the cosine of the angle between the velocity and Alfvén velocity jumps. All missions exhibit clustering near 0$^\circ$ and 180$^\circ$, corresponding to correlated and anti-correlated states, respectively. PSP observations exhibit stronger alignment (cosine values closer to $\pm1$), whereas WIND and ARTEMIS data show a broader angular spread. +Panels (c) and (d) present the fractional change in magnetic field magnitude across current sheets. These variations are small (about 0.1 on average) for all missions, with PSP showing slightly smaller changes than WIND and ARTEMIS. This indicates that most current sheets are nearly incompressible. + +We also calculate the Alfvén velocity, including the anisotropy factor $\Lambda = \mu(p_{\parallel,e} - p_{\perp,e} + p_{\parallel,i} - p_{\perp,i})/B^2$, for PSP and WIND observations when the necessary plasma data are available. In most cases, including this anisotropy correction does not significantly alter the velocity-jump ratio. The persistent discrepancy between the observed ratios ($R < 1$) and the predictions of anisotropic MHD theory may have several explanations. First, the limited temporal resolution of the plasma anisotropy measurements—typically much lower than that of the magnetic field data—may cause underestimation of instantaneous variations at current sheet crossings [@huangProtonTemperatureAnisotropy2020; @shenComparingPlasmaAnisotropy2024]. Second, kinetic-scale magnetic discontinuities often deviate from the assumptions of MHD theory: their non-Maxwellian particle velocity distributions, including plasma agyrotropies and field-aligned beams [@artemyevIonNongyrotropySolar2020; @neukirchKineticModelsTangential2020], are not accounted for in the fluid equations but can strongly influence the measured velocity ratios. Finally, uncertainties in deriving the parallel and perpendicular temperatures—typically obtained from nonlinear fits to complex velocity distribution functions—may introduce additional errors, especially when multiple particle populations are present [@huangProtonTemperatureAnisotropy2020; @halekasElectronsYoungSolar2020]. + +::: {#fig-Alfvenicities} + +![](figures/psp/Alfvenicities.pdf) + +Probability density functions of parameters used to characterize the Alfvénicity of current sheets: (a) ratio of velocity-jump magnitude to Alfvén velocity-jump magnitude, (b) cosine of the angle between the two vectors, (c) the variation in magnetic field magnitude between the leading and trailing edges of each current sheet; and (d) the maximum variation in magnetic field magnitude observed within each current sheet. + +::: + +To further investigate the physical origins of the observed deviations from Alfvénic behavior, we use a single scalar parameter, $Q^{\pm}$, introduced by [@sonnerupQualityMeasureWalen2018], to quantify the Alfvénicity of current sheets and examine how $Q^{\pm}$ correlates with key current sheet and plasma parameters. + +The parameter $Q^{\pm}$ is defined as $Q^{\pm} = \pm \left(1 - \frac{|\Delta \mathbf{V} \mp \Delta \mathbf{V}_A|}{|\Delta \mathbf{V}| + |\Delta \mathbf{V}_A|}\right)$, which combines both magnitude and directional discrepancies between the velocity and Alfvén velocity jumps, and satisfies $-1 < Q^{\pm} < 1$. The upper sign is used for cases where $\cos \theta > 0$ and the lower sign for $\cos \theta < 0$. The values of $Q^{\pm}$ approaching ±1 correspond to a highly Alfvénic behavior. The joint distribution of $|\Delta \mathbf{V}| / |\Delta \mathbf{V}_A|$ and $\cos \theta$ is shown in the first panel of @fig-Q_sonnerup_joint_dist_den. + +::: {#fig-Q_sonnerup_joint_dist_den} + +![](figures/psp/Q_sonnerup_joint_dist_den.pdf) + +(a) Joint distribution of the velocity-jump ratio ($|\Delta \mathbf{V}| / |\Delta \mathbf{V}_A|$) and the cosine of the alignment angle ($\cos \theta$), used in computing the Sonnerup ($Q^{\pm}$) parameter. (b) Joint distribution of cross helicity $\sigma_c$ and $Q^{\pm}$. (c) Joint distribution of residual energy $\sigma_r$ and $Q^{\pm}$. (d) Joint distribution of temporal duration and $Q^{\pm}$. + +::: + +We examine the relationship between $Q^{\pm}$ and various current sheet and ambient solar wind properties. Most parameters show little or no correlation with $Q^{\pm}$; however, two quantities exhibit strong correlations: the cross helicity, $\sigma_c$, and residual energy, $\sigma_r$, of the surrounding solar wind. These parameters respectively describe the degree of velocity–magnetic field correlation and the partition of energy between velocity and magnetic field fluctuations. Both serve as indicators of the Alfvénic character of the background solar wind rather than of individual current sheets. + +Panels (b) and (c) of @fig-Q_sonnerup_joint_dist_den show that the Alfvénicity of the current sheets, represented by $Q^{+}$, increases with greater cross helicity and greater residual energy (more positive), while $Q^{-}$ increases with greater cross helicity and smaller residual energy (more negative). This trend suggests that current sheets embedded within highly Alfvénic solar wind streams tend to exhibit more substantial velocity–magnetic field alignment, whereas those located in more turbulent or energy-imbalanced regions—where magnetic energy ($E_B$) exceeds kinetic energy ($E_V$)—exhibit greater departures from Alfvénic expectations. + +Additionally, observations reveal a weak positive correlation between Alfvénicity ($Q^{\pm}$) and the temporal duration of current sheets, as shown in Panel (d) of @fig-Q_sonnerup_joint_dist_den. In contrast, no clear correlation is found between current-sheet thickness and Alfvénicity. However, the correlation between Alfvénicity and temporal duration may be partially influenced by instrumental limitations, as longer-duration events are more reliably resolved in plasma velocity measurements. This effect is evident in the spacecraft dependence: PSP, with its higher cadence, shows a much weaker correlation, whereas ARTEMIS and WIND, which operate at lower cadence, display a stronger apparent correlation. +Other plasma parameters, such as plasma beta and alpha-particle abundance, show no significant correlation with $Q^{\pm}$, implying that the Alfvénic nature of current sheets is primarily governed by the local turbulent environment rather than by large-scale plasma conditions. + +Since the normal component of the magnetic field plays a key role in determining both the spatial scale and the associated current density, we next examine the rotational characteristics of the current sheets and their relationship with the normalized normal magnetic field component, $B_N/B$. +@fig-B_n_ω shows the distributions of the in-plane rotation angle (also referred to as the shear angle) and the normalized normal component $B_N/B$, along with their joint distributions for different missions. Panel (a) reveals that PSP observes a distinctly different distribution of rotation angles compared to the WIND and ARTEMIS missions, while the distributions of $B_N/B$ are largely similar, as shown in Panel (b). This discrepancy suggests that the rotational structure of current sheets evolves significantly as the solar wind propagates from the near-Sun environment to 1 AU. +The multiple peaks observed in the PSP shear-angle distribution likely reflect the coexistence of several current sheet populations. %possibly formed through different mechanisms or originating from different solar wind sources. +As the solar wind expands and mixes, this structured distribution becomes smoother, as observed by WIND and ARTEMIS near Earth. We note that the identification criteria and additional constraints imposed to ensure the reliability of the minimum variance analysis exclude some current sheets with small shear angles, which may contribute to the apparent single peak in the PDFs [@liuFailuresMinimumVariance2023]. However, current sheets with shear angles exceeding $\approx 30^\circ$ necessarily satisfy the selection criteria and are therefore unaffected. Consequently, the PDFs are considered reliable for $\omega_{\text{in}} > 30^\circ$. + +The distribution of $B_N / B$ is clearly bimodal, with one population corresponding to nearly field-aligned normals and another to nearly perpendicular normals. Only a small fraction of events fall in the intermediate range. Such bimodal behavior is consistent with the coexistence of different types of discontinuities—most notably rotational and tangential discontinuities [@madarDirectionalDiscontinuitiesInner2024]. We note, however, that our selection criteria may have different passing rates for different values of $B_N/B$, which could affect the relative amplitudes of the PDFs, though unlikely the presence of the bimodal structure itself [@liuFailuresMinimumVariance2023]. + +Panel (c) shows the joint distribution of the in-plane rotation angle and $B_N / B$, where each column has been normalized to highlight how the rotation angle varies with $B_N / B$. The distinct distribution observed by PSP again indicates an evolutionary trend: current sheets closer to the Sun exhibit a more structured and varied rotational behavior, and they are smoother and more uniform at 1AU. + +::: {#fig-B_n_ω} + +![](figures/psp/B_n_w_-mva-subset=true.pdf) + +(a) Probability density functions of the in-plane rotation angle ($\omega_{\text{in}}$) for PSP and ARTEMIS + WIND. (b) Probability density functions of the normalized normal magnetic field component ($B_N / B$). (c) Joint distributions of $\omega_{\text{in}}$ and $B_N / B$ for PSP (left) and ARTEMIS + WIND (right). + +::: + +Finally, @fig-duration presents the PDFs of current sheet temporal durations, $\Delta t$, observed by different missions. Under the assumption that current sheets propagate without significant relaxation, the temporal separation between different parts of a structure should remain unchanged between inner and outer spacecraft, independent of solar-wind acceleration [@berriotIdentificationSinglePlasma2024]. As such, temporal duration provides additional diagnostic information on the nature and evolution of current sheets. The resulting distributions differ noticeably in shape, reflecting variations in the relative occurrence of short- and long-duration events across the datasets. In particular, changes in slope with increasing duration suggest the presence of at least two, and possibly three, distinct regimes contributing to the observed range of current sheet durations. +The shortest-duration events are likely associated with kinetic-scale current sheets, while the longer-duration events may correspond to flux-tube boundaries, large-scale magnetic structures, or even heliospheric current sheet crossings [@borovskyFluxTubeTexture2008; @liouCharacteristicsHeliosphericCurrent2021]. The similar slopes observed at longer durations across missions indicate that large-scale structures are relatively stable with heliocentric distance. In contrast, the short-duration population evolves more rapidly as the solar wind expands, either because such structures are generated locally by turbulence or because they are advected with the flow after being generated by turbulence much closer to the Sun. + +Assuming that current sheets propagate radially without significant distortion or dissipation, the measured temporal duration can be interpreted as a proxy for the spatial thickness of the structure. This interpretation is modulated by the local magnetic field orientation but is not affected by solar wind acceleration. To account for geometric effects, we apply a correction based on the Parker spiral model by multiplying the PSP durations by a factor $\sin(\Theta_{\text{Earth}}) / \sin(\Theta_{\text{PSP}})$. Here, $\Theta \approx \arctan(B_T / B_R)$ denotes the angle between the magnetic field and the radial direction, where $B_R$ and $B_T$ are the radial and transverse components of the magnetic field, respectively. Although this correction is a simplified approximation, it provides a useful scaling comparison: after adjustment, the PSP duration distribution aligns closely with those of WIND and ARTEMIS near 1 AU for short-duration current sheets. This agreement suggests that the shortest-duration (kinetic-scale) current sheets are predominantly oriented perpendicular to the mean magnetic field and evolve substantially with distance from the Sun. + +::: {#fig-duration} + +![](figures/psp/duration_dist.pdf) + +Probability density functions of current sheet durations (in seconds) observed by different missions. Grey vertical dashed lines indicate durations of 1, 2, 4, 8, 16, and 32 s (from left to right). + +::: + +## Conclusion + +1. We investigated three Parker Solar Probe (PSP) encounters, complemented by ARTEMIS and WIND measurements, to identify and characterize current sheets across different heliocentric distances. The results indicate that variations in the selected time intervals exert only a minor influence on key current sheet properties such as thickness, current density, and Alfvénicity. This suggests that the observed evolution of current sheet characteristics primarily reflects spatial effects associated with radial expansion rather than temporal variability. + +2. Our analysis confirms that current sheet properties are closely linked to local plasma conditions. In particular, the results extend previously reported scale-dependent relationships between current-sheet thickness and current density by more than two orders of magnitude, encompassing structures ranging from kinetic to large magnetohydrodynamic (MHD) scales. Despite substantial differences in absolute scales, the normalized current density remains nearly constant across radial distances, while the normalized current-sheet thickness exhibits only a modest increase with heliocentric distance. Variations in the median values are minor compared to the widths of the corresponding probability density functions, indicating that the intrinsic spread of current sheet properties dominates over systematic radial trends. The pronounced anti-correlation between current density and spatial scale—stronger current densities corresponding to thinner sheets—and the similarity of normalized current densities across radial distances indicate a close connection between current sheets and MHD turbulence, since localized, non-Gaussian current density enhancements are a hallmark of intermittency in turbulent plasmas. + +3. The Alfvénicity of current sheets, quantified by the parameter $Q$, remains below unity for the majority of events. The magnitude ratios reported by PSP and WIND are broadly consistent, while PSP observations show smaller angular deviations than those from WIND and ARTEMIS. Pressure anisotropy alone does not account for the observed velocity and magnetic field jumps. Instead, the degree of Alfvénicity appears to be primarily controlled by the surrounding turbulent plasma state—specifically, by cross helicity and residual energy. Most current sheets exhibit low compressibility (typically $<$0.1). These findings support a picture in which current sheets are embedded within, and shaped by, the local turbulent cascade. In other words, the local macroscopic turbulence environment determines whether a boundary appears Alfvénic or non-Alfvénic when sampled in situ. + + +4. A comparison between PSP and near-Earth missions reveals that the statistical properties of solar wind current sheets undergo systematic evolution with heliocentric distance. Apparent differences are observed in the distributions of unnormalized current density, thickness, magnetic field jump, rotation angle, and temporal duration. Processes that may drive this evolution include global solar wind expansion [@matteiniIonKineticsSolar2012], nonlinear dynamics of Alfvén waves and Alfvénic turbulence [@medvedevDissipativeDynamicsCollisionless1997], and interactions with other coherent structures. This evolution has important implications for the role of current sheets in plasma heating [@sioulasStatisticalAnalysisIntermittency2022], particle energization and transport [@zhangQuantificationIonScattering2025], which are expected to depend strongly on radial distance. + +Taken together, our results support a picture where current sheets are predominantly turbulence-generated, dynamically evolving structures whose Alfvénic character are governed by the ambient solar wind turbulence properties—through cross helicity and residual energy. The tight coupling between current sheet intensity and spatial scale, the coexistence of at least two structural regimes, and the systematic radial evolution of their properties collectively point to a complex interplay between intermittent turbulence, solar wind expansion, and kinetic processes. Resolving these features will require higher-resolution particle measurements, multi-spacecraft observations, and dedicated kinetic modeling. \ No newline at end of file diff --git a/docs/others/phd/2026_grad/_review_current_sheet.qmd b/docs/others/phd/2026_grad/_review_current_sheet.qmd new file mode 100644 index 0000000..ec757f6 --- /dev/null +++ b/docs/others/phd/2026_grad/_review_current_sheet.qmd @@ -0,0 +1,248 @@ + + +# Observations of Solar Wind Current Sheets + +This chapter reviews the observational landscape of solar wind current sheets [@tsurutaniReviewDiscontinuitiesAlfven1999; @neugebauerProgressStudyInterplanetary2010; @khabarovaCurrentSheetsPlasmoids2021], tracing their study from the earliest spacecraft measurements to the high-cadence, multi-point observations available today. The review is organized as follows. We begin by recounting how abrupt magnetic field rotations were first recognized as magnetohydrodynamic discontinuities and how subsequent high-resolution measurements revealed their fundamentally kinetic character. We then discuss the analysis methods used to determine current sheet orientation and thickness, with particular attention to the strengths and limitations of single-spacecraft and multi-spacecraft techniques. The bulk of the chapter is devoted to the statistical properties of current sheets — their identification, magnetic field configuration, spatial scales, current densities, occurrence rates, Alfvénicity, and orientation — as established by surveys spanning from the inner heliosphere to beyond 5 AU. Throughout, we emphasize the extent to which reported statistics depend on the identification method employed, a recurring theme that must be kept in mind when comparing results across studies. The chapter concludes by identifying several open questions that motivate the work presented in subsequent chapters. + +## From MHD Discontinuities to Kinetic-Scale Current Sheets + +Early observations from the Pioneer 6 mission revealed that the direction of the IMF is highly variable, an unexpected finding at the time [@nessPreliminaryResultsPioneer1966]. As shown in @fig-ness1966-fig6, on hour-long timescales, these abrupt directional changes are clearly distinguishable from the comparatively quiet background in which the magnetic field evolves slowly. Such rapid variations in the magneto-plasma parameters were recognized as fundamental solar wind features [@colburnDiscontinuitiesSolarWind1966] and were identified as magnetohydrodynamic discontinuities—spatial boundaries separating two distinct plasma regions. + +MHD theory permits such discontinuities but constrains the changes allowed across them through the Rankine–Hugoniot jump conditions. The early solar wind measurements spurred the development of theory for anisotropic plasmas [@hudsonDiscontinuitiesAnisotropicPlasma1970]. Five distinct types are possible, the most relevant here being tangential discontinuities (TDs), rotational discontinuities (RDs), and shocks (relatively rare in the solar wind). Classifying observed discontinuities as RDs or TDs attracted considerable early research interest because the distinction carries physical implications for the topology of the IMFs [@knetterNewPerspectiveSolar2005]. A TD separates two topologically distinct plasma regions with no field-normal component, whereas an RD is a propagating structure that connects magnetically linked regions. This distinction has consequences for energetic particle diffusion coefficients and bears on possible generation mechanisms operating in the solar corona. The relative abundance of RDs and TDs in the solar wind has been the subject of longstanding debate [@smithIdentificationInterplanetaryTangential1973; @neugebauerReexaminationRotationalTangential1984; @neugebauerCommentAbundancesRotational2006]. + +![Two-hour example of 1-minute averages of the interplanetary magnetic field for which the magnitude is average, but the direction is highly variable and principally inclined at large angles ($\theta \approx 90^\circ$) to the ecliptic plane [@nessPreliminaryResultsPioneer1966]](figures/ref/nessPreliminaryResultsPioneer1966-fig6.png){#fig-ness1966-fig6} + +At smaller scales (on the order of seconds), many discontinuities retain their sharp character, though some reveal resolvable internal structure even in early measurements [@siscoePowerSpectraDiscontinuities1968; @burlagaDirectionalDiscontinuitiesInterplanetary1969]. At these scales, the term "discontinuity" becomes inappropriate: the structure width is comparable to the ion inertial length, and MHD theory may no longer be applicable. The term *current sheet* is adopted instead, emphasizing the current layer that maintains the field rotation. In this dissertation, we use "current sheet" throughout to emphasize the kinetic nature of these structures. The kinetic scale generally refers to structures with widths comparable to the ion inertial length, corresponding roughly to temporal scales below 30 seconds at 1 AU. (Four scale regimes were introduced by @burlagaDirectionalDiscontinuitiesInterplanetary1969: macro-scale (> 100 h), meso-scale (1–100 h), micro-scale (30 s – 1 h), and kinetic-scale (< 30 s).) + +Recent observations from multiple missions have confirmed that many solar wind current sheets possess fundamentally kinetic properties that defy a purely MHD description (see @fig-artemyevKineticNatureSolar2019-fig3 for an example). @artemyevKineticNatureSolar2019 analyzed discontinuities observed by the ARTEMIS and MMS missions and found that these structures exhibit characteristics of both TDs and RDs simultaneously: tangential velocity jumps correlate well with Alfvén speed jumps (an RD signature), yet electron density and temperature vary significantly across them (a TD signature). + +![Discontinuity observations by two ARTEMIS (at ∼ 07:38:00 and ∼ 07:38:30) and the MMS 1 (at ∼ 07:53:00) spacecraft. (a) The magnetic field $B_l$ (left axis) and plasma velocity $v_l$ (right axis). (b1 and b2) The electron density $n_e$ (left axis) and temperature $T_e$ (right axis). (c1, c2, d1, and d2) The electron pitch angle distributions for two energy ranges. (e1 and e2) The electron flux anisotropy.](figures/ref/artemyevKineticNatureSolar2019-fig3.jpg){#fig-artemyevKineticNatureSolar2019-fig3} + +Complementary statistical work by @artemyevKineticPropertiesSolar2019 using ARTEMIS data revealed that ion-scale discontinuities are accompanied by density and temperature variations extending over tens of ion inertial lengths. The inversely correlated density and temperature variations suggest a nearly force-free configuration. These structures exhibit two characteristic spatial scales: an intense inner current layer (> 1 nA/m²) enveloped by a broader outer structure. The magnetic field rotation occurs at the outer scale, while plasma pressure gradients provide pressure balance at the inner scale. The electron kinetic behavior around these structures is strongly energy-dependent: near-thermal electrons (10–30 eV) exhibit significant pitch-angle changes across the discontinuity, whereas hot electrons (100–1000 eV) retain their distribution properties on both sides, suggesting they can cross the structure freely. + +Further insight into the kinetic nature of current sheets was provided by @artemyevIonNongyrotropySolar2020. Rotational discontinuities are typically accompanied by velocity jumps $\Delta v_l$ that are systematically smaller than the corresponding Alfvén speed jumps $\Delta v_A$, contrary to the stationary MHD prediction of equality. Previous explanations invoked either pressure anisotropy reducing $\Delta v_A$ or non-stationarity from residual magnetic energy. @artemyevIonNongyrotropySolar2020 proposed an alternative: nonadiabatic ion interactions with intense thin discontinuities produce nongyrotropic ion distributions with a nondiagonal pressure tensor component whose cross-discontinuity gradient reduces $\Delta v_A$. ARTEMIS observations confirmed the existence of such an ion population with sufficient amplitude and spatial profile to account for the discrepancy, demonstrating that ion kinetic effects fundamentally shape the internal structure of solar wind current sheets. + +Consequently, investigating the formation, evolution, and particle interactions of current sheets requires moving beyond MHD theory and into the realm of ion and electron kinetics. + +## Analysis Methods: Normal Determination and the Planar Approximation + +Solar wind current sheets are commonly approximated as one-dimensional structures in which the dominant variation occurs along the direction normal to the sheet. Under this approximation, the most important spatial parameter is the thickness—the scale over which the magnetic field rotates across the structure. Estimating the thickness requires knowledge of the current sheet normal, and the accuracy of this estimate depends critically on the method used to determine it. +A detailed treatment and comprehensive review of both single-spacecraft and multi-spacecraft analysis techniques can be found in @knetterNewPerspectiveSolar2005. In this section, we do not attempt to reproduce that earlier review. Instead, we focus on developments that have emerged since then. + +### Single-Spacecraft Methods + +For single-spacecraft observations, two methods are most widely used. The first is minimum variance analysis (MVA) of the magnetic field, based on the continuity of the normal component of the magnetic field. This method identifies the direction of minimum magnetic field variance in the transition layer and interprets it as the sheet normal [@sonnerupMagnetopauseStructureAttitude1967; @sonnerupMinimumMaximumVariance1998]. The second is the cross-product method, which estimates the normal from the cross product of the upstream and downstream magnetic field vectors [@burlagaTangentialDiscontinuitiesSolar1969; @wangSolarWindCurrent2024]. + +MVA has been the dominant technique in the literature, and much of the global statistical picture of solar wind discontinuities has been built upon it. A key advantage of MVA is that it makes no a priori assumption about the magnetic field geometry—it can in principle recover the normal direction for both TDs and RDs. The method is designed to handle realistic deviations from an ideal one-dimensional layer—including 2D or 3D internal structure, temporal fluctuations in normal orientation, and measurement uncertainties—provided there is no systematic change in the normal direction during the spacecraft traversal [@sonnerupMinimumMaximumVariance1998]. In practice, however, these non-ideal effects can be severe enough to cause MVA to fail. Systematic comparisons with multi-spacecraft determinations have revealed that MVA can be inaccurate for solar wind current sheets. @horburyThreeSpacecraftObservations2001, using three-spacecraft timing with Geotail, Wind, and IMP-8, found that many MVA normal estimates lie far from the timing-derived normals. In their dataset, 77% of events were likely TDs based on timing, yet single-spacecraft MVA would have classified a much larger fraction as RDs—a discrepancy they attributed partly to surface waves on the discontinuities (which may cause the minimum variance directions approximately perpendicular to discontinuity normals). + +The nature of these MVA failures has been clarified by subsequent work. @tehLocalStructureDirectional2011, applying Grad–Shafranov reconstruction (based on the ideal 2D MHD equations in steady state) to Cluster observations, showed that internal structures such as magnetic islands (flux ropes) within the transition layer can cause MVA to fail as a predictor of the normal direction. Since MVA assumes a planar, one-dimensional geometry, any two-dimensional substructure may shift the minimum variance direction away from the true normal. (Good agreement with timing methods can be achieved by imposing the additional constraint $\langle B_N \rangle = 0$ on MVA.) @liuFailuresMinimumVariance2023 conducted a comprehensive statistical assessment using 6,752 Cluster discontinuities with timing-derived normals as a benchmark. They found that while increasing the eigenvalue ratio $\lambda_2/\lambda_3$ and narrowing the analysis window can reduce scatter, MVA suffers from an inherent geometric defect: discontinuities with small normal magnetic field component ($|B_N|/|B| < 0.2$) and small magnitude change ($\Delta|B|/|B| < 0.05$) are systematically misidentified. In these cases, MVA confuses the dominant in-plane magnetic field component with the normal component, producing a spurious large $|B_N|$ and an apparent in-plane rotation angle of $\sim 180°$. This mechanism causes genuine TDs and EDs to be misclassified as RDs with dominant normal fields, explaining the false RD predominance reported in many earlier MVA-based studies. Despite these limitations, @liuFailuresMinimumVariance2023 showed that MVA can achieve acceptable accuracy under favorable geometric conditions. Since the magnitude change $\Delta|B|/|B|$ and the rotation angle $\omega$ do not depend on knowledge of the normal and are known a priori, they can serve as pre-selection criteria: MVA errors remain generally below $30°$ when either $\Delta|B|/|B| > 0.05$ or $\omega > 60°$. The problematic population is therefore confined to small-angle, nearly constant-magnitude current sheets. + +The cross-product method, by contrast, has been shown to perform substantially better in accuracy. @wangSolarWindCurrent2024, comparing single-spacecraft estimates against four-spacecraft Cluster timing for 1,831 current sheets, demonstrated that the cross-product normal agrees with the timing normal to within $15°$ at the 90% confidence level, whereas MVA normals frequently deviate by more than $60°$—likely due to contamination by *anisotropic* turbulent fluctuations. Combined with the Taylor frozen-in hypothesis (validated by the finding that current sheet propagation velocities agree with local ion flow velocities within ~20%), the cross-product method delivers current sheet thickness and current density amplitude within 20% of their multi-spacecraft values at the 90% confidence level. However, the cross-product method has an important limitation: it assumes a vanishing normal magnetic field component ($B_N = 0$), which is strictly valid only for tangential discontinuities. For structures with a significant normal field component, the cross product of the boundary fields does not coincide with the true normal. In practice, this limitation is mitigated by the observational finding that the vast majority of kinetic-scale current sheets have very small $|B_N|/|B|$ [@erdosDensityDiscontinuitiesHeliosphere2008; @wangSolarWindCurrent2024], making the assumption a good approximation for the bulk of the population. This single-spacecraft methodology has been widely adopted in recent statistical studies of kinetic-scale current sheets [@vaskoKineticscaleCurrentSheets2021; @vaskoKineticscaleCurrentSheets2022; @lotekarKineticscaleCurrentSheets2022; @vaskoKineticScaleCurrentSheets2024]. + +@erdosDensityDiscontinuitiesHeliosphere2008 reached a partially reconciling conclusion from the Ulysses dataset: for the subset of well-defined discontinuities with reliable normals, MVA and the cross-product technique yield consistent results. They advocated retaining only such well-defined events for further analysis, noting that the vast majority of these have a small magnetic field component parallel to the normal. The practical implication is that MVA can be used reliably when applied with appropriate quality filters, but uncritical application to the full population of discontinuities introduces systematic biases that have historically distorted the statistical picture—particularly the RD/TD classification. + +#### Multi-Spacecraft Methods and Higher-Dimensional Structure + +Multi-spacecraft observations have played a crucial role in advancing our understanding of current sheets, as they allow improved determination of the normal direction and direct separation of spatial structure from temporal evolution [@knetterNewPerspectiveSolar2005; @knetterFourpointDiscontinuityObservations2004; @knetterDiscontinuityObservationsCluster2003; @horburyThreeSpacecraftObservations2001; @nessSimultaneousMeasurementsInterplanetary1966]. The timing method compares the arrival times of the same structure at several spacecraft with known relative positions, enabling geometric inference of the sheet orientation and propagation velocity [@vogtAnalysisDataMultisatellite2020; @paschmannMultispacecraftAnalysisMethods2008; @knetterNewPerspectiveSolar2005; @paschmannAnalysisMethodsMultispacecraft2000]. Missions such as Cluster and MMS have greatly expanded the use of these techniques. + +Beyond providing more accurate normals, multi-spacecraft observations have revealed that many solar wind current sheets are not perfectly planar. @leppingTwodimensionalCurvatureLarge2003, using Wind and IMP-8 data for 134 large-angle ($\omega > 90°$) discontinuities, estimated a weighted-average radius of curvature of $\sim 380\,R_E$ (with a most probable value of $\sim 290\,R_E$; however, their analysis is unable to distinguish real curvature from shorter-scale surface variations using only two-spacecraft data sets), and an average thickness of $\sim 14\,R_E$ (most probable $\sim 6\,R_E$). These results caution against the simplistic use of the planar assumption when projecting a distantly observed discontinuity to predict its arrival characteristics at a downstream location. @malaspinaTwoSpacecraftObservations2012, exploiting the variable separation of the twin STEREO spacecraft, studied tens of thousands of discontinuities and found that the distributions of thickness, normal orientation, shear angle, and waiting times differ systematically between discontinuities observed by both spacecraft and those seen by only one. The population observed by both spacecraft—those with sufficient lateral extent to be intercepted at two separated points—was most consistently interpreted as the walls of solar wind flux tubes. + +@sodingMinimumVarianceAnalysis1999 proposed a method to determine the orientation and propagation velocity of two-dimensional structures using two-spacecraft data under the assumption of a steady-state, divergence-free magnetic field. While no clear 2D structures were identified on the ~10-hour scales examined with Wind and IMP-8, the method established a framework for probing departures from planarity at smaller scales. @tehLocalStructureDirectional2011 subsequently demonstrated with Grad–Shafranov reconstruction that directional discontinuities can contain internal magnetic islands, making them irreducible to simple TD or RD classifications and underscoring the importance of accounting for multidimensional geometry when interpreting spacecraft crossings. + +## Statistical surveys and identification methods + +Following the initial discovery of solar wind current sheets, research shifted toward systematic statistical surveys [@tsurutaniReviewDiscontinuitiesAlfven1999; @neugebauerProgressStudyInterplanetary2010]. This section reviews general statistical properties and the methods used to identify current sheets. Parameters central to understanding both their physical nature and their dynamical influence on energetic particles — magnetic field configuration, spatial scale [@sec-scale_density], and occurrence rate [@sec-occurrence-rate] — are discussed in the subsequent sections. + +A key point that must be emphasized at the outset is that statistical properties depend critically on the identification method. Different selection criteria—thresholds on magnetic field rotation angle, magnetic field increments, partial variance of increments (PVI), or relative standard deviation—introduce systematic biases into the sampled population. Methods optimized for large-amplitude, well-defined discontinuities preferentially select broader, MHD-scale structures, whereas gradient-based or increment-based approaches are more sensitive to thinner, kinetic-scale current sheets. Reported distributions of thickness, current density, and occurrence rate are therefore inherently method-dependent, and care must be taken when comparing results across studies. + +To process the vast amounts of spacecraft data, various automated identification algorithms have been developed. @tbl-identification-methods summarizes the primary quantitative criteria utilized in the literature. + +| Method | Description | Method Reference | Applications | +|--------|------------|-----------------|--------------| +| Directional change | Change in the direction of **B** | @burlagaDirectionalDiscontinuitiesInterplanetary1969 | @sodingRadialLatitudinalDependencies2001 | +| Relative field change | Relative change in magnetic field **B** | @tsurutaniInterplanetaryDiscontinuitiesTemporal1979 | @sodingRadialLatitudinalDependencies2001 | +| Correlation / angle distribution | Two-time correlation functions and distribution of angle change over a time lag | @liIdentifyingCurrentSheetlikeStructures2007 | @liAreThereCurrentsheetlike2008 | +| PVI | Partial Variance of Increments | @grecoPartialVarianceIncrements2017 | @vaskoKineticscaleCurrentSheets2021; @vaskoKineticscaleCurrentSheets2022; @vaskoKineticScaleCurrentSheets2024 | +| Relative standard deviation | Relative standard deviation of **B** | @liuMagneticDiscontinuitiesSolar2022 | @zhangSolarWindDiscontinuities2025a | + +: Summary of current sheet identification methods used in the literature. {#tbl-identification-methods} + +Most early statistical studies focused on large-scale or mesoscale current sheets that are readily identifiable in lower-cadence data. While these works provide essential context, their results cannot be directly compared with statistics derived from high-cadence measurements targeting kinetic-scale structures. Differences in scale selection, detection thresholds, and instrumental resolution introduce subtleties that must be carefully navigated. For instance, as demonstrated by @vasquezNumerousSmallMagnetic2007, current sheets become exponentially more numerous at smaller spread angles, a population often missed by earlier methods which focus on isolated, large-angle events and exclude structures in close proximity to one another. Therefore, while prior studies inform the broader landscape, the results presented in this thesis pertain specifically to the kinetic-scale population and should be interpreted within the framework of the identification methodology employed herein. + +### Small Intensity jump, Δ|B| + +While it is possible for a current sheet to exhibit a large jump in magnetic field magnitude, the vast majority are predominantly characterized by a rotation of the magnetic field across the sheet, with the magnitude remaining nearly constant. This was first recognized by @burlagaDirectionalDiscontinuitiesInterplanetary1969, who observed that most sharp changes in the IMF are primarily directional and introduced the term *directional discontinuity* (DD). Quantitatively, the change in $|B|$ is less than 20% for approximately 75% of the discontinuities in their study. + +More recent high-resolution analyses confirm this near-constant magnitude at smaller scales. Using 1/3-second resolution ACE data, @vasquezNumerousSmallMagnetic2007 found that most discontinuities have ramp-like internal profiles—the field varies nearly monotonically within the layer, with no evidence of rapid compression or dissipation [@tsurutaniRapidEvolutionMagnetic2005]. Furthermore, the intensity jump distribution for solar wind discontinuities is best fit with a lognormal function and is narrowly confined about unity, in contrast to the much broader distribution found in phase-randomized surrogate fields. This difference implies that the layer is regulated by specific physical processes (e.g., nonlinear wave magnetic pressure and Landau damping) rather than random superposition of fluctuations. + +This conclusion is further reinforced by @lotekarKineticscaleCurrentSheets2022 and @vaskoKineticScaleCurrentSheets2024, who analyzed 11200 proton kinetic-scale current sheets near the sun By Parker Solar Probe and 16903 current sheets at 5 AU observed by Ulysses. The magnetic field rotates through a shear angle with only weak magnitude variation. The maximum variation of $|B|$ within a current sheet is statistically larger than the variation between its boundaries, and larger magnitude variations are typical at higher plasma $\beta$ (@fig-vasko2024-fig4). + +![Probability and cumulative distributions of parameters $ΔB/〈B〉$, $ΔB_{\max}/〈B〉$ and $ΔB/〈B〉Δθ$ for subsets of the current sheets (CSs) observed at different plasma betas, β < 1 and β > 3. The bottom panels also present the cumulative distributions corresponding to all the CSs in our data set. Note that parameter $ΔB/〈B〉Δθ$ quantifies the ratio between average perpendicular and parallel current densities within CS.](figures/ref/vaskoKineticScaleCurrentSheets2024-fig4.png){#fig-vasko2024-fig4} + +### Field Rotation Angle + +The field rotation angle $\omega$ (also referred to as the spread angle, shear angle, or directional change) is one of the most fundamental parameters characterizing current sheets. Its distribution depends on the statistical ensemble (i.e., the identification criteria), solar activity level, radial distance from the Sun, and the type of discontinuity. A robust finding across all studies is that discontinuities become more abundant at smaller spread angles. + +@burlagaDirectionalDiscontinuitiesInterplanetary1969 first showed that the number of discontinuities falls off rapidly with increasing $\omega$. Subsequent studies confirmed this behavior and characterized the distribution quantitatively [@burlagaHydromagneticWavesDiscontinuities1971; @burlagaNatureOriginDirectional1971 @marianiVariationsOccurrenceRate1973;]. For $\omega \geq 30°$, the distribution is well described by $N(\omega) \propto \exp\left[-\left(\frac{\omega}{\omega_s}\right)^2\right]$, where the scale parameter $\omega_s$ encodes the characteristic width of the distribution. @burlagaDirectionalDiscontinuitiesInterplanetary1969 found $\omega_s = 75°$ during a solar minimum period (December 1965–January 1966), while @marianiVariationsOccurrenceRate1973 obtained a significantly smaller $\omega_s = 44°$ during a period of higher solar activity. This difference likely reflects a dependence on solar cycle phase [@marianiVariationsOccurrenceRate1973]. @knetterNewPerspectiveSolar2005, using our coordinated spacecraft (Cluster) for four different periods between 2001 and 2003, further demonstrated that $\omega$ depends on solar wind type: the spread angle tends to be smaller in slow solar wind from active regions and larger in fast solar wind originating from coronal holes. + +Extending the analysis to small rotation angles, @vasquezNumerousSmallMagnetic2007 showed that the small-spread-angle population forms a smooth continuation of the larger-angle distribution, which is best fit by a *lognormal* function. For most discontinuities, the maximum spread angle within the layer is nearly equal to the net edge-to-edge value, confirming that the field rotation is approximately monotonic across the structure. This finding was subsequently cited by @neugebauerProgressStudyInterplanetary2010 as evidence that the method of @vasquezNumerousSmallMagnetic2007 captures a previously unexamined but physically continuous population. + +A radial dependence of $\omega_s$ was established by @sodingRadialLatitudinalDependencies2001, who found that the distribution steepens with increasing heliocentric distance: $\omega_s$ decreases from ~82° to ~50° between the inner heliosphere and several AU (see @fig-soding2001-fig11). Fewer events with $\omega > 60°$ are observed at larger distances, indicating that discontinuities evolve during their outward propagation. Whether this evolution leads to eventual annihilation of current sheets remains unclear. Notably, the radial dependence of $\omega_s$ differs depending on the identification criterion: it is present for the Tsurutani–Smith (TS) criterion but absent for the Burlaga (B) criterion inside 2.3 AU, suggesting that the evolution is driven by the additionally identified population of anisotropic RDs. For TDs alone, the mean rotation angle $\langle\omega\rangle \approx 78°$ shows no radial dependence, whereas for RDs, smaller $\omega$ is more probable and fewer large-$\omega$ events survive at greater distances. + +![Relative frequency of $\omega$ for Helios 2 (top) and Voyager 2 (bottom) as a histogram; thin solid line is a fit to the distribution proportional to $\exp\left[-\left(\frac{\omega}{\omega_s}\right)^2\right]$](figures/ref/sodingRadialLatitudinalDependencies2001-fig11.png){#fig-soding2001-fig11} + + + +### Occurrence rate {#sec-occurrence-rate} + +The occurrence rate of current sheets is important from both plasma-physics and particle-transport perspectives. From the standpoint of solar wind physics, occurrence statistics constrain the generation mechanisms of current sheets—including their relation to turbulence intermittency, flux-tube boundaries, and large-scale solar wind structuring—and provide information about their stability and evolution during outward propagation. From the standpoint of energetic particle transport, the frequency with which particles encounter current sheets determines the cumulative scattering rate and therefore influences large-scale diffusion properties in both momentum and configuration space. + +A central question is whether current sheets are formed close to the Sun and subsequently convected outward by the solar wind, or whether they are generated in situ at all heliocentric distances, for example in colliding solar wind streams. As summarized by @knetterDiscontinuityObservationsCluster2003, early radial surveys spanning approximately 0.3 to 10 AU consistently reported a decrease in occurrence rate with increasing radial distance. However, interpreting this trend is not straightforward. The observed decrease may indicate genuine disintegration of current sheets during their outward propagation. Alternatively, it could reflect a changing balance between local generation and destruction processes. It may also arise from observational effects: as structures evolve, their orientation relative to the radial direction may change, reducing the locally detected occurrence rate. In addition, current sheets may thicken with increasing distance, causing them to fall below instrumental detection thresholds and thereby introducing an observational bias [@leppingMagneticFieldDirectional1986]. + +The earliest radial studies established the basic phenomenology. @burlagaNatureOriginDirectional1971, using Pioneer 6 data, found that the occurrence rate at 0.82 AU is only slightly lower than at 1 AU ("this may be due to the lower data quality and increase in the number of data gaps when the spacecraft is far from the earth"), and that the distributions of rotation angle and discontinuity normals are very similar across the range 0.8–1.0 AU. This suggested that most discontinuities originate inside 0.8 AU and do not evolve appreciably over this distance range. Importantly, the occurrence rate in regions of increasing bulk speed was only slightly higher than elsewhere, arguing against stream collision as the primary generation mechanism. @marianiVariationsOccurrenceRate1973, analyzing over 16,000 events from Pioneer 8, reported an average occurrence rate of approximately 3.6 per hour near 1 AU (with ~1.6 per hour identified as TD-like) and found a correlation with the directional change $\omega$ and a decrease with increasing heliocentric distance. However, they also noted a possible dependence on heliographic latitude. + +@tsurutaniInterplanetaryDiscontinuitiesTemporal1979 made a major contribution by using simultaneous Pioneer 10 and 11 data to separate spatial from temporal variations. This distinction was essential, as occurrence rates display substantial day-to-day and solar-rotation-scale fluctuations well outside *Poisson* expectations. They found that the rates averaged over Bartels rotations were strongly correlated between the two spacecraft despite their ~2 AU separation, and that the statistical properties of discontinuities at 1 and 5 AU were remarkably similar. Both findings support a scenario in which discontinuities originate within 1 AU and are subsequently convected outward by the solar wind. The radial dependence of the occurrence rate follows $\rho = 50\,e^{-(R-1)/4}$ per day, corresponding to an apparent decrease of about 25% per AU. However, they argued that this radial gradient may not represent true physical decay: it could arise from progressive thickening of current sheets such that they no longer satisfy identification criteria. Finally, they demonstrated that temporal variations, persisting over several months, had likely been misinterpreted as latitudinal gradients in earlier Pioneer 8 results [@marianiVariationsOccurrenceRate1973]. + +The picture was enriched by Ulysses observations at high heliographic latitudes. @tsurutaniInterplanetaryDiscontinuitiesAlfven1996 found a radial decrease from 1 to 5 AU following $e^{-(r-1)/5}$. More strikingly, the occurrence rate increased by a factor of ~5 as Ulysses moved from Jupiter at 5 AU to 2.5 AU over the south pole (from the ecliptic plane to −80° heliographic latitudes), with a one-to-one correspondence between high occurrence rates and high-speed streams from coronal holes. In these streams, nonlinear outward-propagating Alfvén waves with large transverse fluctuations are ubiquitous, and rotational discontinuities frequently form the edges of phase-steepened Alfvén waves—offering a natural explanation for the elevated occurrence rates. + +@sodingRadialLatitudinalDependencies2001 synthesized data from five missions spanning 0.3–19 AU and $-80°$ to $+10°$ latitude during solar minimum. They found that the occurrence rate depends linearly on solar wind velocity (a geometric effect: faster wind sweeps more plasma volume past the spacecraft per unit time) and decreases radially as $r^{-0.78}$ (TS criterion) or $r^{-1.28}$ (B criterion). After normalizing to 400 km/s and 1 AU, approximately 64 discontinuities per day were identified with both criteria, and no residual dependence on heliographic latitude or solar wind structure type was observed—indicating that current sheets are uniformly distributed on spherical shells. Nonetheless, large day-to-day variations persisted even after normalization. The RD-to-TD ratio depended on solar wind structure, with relatively more RDs in high-speed streams, but showed no radial or latitudinal dependence in the inner heliosphere ($r < 10$ AU). + +@vasquezNumerousSmallMagnetic2007, using their method sensitive to small-spread-angle events, found dramatically higher rates than earlier surveys: an average exceeding 243 per day for all discontinuities, 117 per day above $15°$, and 52 per day above $30°$ (comparable to the ~30 per day above $30°$ reported by classical methods). These rates exhibit pronounced temporal variability on both daily and hourly timescales, and discontinuities occur in spatial groupings with *lognormally* distributed separations. Even excluding active periods (interplanetary shocks, solar ejections), the rates were only weakly correlated with solar wind speed. + +A methodological subtlety that pervades all occurrence rate studies was highlighted by @erdosDensityDiscontinuitiesHeliosphere2008, who used the extensive Ulysses magnetometer dataset to critically examine the role of the identification method. They showed in @fig-erdos2008-fig4 that occurrence rates differ dramatically depending on whether events are selected by their temporal rate of change (in the spacecraft frame) or by their spatial gradient (transformed into the solar wind frame): the temporal criterion systematically overestimates the number of discontinuities in fast solar wind, because structures convect more rapidly past the observer. After correcting for this bias, they confirmed the radial decrease in spatial density with increasing distance from the Sun. And surprisingly, they found that at a given radial distance, periods with slower solar wind tended to contain more discontinuities. + +![The number of discontinuities as a function of the distance from the Sun (horizontal scale) and the velocity of solar wind (color coded). Upper panel: selection of events by time rate of change. Lower panel: selection of events by spatial gradients.](figures/ref/erdosDensityDiscontinuitiesHeliosphere2008-fig4.png){#fig-erdos2008-fig4} + +Recent inner-heliosphere measurements from Parker Solar Probe and Solar Orbiter have extended these statistics closer to the Sun than previously possible. @liuCharacteristicsInterplanetaryDiscontinuities2021 analyzed 3,948 discontinuities between 0.13 and 0.9 AU and found a steep radial decrease following $r^{-2.00}$. A particularly interesting finding was that the RD occurrence rate decreases more steeply ($r^{-2.17}$) than the TD rate, so that the RD-to-TD ratio drops sharply from ~8 at $r < 0.3$ AU to ~1 at $r > 0.4$ AU, exhibiting distinct evolution with distance. + +@madarDirectionalDiscontinuitiesInner2024, combining Solar Orbiter and Parker Solar Probe data, identified over 140,000 discontinuities between 0.06 and 1.01 AU and confirmed the power-law decrease in spatial density, fitting an exponent of $-0.93$—somewhat shallower than the $r^{-2.00}$ of @liuCharacteristicsInterplanetaryDiscontinuities2021, likely reflecting differences in identification criteria and the correction for solar wind velocity effects. They identified several competing mechanisms that shape the radial profile: the increasing Parker spiral angle with distance affects how many TD-like flux-tube boundaries are swept past the spacecraft; the smaller cross-section of flux tubes near the Sun makes boundary crossings more probable; and any radial evolution of current sheet thickness introduces selection biases for gradient-based detection methods. + +Taken together, these studies establish that current sheet occurrence rates decrease with heliocentric distance, but the precise radial scaling depends sensitively on identification criteria and the population of discontinuity types sampled with corrections for solar wind velocity effects. The much higher rates found by methods sensitive to small rotation angles [@vasquezNumerousSmallMagnetic2007] underscore that the total population of current sheets is substantially larger than suggested by classical surveys restricted to large-angle events. The differential radial evolution of RDs and TDs points to fundamentally different origins and lifetimes for these two populations—a distinction with direct implications for understanding how they are generated and sustained by solar wind turbulence. + +### Spatial Scale and Current Density {#sec-scale_density} + +In the classical MHD framework, current sheets are treated as infinitely thin discontinuities—mathematical step functions across which plasma parameters change instantaneously. In reality, however, the transition between upstream and downstream regions occurs over a finite thickness, requiring a treatment beyond the MHD approximation. The impact of a current sheet on the plasma — and specifically on the dynamical process of particles within the sheet [@yamadaMagneticReconnection2010] — is fundamentally governed by this spatial scale [@sergeevCurrentSheetThickness1990] and the associated internal current density (the ratio between gyroradius and the characteristic scale of magnetic inhomogeneity). These two closely coupled parameters (usually compared to local proton inertial length and Alfvén current density) dictate the transition from fluid-like MHD behavior to kinetic physics and are therefore central to understanding the role of current sheets in both turbulence dissipation and particle transport. + +The thickness of a current sheet determines the physical regime in which the structure operates. The critical threshold occurs when the thickness approaches fundamental kinetic length scales: the ion inertial length $d_i = c/\omega_{pi}$ or the thermal ion gyroradius $\rho_i$. When $\lambda \sim \rho_i$, the assumptions of ideal MHD break down: ions become demagnetized within the sheet while electrons—with their much smaller gyroradius—remain magnetized. Test particle simulations reveal the kinetic consequences of this intermittent structure: @dmitrukTestParticleEnergization2004 found that the current sheets spontaneously formed by MHD turbulence produce differential energization, with electrons developing large parallel velocities within current sheets while protons are energized preferentially in the perpendicular direction by nonuniform electric fields varying on proton kinetic scales. This differential response generates Hall electric fields and enables the onset of collisionless magnetic reconnection, which requires current sheet thicknesses comparable to $d_i$ to proceed at sufficiently fast rates. @cassakModelSpontaneousOnset2006 showed that as a Sweet–Parker dissipation region dynamically thins during reconnection, a critical transition occurs when its width drops below $d_i$: the Hall effect becomes dominant, the resistive MHD solution ceases to exist, and the system transitions abruptly to fast collisionless reconnection with rates orders of magnitude higher. @papiniFastMagneticReconnection2019 extended this picture by investigating the tearing instability in both the MHD and Hall-MHD regimes. They showed that when a current sheet achieves a sufficiently small aspect ratio ($a/L \sim S^{-1/3}$ for Lundquist number $S \gg 1$), reconnection proceeds on ideal Alfvén timescales independent of $S$. In the nonlinear phase, secondary current sheets spontaneously form and, at high $S$, naturally adjust to this critical aspect ratio, driving very rapid reconnection. When the Hall term is included—appropriate once the resistive layer width $\delta$ becomes comparable to $d_i$—the secondary reconnection rate is enhanced by up to a factor of two relative to the pure MHD case and up to ten times faster than the linear phase, leading to explosive energy release on super-Alfvénic timescales. + +The interplay between reconnection and the turbulent cascade is now recognized as fundamental [@boldyrevTearingInstabilityAlfven2020; @boldyrevRoleReconnectionInertial2019]. In dynamically aligned Alfvénic turbulence, magnetic fluctuations naturally form progressively thinner sheet-like structures at smaller scales [@boldyrevSpectrumMagnetohydrodynamicTurbulence2006]. Analytic theories predict that below a critical thickness these sheets become tearing-unstable, disrupting the classical cascade. @malletDisruptionAlfvenicTurbulence2017 calculated the disruption scale $\lambda_D$ at which this onset occurs in a low-$\beta$ collisionless plasma, showing that $\lambda_D$ can exceed the ion sound scale $\rho_s$ and produce a spectral break at $\lambda_D$ rather than at $\rho_s$, with a steepened "transition range" between them—a feature sometimes observed in solar wind turbulence intervals. @boldyrevMagnetohydrodynamicTurbulenceMediated2017 proposed a complementary picture in which the tearing instability modifies the effective alignment of field lines, balancing the eddy turnover rate at all scales below the critical threshold and yielding a reconnection-mediated energy spectrum steeper than the classical prediction. These theoretical expectations have been confirmed numerically. @dongRolePlasmoidInstability2018, in high-resolution 2D MHD simulations at magnetic Reynolds number $R_m = 10^6$, showed that the combined effects of dynamic alignment and turbulent intermittency produce copious plasmoid formation in intense current sheets; the resulting disruption steepens the energy spectrum toward a spectral index near $-2.2$, consistent with the analytic predictions. @dongReconnectiondrivenEnergyCascade2022 extended this to three dimensions, demonstrating that rapid reconnection breaks elongated current sheets into chains of plasmoids and opens a previously unrecognized range of energy cascade in which the transfer rate is controlled by plasmoid growth, again producing a $-2.2$ spectral index accompanied by modified turbulence anisotropy. @franciMagneticReconnectionDriver2017, using high-resolution hybrid-kinetic simulations that retain ion kinetic effects, provided further confirmation: reconnection at current sheets with $a \simeq d_i$ actively drives the sub-ion-scale cascade, generating a stable power-law spectrum below the ion break as soon as the first reconnection events occur—regardless of the state of the large-scale cascade. Taken together, these results establish that current sheets at kinetic scales actively shape the turbulence spectrum, mediate the energy cascade across the ion break, and control the pathway by which magnetic energy is ultimately converted into particle heating. + +For energetic particle transport, the spatial scale is equally decisive. When particles encounter a broad MHD-scale structure ($\lambda \gg \rho_{\mathrm{SEP}}$), their motion remains adiabatic and they smoothly follow guiding-center trajectories. When $\lambda \sim \rho_{\mathrm{SEP}}$, however, the magnetic field changes too abruptly for adiabaticity to be maintained, leading to strong pitch-angle scattering, temporary trapping, or reflection. The implications of this resonance condition for SEP transport will be examined in detail in the following sections. + +Current density is inextricably linked to spatial scale through Ampère's law: $\mathbf{J} = \nabla \times \mathbf{B}/\mu_0$. A thin magnetic field rotation necessarily implies an intense current layer. In the context of magnetic turbulence, the energy cascading from large to small scales is not dissipated uniformly but is concentrated within coherent structures—predominantly thin, high-current-density sheets. Numerical simulations have quantified this intermittency in detail. @zhdankinStatisticalAnalysisCurrent2013, analyzing current sheets in driven reduced-MHD turbulence, found that structures with peak current density exceeding eight times the rms value occupy less than 1% of the simulation volume yet account for more than 25% of all Ohmic dissipation. They also showed that while not all intense current sheets contain magnetic X-points (about 55% do not), the most intense structures are preferentially reconnecting ones, with the probability of containing an X-point rising to ~90% for the strongest events. + +Spacecraft observations corroborate this picture. Current sheets are correlated with enhanced electron and ion temperatures [@osmanEvidenceInhomogeneousHeating2011], and these structures, while constituting only ~19% of the data, can account for ~50% of the total plasma internal energy [@osmanIntermittencyLocalHeating2012]. If reconnection is triggered within an intense current sheet, the contracting magnetic islands can further accelerate particles to high energies. Recent Parker Solar Probe observations have provided direct evidence of proton acceleration up to ~400 keV within the reconnection exhaust of the heliospheric current sheet at ~16 $R_\odot$ [@desaiMagneticReconnectionDriven2025]. + +Understanding how thickness and current density of current sheets evolve with heliocentric distance is therefore crucial for revealing their role in the thermodynamics of solar wind turbulence and dynamics of energetic paticles: whether these structures maintain their kinetic-scale character and whether their current density weakens in tandem with the radial drop in magnetic field constrains theories of their local generation, evolution, and overall contribution to energy dissipation and particle transport throughout the heliosphere. + +#### Thickness + +The earliest thickness estimates, necessarily limited by instrumental resolution, revealed structures with spatial scales of thousands of kilometers or tens of proton inertial lengths [@burlagaDirectionalDiscontinuitiesInterplanetary1969]. + +The radial evolution of current sheet thickness was first systematically studied by @sodingRadialLatitudinalDependencies2001, who analyzed discontinuities from five missions spanning 0.3–19 AU. They found that the mean thickness in kilometers increases with heliocentric distance, as expected from the radial decrease in magnetic field strength and the associated expansion of kinetic length scales. However, when normalized to the local proton gyroradius, the thickness decreases dramatically—by a factor of ~50, from $\langle d_{\rho_g}\rangle \approx 201\,\rho_g$ at 0.3 AU down to $\langle d_{\rho_g}\rangle \approx 4.3\,\rho_g$ at 19 AU. A similar decrease, from $127\,d_i$ to $2.6\,d_i$, was found when normalizing to the ion inertial length. RDs were consistently thicker than TDs by a factor of ~1.5. This strong radial thinning in normalized units indicates that current sheets do not simply expand passively with the solar wind but evolve dynamically, progressively approaching kinetic scales at larger distances. + +Parker Solar Probe has extended these measurements into the pristine inner heliosphere. @liuCharacteristicsInterplanetaryDiscontinuities2021, analyzing discontinuities between 0.13 and 0.9 AU, found distinct behavior for the two types: TD thicknesses normalized by $d_i$ show no clear spatial scaling and range broadly from 5–35 $d_i$, whereas RD thicknesses decrease as $r^{-1.09}$ in normalized units. In absolute terms, the average RD thickness of ~574 km changes little with distance, implying that the normalized thinning reflects the radial increase of $d_i$ itself. @madarDirectionalDiscontinuitiesInner2024, using combined Solar Orbiter and PSP data from 0.06 to 1.01 AU, reported a more nuanced picture: RD thickness first *decreases* between 0.06 and 0.30 AU, then increases beyond 0.30 AU in proportion to the local ion inertial length. TD thickness, by contrast, scales with $d_i$ throughout the entire distance range. The authors interpreted these trends as evidence for different physical origins of the two types of discontinuities. RDs are thought to arise from the nonlinear steepening of Alfvén waves, a process that tends to generate structures with a significant magnetic-field component normal to the discontinuity surface. As the steepening progresses, the characteristic thickness decreases until it approaches ion kinetic scales, where dispersive and kinetic effects limit further steepening. TDs, on the other hand, are more likely associated with boundaries between magnetic flux tubes, whose widths tracks the local kinetic scale. + +High-cadence measurements have made it possible to resolve the kinetic-scale population directly. @vaskoKineticscaleCurrentSheets2022, using Wind data at 11 samples/s, characterized 17,043 current sheets at 1 AU with thicknesses from a few tens to ~1,000 km, corresponding to $\sim$0.1–10$\,\lambda_p$ with typical values around 100 km (~a few $\lambda_p$). Near the Sun, @lotekarKineticscaleCurrentSheets2022 analyzed 11,200 current sheets around PSP's first perihelion, finding thicknesses from a few to ~200 km (typical value ~30 km), or $\sim$0.1–10 $\lambda_p$ with a typical value of ~2 $\lambda_p$. At 5 AU, @vaskoKineticScaleCurrentSheets2024 found half-thicknesses of 200–2,000 km for non-bifurcated current sheets and 500–5,000 km for bifurcated ones, corresponding to 0.5–5 $\lambda_p$ and 0.7–15 $\lambda_p$ respectively. Despite the enormous difference in absolute scale across these distances, the remarkable consistency in normalized thickness—typically a few proton inertial lengths—indicates that current sheets at all heliocentric distances are predominantly kinetic-scale structures whose width is set by the local plasma conditions. + +#### Current Density and Scale-Dependent Properties + +The current density within kinetic-scale current sheets is not independent of their spatial scale. @vaskoKineticscaleCurrentSheets2022 found at 1 AU that the current density increases systematically for thinner structures, following $J_0 \approx 6\;\mathrm{nA/m^2}\cdot(\lambda/100\;\mathrm{km})^{-0.56}$, but does not statistically exceed a critical value $J_A$ corresponding to an ion-electron drift at the local Alfvén speed. In normalized units, this becomes $J_0/J_A \approx 0.17\cdot(\lambda/\lambda_p)^{-0.51}$. A corresponding power-law correlation was observed near the Sun by @lotekarKineticscaleCurrentSheets2022, who found $J_0 \approx 0.15\;\mu\mathrm{A/m^2}\cdot(\lambda/100\;\mathrm{km})^{-0.76}$ with current densities in the range 0.1–10 $\mu$A/m², and at 5 AU by @vaskoKineticScaleCurrentSheets2024, who reported $J_0/J_A \approx 0.14\cdot(\lambda/\lambda_p)^{-0.66}$ with typical current densities of 0.05–0.5 nA/m². + +These current sheets are statistically force-free: the current density is dominated by its magnetic-field-aligned component, consistent with the observation that $|B|$ does not vary substantially across them. The magnetic shear angle is also correlated with spatial scale: @vaskoKineticscaleCurrentSheets2022 found $\Delta\theta \approx 19°\cdot(\lambda/\lambda_p)^{0.5}$ at 1 AU, while @vaskoKineticScaleCurrentSheets2024 found $\Delta\theta \approx 16.6°\cdot(\lambda/\lambda_p)^{0.34}$ at 5 AU. These scale-dependent correlations—thinner sheets carrying proportionally stronger currents with smaller shear angles—are a natural consequence of the turbulent cascade, in which magnetic field gradients steepen as energy is transferred to smaller scales. The approximate scale-invariance of these relationships across heliocentric distances (from 0.17 AU to 5 AU), together with the matching of magnetic field rotation and compressibility between current sheets and the ambient turbulence, provides strong evidence that the majority of kinetic-scale current sheets are produced by the turbulent cascade. + +The observation that current density does not exceed the Alfvén current density $J_A$ is physically significant but not yet fully understood. From the standpoint of reconnection, @vaskoKineticscaleCurrentSheets2021 showed that essentially all 18,785 kinetic-scale current sheets in their dataset satisfy the necessary condition for reconnection not to be suppressed by diamagnetic drift of the X-line. This condition, $\Delta\beta \lesssim 2(L/\lambda_p)\tan(\Delta\theta/2)$, is automatically met due to the geometry of the current sheets as dictated by the turbulent cascade, rather than being a coincidence of local plasma parameters. The same conclusion was reached near the Sun [@lotekarKineticscaleCurrentSheets2022] and at 5 AU [@vaskoKineticScaleCurrentSheets2024]. + + +### Alfvénicity, Walén Relation and Propagation Direction + + + +The relationship between velocity and magnetic field variations across current sheets—their degree of Alfvénicity—has been a central and persistently debated topic, bearing directly on the classification of discontinuities as RDs or TDs and on their dynamical role in the solar wind. + +The pioneering observation by @neugebauerAlignmentVelocityField1985, using IMP 8 and Voyager 2 data, revealed that velocity and magnetic field jumps ($\Delta\mathbf{v}$ and $\Delta\mathbf{B}/\sqrt{\rho}$) across tangential discontinuities (a large change in magnetic field strength) are closely aligned—either parallel or antiparallel—in the sense associated with outward-propagating Alfvén waves. This alignment was found to be independent of solar wind stream structure and heliocentric distance between 1 and 2.2 AU. The result was unexpected for structures classified as TDs, and several explanations were proposed, including interplanetary turbulence, large-amplitude Alfvénic fluctuations propagating independently on both sides of the discontinuity, and surface waves on TDs. + +@neugebauerTangentialDiscontinuitiesSolar1986 confirmed using Helios data that this alignment is already well established inside 0.4 AU, suggesting it is not a product of in situ evolution but may instead reflect a selection effect: TDs for which $\Delta\mathbf{v}$ and $\Delta\mathbf{B}$ are not aligned are destroyed by the Kelvin–Helmholtz instability, so that only Alfvénically aligned TDs survive. The observed decrease in the total number of discontinuities with increasing heliocentric distance may be associated with the growth of this instability as the Alfvén speed declines. + +An earlier comprehensive study by @neugebauerReexaminationRotationalTangential1984, using ISEE 3 data, showed that the relative directions of velocity and field changes across all three discontinuity types (RD, TD, and the intermediate "either" category, ED) are consistent with outward propagation. The magnitude of the velocity change at RDs was found to be systematically smaller than the MHD prediction — a discrepancy only partially reduced by using a two-stream proton fit — foreshadowing the broader $R < 1$ puzzle discussed below. Further, the plasma jump conditions at EDs showed closer resemblance to RDs than to TDs. + +Quantitative assessment of Alfvénicity relies on the Walén relation, which states that the velocity jump across an RD should equal the corresponding Alfvén velocity jump. Two complementary approaches have been developed (see the bottom panels of @fig-paschmann2013-fig2). The first evaluates the Walén relation as a jump condition by comparing velocity and Alfvén velocity changes between two carefully chosen measurement times on opposite sides of the discontinuity. The second checks the level of Alfvénicity continuously for all measurements between those two points: plasma velocity components, after transformation into the de Hoffmann–Teller (HT) frame, are plotted against the corresponding Alfvén velocity components, and the slope of the regression line, $W_{\mathrm{sl}}$, serves as the quality index, with $W_{\mathrm{sl}} = \pm 1$ indicating perfect Alfvénic agreement. + +![Overview plots for DD crossings. For each case, the five panels at the top show the magnetic field magnitude, the plasma density, followed by a comparison between the three components of $\mathbf{v}' = (\mathbf{v} - \mathbf{V}_{\mathrm{HT}})$ (in black) and (in red) the three components of $-\mathbf{V}_A$ or $\mathbf{V}_A$ (depending on the sign of the Walén slope), all from Cluster C1, with the DD at the center of the time series. The panels along the bottom show the HT scatterplot for the 10 min interval, and the Walén scatterplots for the full 10 min and for the 1 min interval centered on the DD. In these scatterplots the vector components are distinguished by their color (black for x, red for y, and green for z).](figures/ref/paschmannDiscontinuitiesAlfvenicFluctuations2013-fig2case1.png){#fig-paschmann2013-fig2} + +@neugebauerCommentAbundancesRotational2006 reported that the magnitude ratio $R = |\Delta\mathbf{v}|/|\Delta\mathbf{v}_A|$ from the jump approach is commonly around 0.6—systematically less than unity. @paschmannDiscontinuitiesAlfvenicFluctuations2013, using Cluster data, performed a comprehensive Walén analysis on 188 directional discontinuities and found that a substantial fraction (77 out of 127 with a good de Hoffmann–Teller frame) exhibited plasma flow speeds exceeding 80% of the Alfvén speed, with 33 cases exceeding 90%. Their analysis also established that the degree of Alfvénicity of the coherent current sheets is nearly the same as that of the fluctuations in which they are embedded, suggesting that whatever process causes deviations from ideal Alfvénicity operates equally on both. This result places current sheets on a continuum with the ambient Alfvénic turbulence rather than as dynamically distinct structures. + +A critical complication in using the Walén test for RD/TD classification was identified by @madarDirectionalDiscontinuitiesInner2024, who analyzed over 140,000 directional discontinuities between 0.06 and 1.01 AU using Parker Solar Probe and Solar Orbiter data. They showed that Alfvén waves propagating along the surface of TDs can produce positive Walén test results, mimicking RD signatures. To disentangle the two populations, they examined the velocity in the HT frame: for surface waves on TDs, the residual velocity $(\mathbf{V} - \mathbf{V}_{\mathrm{HT}})$ lies close to the discontinuity plane, whereas for genuine RDs it is quasi-perpendicular to the surface. A scatter plot of $B_n/B_{\max}$ against $(\mathbf{V} - \mathbf{V}_{\mathrm{HT}}) \cdot \hat{n} / |\mathbf{V} - \mathbf{V}_{\mathrm{HT}}|$ revealed two clearly distinct populations, confirming that many apparent RD candidates are in fact TDs with surface Alfvén waves. After this reclassification, they found that most discontinuities with small normal magnetic field components are TDs, regardless of the jump in field magnitude. + +The systematic shortfall $R < 1$ has remained a longstanding puzzle. As noted by @paschmannDiscontinuitiesAlfvenicFluctuations2013, this deficiency mirrors the behavior of Alfvénic fluctuations more broadly: the Alfvén ratio $r_A = \delta v^2/\delta v_A^2$ is known to decrease systematically with heliocentric distance, reaching approximately 0.5 at 1 AU [@belcherLargeamplitudeAlfvenWaves1971; @borovskyVelocityMagneticField2012]. + +Refined scalar measures for evaluating Alfvénicity have been developed to disentangle directional and magnitude deviations. @sonnerupQualityMeasureWalen2018 introduced a quality index $Q$ that incorporates both the angular deviation and the magnitude ratio between $\Delta\mathbf{v}$ and $\Delta\mathbf{v}_A$, with $Q = \pm 1$ indicating perfect agreement. @paschmannComparisonQualityMeasures2020 systematically compared the jump-based index $Q$ with the regression slope $W_{\mathrm{sl}}$ across nearly 1,000 magnetopause crossings, finding that a substantially higher threshold is needed for $|Q|$ than for $|W_{\mathrm{sl}}|$ to yield comparable numbers of RD candidates, and that the events selected by the two methods are not identical. They concluded that a complete evaluation of Alfvénicity requires two scalar quality measures: the magnitude ratio $R = |\Delta\mathbf{v}|/|\Delta\mathbf{v}_A|$ and the angle $\Theta$ between $\Delta\mathbf{v}$ and $\Delta\mathbf{v}_A$ [@paschmannLargeScaleSurveyStructure2018]. + +### Additional Statistical Properties + +This subsection briefly summarizes several additional properties of solar wind current sheets that, while not the primary focus of this thesis, form an important part of the broader observational picture and provide useful context for interpreting the current sheet population. + +#### Orientation + +The orientation of current sheet normals relative to the local magnetic field depends on both the type of discontinuity and heliocentric distance. In the inner heliosphere, @marianiVariationsOccurrenceRate1973 analyzed Pioneer 8 data (10 s cadence) and found that TD normals are predominantly perpendicular to the local Parker Archimedean spiral field, consistent with TDs separating adjacent flux tubes. @sodingRadialLatitudinalDependencies2001 corroborated this and highlighted a contrasting behavior for rotational discontinuities (RDs): inner heliospheric RD normals are primarily parallel to the mean field, $\mathbf{B}_0$ (i.e., the normal-to-field angle $\gamma \approx 0^\circ$). This aligns with Alfvén waves propagating along the field, although obliquely propagating RDs are also present. Conversely, in the middle heliosphere (10–40 AU), the distribution of $\gamma$ becomes nearly uniform across all directions, with a notable depletion of RDs propagating parallel to $\mathbf{B}_0$. To explain this radial evolution, @sodingRadialLatitudinalDependencies2001 suggested that field-aligned RDs are unstable over large spatial and temporal scales so the effects of this instability are absent in the inner heliosphere (0.3–2.3 AU) but become pronounced at greater distances. + +#### Plasma Beta Dependence + +The properties of current sheets depend on the local plasma $\beta$. @shaikhBetadependentPropertiesSolar2026, using interplanetary coronal mass ejections as a natural laboratory spanning a broad range of beta ($10^{-2} \lesssim \beta_e \lesssim 10$, $10^{-3} \lesssim \beta_p \lesssim 10$), showed that both the shear angle $\Delta\theta$ and the normalized thickness $\lambda/\lambda_p$ of current sheets depend on electron and proton beta. They argued that the beta dependence of the shear angle is an intrinsic feature of solar wind turbulence arising from the natural correlation between turbulence intensity and plasma beta. According to recent theory, current sheets formed in turbulence are disrupted by the electron tearing instability once their thickness falls below a critical scale, mediating the transition from the inertial to the kinetic cascade [@malletDisruptionSheetlikeStructures2017]. @shaikhBetadependentPropertiesSolar2026 demonstrated that normalizing current sheet thickness by this critical scale eliminates the beta dependence across the entire measured range. + +#### Bifurcated Current Sheets and Reconnection Exhausts + +A distinctive subclass of solar wind current sheets exhibits bifurcated structure—a double-step magnetic field rotation rather than as a single monotonic rotation [@goslingMagneticReconnectionSolar2012; @neugebauerProgressStudyInterplanetary2010]. As reviewed by @goslingMagneticReconnectionSolar2012, these bifurcated current sheets are the observational signature of magnetic reconnection in the solar wind. When reconnection occurs, the reconnecting current sheet splits into a pair of back-to-back rotational discontinuities that bound a wedge of accelerated plasma—the reconnection exhaust. The exhaust plasma flows at roughly the local Alfvén speed, with correlated changes in $\mathbf{V}$ and $\mathbf{B}$ at one boundary and anti-correlated changes at the other, reflecting the oppositely propagating Alfvénic disturbances generated by the reconnection process. + +The observational picture of reconnection in the solar wind has developed rapidly since the first unambiguous identification of reconnection exhausts by @goslingDirectEvidenceMagnetic2005. Using Wind 3-second data, @goslingMagneticReconnectionSolar2012 reported typical occurrence rates of 40–80 exhausts per month at 1 AU near solar minimum, with most events having temporal widths of tens of seconds (local widths of order $10^4$ km). Reconnection occurs most frequently at current sheets with field shear angles below $90°$—simply because such current sheets are the dominant type in the solar wind—and has been observed at shear angles as small as $11°$ [@goslingBifurcatedCurrentSheets2008]. The narrowest exhaust identified had a local width of $\sim 10^3$ km ($\sim 18\,\lambda_i$), and current sheets thinner than $\sim 3\,\lambda_i$ were absent from the dataset, suggesting that such ultrathin structures are quickly disrupted by reconnection. + +Multi-spacecraft observations have revealed that reconnection in large-scale current sheets typically occurs in a quasi-stationary fashion at a single dominant, extended X-line. The most extensive event documented involved five spacecraft and demonstrated continuous reconnection persisting for over 5 hours along an X-line extending at least $4.26 \times 10^6$ km [@goslingMagneticReconnectionSolar2012]. Exhaust boundaries are roughly planar on large scales, though finer-scale corrugations are sometimes observed. The occurrence of reconnection depends on a combination of magnetic shear angle and the plasma $\beta$ difference across the current sheet: for low $\beta$, reconnection occurs at essentially all shear angles, whereas for high $\beta$ it is restricted to large shear angles—consistent with the theoretical prediction that diamagnetic drift of the X-line suppresses reconnection at low-shear, high-$\beta$ current sheets. + +@goslingBifurcatedCurrentSheets2008, examining an interval containing 11 reconnection exhausts and 27 thin current sheets within a magnetic cloud and its trailing high-speed stream, found that at least three of the thin sheets also exhibited bifurcated structure, indicating that they had been disrupted by reconnection. The relative absence of ultrathin current sheets was interpreted as evidence that such structures are rapidly disrupted once they form. More recently, @yoonCollisionlessRelaxationDisequilibrated2021 showed through particle-in-cell simulations that bifurcated current sheets can also arise naturally from the collisionless equilibration of a disequilibrated current sheet, through transitions among single-particle orbit classes, without requiring active reconnection. This suggests that not all observed bifurcated structures necessarily indicate ongoing or recent reconnection, and that collisionless relaxation may be an additional pathway to bifurcation. + +It should be noted that the reconnection exhausts described above are observed at relatively large-scale current sheets resolvable with 3-second plasma cadence ($\gtrsim 10^3$ km). Whether reconnection at kinetic-scale current sheets—the focus of this thesis—produces qualitatively similar or distinct signatures remains an active area of investigation, as discussed in the context of reconnection onset conditions earlier in this chapter. + +#### Contribution to the Magnetic Fluctuation Spectrum + +Current sheets contribute significantly to the power of magnetic field fluctuations in the solar wind. @borovskyContributionStrongDiscontinuities2010, analyzing 8.5 years of ACE magnetometer data, constructed an artificial time series preserving only the timing and amplitudes of strong (large-rotation-angle) discontinuities. The power spectrum of this discontinuity series follows a power law in the inertial subrange with spectral index near the Kolmogorov $-5/3$ value, and accounts for approximately half of the total spectral power of the solar wind magnetic field over this range. This result warns that the measured spectral properties of solar wind turbulence are heavily influenced by the discrete contribution of current sheets, complicating the interpretation of spectral indices. + +@liEffectCurrentSheets2011 provided complementary evidence from three years of Ulysses data in which over 28,000 current sheets were identified. They showed that during the longest current-sheet-free intervals, the magnetic field power spectra are consistently described by the Iroshnikov–Kraichnan $k^{-3/2}$ scaling, whereas during the most current-sheet-abundant intervals, the spectra exhibit Kolmogorov $k^{-5/3}$ scaling. This finding implies that the commonly observed Kolmogorov scaling in the solar wind may be a consequence of the ubiquitous presence of current sheets rather than an intrinsic property of the underlying turbulent cascade, and that a proper analysis of solar wind power spectra must account for the contribution of intermittent structures. + +#### Other Plasma Jump Conditions + +Beyond the magnetic field signatures discussed above, the behavior of plasma parameters across current sheets provides additional constraints on their nature and on the validity of the RD/TD classification. + +@neugebauerReexaminationRotationalTangential1984 conducted a comprehensive examination of plasma jump conditions across 221 discontinuities using ISEE 3 magnetic field and proton data. They found that the first and second adiabatic invariants ($T_{p\perp}/B$ and $T_{p\parallel}B^2/n^2$) are approximately conserved across RDs but not across TDs, confirming that the MVA-based classification into these two types captures a genuine physical distinction. The product of plasma density and the anisotropy factor, $\rho A$, tends to be conserved across all three discontinuity types (RDs, TDs, and EDs)—a result expected for RDs but not required by MHD theory for TDs. The helium abundance $n_\alpha/n_p$ is generally conserved across RDs but can change substantially across TDs; however, a broad tail in the $n_\alpha/n_p$ distribution for RDs indicates that the helium abundance does change at a small fraction of them. + +The multi-species dynamics at RDs proved particularly revealing. @neugebauerReexaminationRotationalTangential1984 showed that the primary and secondary proton beams flow through RDs in opposite directions with oppositely directed velocity changes, while alpha particles interact much more weakly—with velocity changes clustered near zero and no preferred direction. They proposed a simple model in which the RD moves through the primary proton fluid at slightly less than the local Alfvén speed; because the alpha particles drift relative to the primary protons at approximately this same speed, they effectively co-move with the RD and do not cross it. Under these conditions, the jump conditions for alphas resemble those across a contact discontinuity, explaining how the helium abundance can change even across a genuine RD. When this multi-stream model was used to recalculate the Walén ratio $R_{VB}$, it increased from $0.59 \pm 0.03$ (single-stream proton moments) to $0.74 \pm 0.02$, and further inclusion of alpha particle anisotropy and estimated electron contributions raised it to $0.77 \pm 0.03$—diminishing but not eliminating the longstanding $R_{VB} < 1$ discrepancy. + +The properties of the magnetically ambiguous EDs were found to resemble those of RDs much more closely than those of TDs across essentially all parameters: adiabatic invariants, density conservation, helium abundance, Walén ratio, and mean solar wind speed. This led @neugebauerReexaminationRotationalTangential1984 to conclude that the ED population consists predominantly of obliquely propagating RDs with small but finite $B_n$, rather than TDs with coincidentally small $\Delta|B|$. + +## Summary and Open Questions + +The body of work reviewed in this chapter establishes a rich but incomplete picture of solar wind current sheets. Their basic phenomenology is well characterized: the magnetic field rotates through a shear angle with only weak magnitude variation; the thickness of kinetic-scale current sheets is typically a few proton inertial lengths regardless of heliocentric distance; current density scales inversely with thickness in a manner consistent with the turbulent cascade; and occurrence rates decrease with radial distance from the Sun, though the precise scaling depends sensitively on identification criteria and corrections for solar wind speed and current sheet orientation. The majority of these structures are statistically force-free, satisfy the necessary conditions for magnetic reconnection onset, and share the Alfvénic character of the ambient turbulence in which they are embedded. + +Despite this progress, several important questions remain open and motivate the investigations presented in subsequent chapters of this dissertation. + +**Unified method across scales and distances.** Previous studies have typically examined either large-scale discontinuities or kinetic-scale current sheets, often with different identification methods, instrumental cadences, and spacecraft datasets. As a result, a coherent picture of how the current sheet population evolves across spatial scales and heliocentric distances has not yet emerged. Developing a detection framework capable of capturing current sheets over a broad range of scales — from the kinetic regime to the MHD regime — and applying it consistently across multiple radial distances would therefore be highly valuable. + +**Separation of temporal variability from radial evolution.** A persistent difficulty in interpreting radial trends is the contamination by temporal variability. Occurrence rates fluctuate substantially on timescales from hours to solar rotations, and single-spacecraft surveys at different heliocentric distances inevitably sample different solar wind conditions at different times. Multi-spacecraft observations and extended orbital coverage offer opportunities to separate genuine spatial evolution from temporal modulation, but this separation has rarely been achieved systematically for the kinetic-scale population. + +**Multi-scale nature, distinct sub-populations, and their origins.** Current sheets span a vast range of spatial scales, and it is plausible that structures at different scales originate from different physical mechanisms — for example, coronal flux-tube boundaries, nonlinear Alfvén wave steepening, or turbulence-driven intermittency. Furthermore, these distinct populations may follow different evolutionary paths as they propagate outward with the solar wind. Observational evidence already hints at such differentiation: the RD-to-TD ratio declines steeply inside 0.5 AU, pointing to fundamentally different generation mechanisms and lifetimes for the two classes. Yet at kinetic scales, the classical TD/RD distinction becomes blurred, as individual structures frequently exhibit signatures of both types simultaneously. Identifying physically meaningful sub-populations within the current sheet ensemble, characterizing their properties, and tracking their evolution with heliocentric distance remains an important challenge. + +**Interaction with energetic particles.** The spatial scales of kinetic-scale current sheets — comparable to the gyroradii of energetic particles in the keV to MeV range — place them in a regime where adiabatic particle motion breaks down. The cumulative effect of encounters with such structures on pitch-angle scattering, cross-field transport, and particle energization is not yet quantitatively understood from an observational standpoint. Establishing the statistical framework of current sheet properties is a necessary prerequisite for addressing these transport questions, which will be taken up in later chapters. + +These questions collectively define the motivation for the analyses presented in the following chapters, where we develop and apply a consistent methodology for identifying and characterizing current sheets across heliocentric distances, and examine their implications for both solar wind turbulence and energetic particle dynamics. + + diff --git a/docs/others/phd/2026_grad/_review_energetic_particles.qmd b/docs/others/phd/2026_grad/_review_energetic_particles.qmd new file mode 100644 index 0000000..169d8b2 --- /dev/null +++ b/docs/others/phd/2026_grad/_review_energetic_particles.qmd @@ -0,0 +1,350 @@ +# Energetic Particle Interaction With Solar Wind Current Sheets + + + +The transport of energetic particles through the heliosphere is governed not only by the large-scale structure of the interplanetary magnetic field, but also by the small-scale, intermittent structures embedded within it [@ewartCosmicrayTransportInhomogeneous2025; @engelbrechtTheoryCosmicRay2022; @oughtonSolarWindTurbulence2021; @vandenbergPrimerFocusedSolar2020]. Chief among these are current sheets—thin layers of intense current and rapid magnetic field rotation that occupy a small fraction of the heliospheric volume yet exert a disproportionate influence on particle dynamics. This chapter reviews the theoretical and observational foundations necessary to understand how energetic particles interact with these structures, with particular focus on the mechanisms by which current sheets scatter particles in pitch angle and modulate their transport through the heliosphere. + +We begin with the heliospheric context: the sources and classification of solar energetic particles, and the observational phenomena—reservoir effects and dropout events—that reveal the dual role of current sheets as both barriers and facilitators of transport. We then describe the standard transport frameworks within which pitch-angle scattering is parameterized, and discuss how solar wind turbulence intermittency motivates going beyond classical wave-based scattering theories. The core of the chapter develops the quasi-adiabatic theory of charged particle dynamics near magnetic field reversals, tracing how the adiabatic invariant breaks down at separatrix crossings through geometrical and dynamical jumps—a scattering mechanism qualitatively distinct from, and potentially more efficient than, wave–particle resonance [@artemyevElectronPitchangleDiffusion2013]. The chapter closes with a summary of the key results and a discussion of the open questions that motivate the quantitative program developed in subsequent chapters. + +## Energetic Particles in the Heliosphere + +### Sources of Energetic Particles + +The heliosphere hosts several distinct populations of energetic particles spanning a vast range of energies and origins [@simnettEnergeticParticlesHeliosphere2017]. _Galactic cosmic rays_ (GCRs) are high-energy particles originating from outside the solar system—primarily accelerated by supernova shocks—that propagate inward through the heliosphere and are modulated by the solar wind and heliospheric magnetic field [@prantzosOriginCompositionGalactic2012; @moraalCosmicRayModulationEquations2013]. _Anomalous cosmic rays_ (ACRs) represent a population of singly charged ions that originate as interstellar neutral atoms, become ionized upon entering the heliosphere (forming pickup ions), and are subsequently accelerated to energies of tens of MeV at the heliospheric termination shock [@giacaloneAnomalousCosmicRays2022]. _Solar energetic particles_ (SEPs), the population most relevant to this thesis, are accelerated at or near the Sun during transient eruptive events and span energies from suprathermal (a few keV) to relativistic ($\sim$few GeV) [@reamesTwoSourcesSolar2013; @desaiLargeGradualSolar2016; @kleinAccelerationPropagationSolar2017; @anastasiadisSolarEnergeticParticles2019]. + +Unlike the quasi-steady GCR background, SEP events are episodic and highly variable. Intensities can increase by several orders of magnitude within minutes and exhibit dramatic event-to-event variations in spectral shape, heavy-ion composition, charge states, and spatial distribution [@anastasiadisSolarEnergeticParticles2019]. This variability reflects both the diverse acceleration mechanisms at the Sun and the complex transport processes that particles undergo as they propagate through the structured, turbulent interplanetary medium [@kleinAccelerationPropagationSolar2017]. + +### Two Classes of Solar Energetic Particle Events + +SEP events are broadly classified into two categories based on their acceleration mechanism and observational characteristics [@reamesTwoSourcesSolar2013; @desaiLargeGradualSolar2016; @vlahosSourcesSolarEnergetic2019]: + +![The two-class picture for SEP events. @desaiLargeGradualSolar2016](figures/ref/desaiLargeGradualSolar2016-fig3.png) + +_Impulsive SEP events_ are produced by magnetic reconnection-driven processes during solar flares. In the standard picture, reconnection in the solar corona—occurring along open magnetic field lines—accelerates particles through stochastic processes related to turbulence and fragmented current sheets in the reconnection region [@vlahosSourcesSolarEnergetic2019]. Impulsive events are typically short-lived (minutes to hours) and are characterized by several distinctive compositional signatures: large enhancements of $^3\text{He}/^4\text{He}$ (by factors up to $\sim$ 1000 relative to solar wind values), elevated heavy-ion ratios such as Fe/O, high charge states indicating source temperatures of $\sim 3 \times 10^7$ K, and relatively high electron-to-proton ratios [@reamesTwoSourcesSolar2013]. Because the accelerated particles are injected onto a narrow range of magnetic flux tubes rooted in the compact flare site, impulsive events are observed only by spacecraft that are magnetically well-connected to the source—typically spanning only a few tens of degrees in solar longitude. + +_Gradual SEP events_ are driven by shock waves propagating through the corona and interplanetary space, typically associated with fast coronal mass ejections (CMEs) [@leeShockAccelerationIons2012]. The CME-driven shock provides a spatially extended acceleration region that can fill a broad range of magnetic field lines as it expands outward, beginning when the shock reaches approximately 2 solar radii [@desaiLargeGradualSolar2016]. Particles are accelerated through diffusive shock acceleration (DSA), in which ions are scattered back and forth across the shock front by self-generated Alfvén waves amplified by the streaming accelerated protons themselves [@kleinAccelerationPropagationSolar2017]. Gradual events typically last for several days, are proton-dominant, and exhibit elemental abundances similar to the solar corona and charge states corresponding to source temperatures of $\sim 3 \times 10^6$ K. These events produce by far the highest SEP intensities observed near Earth and represent the primary source of radiation hazard to astronauts, spacecraft electronics, and passengers on high-altitude polar flight routes [@anastasiadisSolarEnergeticParticles2019]. + +While this two-class paradigm provides a useful organizational framework, it is important to note that the boundary between the two classes is not always sharp [@kleinOriginSolarEnergetic2001]. Many large gradual events show elevated $^3$He and Fe/O ratios at high energies, suggesting that residual suprathermal ions from previous impulsive events can serve as a seed population for subsequent shock acceleration [@desaiLargeGradualSolar2016]. Furthermore, recent multi-spacecraft observations from widely distributed vantage points (STEREO, ACE, SOHO, and more recently Parker Solar Probe and Solar Orbiter) have revealed that SEPs can fill remarkably broad regions of the heliosphere—often extending over more than 180° in longitude—challenging simple models of localized injection and raising fundamental questions about the relative roles of extended shock geometry, coronal transport, and interplanetary cross-field diffusion in distributing particles throughout the inner heliosphere [@anastasiadisSolarEnergeticParticles2019; @kleinAccelerationPropagationSolar2017]. + +### Reservoir and Dropout Phenomena + +Two contrasting observational signatures of SEP transport provide particularly compelling evidence for the role of magnetic structure in controlling particle propagation: + +The **reservoir effect** (or invariant spectra) is frequently observed during the decay phase of large gradual SEP events. Particle intensities measured by widely separated spacecraft (sometimes at different heliocentric distances and heliolatitudes) become nearly equal and exhibit similar temporal decay profiles, with energy spectra that maintain an invariant shape as overall intensities decrease [@reamesTwoSourcesSolar2013; @desaiLargeGradualSolar2016]. The three-dimensional character of the reservoir, revealed by high-heliolatitude observations from the Ulysses mission [@dallaPropertiesHighHeliolatitude2003], favors explanations involving substantial cross-field transport that distributes particles throughout a large volume of the inner heliosphere, rather than simple trapping behind an expanding magnetic-bottle-like structure. The velocity-dependent migration of particles through the tangled interplanetary field—potentially mediated by interactions with current sheets and other coherent structures—is a natural candidate mechanism for establishing such spatially uniform distributions. + +In stark contrast, **dropout events** are characterized by abrupt, sharp variations (depletions) in energetic particle intensity, often observed during impulsive SEP events [@mazurInterplanetaryMagneticField2000; @tesseinEffectCoherentStructures2015]. Spacecraft crossing from one magnetic flux tube into an adjacent one observe sudden drops (or increases) in particle count rates, with the intensity boundaries occurring on spatial scales comparable to particle gyroradii [@neugebauerEnergeticParticlesTangential2015]. This behavior indicates that particles are effectively confined within distinct flux tubes with minimal lateral transport. The sharp boundaries between particle-filled and particle-empty flux tubes are frequently identified with tangential discontinuities in the solar wind magnetic field, which act as barriers to cross-field propagation because no magnetic field component threads through their surface ($B_n = 0$). + +Together, these observations highlight a fundamental duality: current sheets can both _impede_ particle transport (when acting as flux tube boundaries at tangential discontinuities) and _facilitate_ it (through scattering that decouples particles from their field lines at rotational discontinuities). The relative importance of these two effects depends on the internal structure of the current sheets and the particle energy—a theme developed quantitatively in this thesis. + +### Turbulence Transport Frameworks + +The large-scale transport of energetic particles through the heliosphere is governed by a competition between parallel streaming along magnetic field lines, scattering by magnetic fluctuations, adiabatic focusing in the diverging interplanetary magnetic field, convection with the solar wind, and gradient-curvature drifts [@dallaSolarEnergeticParticle2013]. The foundational framework is the Parker transport equation [@parkerPassageEnergeticCharged1965], which describes the evolution of the nearly isotropic part of the particle distribution using a diffusive approximation (justified when the particle scattering time is short compared to the timescale of interest): + +$$ +\frac{\partial f}{\partial t} = \frac{\partial}{\partial x_i}\left[\kappa_{ij}\frac{\partial f}{\partial x_j}\right] - U_i \frac{\partial f}{\partial x_i} - V_{d,i}\frac{\partial f}{\partial x_i} + \frac{1}{3}\frac{\partial U_i}{\partial x_i}\frac{\partial f}{\partial \ln p} + \text{Sources} - \text{Losses}, +$$ + +where $f$ is the omnidirectional distribution function, $\kappa_{ij}$ is the symmetric part of the diffusion tensor, $U_i$ is the solar wind velocity, $V_{d,i}$ is the gradient-curvature drift velocity, and $p$ is the particle momentum. The drift velocity can be formally derived from the guiding center approximation averaged over a nearly isotropic distribution, and can be included as the antisymmetric part of the diffusion tensor. The diffusion tensor can be decomposed into components parallel ($\kappa_\parallel$) and perpendicular ($\kappa_\perp$) to the mean magnetic field: $\kappa_{ij} = \kappa_\perp \delta_{ij} + (\kappa_\parallel - \kappa_\perp) B_i B_j / B^2$. + +The parallel diffusion coefficient is related to the pitch-angle diffusion coefficient $D_{\mu\mu}$ through the quasilinear theory (QLT) framework [@jokipiiCosmicRayPropagationCharged1966]: + +$$ +\kappa_\parallel = \frac{v^2}{8}\int_{-1}^{1}\frac{(1-\mu^2)^2}{D_{\mu\mu}(\mu)}\,d\mu, +$$ + +where $\mu = \cos\alpha$ is the pitch-angle cosine and $v$ is the particle speed. This relation makes $D_{\mu\mu}$ the central microphysical quantity controlling parallel transport: all the physics of particle–field interaction is encoded in this single function of pitch angle. + +When significant anisotropy is present—as during the early phases of SEP events, near interplanetary shocks, or when focusing effects are important—the focused transport equation [@roelofPropagationSolarCosmic1969; @earlEffectAdiabaticFocusing1976] retains the explicit pitch-angle dependence: + +$$ +\frac{\partial f}{\partial t} + \mu v \frac{\partial f}{\partial z} + \frac{v}{2L}(1-\mu^2)\frac{\partial f}{\partial \mu} = \frac{\partial}{\partial \mu}\left(D_{\mu\mu}\frac{\partial f}{\partial \mu}\right), +$$ + +where $L = -B(dB/dz)^{-1}$ is the focusing length characterizing the divergence of the magnetic field. + +In both frameworks, the pitch-angle diffusion coefficient $D_{\mu\mu}$ is the quantity most directly shaped by magnetic turbulence. Its value is governed by several key turbulence parameters — spatial inhomogeneity, fluctuation level ($\delta B/B_0$), spectral index, and wave-vector anisotropy [@pucciEnergeticParticleTransport2016; @chandranScatteringEnergeticParticles2000]. Solar wind turbulence spans an enormous range of scales, from the large-scale coherence length (~0.01 AU) down to kinetic dissipation scales near the thermal ion gyroradius (~100 km). The most relevant portion for energetic particle transport is the inertial range: for a 5 nT background field, this range corresponds to proton gyroradii from roughly 1 keV to 1 GeV, a window encompassing nearly all SEPs. + +Classical scattering theory evaluates $D_{\mu\mu}$ by modeling the turbulence as a superposition of random, low-amplitude, phase-uncorrelated waves with power-law spectra, in which particles scatter via cyclotron resonance with fluctuations at scales matching their gyroradius [@jokipiiCosmicRayPropagationCharged1966; @jokipiiCosmicRayPropagationIi1967]. Many numerical models adopt the same idealization, constructing magnetic fluctuations — such as the slab component $\delta\mathbf{B}^s$ and two-dimensional component $\delta\mathbf{B}^{2D}$ (@eq-δ𝐁) — without accounting for the intermittent, structured nature of real solar wind turbulence. This approach has well-known limitations: it fails to scatter particles through $\mu = 0$ (the 90° pitch-angle problem) and produces mean free paths that disagree with observational inferences. + +$$ +\begin{aligned} +& δ𝐁^s=\sum_{n=1}^{N_m} A_n\left[\cos α_n\left(\cos \phi_n \hat{x}+\sin \phi_n \hat{y}\right)+i \sin α_n(-\sin \phi_n \hat{x}+\cos \phi_n \hat{y})\right] \times \exp \left(i k_n z+i β_n\right) \\ +& δ𝐁^{2 D}=\sum_{n=1}^{N_m} A_n i\left(-\sin \phi_n \hat{x}+\cos \phi_n \hat{y}\right) \times \exp \left[i k_n\left(\cos \phi_n x+\sin \phi_n y\right)+i β_n\right] +\end{aligned} +$$ {#eq-δ𝐁} + +### New model of particle transport in reversing magnetic fields + +The standard paradigm of cosmic ray transport assumes small-amplitude fluctuations ($\delta B / B_0 \ll 1$) about a strong mean magnetic field and relies on quasi-linear theory to relate the pitch-angle diffusion rate to the magnetic power spectrum. @kempskiCosmicRayTransport2023 and @lemoineParticleTransportLocalized2023 challenge this paradigm by studying particle transport in the opposite asymptotic limit: large-amplitude MHD turbulence with $\delta B/B_0 \gg 1$, where the turbulence stretches and folds field lines into highly intermittent structures with ubiquitous small-scale reversals. + +The magnetic field geometry is characterized by the field-line curvature $K_\parallel \equiv |\hat{\mathbf{b}} \cdot \nabla \hat{\mathbf{b}}|$ and the inverse perpendicular reversal scale $K_\perp \equiv |\hat{\mathbf{b}} \times (\hat{\mathbf{b}} \times \nabla \ln B)|$. +The probability density functions of $K_\parallel$ and $K_\perp$ extracted from the simulations show that the perpendicular reversal scale is systematically smaller than the parallel coherence length, meaning that the magnetic field is organized into *fold*-like structures: elongated regions of roughly straight field with small perpendicular reversal scales ($K_\perp^{-1} \ll l_B$, the parallel coherence length), connected by tight bends where $K_\parallel \sim K_\perp$. Regions of large curvature have systematically weaker fields, scaling as $\langle B(K_\parallel)\rangle \propto K_\parallel^{-1/2}$. (Similar curvature power-law indices have been measured in other simulation settings [@schekochihinSimulationsSmallScaleTurbulent2001], and *in situ* in the Earth's magnetosheath [@bandyopadhyayMeasurementCurvatureMagnetic2020; @huangObservationsMagneticField2020] and solar wind [@] —suggesting that this property is robust.) + +![Schematic of particle transport in a magnetic fold. The fold has parallel coherence length $\sim l_B$ and perpendicular reversal scale $K_\perp^{-1} \ll l_B$. Along the straight segments $K_\parallel \ll K_\perp$; at the bend, $K_\parallel \sim K_\perp$. The magenta trajectory illustrates a particle whose gyroradius grows as $B$ weakens near the bend [@kempskiCosmicRayTransport2023].](figures/ref/kempskiCosmicRayTransport2023-fig7.jpeg){#fig-kempski-fold} + +The transport of a particle through such a fold depends on its gyroradius relative to the perpendicular reversal scale. A particle with $r_{L0} \gg K_\perp^{-1}$ is not magnetized on the scale of the fold: it samples many field reversals within a single gyro-orbit, so the opposing field contributions partially cancel and the effective magnetic force on the particle is suppressed. Such high-energy particles are largely insensitive to the fold structure. For $r_{L0} < K_\perp^{-1}$, the particle can gyrate around the field line along the straight segments where $K_\parallel \ll K_\perp$, but as it approaches the bend the weakening field increases its gyroradius and decreases its gyrofrequency. Three outcomes then result: (i) if $\Omega \gg K_\parallel c$ at the bend, the particle adiabatically follows the reversing field, undergoing effective spatial 'scattering' through the field-line random walk without pitch-angle change—a mechanism with no analogue in strong guide-field transport, where a particle perfectly following a field line does not random walk; (ii) if $\Omega \sim K_\parallel c$, the particle experiences 'resonant curvature'—order-unity pitch-angle scattering analogous to the $\kappa_l r_g \gtrsim 1$ criterion of @lemoineParticleTransportLocalized2023; (iii) if $\Omega \ll K_\parallel c$, the particle demagnetizes and drifts out of the fold via $\nabla B$ drift before completing the reversal. The resulting heuristic scattering frequency is + +$$ +\nu(r_{L0}) \sim \frac{c}{l_B} \int^{K_\mathrm{max}} P(K_\perp)\,\mathrm{d}K_\perp, +$$ {#eq-kempski-nu} + +where $l_B = \int K_\parallel^{-1} P(K_\parallel)\,\mathrm{d}K_\parallel$ is the characteristic parallel coherence length. The upper limit $K_\mathrm{max}(r_{L0})$ encodes the magnetization condition at the bend: $K_\mathrm{max}\,r_{L0} = f_r\,\langle B(K_\parallel = K_\mathrm{max})\rangle / B_\mathrm{rms}$, with $f_r \sim \mathcal{O}(1)$. Using $\langle B\rangle \propto K_\parallel^{-1/2}$ yields $K_\mathrm{max} \propto r_{L0}^{-2/3}$, so that lower-energy particles can traverse a larger fraction of folds and are better confined. + +The perpendicular reversal scale $K_\perp$ plays a role with no analogue in quasi-linear theory. In effect, it controls the fraction of the fold population that contributes to the effective scattering frequency. It does not directly scatter particles, but sets the curvature of the bend at the end of each fold, thereby determining whether particles satisfy the resonant-curvature condition or follow the reversal adiabatically. + +### Pitch-Angle Scattering by Current Sheets + +Solar wind turbulence is highly intermittent: magnetic energy and current density concentrate into thin, intense current sheets that occupy a small fraction of the volume yet account for a disproportionate share of the dissipation [@borovskyContributionStrongDiscontinuities2010]. The scattering produced by these coherent structures is fundamentally different from wave–particle resonance. When a particle's gyroradius is comparable to the sheet thickness ($\rho \sim L$), the interaction involves separatrix crossings in phase space and the associated destruction of the quasi-adiabatic invariant $I_z$ (discussed in detail in the following sections). The resulting pitch-angle changes are large—$\Delta\alpha \sim \mathcal{O}(1)$ in the case of geometrical chaotization—and occur on timescales comparable to a single gyroperiod, far faster than the gradual diffusion assumed in quasilinear theory. Because kinetic-scale current sheets are abundant in the solar wind [@vaskoKineticscaleCurrentSheets2022; @vasquezNumerousSmallMagnetic2007], and their thickness overlaps with the gyroradii of energetic particles at typical field strengths, a propagating particle will encounter many such structures en route from the Sun to Earth. The cumulative pitch-angle scattering from these encounters contributes to an effective $D_{\mu\mu}$ through a mechanism that is qualitatively distinct from—and potentially more efficient than—classical wave–particle resonance. + +A complementary perspective on structure-mediated scattering comes from @lemoineParticleTransportLocalized2023, who studied the role of sharp magnetic field bends—characterized by the local curvature $\kappa \equiv B^{-1}|\mathbf{b} \times (\mathbf{b} \cdot \nabla)\mathbf{B}|$—in MHD simulations without a mean field. The key insight of @lemoineParticleTransportLocalized2023 is that, while such regions are rare in a root-mean-square sense—the characteristic curvature on scale $l$ scales as $\kappa_l \sim (l/\ell_c)^{-1/3}\,\ell_c^{-1}$ following the Goldreich--Sridhar cascade, giving $\kappa_l r_g \ll 1$ for $r_g \ll \ell_c$—the non-Gaussian, power-law tails of the curvature p.d.f. guarantee that sufficiently many scattering-strength bends exist on all scales $l < \ell_c$. This yields a mean free path $\lambda_s \sim \ell_c^{0.7} \bar{r}_g^{0.3}$, a scaling that is distinct from quasilinear predictions and arises from the intermittent structure of the turbulence. Importantly, particle tracking in that simulation confirmed that magnetic moment diffusion proceeds through localized, violent interactions rather than continuous weak scattering, producing non-Brownian transport on scales $\lesssim \ell_c$. + +a scaling that is distinct from quasilinear predictions and arises entirely from the intermittent structure of the turbulence. Particle tracking in the same simulation confirms that the magnetic moment diffuses through localized, violent interactions rather than continuous weak scattering: the p.d.f. of the normalized magnetic moment $\hat{M}(t) \equiv M(t)/M(0)$ broadens through the development of power-law tails, and the cumulative distribution shows that roughly 25% of particles experience order-unity changes in $M$ after propagating $\sim 2\ell_c$. The inferred diffusion coefficient $D_{\hat{M}} \simeq 0.4\,c/\ell_c$ is consistent with the mean free path predicted by the curvature statistics to within a factor of a few. Correspondingly, the overall transport process is non-Brownian on scales $\lesssim \ell_c$, converging to standard diffusive behaviour only on asymptotically large scales. + +### Perpendicular Transport + +While parallel transport is relatively well understood, perpendicular (cross-field) transport remains more elusive due to its nonlinear and non-resonant nature [@shalchiPerpendicularDiffusionEnergetic2021; @costajr.CrossfieldDiffusionEnergetic2013]. In the classical picture, cross-field diffusion arises from two mechanisms: the random walk of magnetic field lines, which carries particles across the mean field, and the decorrelation of particles from their original field lines through scattering. The perpendicular diffusion coefficient $\kappa_\perp$ is typically assumed to be a small fraction of $\kappa_\parallel$ [@giacaloneTransportCosmicRays1999], but this assumption is challenged on two fronts: observations—such as the reservoir effect and the broad longitudinal spread of SEP events—demand significant cross-field transport, and recent simulations show that $\kappa_\perp$ can be substantial and strongly dependent on particle energy and turbulence structure [@dundovicNovelAspectsCosmic2020]. The dimensionality of the turbulence also matters [@giacaloneChargedParticleMotionMultidimensional1994]: in models with at least one ignorable spatial coordinate (e.g., slab geometry), cross-field diffusion is artificially suppressed, omitting essential physics. + +The intermittent-scattering framework introduces cross-field mechanisms that go beyond the classical field-line random walk, which dominates cross-field motion in the quasi-linear regime with a strong guide field. In the fold-propagation model of @kempskiCosmicRayTransport2023, particles that demagnetize at fold bends ($\Omega \ll K_\parallel c$) drift out via $\nabla B$ drift, producing cross-field displacements of order $K_\perp^{-1}$. For particles that do traverse the bend, the field-line random walk provides rigidity-independent perpendicular diffusion with $\kappa_B \sim c\,l_B$. The relative importance of these two channels depends on the particle energy through the hierarchy of $K_\mathrm{max}(r_{L0})$ and $K_\mathrm{peak}$, the latter defined as the $K_\perp$ at which $K_\perp P(K_\perp)$ is maximal. For large $r_{L0}$ with $K_\mathrm{max} \gg K_\mathrm{peak}$, resonant-curvature scattering dominates and cross-field transport is rigidity-dependent; for small $r_{L0}$ with $K_\mathrm{max} \ll K_\mathrm{peak}$, the field-line random walk takes over and the perpendicular diffusion becomes independent of particle energy. The transition between these regimes, and the interplay of the two contributions, may produce a $\kappa_\perp$ with complex energy dependence qualitatively different from the standard $\kappa_\perp \propto \kappa_\parallel$ scaling. + +@lemoineParticleTransportLocalized2023 emphasizes a complementary mechanism rooted in the same intermittent structures. The systematic anti-correlation between field-line curvature and magnetic field strength—$\kappa \propto B^{-2}$ approximately [@yangRoleMagneticField2019; @yuenCurvatureMagneticField2020]—means that non-adiabatic, magnetic-moment-violating interactions occur precisely where the field is weak and inhomogeneous on gyroradius scales. During such interactions, the particle is effectively demagnetized and its guiding centre can jump to a neighbouring field line. Even outside these non-adiabatic regions, strong curvature and $\nabla B$ gradients drive perpendicular drifts with velocities $v_D \sim v\,r_g/(3L)$ for modes on scale $L > r_g$. While the displacement per interaction is small ($\sim r_g l / L $), the cumulative effect over many scattering events provides an additional source of cross-field transport. + +Current sheets deserve particular attention in this context, as they concentrate several of these mechanisms simultaneously. For an idealized one-dimensional force-free current sheet (e.g., a force-free Harris sheet), however, there is no $\nabla B$ drift and no field-line random walk—the classical perpendicular transport channels are absent. Nonetheless, the field becomes highly inhomogeneous on scales comparable to or smaller than the gyroradius: particles become unmagnetized and escape their initial field lines. This current-sheet-driven mechanism may constitute a genuinely non-diffusive contribution to cross-field transport. + +A quantitative theory of perpendicular transport in the intermittent-scattering regime remains to be developed. The joint statistics of the curvature $\kappa$ and the mirror term $m \equiv B^{-1}\hat{\mathbf{b}} \cdot (\hat{\mathbf{b}} \cdot \nabla)B$ along field lines are likely correlated—both are large at current sheets—yet their combined influence on perpendicular decorrelation has not been quantified [@lemoineParticleTransportLocalized2023]. The topology of high-curvature regions, whether they form isolated patches or connected networks, will determine how efficiently successive scattering events translate into macroscopic cross-field diffusion. Furthermore, the non-Brownian character of parallel transport on scales $\lesssim \ell_c$ identified by @lemoineParticleTransportLocalized2023 suggests that perpendicular transport may also exhibit anomalous features on sub-coherence-length scales. + + +### Evidence for Current Sheet Modulation of SEP Intensity + +Direct observational evidence for the influence of current sheets on SEP transport comes from studies at both large and small scales. At the large scale of the heliospheric current sheet (HCS), @liouSolarEnergeticParticle2024 performed a superposed epoch analysis of 319 HCS crossings observed by the Wind spacecraft, finding a systematic drop in 2–10 MeV/nucleon helium flux at the HCS that was strongest at low energies and diminished at higher energies. They identified 15 individual SEP flux dropout events coinciding with HCS crossings, all originating from western-hemisphere sources at longitudes far from the crossing location—indicating that the HCS severed the magnetic connection between the particle source and the observer. The energy dependence of the dropout fraction is consistent with more effective scattering or blocking of lower-energy particles whose gyroradii are comparable to the current sheet thickness. The transport and drift effects of the HCS have also been characterized through simulation. @battarbeeSolarEnergeticParticle2017 integrated fully three-dimensional proton trajectories near an analytically defined flat HCS in the 1–800 MeV range, finding that gradient and curvature drifts along the sheet can carry protons to longitudes far from their injection site—producing multi-component intensity profiles that could be misinterpreted as evidence for multiple injection events—and confirming that the HCS acts as an effective barrier to cross-hemisphere transport. Extending this work to a more realistic geometry, @battarbeeModelingSolarEnergetic2018 modeled SEP propagation near a wavy HCS whose position was constrained by fits to magnetic source surface maps. They found that the waviness of the sheet introduces longitudinally periodic enhancements in particle fluence and enables efficient longitudinal transport along the sheet, with the magnitude and spatial distribution of these effects depending sensitively on the IMF polarity configuration (A+ vs. A−) and the HCS tilt angle. + +At smaller scales, @tesseinEffectCoherentStructures2015 analyzed over 12 years of ACE observations and found a strong statistical correlation between coherent structures—identified using the partial variance of increments (PVI) method, which detects current sheets and sharp magnetic field gradients—and energetic particle intensity variations in the 0.047–4.78 MeV range. Local PVI maxima frequently coincided with regions of rising or falling particle intensity, suggesting that magnetic discontinuities act as local barriers or modulators of transport. This correlation persisted after removing shock-associated intervals, confirming that the effect is intrinsic to the current sheets rather than a byproduct of shock-related enhancements. + +### Non-Diffusive Transport Effects + +Beyond classical diffusion, SEP observations and measurements near interplanetary shocks frequently reveal anomalous transport behavior, characterized by subdiffusive or superdiffusive scaling of mean-square displacement with time, $\langle\Delta x^2(t)\rangle \propto t^\alpha$ with $\alpha \neq 1$ [@zimbardoSuperdiffusiveSubdiffusiveTransport2006; @zimbardoSuperdiffusiveTransportLaboratory2015]. These deviations from normal diffusion ($\alpha = 1$) are attributed to the intermittent, structured nature of solar wind turbulence and are better captured by generalized frameworks such as fractional diffusion models [@del-castillo-negreteNondiffusiveTransportPlasma2005] or Lévy statistics [@zaburdaevLevyWalks2015]. + +The interactions with current sheets have implications that extend beyond modifications to the diffusion coefficients. When pitch-angle changes are dominated by rare but intense jumps rather than continuous weak scattering, the statistical assumptions underlying the diffusion approximation can break down [@lemoineParticleTransportLocalized2023]. Current sheets can induce non-Markovian memory effects [@zimbardoNonMarkovianPitchangleScattering2020]: correlations in the spatial arrangement and properties of current sheets mean that scattering at one structure influences both the likelihood and character of scattering at the next. The intermittent, clustered distribution of current sheets in the solar wind naturally generates the heavy-tailed step-size distributions that underlie anomalous transport in both regimes—subdiffusive ($\alpha < 1$) when sheets trap particles, and superdiffusive ($\alpha > 1$) when intense encounters produce large pitch-angle jumps and correspondingly large spatial displacements. + +## Quasi-Adiabatic Dynamics of Charged Particles and Its Destruction in Current Sheets + +### Adiabatic Motion and Its Breakdown + +A charged particle moving in a magnetic field that varies slowly in space and time possesses an approximate conservation law. The particle gyrates around the magnetic field line with a characteristic frequency—the cyclotron frequency $\Omega_c = qB/mc$—and a characteristic radius—the gyroradius (or Larmor radius) $\rho = v_\perp / \Omega_c$, where $v_\perp$ is the velocity component perpendicular to the magnetic field. If the magnetic field changes only gradually over the spatial scale of the gyro-orbit, the particle's magnetic moment + + +$$ +\mu = \frac{mv_\perp^2}{2B} +$$ + +is approximately conserved. This quantity, the first adiabatic invariant, is the foundation of guiding-center theory [@northropAdiabaticChargedparticleMotion1963], in which the rapid gyromotion is averaged out and the particle is described by the drift of its guiding center along and across the magnetic field. The conservation of $\mu$ underlies many fundamental phenomena in space physics, including magnetic mirroring, radiation belt trapping, and the confinement of particles in magnetic bottles. + +The key assumption of guiding-center theory is that the magnetic field varies on spatial scales much larger than the particle's gyroradius: $L_B \gg \rho$, where $L_B$ is the characteristic length scale of the field gradient. When a particle encounters a magnetic structure whose thickness $\lambda$ is comparable to or smaller than its gyroradius—that is, when $\lambda \sim \rho$—this assumption breaks down. The particle can no longer complete a full gyration within a region of approximately uniform field. The magnetic moment ceases to be conserved, and the guiding-center description fails [@escandeBreakdownAdiabaticInvariance2021]. This is precisely the situation that arises when energetic particles interact with kinetic-scale current sheets in the solar wind, where the structure thickness is on the order of the ion inertial length $d_i$ or a few ion gyroradii. + +A different theoretical framework is needed to describe particle dynamics in this regime. Rather than averaging over the gyromotion (as in guiding-center theory), one must identify the appropriate fast periodic motion for the specific field geometry and construct the corresponding adiabatic invariant. This is the domain of the quasi-adiabatic theory [@zelenyiQuasiadiabaticDynamicsCharged2013; @whippleAdiabaticTheoryRegions1986], which we now describe. + +### Adiabatic Invariants in Slow–Fast Hamiltonian Systems + +The mathematical foundation for understanding particle motion in current sheets lies in the general theory of slow–fast Hamiltonian systems and their adiabatic invariants. We follow the review by @neishtadtMechanismsDestructionAdiabatic2019, which provides a comprehensive treatment of the mechanisms by which adiabatic invariance can be destroyed. + +Consider a Hamiltonian system in which the variables naturally separate into fast and slow degrees of freedom. The fast variables $(p, q)$ oscillate rapidly, while the slow variables $(y, x)$ evolve on a much longer timescale, controlled by a small parameter $\varepsilon \ll 1$. The equations of motion take the form + + +$$ +\dot{p} = -\frac{\partial E}{\partial q}, \quad \dot{q} = \frac{\partial E}{\partial p}, \quad \dot{y} = -\varepsilon \frac{\partial E}{\partial x}, \quad \dot{x} = \varepsilon \frac{\partial E}{\partial y}. +$$ + +For frozen values of the slow variables, the fast system describes periodic motion. One can then introduce _action-angle variables_ $(I, \varphi)$ for this periodic motion, where the action + +$$ +I = \frac{1}{2\pi} \oint p \, dq +$$ + +is the area enclosed by the orbit in the fast phase plane, divided by $2\pi$. Averaging the equations of motion over the fast phase $\varphi$ yields the _adiabatic approximation_: + +$$ +I = \text{const}, \quad \dot{y} = -\varepsilon \frac{\partial H_0}{\partial x}, \quad \dot{x} = \varepsilon \frac{\partial H_0}{\partial y}, +$$ + +where $H_0(I, y, x)$ is the Hamiltonian expressed in terms of the action and slow variables. The action $I$ is an _adiabatic invariant_: it is conserved with accuracy $O(\varepsilon)$ over time intervals of order $1/\varepsilon$. + +Under favorable conditions, conservation can be much better than this. When the system has two degrees of freedom (one fast, one slow) and the phase portrait of the fast system is everywhere filled by closed trajectories (no separatrices), Arnold showed that invariant tori of the exact system fill the energy surface up to a residue of small measure, and the adiabatic invariant is conserved perpetually: $|I(t) - I(0)| = O(\varepsilon)$ for all time [@arnoldSmallDenominatorsProblems1963]. This result has direct physical consequences: it implies, for example, that charged particles can be confined indefinitely in axisymmetric magnetic traps. + +### Separatrix Crossing and Destruction of Adiabatic Invariance + +The situation changes fundamentally when the phase portrait of the fast system contains a _separatrix_—a trajectory that separates topologically distinct regions of phase space. @fig-neishtadt2019-fig3 illustrates the generic structure: a saddle point $C$ in the fast phase plane generates separatrices $l_1$ and $l_2$ that divide the portrait into three domains $G_1$, $G_2$, and $G_3$, each corresponding to a different type of periodic motion. For example, $G_1$ and $G_2$ might correspond to oscillations in two separate potential wells, while $G_3$ corresponds to oscillations spanning both wells. + +![Phase portrait of the fast system [@neishtadtMechanismsDestructionAdiabatic2019].](figures/ref/neishtadtMechanismsDestructionAdiabatic2019-fig3.png){#fig-neishtadt2019-fig3} + +As the slow variables evolve, the phase portrait of the fast system changes shape, and a trajectory that was initially in one domain can be driven toward the separatrix and cross into another domain. This crossing has profound consequences for the adiabatic invariant [@tennysonChangeAdiabaticInvariant1986]. There are two distinct contributions to its change: + +1. **Geometrical jump.** When a trajectory crosses from, say, domain $G_3$ (the large outer domain) to $G_1$ (one of the inner domains), the area enclosed by the orbit changes discontinuously—from $S_3 = S_1 + S_2$ to $S_1$. This purely geometric change $\Delta I^{\text{geom}}$ is independent of $\varepsilon$ and is determined entirely by the relative areas of the separatrix loops at the moment of crossing. + +2. **Dynamical jump.** In addition to the geometric change, the logarithmic divergence of the oscillation period $T$ near the separatrix produces a correction $\Delta I^{\text{dyn}} \sim \varepsilon \ln \varepsilon$. The precise value of this correction depends sensitively on the phase of the fast motion at the moment of crossing, parameterized by a variable $\xi$. This quantity varies rapidly ($\sim 1/\kappa$ times faster than the slow evolution) and can be treated as a uniformly distributed random variable $\xi \in (0,1)$. + +Because the dynamical jump depends sensitively on initial conditions ($O(\varepsilon)$ changes in initial data produce $O(1)$ relative changes in the jump), the outcome of separatrix crossing is effectively probabilistic. The probability of entering domain $G_i$ after crossing from $G_3$ is $P_i = \Theta_i / \Theta_3$, where $\Theta_i = \{S_i, h_c\}$ is the Poisson bracket of the area of domain $G_i$ with the value of the Hamiltonian $E$ (energy) at the saddle point $C$. The accumulation of geometrical and dynamical jumps over multiple separatrix crossings destroys the adiabatic invariance and leads to chaotic dynamics. + +### Application to Current Sheets: the Quasi-Adiabatic Invariant + +The motion of charged ions in a magnetic field reversal—the configuration of a current sheet—provides the most physically important application of this theory. Consider a current sheet with a reversing field $B_x(z) = B_0(z/L)$ (where $L$ is the half-thickness), a small normal component $B_z$, and an optional guide field $B_y$. In normalized variables, the Hamiltonian for ion motion takes the form [@artemyevIonMotionCurrent2013] + +$$ +H = \frac{1}{2}p_z^2 + \frac{1}{2}(p_x - sz)^2 + \frac{1}{2}\left(\kappa x - \frac{1}{2}z^2\right)^2, +$$ + +where the two key dimensionless parameters are + +$$ +\kappa = \frac{B_z}{B_0}\sqrt{\frac{L}{\rho_0}}, \qquad s = \frac{B_y}{B_0}\sqrt{\frac{L}{\rho_0}}, +$$ + +with $\rho_0 = \sqrt{2hm}c/(qB_0)$ being the characteristic Larmor radius for a particle of energy $h$. The parameter $\kappa$ controls the separation of timescales: $\kappa \ll 1$ means the coordinate $z$ (normal to the current sheet) oscillates rapidly while $\kappa x$ (along the reversing field direction) evolves slowly. For typical ions in the Earth's magnetotail current sheet, $\kappa \in [0.01, 0.1]$; for energetic ions interacting with solar wind current sheets, $\kappa$ can range from $\sim 0.01$ to $\sim 0.3$ depending on the particle energy and current sheet thickness. + +The structure of the fast motion in the $(z, p_z)$ plane depends on the instantaneous values of the slow variables $(\kappa x, p_x)$. The effective potential energy is + +$$ +U(z) = \frac{1}{2}(p_x - sz)^2 + \frac{1}{2}\left(\kappa x - \frac{1}{2}z^2\right)^2. +$$ + +This potential can have either two local minima separated by a maximum (forming two potential wells) or a single minimum (a single well), depending on $(\kappa x, p_x)$. The two-well structure corresponds physically to the particle oscillating about a magnetic field line on one side of the current sheet, while the single-well structure corresponds to the particle oscillating across the neutral plane $z = 0$, executing figure-eight-like orbits that span both sides of the sheet. + +Since the fast motion is periodic at any frozen value of the slow variables, one can define the quasi-adiabatic invariant + +$$ +I_z = \frac{1}{2\pi} \oint p_z \, dz, +$$ + +which is the action variable of the fast oscillation. This quantity was first introduced in this context by Büchner and Zelenyi (1986, 1989) and plays a role analogous to the magnetic moment $\mu$ of guiding-center theory, but is appropriate for the regime $\lambda \sim \rho$ where guiding-center theory fails. In the adiabatic approximation, $I_z = \text{const}$, and the equation $I_z(\kappa x, p_x) = \text{const}$ (at fixed energy $H = h$) determines closed trajectories in the slow-variable plane $(\kappa x, p_x)$. These trajectories describe the slow drift of the particle as it oscillates back and forth in $z$. + +The boundary between regions of the slow-variable plane corresponding to the two types of fast motion (two wells vs. one well) is called the _uncertainty curve_. When a particle's trajectory in the $(\kappa x, p_x)$ plane reaches the uncertainty curve, the corresponding trajectory in the $(z, p_z)$ plane crosses the separatrix—the particle switches from oscillating on one side of the current sheet to oscillating across both sides, or vice versa. It is at these crossings that the quasi-adiabatic invariant $I_z$ undergoes jumps and the adiabatic approximation breaks down. + +### Symmetric Current Sheets: Slow Diffusion + +In the simplest and most-studied case—a current sheet without a guide field ($s = 0$)—the phase portrait in the $(z, p_z)$ plane is symmetric about $z = 0$. The two separatrix loops enclose equal areas, $S_l = S_r$, and they grow and shrink in synchrony as the slow variables evolve. As a consequence, the geometrical jumps at two successive separatrix crossings (one entering the single-well regime, one returning to a double-well regime) exactly cancel and do not contribute to stochastization (in other words, geometrical jumps can be absorbed into a factor-of-two renormalization of $I_z$). Only the dynamical jumps remain, and for the symmetric system these take the well-known form [@caryAdiabaticinvariantChangeDue1986; @buchnerRegularChaoticCharged1989] + +$$ +\Delta I_z^{\text{dyn}} = -\frac{2}{\pi} \kappa p_x \ln(2 \sin \pi \xi), +$$ + +where $\xi \in (0, 1)$ is the quasi-random phase variable. Since the average over $\xi$ vanishes—$\langle \Delta I_z^{\text{dyn}} \rangle_\xi = 0$—there is no directed drift in the invariant space. The destruction of $I_z$ proceeds as a diffusive random walk: each jump has amplitude $\sim \kappa$, jumps occur at intervals $\sim 1/\kappa$ (one period of slow motion), and the variance accumulates as $\langle (\Delta I_z)^2 \rangle \sim \kappa^2 \cdot (\kappa t)$. A substantial (order-unity) change in $I_z$ therefore requires a time $\sim \kappa^{-3}$. For typical magnetotail parameters ($\kappa \sim 0.05$–0.1), this corresponds to stochastization timescales of tens of minutes—slow, but physically relevant for the lifetime of thin current sheet configurations. + +An important phenomenon in the symmetric system is _resonant interaction_. If the phase accumulated between two successive separatrix crossings satisfies a specific condition ($W = \pi$, where $W$ is determined by the integral of the fast frequency $\Omega_z$ along the adiabatic trajectory), then the two dynamical jumps can exactly compensate: $\sum \Delta I_z^{\text{dyn}} = 0$. This condition depends on $\kappa$ but not on the random variable $\xi$, so it can be simultaneously satisfied for an entire population of particles with the appropriate energy. The resulting coherent particle beams ("beamlets") have been observed in the Earth's magnetotail and studied extensively [@buchnerRegularChaoticCharged1989]. + +### Guide Field Effects on the Quasi-Adiabatic Theory + +The introduction of a guide field ($B_y \neq 0$, corresponding to $s \neq 0$ in the Hamiltonian) breaks the symmetry of the system and profoundly alters the quasi-adiabatic dynamics. @artemyevIonMotionCurrent2013 developed the complete quasi-adiabatic theory for arbitrary values of $s$, identifying new types of particle trajectories and new mechanisms of invariant destruction that have no counterpart in the symmetric ($s = 0$) case. We summarize the key results here; the reader is referred to that paper for full details. + +#### Modified phase-space structure + +In the symmetric system ($s = 0$), the slow-variable plane $(\kappa x, p_x)$ is divided into two domains by the uncertainty curve (a half-circle $(\kappa x)^2 + p_x^2 = 1$, $\kappa x > 0$): a domain (t1) where the particle oscillates in one of two symmetric potential wells (motion along field lines on one side of the sheet), and a domain (t2) where the particle oscillates in a single well spanning both sides of the sheet (motion across the neutral plane). The separatrix in the $(z, p_z)$ plane demarcates these two types of motion, and the uncertainty curve is its projection onto the slow-variable plane. + +When $s \neq 0$, the potential $U(z)$ loses its symmetry about $z = 0$: the saddle point shifts to $z = z_c \neq 0$, and the two separatrix loops enclose unequal areas, $S_l \neq S_r$ (see @fig-artemyev2013-fig3, schematic). Correspondingly, the slow-variable plane acquires a richer structure. Two new domains appear in addition to (t1) and (t2): domain (t2r), where the particle oscillates in a single well _above_ the neutral plane ($z > 0$, both solutions of $U = H$ are positive), and domain (t2l), where it oscillates _below_ the neutral plane ($z < 0$). In neither of these new domains does the particle cross $z = 0$. Meanwhile, the (t1) domain—the region with two potential wells and a separatrix—shrinks, and the uncertainty curve contracts from a half-circle to a shorter arc. For $s \geq 1$, the uncertainty curve (and with it the separatrix) vanishes entirely. + +![The phase plane of slow variables $(\kappa x, p_x)$ is shown for two values of the parameter $s$. Various colours are used for domains with different types of particle motion. Dotted grey lines show the position of energy level $U = 1/2$.](figures/ref/artemyevIonMotionCurrent2013-fig3.png){#fig-artemyev2013-fig3} + +#### Trajectory types in the current sheet with $B_y$ + +The trajectories of trapped particles in the current sheet without a guide field are simple: the particle oscillates in one potential well, crosses the separatrix into the single well (performing a half-rotation around $B_z$ in the neutral plane), then crosses back into a potential well. In the $(\kappa x, p_x)$ plane this appears as a single closed curve, determined by $I_z = \text{const}$ with a factor-of-two renormalization at the separatrix to account for the doubling of enclosed area ($S_l = S_r$, so $S = 2S_l$). The motion is simple because the equal areas and synchronous evolution of the two loops ensure that geometrical jumps cancel exactly. + +In the system with $B_y \neq 0$, particle trajectories become qualitatively more complex because the areas $S_l$ and $S_r$ are no longer equal and their rates of change $\Theta_l$ and $\Theta_r$ are no longer identical. @artemyevIonMotionCurrent2013 showed that, as a consequence, the trajectory in the $(\kappa x, p_x)$ plane is no longer a single closed curve but a composite object: within each domain, the particle follows a segment of an adiabatic trajectory $I_z(\kappa x, p_x) = \text{const}$, but the value of $I_z$ differs from one domain to the next because of the geometrical jumps $\Delta I_z^{\text{geom}}$ that occur at each separatrix crossing. The full trajectory is constructed by matching these segments at the uncertainty curve. + +A further complication arises from the asynchronous evolution of $S_l$ and $S_r$. At certain points along the uncertainty curve, one loop area is increasing ($\Theta_l > 0$) while the other is decreasing ($\Theta_r < 0$). At such points, a particle arriving from the (t1) domain has _two possible continuations_: it can cross into the single-well domain (t2), or it can remain in the (t1) domain but switch from the right potential well to the left one (or vice versa). This phenomenon—called _trajectory splitting_—is unique to the asymmetric system. The choice between the two continuations depends on the precise phase of the fast motion at the separatrix crossing, which is effectively random. The probabilities of the two outcomes, $P_l$, $P_r$, and $P = 1 - P_l - P_r$ (capture into the left well, right well, or single well respectively), can be calculated analytically from the rates $\Theta_l$ and $\Theta_r$ at the crossing point [@artemyevIonMotionCurrent2013]. + +Additionally, when $s \neq 0$, a new type of transition appears that has no counterpart in the symmetric system: the particle can switch between the left and right potential wells _without crossing the separatrix at all_. This occurs when one well disappears as the slow variables evolve, the particle is carried smoothly to the position of the other well, and a new well reappears. In the $(\kappa x, p_x)$ plane, this corresponds to the trajectory going around the end of the uncertainty curve rather than crossing it. + +![Trajectory splitting in the $(\kappa x, p_x)$ plane for $s=0.5$, showing two possible continuations at the uncertainty curve. Schemes of particle trajectories in systems with $s = 0$ and with $s = 0.5$ are shown in the phase plane $(\kappa x, p_x)$. Fragment of $(\kappa x, p_x)$ plane with trajectory splitting is shown in separated panel. Bottom schemes (C1, C2, C3) show particle trajectories before (dotted curves) and after (solid curves) separatrix crossings in the plane $(z, p_z)$.](figures/ref/artemyevIonMotionCurrent2013-fig4.png) + +#### Four regimes of particle motion + +@artemyevIonMotionCurrent2013 identified four distinct regimes of particle dynamics, controlled by the value of $s$: + +1. **$0 < s < s_{\text{bif}} \approx 0.25$**: Only one type of trajectory exists, analogous to (but more complex than) the trajectories of the symmetric system. Particles cross the uncertainty curve multiple times per period of slow motion, with the number of crossings increasing as $s \to 0$. At each crossing, geometrical jumps modify $I_z$, and the trajectory splits into segments matched at the uncertainty curve. Despite this splitting, the total number of crossings is finite and well-prescribed for a given $s$. + +2. **$s_{\text{bif}} < s < \bar{s} \approx 0.35$**: A second type of trajectory appears in addition to the first. Trajectories of this new type cross the uncertainty curve only twice (once for $p_x > 0$ and once for $p_x < 0$), and the transition between wells occurs without separatrix crossing. Both trajectory types coexist in this range. + +3. **$\bar{s} < s < 1$**: Only the second type of trajectory survives. The rate $\Theta_r$ is negative everywhere along the uncertainty curve, meaning that capture into the right well at the separatrix is impossible. All well-switching occurs without separatrix crossing. + +4. **$s \geq 1$**: The separatrix vanishes entirely ($\ell(s) = 0$), and with it all geometrical and dynamical jumps. The quasi-adiabatic invariant is exactly conserved in the adiabatic approximation, and the motion is regular. This critical value corresponds to $B_y > B_0\sqrt{L/\rho_0}$, the same magnetization criterion derived independently by Galeev and Zelenyi (1978). For $s \gg 1$, the quasi-adiabatic invariant $I_z$ reduces to the ordinary magnetic moment, and the guiding-center theory becomes applicable. + +These four regimes are summarized in Table 1 of @artemyevIonMotionCurrent2013. The threshold values $s_{\text{bif}} \approx 0.25$ and $\bar{s} \approx 0.35$ are determined by the geometry of the separatrix loop areas and their evolution rates along the uncertainty curve, while $s = 1$ corresponds to the vanishing of the separatrix itself (the uncertainty curve length $\ell(s) = 2\arctan\sqrt{s^{-4/3} - 1} - s^{1/3}\sqrt{s^{2/3} - s^2}$ reaches zero). + +#### Importance of geometrical jumps + +A key result of @artemyevIonMotionCurrent2013 is the demonstration that, for $s > (2/\pi)\kappa \ln 2$ (i.e., $B_y > 0.44 B_z$), the geometrical jumps dominate over the dynamical jumps in shaping particle trajectories. In this regime, the adiabatic trajectories in the $(\kappa x, p_x)$ plane are determined primarily by the geometrical jumps—which are of order unity and independent of $\kappa$—rather than by the small dynamical jumps $\sim \kappa \ln \kappa$. The dynamical jumps produce only a slow diffusion across these geometrically determined trajectories. For $s < (2/\pi)\kappa \ln 2$, the geometrical jumps become smaller than the dynamical ones and the system behaves effectively like the $s = 0$ case. This separation is the gross features of particle dynamics in a current sheet, with geometrical jumps defining the trajectory structure and dynamical jumps providing the stochastic spreading. + +It is important to note that, however, the remaining $z \to -z$, $p_x \to -p_x$ symmetry constrains the possible values of $I_z$ to a finite set: the particle's invariant cycles through a finite number of discrete values, and the trajectories remain closed (though more complex than in the $s = 0$ case). The number of distinct $I_z$ values depends on $s$ and corresponds to the number of uncertainty curve crossings per period of slow motion (which can be counted using the graphical construction in Appendix C of @artemyevIonMotionCurrent2013). + +### Non-adiabatic Effects: Destruction of the Quasi-adiabatic Invariant + +The quasi-adiabatic theory developed in the previous subsections treats the invariant $I_z$ as exactly conserved between separatrix crossings and accounts only for the geometrical jumps at crossings. In reality, $I_z$ is only an approximate invariant: it oscillates with amplitude $\sim \kappa$ about its mean value even far from the separatrix, and it undergoes both geometrical and dynamical jumps at each separatrix crossing. @artemyevIonMotionCurrent2013a provided a comprehensive analysis of these non-adiabatic effects for the current sheet with sheared magnetic field, deriving analytical expressions for the jumps and quantifying their consequences for particle stochastization. We summarize the key results here. + +Understanding the nature, magnitude, and statistics of these jumps is essential for this thesis, because the destruction of $I_z$ is physically equivalent to pitch-angle scattering: $I_z$ determines the particle's oscillation amplitude in the current sheet, which in turn determines its pitch angle relative to the local magnetic field. A jump in $I_z$ therefore corresponds directly to a change in pitch angle, and the rate of $I_z$ destruction determines the pitch-angle diffusion rate. + +#### The improved quasi-adiabatic invariant and its jumps + +Far from the separatrix, one can construct an _improved quasi-adiabatic invariant_ $J = I_z + \kappa u(z, p_z, \kappa x, p_x)$, where $u$ is a correction function defined at each point of the four-dimensional phase space (except on the separatrix itself). The improved invariant $J$ is conserved with accuracy $\sim \kappa^2$, compared to the accuracy $\sim \kappa$ of the original $I_z$. This improvement is important because it isolates the true non-adiabatic behavior—the jumps at separatrix crossings $\Delta J = \Delta J^{\text{geom}} + \Delta J^{\text{dyn}}$—from the oscillatory variations that occur during regular motion. It is through the accumulation of these jumps over multiple separatrix crossings that the quasi-adiabatic invariant is destroyed and particle motion becomes chaotic. + +#### Nonzero mean dynamical jump and accelerated stochastization + + + +For the symmetric system ($s = 0$), the well-known expression for the dynamical jump is $\Delta J^{\text{dyn}} = -\frac{2}{\pi} \kappa p_x \ln(2\sin\pi\xi).$ For the asymmetric system ($s \neq 0$), the expressions for $\Delta J^{\text{dyn}}$ are considerably more complex. @artemyevIonMotionCurrent2013a derived the full expressions for the dynamical jumps using the general formulas of @neishtadtChangeAdiabaticInvariant1987 (the derivation is detailed in Appendix A of that paper). The key difference from the symmetric case is that the expressions involve the _individual_ rates of evolution $\Theta_l$ and $\Theta_r$ (rather than their sum) and the ratio $\tilde{\theta} = \Theta_r/\Theta_l$, which parameterizes the asymmetry. All these parameters are functions of the crossing point along the uncertainty curve, and their asymmetry produces a nonzero $\langle \Delta J^{\text{dyn}} \rangle_\xi$ at each crossing. + +The physical consequence of $\langle \Delta J^{\text{dyn}} \rangle_\xi \neq 0$ is profound: it introduces a _directed drift_ in the invariant space, superimposed on the diffusive spreading from the random component. The stochastization timescale is then set by the drift rather than the diffusion. Each separatrix crossing changes $I_z$ by $\sim \kappa$ on average (rather than zero on average), and crossings occur at intervals $\sim 1/\kappa$ (one period of slow motion in the $(\kappa x, p_x)$ plane). An order-unity change in $I_z$ therefore requires only $\sim 1/\kappa$ crossings, taking a total time $\sim \kappa^{-2}$—compared to $\sim \kappa^{-3}$ for the symmetric system where only diffusion operates. For the Earth's magnetotail with $\kappa \sim 0.05$–0.1, this reduces the stochastization timescale from tens of minutes to a few minutes. It should be noted that the conservation of phase-space volume (Liouville's theorem) imposes a constraint: the double average $\langle\langle \Delta J^{\text{dyn}} \rangle\rangle_{I_z}$ over the entire particle population must vanish. That is, while individual values of $I_z$ experience a nonzero mean drift, the net effect over the full distribution is a _redistribution_ of invariants without the appearance of net fluxes in phase space. The presence of current sheet boundaries at $|z| = \lambda$ can, however, break this constraint and produce a net drift in the invariant space [@zelenyiSplittingThinCurrent2003]. + +#### Geometrical chaotization: rapid destruction independent of $\kappa$ + +While the nonzero mean dynamical jump accelerates stochastization from $\kappa^{-3}$ to $\kappa^{-2}$, an even faster mechanism exists when the system possesses a fully broken symmetry. As discussed in the quasi-adiabatic theory, the Hamiltonian with $s \neq 0$ retains a residual symmetry: invariance under the combined transformation $z \to -z$, $p_x \to -p_x$. As a result, the net change in $I_z$ after one full period of slow motion is determined by the dynamical jumps alone. + +@artemyevRapidGeometricalChaotization2014 showed that when this residual symmetry is also broken—for example, by a cross-sheet electric field $\varepsilon$ that adds a term $\sim -szu_d$ to the Hamiltonian (where $u_d = \varepsilon/\kappa$ is the drift velocity)—the trajectories in the $(\kappa x, p_x)$ plane become asymmetric about $p_x = 0$. The surfaces of constant energy in the four-dimensional phase space that correspond to the two separatrix loops now project onto _different_ (shifted) domains in the $(\kappa x, p_x)$ plane. Consequently, the two successive geometrical jumps no longer compensate: $\sum \Delta I_z^{\text{geom}} \neq 0$. + +The critical feature of this mechanism is that each geometrical jump $\Delta I_z^{\text{geom}}$ is of order unity—it equals the difference of the areas of the two separatrix loops, which is a geometric property of the phase portrait and does not depend on $\kappa$. This means that even for arbitrarily small $\kappa$ (arbitrarily thin current sheets or arbitrarily energetic particles), a _single pair_ of separatrix crossings produces an $O(1)$ change in $I_z$. The particle trajectory in the slow-variable plane jumps between widely separated adiabatic curves, and after only a few separatrix crossings the trajectory fills a large domain, with the adiabatic invariant completely destroyed. + +@artemyevRapidGeometricalChaotization2014 termed this mechanism _geometrical chaotization_ (an immediate, large-scale redistribution of $I_z$ values) to distinguish it from the much slower _diffusive chaotization_ (a gradual Gaussian spreading of $I_z$ distributions with width growing as $\sqrt{t}$) produced by dynamical jumps. Numerical integration of $10^5$ test particle trajectories confirmed that the width of the $I_z$ distribution after ten separatrix crossings is dramatically broader in the fully asymmetric system (3) than in either the symmetric case (1) or the partially symmetric case (2; $s \neq 0$ but no electric field). As shown in @fig-Iz-distributions-comparison, system (3) has a broad, nearly uniform distribution over a wide range of $I_z$ values (complete invariant destruction) while system (1) peaks narrowly at $I_z/I_{z,\text{init}} = 1$ (slow diffusion) and system (2) splits into a few discrete peaks (the finite number of possible $I_z$ values due to geometrical jumps with compensating sums). + +![Distribution of $I_z$ values for an ensemble of $10^5$ trajectories after 10 separatrix crossings: (a) $s=0$, $\varepsilon=0$; (b) $s=0.15$, $\varepsilon=0$; (c) $s=0.15$, $\varepsilon=0.003$. All with $\kappa=0.01$. From @artemyevRapidGeometricalChaotization2014.](figures/ref/artemyevRapidGeometricalChaotization2014-fig3.png){#fig-Iz-distributions-comparison} + +#### Current sheet dynamics: particle reflection, transmission and resonance + +These non-adiabatic effects also have several physically important consequences for current sheet dynamics. The guide field introduces an asymmetry in particle reflection and transmission at the current sheet boundaries. @artemyevIonMotionCurrent2013a showed that for $B_y > 0$, particles approaching from the Southern Hemisphere ($z > 0$) are increasingly likely to transit through the sheet rather than be reflected, with transition becoming certain for $s > 0.35$ (i.e., $B_y > 0.35 B_0 \sqrt{L/\rho_0}$). Particles from the Northern Hemisphere ($z < 0$), by contrast, cross the sheet without any scattering for $s > (\pi^{-1} \ln 2)\kappa$, because the uncertainty curve shrinks and these particles never encounter the separatrix. This directional asymmetry of current sheet interaction has implications for asymmetric auroral precipitation and for the self-consistent equilibrium of current sheets with finite guide fields. + +The finite guide field also destroys the resonant condition under which two successive dynamical jumps can exactly compensate—a condition that, in the symmetric case, produces coherent beamlets of accelerated particles. For $s > (\pi^{-1} \ln 2)\kappa$, the resonance condition, $\sum \Delta I_z^{\text{dyn}} = 0$, cannot be simultaneously satisfied for a large particle population. Moreover, the dependence of the integral $W$ on $I_z$ steepens with increasing $s$, so that even if the resonance condition is satisfied at one value of $I_z$, particles with slightly different invariants are far from resonance. The coherent beamlet structures predicted for the symmetric system are thus disrupted. + +### Application to Force-Free Solar Wind Current Sheets: Superfast Scattering + +The theoretical framework described above was applied to energetic ion scattering by solar wind discontinuities by @artemyevSuperfastIonScattering2020. The key observation motivating that work is that the internal magnetic field structure of observed solar wind current sheets differs in an important way from the idealized models previously considered. In observed compressionless (force-free) discontinuities, the reversal of the maximum-variance component $B_l$ is accompanied by a peak in the intermediate-variance component $B_m$, such that $|B| \approx \text{const}$ across the structure (as shown earlier in @fig-ness1966-fig6 and confirmed statistically by @vaskoKineticscaleCurrentSheets2022). This $B_m$ peak was absent from earlier Hamiltonian models of ion–current sheet interaction, which assumed either $B_m = 0$ (pure field reversal) or $B_m = \text{const}$ (uniform guide field). + +To account for this observed field configuration, @artemyevSuperfastIonScattering2020 modeled the discontinuity magnetic field as $B_l \approx B_0(r_n/L)$, $B_n = \text{const}$, $B_m = \sqrt{B_0^2 - B_l^2} \approx B_0(1 - r_n^2/2L^2)$, which in the normalized Hamiltonian introduces a cubic term: + +$$ +H = \frac{1}{2}p_z^2 + \frac{1}{2}\left(p_x - z + \frac{z^3}{6}\right)^2 + \frac{1}{2}\left(\kappa x - \frac{z^2}{2}\right)^2. +$$ + +The $z^3/6$ term, which encodes the $B_m$ peak, breaks the symmetry of the phase portrait in the $(z, p_z)$ plane. This is the essential difference from the $p_x^2/2$ term (no $B_m$; pure compressional discontinuity) or the $(p_x - z)^2/2$ term (constant $B_m$; uniform guide field) used in previous studies. As demonstrated by the theory of geometrical chaotization [@artemyevRapidGeometricalChaotization2014], this asymmetry generates large geometrical jumps $\Delta I_z \sim 1$ at each separatrix crossing, producing very fast destruction of the adiabatic invariant. + +This result has several important implications for energetic particle transport in the solar wind: + +First, the scattering rate (measured in the slow time $\kappa t$) is independent of $\kappa = (B_n/B_0)\sqrt{L/\rho}$. Since $\kappa$ determines whether a discontinuity is classified as rotational (finite $B_n$) or tangential ($B_n \to 0$), this means that the distinction between RDs and nearly-TDs is irrelevant for the efficiency of ion scattering, provided the $B_m$ peak is present. In a solar wind containing multiple discontinuities, the timescale between successive scattering events is determined by the occurrence rate of discontinuities rather than by $\kappa$, further reducing the importance of $B_n$. + +Second, the condition for strong scattering is $L \sim \rho$, which can be rewritten in terms of particle energy as $h/T_i \sim \beta_i(L/d_p)^2$. For the most intense kinetic-scale discontinuities observed in the solar wind ($L/d_p \in [1, 10]$) and typical plasma conditions ($\beta_i \in [0.1, 10]$), this condition is satisfied for essentially the entire suprathermal ion population. The mechanism is therefore expected to operate broadly. + +Third, test particle simulations confirmed that in the presence of the $B_m$ peak, initially narrow distributions in both $I_z$ and pitch angle broaden rapidly—within a few separatrix crossings—to fill a broad range. In contrast, when the $B_m$ peak is absent ($B_m = 0$), the distributions remain narrowly peaked around their initial values, consistent with the much slower diffusive destruction of $I_z$ in the symmetric system. The effect is robust for $B_m/B_0 > 0.75$, which encompasses the majority of observed force-free discontinuities. + +The identification of this "superfast" scattering mechanism establishes that kinetic-scale current sheets in the solar wind are not merely passive structures but active agents of energetic particle scattering. This mechanism is expected to shape the observed low-anisotropy ion distributions at 1 AU and to contribute significantly to cross-field transport of energetic particle populations. The quantitative characterization of this scattering—including the derivation of analytical pitch-angle diffusion coefficients informed by observed current sheet parameters—forms one of the central contributions of this thesis and is developed in the following chapters. + +## Summary and Open Questions + +This chapter has established the theoretical and observational foundation for understanding energetic particle scattering by solar wind current sheets. We summarize the key points and identify the open questions that motivate the work presented in this thesis. + +Solar wind turbulence is highly intermittent, with magnetic energy concentrating into current sheets whose thickness overlaps with the gyroradii of suprathermal and energetic ions. When a particle's gyroradius is comparable to the sheet thickness, guiding-center theory fails and the quasi-adiabatic invariant $I_z$ governs the dynamics instead of the magnetic moment $\mu$. This invariant is destroyed at separatrix crossings through two mechanisms: dynamical jumps, which drive slow diffusion on a timescale $\sim\kappa^{-3}$ (or $\sim\kappa^{-2}$ with a guide field), and geometrical jumps, which in asymmetric field configurations—including the force-free current sheets most commonly observed in the solar wind—produce order-unity changes in $I_z$ within a single gyration. This *geometrical chaotization* constitutes a superfast scattering mechanism for energetic particles. + +Observationally, current sheets modulate SEP intensities at both large and small scales. The heliospheric current sheet acts as a barrier to cross-hemisphere transport and drives systematic flux dropouts whose energy dependence is consistent with gyroradius-scale interactions. At kinetic scales, coherent magnetic structures are statistically correlated with sharp variations in energetic particle intensity. Together, these observations confirm that current sheets are not passive features of the background medium but active agents of particle scattering and transport. + +Several important questions remain open and motivate the work in this thesis. + +**Pitch-angle diffusion coefficients from current sheet parameters.** While the superfast scattering mechanism has been demonstrated theoretically and confirmed in test-particle simulations, a quantitative analytical expression for $D_{\mu\mu}$ as a function of particle energy and current sheet parameters has not yet been derived. Such an expression—informed by the observed statistical distributions of magnetic field configurations—is needed to incorporate current-sheet scattering into heliospheric transport models and constitutes a central goal of this thesis. + +**Perpendicular transport and cross-field diffusion.** The role of current sheets in cross-field transport remains largely unexplored. Near current sheets, the magnetic field varies on scales comparable to or smaller than the gyroradius, enabling enhanced transfer of particles between field lines through a mechanism distinct from field-line random walk. The anti-correlation between large curvature and weak magnetic field strength [@yangRoleMagneticField2019; @kempskiCosmicRayTransport2023] further suggests that particles undergoing magnetic moment-violating interactions are simultaneously more likely to hop between field lines—a potentially important source of perpendicular transport that deserves systematic study in the solar wind context. \ No newline at end of file diff --git a/docs/others/phd/2026_grad/_review_origin.qmd b/docs/others/phd/2026_grad/_review_origin.qmd new file mode 100644 index 0000000..f47305f --- /dev/null +++ b/docs/others/phd/2026_grad/_review_origin.qmd @@ -0,0 +1,46 @@ +## On the origin of solar wind current sheets + +- @matthaeusIntermittencyNonlinearDynamics2015 + +interplanetary field is composed of a large number of separate filamentary tubes that originate at the sun. + +Many such discontinuities are visible on mesoscales (~4 days, ~1 AU). Earlier interpretations had proposed that discontinuities are simply boundaries of spaghetti-like filaments extending from the Sun to the Earth, but @burlagaDirectionalDiscontinuitiesInterplanetary1969 showed that well-ordered filaments with sharp boundaries are not always present even though discontinuities are ubiquitous. This observation favors a _discontinuous_ rather than _filamentary_ model of the interplanetary medium, allowing for the possibility that discontinuities form in situ far from the Sun rather than being exclusively convected from the corona. + +- @neugebauerCommentAbundancesRotational2006 + - Abstract: "Neugebauer and Buti [1990] failed to find the plasma and field signatures predicted for the evolution of Alfve ́n waves into solitons and thence into RDs." + - "A final possibility is that the distinction between RDs and TDs is an oversimplification of reality. Along that line, De Keyser et al. [1998, p. 2652] suggest that ‘‘most solar wind TDs, at least in the fast wind, are a particular case of RDs, that is, phase-steepened large-amplitude Alfve ́n waves [Tsurutani et al., 1994] that happen to propagate quite slowly as their wave vector is nearly perpendicular to the magnetic field.’’" +- @vasquezNumerousSmallMagnetic2007 + - """Association With Discontinuities From Flux Tubes + The tangential discontinuity is a boundary with a normal across the background magnetic field. The TD can support waves but here its formation is limited to a discontinuity which is not the result of Alfve ́n wave evolution or Alfve ́nic turbulence. Instead, we consider cases where the magnetic field develops thin current sheets in association with flux tube dynamics at the Sun. [56] At the photosphere, the magnetic field is strongest in small magnetic bright points [e.g., G-band bright points] between granules in the network which lie along the boundary of supergranules. These bright points have scales of 100 to 200 km in the photosphere and average separations of 350 to 700 km. Cranmer and van Ballegooijen [2005] developed a model of an open-field source region using these bright points which we use here to describe flux tube structure. From these bright points, the magnetic field is constrained on the outside mostly by plasma pressure and is vertical. In the low chromosphere at an altitude of about 2000 km, the lower gas pressure causes the flux tubes to expand until they form boundaries adjacent to one another. Current sheets could characterize all of these boundaries and possibly yield a flux tube structure that could be found in the solar wind. Because collisions are so infrequent, the current sheets can have thicknesses which are as small as the proton inertial length and so correspond to TDs in the solar wind. A possible constraint on whether or not the discontinuities are rooted in the Sun comes from the rapid Figure 13. Histogram of widths based on normal direction in units of (a) proton inertial length c/wpp and (b) proton gyroradius rgp for 203 discontinuities where plasma proton b is less than 0.1. In these plasma conditions, c/wpp is larger than rgp. Part (a) has a similar distribution of widths as is seen in Figure 12a, while in part (b) widths are larger in gyroradii and have not thinned proportionally to rgp. This indicates that c/wpp determines the widths for small b. For large b > 4 (not shown) results are consistent with widths determined by rgp. time-scale of evolution for the solar magnetic field. The magnetic field at the Sun undergoes reconnection on a scale of 1.4 h [e.g., Close et al., 2004]. If the reconnection is an interchange between closed and open field lines, then current sheets may be continuously rooted in the Sun if the result is a braiding of magnetic fields emanating from separate bright points. [57] A number of properties of the observed discontinuities are consistent with TDs. Within measurement uncertainty discontinuities with triangulated normals can have zero normal field components which are consistent with TDs [e.g., Knetter et al., 2004]. The magnetic field intensity and direction typically change across the layer of all discontinuities. TDs are unlimited in these changes and generally such changes would be expected to be observed among a large number of TDs. The lognormal distribution of separations between discontinuities could be due to net fragmentation of magnetic field flux concentrations from advection and reconnection at the Sun. [58] However, there are far more identified discontinuities than expected of solar wind flux tubes. Near 1 AU, z et al. [1966] and McCracken and Ness [1966] inferred that energetic ions follow magnetic field lines with abrupt changes in direction (within the 7.5 min resolution of particle sampling) occurring about every 0.5– 4 h or 0.54 106 km. They interpreted the abrupt changes to be the result of spaghetti-like flux tubes in the solar wind that are rooted in the Sun and referred to these as filaments. Recently, Borovsky [2006] has revived this concept based on the behavior of spread angle on the scale of 128 s noting that spreads larger than 52° have a different distribution from smaller spreads and could mark out a flux tube structure with an average occurrence rate of 25 min. Burlaga and Ness [1968] found that discontinuities do not clearly outline the boundaries of filaments between which magnetic fields and plasma should vary only slowly. Instead, they concluded that the solar wind is discontinuous rather than occupied by filaments. More recent observations find flux dropouts of energetic ions and suprathermal electrons from the Sun which usually have durations on the order of a few hours [e.g., Larson et al., 1997; Mazur et al., 2000; Gosling et al., 2004a, 2004b]. These do indicate the presence of some flux tube structure in the solar wind but boundaries are relatively far apart. Even if all of these flux tubes had very sharp boundaries, they would not match the occurrence rate of identified discontinuities. Discontinuities with dB/B > 0.2, which are classified as TDs apart from EDs, occur on average every 5 h. This rate is lower than the rate associated with the inferred solar wind flux tube structure. However, if some flux tube boundaries are not sharp or if some are bounded by discontinuities with smaller dB/B, then the observed lower rate can be explained. [59] In the extreme, the number of discontinuities is limited up to the point of filling all space. We estimated that about 3% of the solar wind is filled by discontinuities satisfying our prechosen criteria. Thereby, room for more is available. The average separation between magnetic bright points at the Sun (350 to 700 km) can be mapped to 1 AU using a more than radial expansion factor of 3 to 9 and a distance of 212 solar radii. This corresponds to 2.2 105 1.3 106 km at 1 AU which at a solar wind speed of 400 km/s requires 9 to 54 min to convect past the spacecraft. The identified discontinuities have a separation of several minutes and so potentially could correspond to the magnetic field structure associated with boundaries of magnetic bright points but more likely require a smaller solar structure or substructure within the bright points. [60] The most difficult property to explain consistently with TDs concerns the dV and dB correlation which is generally found in the vicinity of discontinuities [e.g., Belcher and Davis, 1971; Belcher and Solodyna, 1975; Neugebauer et al., 1984; Neugebauer, 1985, 2006]. Once velocity is associated with the discontinuities, one needs to know whether or not the discontinuity is and has been a static boundary across which forces are balanced. Correlated dV and dB fluctuations when associated with open magnetic field regions are more likely to lead to propagating waves and Alfve ́nic turbulence [e.g., Hollweg and Lee, 1989; Parker, 1991]. However, we consider two possibilities below that could give TDs that satisfy the correlation. [61] Neugebauer et al. [1986] considered the potential role of Kelvin-Helmholtz instability at TDs. A more parallel or antiparallel orientation of dV to dB acts to stabilize the instability. Neugebauer theorized that this instability operates at the Sun to remove discontinuities with more perpendicular orientations between dV and dB. [62] Another way to account for the dV and dB correlation is to have Alfve ́nic surface waves supported by the TD [e.g., Hollweg, 1982; Neugebauer, 1985; Horbury et al., 2001; Vasquez et al., 2001; Vasquez, 2005]. Assuming that the discontinuity is purely magnetic, then the velocity seen there is from the wave. If these waves have large amplitudes compared to the magnetic field change across the discontinuity, then what is observed at the discontinuity approaches a case where the observed magnetic field and the velocity appear correlated. However, large amplitude surface waves would cause the cross-product method to give errant normals, which is not found for discontinuities which have triangulated normals. In section 3, the results on crossproduct normals infer that fluctuations in the vicinity of discontinuities on average have relative amplitudes of 5% which is too small to explain dV and dB correlation. + """ + - """Association With Discontinuities From Alfve ́n Wave Steepening + The RD can be generated by steepening of an Alfve ́n wave [e.g., Cohen and Kulsrud, 1974; Malara and Elaoufir, 1991; Vasquez and Hollweg, 1996, 1998a, 1998b, 1999a, 1999b, 2001; Medvedev et al., 1997]. The Alfve ́n wave steepens because of the resonance between wave speeds of MHD modes which exists when propagating along the background magnetic field. RDs result when the larger amplitude portion of the wave overruns the smaller amplitude region due to the nonlinear wave magnetic pressure force density r(dB dB)/2m0. The process is halted in a collisionless plasma when ion wave dispersion balances the steepening. This mechanism favors normals which are nearly field aligned. Starting in the corona where waves have small relative amplitudes, nonlinear wave steepening times obtained in the small amplitude and parallel propagating limit from the MHD equations by Cohen and Kulsrud [1974] are approximately 10 h, so that RDs can be generated before transit to 0.2 AU. So RDs should arise during these times, if ever, with large normal magnetic field components. At larger relative amplitudes, the resonance of modes is broadened, and steepening can occur faster and in more oblique directions. The process of RD formation tends to smooth away gradients of B locally in the wave. Time-asymptotically, B shows little change across the discontinuity. + The RD as a product of Alfve ́n wave evolution is consistent with the observed fluctuation dV and dB correlation in the vicinity of discontinuities. Furthermore, the tendency to small dB/B is partly consistent. However, the intensity change should vanish at long times after steepening, but the discontinuities show a distribution of intensity change. Steepening is expected to produce discontinuities with normals in all directions with a preference for field aligned normals. This is inconsistent with the nearly perpendicular normals found. Steepening would also be expected to proceed from the largest waves, overrunning any smaller discontinuities within the steepening region to produce isolated discontinuities. No more than two per wavelength would be expected. The correlation scale of waves in the solar wind is approximately 106 km. At two per wavelength, the average upper limit on occurrence would be about 2 RDs per hour for average VSW = 381 km/s. Locally, other wavelengths could be associated with the fluctuations, which if shorter than the correlation scale could give a larger upper limit on occurrence rate per hour. Additionally, if the steepening occurs progressively starting and completing at larger amplitudes and longer wavelengths and then proceeding to smaller amplitudes and shorter wavelengths after the larger discontinuities are already produced, then more RDs could be present at 1 AU and large and small ones in close proximity could be observed. Moreover, if fast shocks occur around RDs, then groupings and higher occurrence rates of observed discontinuities might be explained. These fast shocks move away from RD layers just after simulated Alfve ́n waves form imbedded RDs [e.g., Vasquez and Hollweg, 1998b]. However, these fast shocks dissipate at the Landau resonance on the order of 600 proton gyroperiods ( 2 h at 1 AU) and so would not be observed at 1 AU in the solar wind unless the steepening has occurred recently. + Alternatively, wave steepening may generate kinetic features other than RDs [e.g., Buti et al., 2001]. Tsurutani et al. [2002a, 2002b, 2005a, 2005b, 2007] observe discontinuities with large dB/B that are associated both with Alfve ́nic fluctuations and with magnetic holes and decreases. As discussed in section 1, Tsurutani et al. find that phase steepening and plasma acceleration through a wave ponderomotive force [e.g., Tsurutani et al., 2002b; Dasgupta et al., 2003] can explain the observations. Therefore discontinuities with large dB/B, often associated with static TDs, may in fact be kinetic structures associated with wave steepening or intermediate shocks [Tsurutani et al., 2005a] + """ + - """Association With Discontinuities From Reconnection + Reconnection between magnetic fields in different directions can result in the formation of a number of discontinuous features including the RD and slow shock [e.g., Yang and Sonnerup, 1977; Heyn et al., 1988] and can be associated with magnetic holes and decreases [e.g., Farrugia et al., 2000, 2001; Zurbuchen et al., 2001]. The RD which forms always has a small normal field component and so is consistent with this important aspect of the solar wind discontinuities. When formed in a fluid, the RD and slow shock often nearly coincide, so that the discontinuity has properties of a slow shock and is often referred to as a pseudoslow shock. Downstream of the propagation direction of the RD, the magnitude of the velocity transverse to the normal direction increases above that found upstream by the amount of the local Alfve ́n speed. This is the plasma jet associated with the reconnection outflow region. [67] If formed from an impulsive reconnection event, the RD can separate from the slow shock, at least in the fluid description, because they travel at different speeds. In the interplanetary medium, this is an important aspect since the time from formation can be long. Once separated, the RD closely obeys the Walen relation jdVj = jdBj/(m0r)1/2 and shows an increase of the magnitude of the velocity by the amount of the local Alfve ́n speed on its downstream side [e.g., Farrugia et al., 2000, 2001]. Most solar wind reconnection regions are identified by pairs of DDs bounding an outflow region with speeds comparable to the local Alfve ́n speed [e.g., Gosling et al., 2005a, 2005b, 2006a, 2006b, 2006c]. In each pair, the DDs have opposite dV dB correlations with respect to one another which is consistent with ingoing and outgoing propagation, respectively. Here the DDs appear to be RDs only or pseudoslow shocks that do not break up into separate propagating RDs and slow shocks, possibly as a result of driven reconnection and also of ion kinetics which allows the reconnection plasma jet to intermix with the inflow, crossing the contact discontinuity that would exist in the fluid description. Most observed DDs are not ordered in the manner of a reconnection region, do not have associated speed increases, and do not closely satisfy the Walen relation. + """ + - """Association With Discontinuities From Alfve ́nic Turbulence + Alfve ́nic MHD turbulence ensues when a large number of Alfve ́n waves travel in many different directions forward and backward with respect to the background magnetic field so that the generalized Reynolds stress r(dV r)dV (dB r)dB/m0 among the waves is nonzero. Analysis and simulations show that fluctuation energy cascades across the background magnetic field to smaller perpendicular scales [e.g., Shebalin et al., 1983; Ng and Bhattacharjee, 1996]. A number of thin current sheets with normals directed across and nearly across the magnetic field develop. In MHD simulations [e.g., Matthaeus et al., 1996; Kinney and MacWilliams, 1998; Cho and Vishniac, 2000; Mu ̈ller and Grappin, 2005], energy dissipates in these sheets due to viscosity and resistivity because of their small scales. The sheets are then transient and are constantly forming and decaying at various positions throughout the simulation box. [69] In our estimation, turbulence gives a convincing explanation for the main properties of observed discontinuities. First, relative intensity changes are mostly small and distributed approximately in accord with a Gaussian distribution suggesting that the changes occur in a random manner. Moreover, Neugebauer [2006] has shown that discontinuities with small intensity changes also have small changes in proton density, alpha concentration, and proton magnetic moment. Small changes in all these quantities are consistent with Alfve ́n waves which are noncompressive wave modes and also with nonlinear interactions among Alfve ́n waves which are nearly incompressible [e.g., Zank and Matthaeus, 1993] or weakly compressive [e.g., Bhattacharjee et al., 1998] in the limit of nearly perpendicular propagation. Second, the dV and dB correlation in the vicinity of discontinuities [e.g., Neugebauer, 2006] is similar to that seen on larger scales associated with Alfve ́nic fluctuations. Third, the high abundance of discontinuities throughout all solar wind sampled is consistent with the nearly ubiquitous presence of Alfve ́nic fluctuations. Since these Alfve ́nic fluctuations have properties consistent with evolving turbulence, we would expect this evolution to result in the formation of current sheets. Fourth, discontinuities have small normal field components as would turbulent-generated current sheets. Fifth, lognormal distributions of separations between discontinuities are consistent with multiplicative random cascades and intermittency which can arise in turbulence (see section 3 for a fuller discussion). Since the discontinuities have a number of properties which are distinct from those in phase-random magnetic fields, they appear to be associated with features generated by the turbulence. [70] Because turbulence is difficult to simulate with current computer resources, a number of details of the evolution of the current sheets are unknown. Most importantly, no simulations including ion kinetics have been done. [71] An outstanding issue concerns whether or not some current sheets persist in plasmas and take on the appearance of TDs but with a dV dB correlation. Instead of dissipating all small scale structure as in fluid simulations, collisionless plasmas can have current sheets which are in equilibrium [e.g., De Keyser and Roth, 1997, 1998]. Persistent sheets from turbulence could evolve if they become close to planar. In this situation, the nonlinear generalized Reynolds stresses nearly vanish because in the limit of one spatial dimension of variation normal to a planar structure, the generalized Reynolds stress is zero if bulk velocity does not change along the normal direction, which is the case for Alfve ́n wave modes. Nonlinearity becomes depleted in the vicinity of the sheet [e.g., Boldyrev, 2005, 2006] and if it does not develop instability within the layer to dissipate its associated energy or current, the sheet will persist. In Knetter’s work, discontinuities are found to be locally planar and may correspond to sheets where nonlinearity is depleted. Thereby, discontinuities may have a close affinity with TDs and yet have the dV-dB correlation associated with an Alfve ́nic feature. Boldyrev [2005, 2006] has theorized that persistent sheets could have important effects on driven perpendicular cascades in that the perpendicular cascade is weakened from a Kolmogorov-like to Kraichnanlike cascade [e.g., Mu ̈ller and Grappin, 2005; Vasquez et al., 2007]. + """ +- @madarDirectionalDiscontinuitiesInner2024 "Directional discontinuities in the inner heliosphere from Parker Solar Probe and Solar Orbiter observations" + - "we observationally confirmed that RDs are produced by Alfvén-wave steepening, while the TDs are most likely the boundaries of flux tubes." + - "On the other hand, RDs are frequently observed at the edges of or embedded in Alfvén-wave trains, and two of the most popular theories about their formation are turbulent processes and Alfvén-wave steepening (Cohen & Kulsrud 1974; Malara & Elaoufir 1991). The former produce RDs with randomly distributed normals relative to the local magnetic field, while the latter tends to produce discontinuities with typically a large normal component (Vasquez et al. 2007). Thus, it can be decided which of the proposed formation processes is more consistent with the measurements." + - "Earlier studies have found that even with the application of the triangulation method, it is not possible to distinguish between a TD and an RD at a heliocentric distance of 1 AU if the RD has small Bn/Bmax parameter (Knetter et al. 2004; Wang et al. 2024). This is mainly due to the low Alfvén velocity at 1 AU, which prevents the triangulation method from detecting whether a DD is propagates relative to the solar wind. We argue that in the vicinity of the Sun, the high propagation velocity of the DDs that are finally classified as RDs is a solid proof of their RD nature. This follows from the requirement that RDs must have a good Walén correlation above the threshold, which (because the Alfvén speed is high near the Sun) also implies a high propagation speed relative to the solar wind. This also means that the uncertainty of the Walén classification decreases closer to the Sun. The clear separation of the two populations shown in Figure 5 strongly suggests that the small Bn/Bmax DDs are mostly TDs everywhere, and thus our new classification scheme, which only uses the magnetic field data, is able to distinguish the two types even for larger solar distances. An important step toward understanding the origin and evolution of the interplanetary discontinuities would be a thorough investigation of their internal structure. Artemyev et al. (2019) demonstrated at a heliocentric distance of 1 AU that the internal structure of DDs is dual, with a thin intense current layer embedded within a thick current sheet for which the Walén relation holds. They stated that this structure endows the discontinuity with both RD and TD-like properties simultaneously, but we argue that these DDs are in fact TDs on which surface Alfvén fluctuations are superposed. This can explain both the jumps in the plasma parameters that (Artemyev et al. 2019) found in the thin current sheets and the RD-like behavior of the outer layer." + - "The classification also provides the opportunity of investigating the angular distribution of the DD normals in the inner heliosphere. Lukács & Erdo ̋s (2013) found that most of the TDs in the 0.3–1.0 AU region tend to have normals that are normal to the local Parker spiral direction. Liu et al. (2021) reported similar results while investigating the inner heliosphere 0.12–0.80 AU and concluded that RD normals are likely be aligned with the local Parker direction. Using PSP and Solar Orbiter data from 0.06 to 1.01 AU, we repeated the same analysis, but with the correction for the plasma flow direction, which is a combination of the solar wind flow direction and the spacecraft velocity direction. The local Parker spiral direction was determined using a 900 s time window centered on the DDs. These results are shown in Figure 8. It is clear that the RD normals are reasonably well aligned with the Parker direction, but the normals of TDs have a large angle with respect to the local Parker spiral direction. The fact that the two populations show such a distinctly different behavior reconfirms our DD classification. This result agrees with the idea that the majority of TDs are boundaries between flux tubes because the flux tube boundaries are aligned with the Parker spiral on average. It is still debated whether these flux tubes originate from the solar photosphere, as the simplified ‘spaghetti solar wind’ paradigm suggests, or if turbulent mixing creates them. Based on our observation, we cannot rule out any of these possibilities. The RD normals, on the other hand, are mostly well aligned with the local magnetic field, which agrees with the Alfvén-wave steepening mechanism." + +### Generation, stability and annihilation + +@hoTangentialDiscontinuitiesHigh1995 + +@tsurutaniInterplanetaryDiscontinuitiesAlfven1995: Abstract. The rate of occurrence of interplanetary discontinuities (ROID) is examined using Ulysses magnetic field and plasma data from 1 to 5 AU radial distance from the Sun and at high heliospheric latitudes. It is found that there are two regions in interplanetary space where the ROID is high: in stream-stream interaction regions and in Alfvén wave trains. This latter feature is particularly obvious at high heliographic latitudes when Ulysses enters a high speed stream associated with a polar coronal hole. These streams are characterized by the presence of continuous, large-amplitude (Δ bar B/l BI - 1 - 2,) Alfvén waves and an extraordinarily high ROID value (∼150 discontinuities/day). In a number of intervals examined, it is found that (rotational) discontinuities are an integral part of the Alfvén wave: they represent ∼ 90° phase rotation of the wave out of the full 360° rotation of the wave. These large amplitude nonlinear Alfvén waves thus appear to be phase steepened. The nonlinear Alfvén waves are spherically polarized, i.e., the tip of the perturbation vector resides on the surface of a sphere (a consequence of constant |B|). The best description of this wave plus discontinuity is a “spherical arc polarization”. + +@tsurutaniRapidEvolutionMagnetic2005 +They find that in 6 of 6 cases the layer width thinned and in 4 of 6 cases the magnetic decrease clearly deepened between ACE and Cluster. They conclude that these discontinuities undergo rapid changes and can be dissipative [e.g., Lin et al., 1995, 1996; Tsurutani et al., 2002b]. + +- @vasquezNumerousSmallMagnetic2007 + - "An outstanding issue concerns whether or not some current sheets persist in plasmas and take on the appearance of TDs but with a dV dB correlation. Instead of dissipating all small scale structure as in fluid simulations, collisionless plasmas can have current sheets which are in equilibrium [e.g., De Keyser and Roth, 1997, 1998]. Persistent sheets from turbulence could evolve if they become close to planar. In this situation, the nonlinear generalized Reynolds stresses nearly vanish because in the limit of one spatial dimension of variation normal to a planar structure, the generalized Reynolds stress is zero if bulk velocity does not change along the normal direction, which is the case for Alfve ́n wave modes. Nonlinearity becomes depleted in the vicinity of the sheet [e.g., Boldyrev, 2005, 2006] and if it does not develop instability within the layer to dissipate its associated energy or current, the sheet will persist. In Knetter’s work, discontinuities are found to be locally planar and may correspond to sheets where nonlinearity is depleted. Thereby, discontinuities may have a close affinity with TDs and yet have the dV-dB correlation associated with an Alfve ́nic feature. Boldyrev [2005, 2006] has theorized that persistent sheets could have important effects on driven perpendicular cascades in that the perpendicular cascade is weakened from a Kolmogorov-like to Kraichnanlike cascade [e.g., Mu ̈ller and Grappin, 2005; Vasquez et al., 2007]." + +- @sodingRadialLatitudinalDependencies2001 + - "To find a mechanism for the evolution, one should look at the stability of RDs. One-dimensional hybrid simulations do not lead to definite results. While Richter and Scholer (1989) observed that RDs with θBn < 45◦ are unstable, where θBn is the angle between the upstream magnetic field and n, Goodrich and Cargill (1991) observed stable RDs with θBn = 30◦." diff --git a/docs/others/phd/2026_grad/_scattering.qmd b/docs/others/phd/2026_grad/_scattering.qmd new file mode 100644 index 0000000..d526f79 --- /dev/null +++ b/docs/others/phd/2026_grad/_scattering.qmd @@ -0,0 +1,182 @@ +# Quantification of particle scattering by solar wind current sheets: pitch-angle diffusion rates + +## Introduction + +The transport of energetic particles within the heliosphere is significantly influenced by the turbulent magnetic field present in the solar wind [@giacaloneTransportCosmicRays1999; @pucciEnergeticParticleTransport2016]. Rather than being a simple superposition of random fluctuations, these turbulent fields exhibit a structured nature, frequently observed in the solar wind magnetic field in the form of current sheets, discontinuities, Alfvén vortices, magnetic holes, and other coherent structures [@perroneCoherentEventsIon2020; @perroneCompressiveCoherentStructures2016]. These structures arise from nonlinear energy cascade processes [@degiorgioCoherentStructureFormation2017; @meneguzziHelicalNonhelicalTurbulent1981] and play a critical role in modulating particle transport. Specifically, current sheets---often manifesting as rotational discontinuities---act as efficient scatterers in collisionless plasmas [@artemyevSuperfastIonScattering2020; @malaraChargedparticleChaoticDynamics2021]. Understanding the interaction between energetic particles and such coherent structures is essential for accurately modeling particle transport in the heliosphere [@desaiLargeGradualSolar2016] and has fundamental implications for particle acceleration in interplanetary shock waves [@leeShockAccelerationEnergetic1982; @crookerCIRMorphologyTurbulence1999; @giacaloneCosmicrayTransportInteraction2013]. + +Despite extensive research on magnetic turbulence [@pucciEnergeticParticleTransport2016; @oughtonSolarWindTurbulence2021], the quantitative impact of coherent structures on particle transport remains insufficiently explored. Previous studies have primarily focused on idealized turbulence models [@giacaloneTransportCosmicRays1999; @matthaeusNonlinearCollisionlessPerpendicular2003], often neglecting the specific role of coherent structures, such as current sheets, in the scattering of energetic particles. Investigations of interactions between rotational discontinuities (or groups of such discontinuities forming so-called switchbacks; see [@baleHighlyStructuredSlow2019; @witSwitchbacksNearsunMagnetic2020]) and energetic particles have demonstrated highly chaotic particle dynamics [@malaraChargedparticleChaoticDynamics2021; @malaraEnergeticParticleDynamics2023]. However, previous studies focused on elucidating the basic mechanisms rather than quantifying scattering using realistic distributions in accordance with observations. This study aims to bridge that gap by providing a detailed analysis and quantification of particle interactions with current sheets. We adopt an analytical model of the magnetic field configuration that incorporates realistic statistical parameters derived from solar wind observations at 1 AU. By leveraging an exact Hamiltonian formulation that accounts for the rotational effects of the magnetic field, alongside test particle simulations, we investigate the evolution of particle pitch angles and the statistical long-term behavior of pitch-angle scattering across different particle energies. This approach allows us to extend prior works [@whitmanReviewSolarEnergetic2023] about energetic particle modeling and references therein by incorporating realistic coherent structures. + + + +The structure of this paper is as follows. In @sec-equations we present a detailed description of the magnetic field model and the fundamental equations governing particle motion. In @sec-adiabatic-invariance we introduce the concept of adiabatic invariance and discuss its violations due to separatrix crossings in phase space. @sec-simulations presents numerical simulation results and statistical observations of current sheets in the solar wind. In @sec-evolution we analyze the long-term evolution of particle pitch angles due to multiple scatterings by current sheets. Finally, a discussion and conclusions are provided in @sec-conclusion. + + +## Basic Equations {#sec-equations} + +We assume the following 1-D force-free ($B=const$) magnetic field configuration where $\mathbf{B}$ depends only on the $z$ coordinate across the current sheet surface: + +$$ +\mathbf{B} = B (\cos θ \ \mathbf{e_z} + \sin θ ( \sin φ(z) \ \mathbf{e_x} + \cos φ (z) \ \mathbf{e_y})), +$$ + +where $B$ is the magnitude of the magnetic field, and $θ$ is the azimuthal angle between the normal $\mathbf{B_z} = B \cos θ \ \mathbf{e_z} \equiv B_n \mathbf{e_z}$ and the magnetic field. More specifically, we assume the following form: $φ(z) = β \tanh(z/L)$, where $L$ is the thickness of the current sheet and the shear half-angle $β$ is one half of the in-plane rotation angle $ω_{in}$. The transverse magnetic field $\mathbf{B_t} = B_t \sin φ(z) \ \mathbf{e_x} + B_t \cos φ (z) \ \mathbf{e_y}$ rotates by an angle $ω_{in}$ from $-∞$ to $+∞$ where $B_t = B \sin θ$. As an example, we present an observation from the ARTEMIS mission [@angelopoulosARTEMISMission2011], which captures a magnetic field transition consistent with our model. @fig-ARTEMIS shows the variations in $\mathbf{B}$, $θ$, and $φ$ across the current sheet. The transverse field rotates smoothly, with $φ$ following a hyperbolic tangent profile (note that time is linearly proportional to the spatial coordinate $z\approx v_n t$ where $v_n$ is the solar wind velocity along the normal to the current sheet surface), while $θ$ remains nearly constant near $\pi/2$, indicating a small normal component $B_n$. (A three-dimensional visualization of a typical current sheet magnetic field structure, represented by the magnetic field line, together with three representative particle trajectories is provided in @fig-B_diagram_particle_trajectory.) + +::: {#fig-ARTEMIS} +![](figures/scattering/thc.pdf) + +Example of a current sheet observed by ARTEMIS[@angelopoulosARTEMISMission2011]. Top: magnetic field in the current sheet **lmn** coordinate system [@sonnerupMinimumMaximumVariance1998] where $l$ represents the maximum variance direcction ($B_x=B_t\sin φ$ in our model), $m$ the intermediate variance direction ($B_y=B_t\cos φ$), and $n$ the minimum variance direction ($B_z=B\cos\theta$). Here, $B_t$ and $B$ represent the tangential and total magnetic fields, respectively (see text for detailed definitions). Bottom: variations of the azimuthal angle $φ$ and the azimuthal angle $θ$ across the current sheet. Vertical lines indicate the current sheet boundaries, and the horizontal line represents $\pi/2$. The analysis is based on magnetic field data with a 0.25-second resolution from the Fluxgate Magnetometer [@austerTHEMISFluxgateMagnetometer2008]. +::: + +The motion of a charged particle in a pure magnetic field is governed by the Hamiltonian: + +$$ + H = \frac{1}{2m} \left( \mathbf{p} - q \mathbf{A} \right)^2 +$$ + +where $\mathbf{A}$ represents the magnetic vector potential and $\mathbf{p}$ the canonical momentum. In the context of our model, the magnetic vector potential $\mathbf{A}$ admits the following integrable form: + +$$ +\begin{aligned} +A_x &= L B_t f_1(z) +\\ +A_y &= L B_t f_2(z) + x B_n +\\ +A_z &= 0 +\end{aligned} +$$ + +$$ +\begin{aligned} +f_1&(z) =\frac{1}{2} \cos β \ \left[\text{Ci}\left(βs_+(z) \right)-\text{Ci}\left(βs_-(z)\right)\right] +\\ +&+ \frac{1}{2} \sin β \ \left[\text{Si}\left(βs_+(z) \right)-\text{Si}\left(βs_-(z)\right)\right], +\\ +f_2&(z) =\frac{1}{2} \sin β \ \left[\text{Ci}\left(β s_+(z) \right)+\text{Ci}\left(β s_-(z)\right)\right] \\ +& -\frac{1}{2} \cos β \ \left[\text{Si}\left(β s_+(z) \right)+\text{Si}\left(β s_-(z)\right)\right] +\end{aligned} +$$ {#eq-Hamiltonian} + +and $s_\pm(z)=1\pm\tanh(z/L)$, $\text{Ci}$ and $\text{Si}$ are the cosine and sine integral functions [@abramowitzHandbookMathematicalFunctions1972]. + +We introduce a dimensionless Hamiltonian $\tilde{H} = H / h$ by normalizing $(x, z) = (\tilde{x} L + C_x, \tilde{z} L)$ and $(p_x, p_z) = p (\tilde{p_x}, \tilde{p_z})$ where $p = q L B_t/c$ and $h = q^2 L^2 B_t^2/m c^2$. Note that because $∂H/∂y = 0$, $p_y = \text{const}$ and we could let $C_x = c p_y/q B_n$ to eliminate $p_y$: + +$$ +\tilde{H} = \frac{1}{2} \left(\left(\tilde{p_x}-f_1(z)\right)^2+\left(\tilde{x} \cot θ + f_2(z)\right)^2+\tilde{p_z}^2\right) +$$ + +This is an exact Hamiltonian of a charged particle, incorporating the effect of magnetic field rotation characterized by the shear angle $\beta$. Unlike previous studies that rely on Taylor expansions near the current sheet center $z=0$ [@vainchteinQuasiadiabaticDescriptionNonlinear2005; @artemyevIonMotionCurrent2013; @zelenyiQuasiadiabaticDynamicsCharged2013], we retain the full, unexpanded form. However, for comparison, we note that expanding the Hamiltonian to third order around the current sheet center and assuming $β = 1$ yields a reduced form structurally similar to that used in @artemyevSuperfastIonScattering2020, albeit with a different normalization scheme. In this study, we quantitatively investigate the influence of shear half-angle ($β$) and particle energy ($E=H/h$) on the efficiency of pitch-angle scattering. For simplicity, the tilde notation over dimensionless variables is omitted in the following discussion. + +## Adiabatic invariance and its violations at separatrix crossings {#sec-adiabatic-invariance} + +For current sheets where $B_n / B_t ≪ 1$, the variables $(κ x, p_x)$ evolve significantly more slowly compared to $(z, p_z)$ along particle trajectories (with $\dot p_x \propto \kappa$, where $κ = \cot θ$). Assuming that $x$ and $p_x$ are effectively frozen, the Hamiltonian describes periodic motion within the $(z, p_z)$ plane, governed by the effective potential energy $U(z) = H - p_z^2/2$. When variations in $(κ x, p_x)$ occur on a timescale much longer than the fast oscillations in $(z, p_z)$, the generalized magnetic moment $I_z = (2 π)^{-1} ∮ p_z dz$ is approximately conserved as an adiabatic invariant with exponential accuracy [@neishtadtAccuracyPersistenceAdiabatic2000]. Far from the current sheet, where the magnetic field is nearly uniform ($|z/L|\gg 1$), the pitch angle defined by $α = \arccos{\mathbf{B} \cdot \mathbf{v} / |\mathbf{B}||\mathbf{v}|}$ remains constant (i.e., $∂ α / ∂ \mathbf{r} = 0$), where $\mathbf{v}$ is the velocity, $\mathbf{B}$ is the magnetic field, and $r$ is the position. Since the simultaneous conservation of energy and $I_z$ fully determines the motion of the particle, the velocity $\mathbf{v}$ can be expressed as a function of position $\mathbf{r}$, invariant $I_z$, and energy $E$: $\mathbf{v} = \mathbf{v}(\mathbf{r}, I_z, H)$. Given that energy is exactly conserved in a static magnetic field, the pitch angle $α$ depends solely on $I_z$. Therefore, in the absence of $I_z$ destruction, there is no pitch-angle scattering across the current sheet. + +![(a) Phase portraits of the Hamiltonian in the plane of $(z,p_z)$ at fixed $(κx, p_x)$ for $β = 1$. Each curve corresponds to a specific $H$, indicated on the plots. The left panel corresponds to $\kappa x = 4$, $p_x = 1$, while the right panel corresponds to $\kappa x = 0$, $p_x = 0.5$. (b) Phase plane of the Hamiltonian in the $(κx,p_x)$ space. The red line represents the uncertainty curve and the blue line delineates the boundary encompassing all possible phase points. (c) Potential energy profiles defined by $U (z) = H − p_z^2 /2$ at different locations in the $(κx, p_x)$ place, corresponding to the labeled positions (\#) in panel (b).](figures/scattering/fig-bcPlot.pdf){#fig-zPz_phase} + +The Hamiltonian (\ref{eq:Hamiltonian}) admits two distinct types of particle motion in the $(z, p_z)$ plane, as illustrated in @fig-zPz_phase(a). In the left panel, there are two potential wells in $U(z)$, resulting in three possible types of orbits in the $(z, p_z)$ plane depending on the value of $H$: particles can move (oscillate) within one of local $U(z)$ minima, or can move outside these minima along *figure-eight* orbits. In contrast, the right panel represents a case with only one possible type of orbits in the $(z, p_z)$ plane for a fixed $H$. As $(κ x, p_x)$ evolve slowly, the particle's trajectory in the $(z, p_z)$ plane undergoes a gradual transformation. Transitions between motion types can occur, accompanied by significant trajectory reconfigurations when the particle crosses the separatrix, the curve separating motion within one of two local minima of $U(z)$ and motion along *eight-like* orbit (separatrix is shown by red bold line in @fig-zPz_phase(a)). Near the separatrix, the instantaneous period of motion in the $(z, p_z)$ plane increases logarithmically, diverging as the trajectory approaches it [@lichtenbergRegularStochasticMotion1983]. When the timescale (period) of fast oscillations in the $(z,p_z)$ plane and variations in the control parameter ($\kappa x$) become comparable, the particle accumulates a nonvanishing change in the adiabatic invariant, resulting in a jump in $I_z$ [@neishtadtChangeAdiabaticInvariant1999; @caryAdiabaticinvariantChangeDue1986]. Concurrently, the projection of the particle's phase point onto the $(κ x, p_x)$ plane lies along the so-called uncertainty curve [shown by red bold line in @fig-zPz_phase(b)], where each point on the curve corresponds to a particle trajectory in ($z,p_z$) coinciding with the separatrix. + +The jump of $I_z$ due to the separatrix crossing comprises two distinct components. The first component, termed the *dynamical jump*, arises from the singularity of the period of motion in the vicinity of the separatrix and, in our asymmetric case of the $(z, p_z)$ plane about the $z$-axis (stemming from the term in $f_1$) is proportional to $\kappa \ln \kappa$ [see the review by @neishtadtMechanismsDestructionAdiabatic2019, and references therein], resulting in slight changes to particle trajectories. The second component, referred to as the *geometric jump*, corresponds to the difference between the areas enclosed by the particle's trajectory within one of the separatrix wells and the area in the external region outside the separatrix. For instance, when a particle drifts from the boundary immediately before crossing, the total area enclosed by the separatrix increases to the value of the adiabatic invariant, $I_0 = (2\pi)^{-1}S(κ x, p_x)$. Upon crossing, the adiabatic invariant $I_z$ undergoes a jump to approximately the area of the left well $S_l(κ x, p_x)$ or the right well $S_r(κ x, p_x)=S(κ x, p_x)-S_l(κ x, p_x)$, reflecting the reduction in the accessible phase space area for the particles. This geometric jump is independent of $\kappa$ and of order unity [see review by @neishtadtMechanismsDestructionAdiabatic2019, and references therein]. For a single separatrix crossing and small values of $κ$, this jump of adiabatic invariance arises primarily from geometric destruction [@artemyevSuperfastIonScattering2020]. In symmetric Hamiltonians -- where $U(-z)=U(z)$ -- the differences in the enclosed areas during two successive crossings effectively cancel: the area deficit incurred when entering one of two wells is offset by the excess upon exiting, so that the net geometric jump over two crossings is zero [@neishtadtChangeAdiabaticInvariant1999; @buchnerRegularChaoticCharged1989]. However, in our case, the term $f_1(z)$ introduces an asymmetry in the $(z, p_z)$ plane [see @fig-zPz_phase (a) and (c)], leading to an asynchronous evolution of the two areas. As a result, the difference in areas at each crossing does not cancel between successive crossings. This produces a nonzero change in $I_z$ that enhances pitch-angle scattering and drives rapid chaotization of the particle dynamics [@artemyevIonMotionCurrent2013; @artemyevSuperfastIonScattering2020]. %Although the dynamical jumps tend to average out to zero [@neishtadtChangeAdiabaticInvariant1999; @buchnerRegularChaoticCharged1989] for two successive crossings occur, the uncompensated geometric jumps accumulate, producing a nonzero net change in $I_z$ that enhances pitch-angle scattering and drives rapid chaotization of the particle dynamics [@artemyevIonMotionCurrent2013]. + + +Due to the symmetry of particle motion about the $p_z = 0$ line, the potential energy $U(z)$ reaches a local maximum along the $z$-direction at the saddle point $z = z_c, p_z = 0$ of the separatrix (shown as $z_c$ in Fig. \ref{fig-zPz_phase} (left panel)). At this point, conditions $∂ U/∂ z = 0$ and $∂^2 U/∂ z^2 < 0$ are satisfied. These conditions allow us to express the slow variables along the uncertainty curve in the $(κ x, p_x)$ plane as functions of $z_c$. When a particle trajectory crosses the $z = 0$ plane, the crossing point is confined within a circular region defined by $(p_x - f_1(0))^2 + (\kappa x + f_2(0))^2 = 2H$. The likelihood of the trajectory intersecting the uncertainty curve is therefore approximately proportional to the uncertainty curve length $L_{\text{uc}}$, normalized by the square root of the particle energy, $\sqrt{H}$, as illustrated in @fig-UCLength. For fixed $H$, $L_{\text{uc}}/\sqrt{H}$ increases with $β$, indicating higher scattering probabilities at larger magnetic field rotation angles. Similarly, at a fixed $β$, $L_{\text{uc}}/\sqrt{H}$ increases with $H$, suggesting that high-energy particles are more susceptible to pitch-angle scattering. Next, we will demonstrate these changes using test particles in realistic fields, to characterize their properties as functions of system parameters. + +::: {#fig-UCLength} +![](figures/scattering/UCLength.pdf) + +The uncertainty curve length $L_{\text{uc}}/\sqrt{H}$ as a function of $β$ and normalized particle energy $H$. +::: + +## Test particle simulations {#sec-simulations} + +To quantitatively analyze how particles are scattered by solar wind current sheets, we conducted extensive test particle simulations using a dataset of solar wind current sheets at 1 AU. Since the primary focus is on protons, the particle mass was set to $m_p$ (proton mass) and the charge to the elementary charge $q=e$. According to the dimensionless Hamiltonian, the critical current sheet parameters that influence particle dynamics are angles $θ$ and $β = ω_{in}/2$, along with the parameter $B_t L$, which appears in the normalization factors $p$ and $h = \frac{q^2 L^2 B_t^2}{m c^2}$. By a simple transformation, $B_t = B \sin θ$, the key current sheet parameters become $θ$, $ω_{in}$, and $\tilde{v}_B \equiv v_B /c$, where $v_B\equiv q B L / (m_p c)=Ω L$ and $Ω$ is the proton gyrofrequency associated with the magnetic field $B$. + +![3D density plots of the azimuthal angle $θ$, in-plane rotation angle $ω_{in}$, and logarithm of the characteristic velocity $\log \tilde{v}_B$. The left panel corresponds to cases where the MVAB accuracy conditions are satisfied, while the right panel represents cases where they are not satisfied.](figures/scattering/wind_hist3d.png){#fig-windHist3D} + +Using data from the ARTEMIS [@angelopoulosARTEMISMission2011] and the Wind mission [@acunaGlobalGeospaceScience1995], we compiled a dataset of 100000 current sheets [see details of procedure of current sheet selection in @zhangSolarWindDiscontinuities2025; @liuMagneticDiscontinuitiesSolar2022]. The orientations of these current sheets were determined using the minimum variance analysis of the magnetic field (MVAB) method [@sonnerupMinimumMaximumVariance1998]. Accurate orientation determination was crucial for estimating the thickness of the current sheets ($L$) and the in-plane magnetic field rotation angle ($ω_{in}$), both of which significantly influence particle scattering. To ensure reliability, only current sheets with $Δ|B|/|B| > 0.05$ or $ω > 60°$ were used in the following analysis, as these conditions improve the accuracy of the MVAB method, as noted by \citet{liuFailuresMinimumVariance2023}. Note, these conditions do not constrain the discontinuity normal. An alternative approach to orientation determination [@wangSolarWindCurrent2024; @knetterFourpointDiscontinuityObservations2004] involves using the cross-product method to estimate the normal orientation and is largely based on the assumption that $B_n$ is effectively zero. +@fig-windHist3D shows 3D density plots of the azimuthal angle $θ$, the in-plane rotation angle $ω_{in}$, and logarithm of the characteristic velocity $\log \tilde{v}_B$, categorized by whether the MVAB accuracy conditions are satisfied (left for accurate, right for not accurate). Current sheets with accurately determined orientations typically have smaller azimuthal angles $θ$, indicating a smaller $B_n$, and moderate in-plane rotation angles $ω_{in}$. In contrast, current sheets with potentially inaccurate normal orientations display larger $θ$ (larger $B_n$) and larger $ω_{in}$. The most probable values observed in the distribution are a characteristic velocity ($v_B$) of approximately 500 km/s (which corresponds to a typical energy $\sim 1$ keV), an in-plane rotation angle ($ω_{in}$) near 100 degrees, and an azimuthal angle ($θ$) around 85-95 degrees. + +For each magnetic field configuration, we initialize an ensemble of particles far away from the current sheet center (i.e., with an initial $z$ position satisfying $|z_0| > 6 L + 2 r_g$, where $r_g$ is the gyro radius). The particles are uniformly binned in pitch angle ($α_0$) from $0^{\circ}$ to $180^{\circ}$ in 180 bins of size $Δα = 1^{\circ}$, with gyro phase ($ψ_0$) uniformly sampled from $0^{\circ}$ to $360^\circ$ in 120 bins of size $Δψ = 3^{\circ}$. In this simulation, the current sheet is consistently configured with a positive $B_z$ component. Consequently, particles with an initial positive pitch-angle cosine $μ \equiv \cos\alpha$ ($α_0<90^\circ$; $μ_0>0$) are interpreted to originate below the current sheet (and move toward the current sheet), while those with a negative cosine ($α_0>90^\circ$; $μ_0<0$) are considered to come from above it (and move toward the current sheet). The particle trajectories are then numerically integrated until each particle fully exits the current sheet ($|z(t)| > |z_0| + 2 r_g$). @fig-B_diagram_particle_trajectory shows three representative particle trajectories in a typical magnetic field configuration characterized by $\beta=75^\circ$ and $\theta=85^\circ$. All particles have the same initial pitch angle $\alpha_0 = 90^\circ$ and velocity $v_p=8v_B$, but differ slightly in their initial gyrophases: $\phi_0 = 163.3^\circ, 164.4^\circ$, and $165.6^\circ$. +These cases illustrate distinct scattering behaviors: one particle exhibits a negligible change in pitch angle, another undergoes a finite pitch-angle deflection, and the third is reflected by the current sheet. For each trajectory, the final pitch angle $\alpha_1$ is recorded. These pitch angles are then organized into bins to construct a transition matrix (TM), also known as a stochastic matrix [@durrettEssentialsStochasticProcesses2016], which represents the probability distribution of pitch-angle changes resulting from a single particle interaction with the current sheet. + +::: {#fig-B_diagram_particle_trajectory} +![](figures/scattering/fig-B_diagram_particle_trajectory.pdf) + +Three particle trajectories (T1, T2, T3) with identical initial pitch angles ($\alpha_0 = 90^\circ$) and velocity $v_p=8v_B$, but slightly different initial gyrophases ($\phi_0 = 163.3^\circ, 164.4^\circ, 165.6^\circ$) in a representative magnetic field profile ($\beta = 75^\circ, \theta = 85^\circ$). The orange star marks the initial particle position. The black and gray lines represent the magnetic field lines to which the reflected particle (T3) is initially and finally attached, respectively, before and after interaction with the current sheet. +::: + +@fig-tm-example illustrates the transition matrix for 100 keV particles (velocity $v_p \approx 4000$ km/s) under four magnetic field configurations: (i) $v_B = 500$ km/s, $θ = 85°$, $β = 50°$; (ii) $v_B = 500$ km/s, $θ = 85°$, $β = 75°$; (iii) $v_B = 500$ km/s, $θ = 60°$, $β = 50°$; and (iv) $v_B = 4000$ km/s, $θ = 85°$, $β = 50°$. Enhanced probability along the diagonal corresponds to weakly scattered particles ($∆α = α_1 - α_0 \approx 0$), while spreading around the diagonal reflects diffusive scattering. Large pitch-angle jumps are represented by non-diagonal elements. + +::: {#fig-tm-example} +![](figures/scattering/example_subset.pdf) + +Transition matrix for 100 keV protons under four distinct magnetic field configurations: (i) $v_p = 8 v_B$, $θ = 85°$, $β = 50°$; (ii) $v_p = 8 v_B$, $θ = 85°$, $β = 75°$; (iii) $v_p = 8 v_B$, $θ = 60°$, $β = 50°$; and (iv) $v_p = v_B$, $θ = 85°$, $β = 50°$. +::: + +The transition matrix color maps for various current sheet configurations reveal that particle pitch-angle evolution during multiple current sheet crossings is determined by a combination of weak/strong diffusion and large jumps. For example, in current sheets with a typical shear half-angle $β=50°$ and azimuthal angle $θ=85°$ (configuration (i)), particles entering from above the sheet ($z>0$; assuming positive $B_n$) often experience significant pitch-angle jumps or strong diffusion. In contrast, particles entering from below the sheet ($z<0$) typically undergo minimal pitch-angle changes. Occasionally, interactions with current sheets having very large shear half-angles ($β$, e.g., configuration (ii)) or smaller azimuthal angles ($\theta$, e.g., configuration (iii)) result in enhanced diffusion and the reflection of certain particles from the current sheet, indicated by a reversal in the sign of $\cos α$. For high-energy particles, interactions with current sheets of comparable characteristic speed ($v_B \sim v_p$, configuration (iv)) are characterized by weak scattering occurring only over a narrow range of pitch angles. The interplay between non-diffusive jumps and continuous diffusive processes drives a dynamic evolution of the particle ensemble. %Over time, this leads to a broadened and redistributed pitch-angle profile, reflecting the statistical nature of interactions with solar wind current sheets. + + +## Long-term pitch angle evolution {#sec-evolution} + +Modeling the long-term effect of particle pitch-angle scattering by current sheets involves simulating the dynamics of a large number of particles through multiple current sheet interactions over extended periods. To reduce computational time and complexity of such simulations and introduce scattering effects into existing transport models, we derive a simplified probabilistic description of pitch-angle evolution. +The scattering process depends on the particle's initial conditions, $(\mathbf{r}, \mathbf{v})$, as well as the configuration of the current sheet, $\mathbf{B}$. In the 1D model, these initial conditions can be expressed in terms of $z, E_0, α_0$, and $ψ_0$. Since our primary interest lies in the pitch angle after the particle exits the current sheet, and the energy $E_0$ remains constant in the absence of an electric field, the initial energy $E_0$ and position $z$ (provided that the particle is sufficiently far from the center of the current sheet) can be omitted from further consideration. + +The final pitch angle, however, is highly sensitive to initial conditions $(α_0, ψ_0)$, where small variations in the gyro phase can lead to significantly different final pitch angles [@malaraChargedparticleChaoticDynamics2021]. Therefore, the scattering process $\Pi : (α_0, ψ_0) \to (α_1, ψ_1)$ is better represented as a probabilistic transition $p(α_1 | α_0, \Pi)$, with the probability derived by numerical interpolation in the $α_1$-space. Here, $\Pi$ characterizes the current sheet configuration. This probabilistic representation is further motivated by the fact that the gyro phase depends on the particle's location and may undergo random shifts during the crossing of a current sheet. Additionally, numerical integration methods, such as the Boris method, introduce a phase error proportional to $Δ t$, making the gyro phase less reliable. Thus, it is more appropriate to model the gyro phase as a random variable. This probabilistic approach eliminates the dependence on the gyro phase, instead focusing on the statistical relationship between the initial pitch angle $α_0$ and the final pitch angle $α_1$, as governed by the properties of the current sheets. + +For particles with a specific energy, a (weighted-average) mixture distribution [@fruhwirth-schnatterFiniteMixtureMarkov2006] can be constructed based on the observed distribution of solar wind discontinuities (SWDs): $p(α_1 | α_0) = \sum_{i} p(α_1 | α_0, \Pi_i) w_i$, where the weight $w_i$ corresponds to the empirical probabilities of specific SWD configurations. After binning, this yields a weighted transition matrix (WTM), which represents the varying likelihoods of particles encountering SWDs with different properties and encapsulates the overall probability of a particle undergoing a pitch-angle jump due to interactions with an ensemble of SWDs at 1 AU. + + +::: {#fig-tm-stats-100keV} +![](figures/scattering/tm_stats_100keV.pdf) + +Weighed transition matrix for 100 keV particles constructed from the observed distribution of current sheet at 1 AU. +::: + +As illustrated in @fig-tm-stats-100keV, the WTM for 100 keV protons at 1 AU shows a strong likelihood of minimal changes in pitch angle, as evidenced by the bright diagonal. However, there exist significant probabilities associated with diffusive scattering and with large pitch-angle changes. The latter arise mainly from interactions with current sheets whose characteristic scales ($L$) are comparable to or smaller than the gyroradius of (in this case) 100 keV protons. Such strong scattering cannot be adequately described by diffusion alone. However, the mapping described in Equation~\ref{eq-mapping} enables the direct application of the mixture distribution for simulating the long-term evolution of particle pitch-angle distributions (such map is a discretized approximation of stochastic differential equations that describe particle dynamics with a given scattering probability; see [@jacodDiscretizationProcesses2012; @ukhorskiyRoleDriftOrbit2011; @artemyevMappingNonlinearElectron2020; @tonoianElectronResonantInteraction2023]): + +$$ +α_{n+1,i} = W_\Pi\left(α_{n,i}, ξ_{n,i}\right) +$$ {#eq-mapping} + +where $n$ is the number of interactions (with SWDs), $i$ is the particle index within the ensemble, and $W_\Pi$ determines the subsequent pitch angle from the mixture distribution, using the previous pitch angle and a uniformly sampled random variable $ξ_{n,i}$. + +@fig-pa-jump-history illustrates two representative solutions of the dynamical pitch-angle mapping equation derived from the mixture distribution at 1 AU for 100 keV and 1 MeV protons. A key feature of the pitch-angle dynamics is the occurrence of infrequent but substantial jumps, including rare, large-angle changes that can lead to particle reflection from the current sheet. + +::: {#fig-pa-jump-history} + +![](figures/scattering/pa_jump_history_high.pdf) + +Examples of particle pitch-angle scattering by solar wind current sheets for 100 keV and 1 MeV protons. +::: + +However, directly incorporating the mapping (or stochastic difference equations) derived from the mixture distribution into classical numerical schemes for transport-diffusion equations is a complex task (see discussion in [@litvinenkoNumericalStudyDiffusive2013; @straussHitchhikersGuideStochastic2017; @mykhailenkoSDEMethodCosmic2024]). To facilitate comparison with other scattering processes and allow inclusion of SWD-induced scattering effects in such models, we evaluate the effective scattering rate, $D_{μμ}$. This rate acts as a global diffusion coefficient, independent of the local pitch angle, since high-energy particles frequently experience large pitch-angle jumps, leading to strong mixing. The scattering rate depends on the particle energy and the mixture distribution (i.e., the distribution of current sheets). Using the mapping described in Equation~\ref{eq-mapping} for an ensemble of particles, we calculate the evolution of the second moment of the pitch-angle distribution for the particle energy range from 100 keV to 1 MeV for current sheets at 1 AU: + +$$ +\begin{aligned} +M_1(n) = N^{-1}∑_{i=1}^N (α_{n,i} - α_{0,i}) +\\ +M_2(n) = N^{-1}∑_{i=1}^N (α_{n,i} - α_{0,i})^2 - M_1^2(n) +\end{aligned} +$$ + +where $M_1(n)$ represents the mean drift in pitch-angle, and $M_2(n)$ quantifies the spreading in pitch-angle. The function $M_2(n)$ increases linearly with the number of interactions with current sheets before reaching saturation, where the pitch-angle distribution becomes totally mixed. This behavior is illustrated in @fig-mixing-rate. The rate of such mixing is described by $D_{μμ}$, which can be evaluated by fitting $M_2(n)$ to an exponential model, $M_2(n) = d - a e^{-D_{μμ} n}$, where $d$ is the saturated second moment and $a$ is the slope of the exponential fit. This model assumes that at early stages (small $n$), particles undergo diffusive mixing, with $M_2(n)\propto D_{μμ} n$. At later stages (large $n$), the distribution becomes fully mixed, and $M_2$ asymptotically approaches a constant value, $M_2\approx M_2(\infty)$. + +To translate the number of interactions $n$ into a physical time, we consider the occurrence rate of SWD [@vaskoKineticscaleCurrentSheets2022; @zhangSolarWindDiscontinuities2025] and the average solar wind speed. Using these parameters, we can estimate the average distance between SWDs. Assuming that protons travel freely between interactions at velocities $v_p \sim \sqrt{H}$, we can then derive the characteristic time interval between consecutive encounters with discontinuities. This enables the transition from a discrete interaction-based framework ($D_{μμ}$) to a continuous temporal description ($\mathcal{D}_{μμ}$), making it suitable for realistic modeling of particle dynamics and enabling direct incorporation of derived diffusion rates into transport-diffusion simulations. +This approach provides a simplified yet effective means to account for current-sheet-induced scattering in broader models of energetic particle dynamics. + +::: {#fig-mixing-rate} +![](figures/scattering/mixing_rate.pdf) +![](figures/scattering/D_μμ.pdf) + +Top: Second moment of the pitch-angle distribution, $M_2(n)$, as a function of interaction number ($n$) for different particle energies (\textasciitilde100 eV, \textasciitilde5 keV, \textasciitilde100 keV, \textasciitilde1 MeV), The estimated mixing rates, $D_{μμ}$, are indicated in the legend. Bottom: Pitch-angle diffusion rates $\mathcal{D}_{μμ}$ as a function of particle energy $E$. +::: + +## Conclusion {#sec-conclusion} + +In this study, we have investigated the scattering of energetic particles by current sheets in the solar wind and developed a model that incorporates the rotational effects $β$, the magnetic field magnitude $B$, and the current sheet thickness $L$. Our findings indicate that current sheets characterized by large rotation angles or relatively small characteristic velocities, defined as $\tilde{v}_B = \frac{q B L}{m c^2}$, can effectively scatter energetic particles. These scattering effects go beyond the conventional diffusion framework, as large pitch-angle jumps lead to rapid particle mixing in pitch-angles. Leveraging extensive observations of solar wind current sheets at 1 AU, we have provided a statistical estimate that quantifies the long-term scattering rate of energetic particles due to current sheets. + +In the heliosphere, beyond 1AU, the background magnetic field strength decreases approximately as $\sim 1/r$ with increasing radial distance, where the azimuthal component $B_{\phi}$ dominates and $B \approx B_{\phi}\sim 1/r$. The magnitude of the current sheet magnetic field ($B$) scales accordingly with the background field [@zhangSolarWindDiscontinuities2025]. Based on recent observations, the current sheet thickness, $L$, scales with the ion inertial length, $d_i \propto 1/\sqrt{N}$, which increases with radial distance due to the decline in plasma density, $N\propto 1/r^2$ [@marucaTransHeliosphericSurveyRadial2023]. Consequently, the characteristic velocity $\tilde{v}_B \propto B L$ remains nearly constant with the radial distance, implying that energetic particles consistently undergo significant scattering by current sheets as they propagate through the heliosphere. The higher occurrence rate of current sheets closer to the Sun [@liuCharacteristicsInterplanetaryDiscontinuities2021; @lotekarKineticscaleCurrentSheets2022] implies an even more pronounced impact on the transport of solar energetic particles. + +These findings may have significant implications for the dynamics of energetic particles in both space and astrophysical plasmas. Efficient pitch-angle scattering enhances the ability of particles to reverse direction along magnetic field lines, thereby reducing their tendency to stream freely and limiting parallel transport. In quasi-linear theory, the parallel spatial diffusion coefficient is approximately inversely proportional to the pitch-angle diffusion coefficient for resonant interactions [@engelbrechtTheoryCosmicRay2022; @jokipiiCosmicRayPropagationCharged1966; @jokipiiAddendumErratumCosmicRay1968]. \rev{As a result, if such scattering will be combined with particle interactions and scattering by shock waves [@burgessDynamicsUpstreamDistributions1984; @gedalinDistributionEscapingIons2008], the stronger pitch-angle scattering should lead to shorter acceleration timescales in diffusive shock acceleration mechanisms [@katouTheoryStochasticShock2019; @malkovNonlinearTheoryDiffusive2001].} Additionally, observations frequently reveal broad spatial distributions of solar energetic particles and the formation of particle reservoirs [@cohenSolarEnergeticParticles2021], suggesting either reduced diffusion rates along magnetic field lines or highly efficient cross-field diffusion [@zhangPropagationSolarEnergetic2009]. Our results indicate that the pitch-angle scattering due to current sheets can significantly affect parallel and perpendicular transport, and this point will be addressed in future work. + +## Data availability + +The code and data used for the findings and figures in this study are available at [https://github.com/Beforerr/ion_scattering_by_SWD](https://github.com/Beforerr/ion_scattering_by_SWD). ARTEMIS and Wind data are available at NASA's Space Physics Data Facility (SPDF) [https://spdf.gsfc.nasa.gov](https://spdf.gsfc.nasa.gov). \ No newline at end of file diff --git a/docs/others/phd/2026_grad/_spedas.qmd b/docs/others/phd/2026_grad/_spedas.qmd new file mode 100644 index 0000000..3d434cf --- /dev/null +++ b/docs/others/phd/2026_grad/_spedas.qmd @@ -0,0 +1,24 @@ + +### Software Development + +**Context**: A central requirement for this thesis is the ability to perform high-performance, interactive, and reproducible analysis of space plasma data and particle dynamics. While the established SPEDAS framework—originally developed in IDL and later ported to Python—remains widely used in the community, its design limitations hinder modern scientific workflows (big data, parallel/distributed computing, etc.). + +**Approach**: To address this, we developed a suite of Julia-based software tools that combine the flexibility and speed of a modern language with the functionality of legacy systems. + +**Results**: The core of this framework is `SPEDAS.jl`, which has interfaces directly with [`pyspedas`](https://github.com/spedas/pyspedas), [`speasy`](https://github.com/SciQLop/speasy), and [`HAPI`](https://hapi-server.org/) while introducing new routines with significantly improved performance. To enable efficient test-particle tracing in both analytic presets and numerical derived electromagnetic fields, we developed `TestParticle.jl`, a lightweight tool for rapid particle trajectory simulations. Additionally, we created [`SpaceDataModel.jl`](https://github.com/beforerr/SpaceDataModel.jl) to implement flexible, standards-compliant data structures aligned with SPASE and HAPI specifications, and contributed physics utilities through [`ChargedParticles.jl`](https://github.com/JuliaPlasma/ChargedParticles.jl) and [`PlasmaFormulary.jl`](https://github.com/JuliaPlasma/PlasmaFormulary.jl). These tools have been integral to the data analysis (e.g., @fig-sp), modeling, and simulation components of this thesis, enabling scalable and transparent research workflows essential for studying particle transport in the heliosphere. + +```julia +f = Figure() +tvars1 = ["cda/OMNI_HRO_1MIN/flow_speed", "cda/OMNI_HRO_1MIN/E", "cda/OMNI_HRO_1MIN/Pressure"] +tvars2 = ["cda/THA_L2_FGM/tha_fgs_gse"] +tvars3 = ["cda/OMNI_HRO_1MIN/BX_GSE", "cda/OMNI_HRO_1MIN/BY_GSE"] +t0,t1 = "2008-09-05T10:00:00", "2008-09-05T22:00:00" +tplot(f[1, 1], tvars1, t0, t1) +tplot(f[1, 2], tvars2, t0, t1) +tplot(f[2, 1:2], tvars3, t0, t1) +f +``` + +![Example code snippet and resulting output from the Julia implementation of the widely used tplot function.](figures/spedas_jl.png){#fig-sp} + +{{< pagebreak >}} \ No newline at end of file diff --git a/docs/others/phd/2026_grad/_summary.qmd b/docs/others/phd/2026_grad/_summary.qmd new file mode 100644 index 0000000..ae8bdec --- /dev/null +++ b/docs/others/phd/2026_grad/_summary.qmd @@ -0,0 +1,118 @@ +# Summary and Future Perspectives + +## Research Summary + +The central scientific contribution of this thesis is a quantitative, observation-informed account of how kinetic-scale solar wind current sheets scatter energetic particles and shape their transport through the inner heliosphere. This is achieved through two interconnected programs—one observational, one theoretical and numerical—whose results build on and reinforce each other directly. + +**Observational characterization across the inner heliosphere.** The first program establishes how current sheet properties vary across heliocentric distances using simultaneous multi-spacecraft measurements. In the inner heliosphere study (Chapter 5), we analyzed three Parker Solar Probe encounter intervals (Encounters 7–9), selected for their contrasting geometric alignments with near-Earth spacecraft ARTEMIS and WIND, allowing us to distinguish spatial evolution from temporal or solar-cycle variability. Using a robust automated detection algorithm applied consistently across all missions, we identified and characterized more than 45,000 current sheets. The use of normalized, resolution-independent detection criteria—moving beyond earlier fixed-threshold approaches—enables meaningful cross-mission comparison. + +The central observational finding is that the majority of solar wind current sheets are kinetic-scale structures across the inner heliosphere and their normalized properties are largely invariant across radial distance. While absolute thickness increases and absolute current density decreases as the solar wind expands, both quantities remain nearly constant when normalized to the local ion inertial length $d_i$ and Alfvén current density $J_A$ respectively—from 0.17 AU (PSP near perihelion) out to 1 AU. The normalized current density, with most probable value around 0.05–0.15 $J_A$, and normalized thickness, centered around 2–4 $d_i$, show distributions whose widths greatly exceed any systematic radial trend, indicating that the intrinsic spread of current sheet properties dominates over the evolution. A pronounced inverse correlation between current density and spatial scale—stronger currents in thinner sheets—extends over nearly two additional orders of magnitude compared to prior studies, establishing this as a universal feature of solar wind turbulence across kinetic to MHD scales. The Alfvénicity of current sheets, quantified through the Sonnerup parameter $Q^\pm$, is found to be primarily controlled not by the sheet's own properties (thickness, current density, plasma $\beta$) but by the surrounding turbulent environment—specifically the cross helicity $\sigma_c$ and residual energy $\sigma_r$ of the ambient solar wind. This demonstrates that current sheets are embedded within, and shaped by, the local turbulent cascade rather than being independent structures. + +The outer heliosphere study (Chapter 4) extends this characterization from 1 to 5 AU using the Juno spacecraft during its cruise phase (2011–2016), complemented by simultaneous 1 AU observations from WIND, ARTEMIS, and STEREO. More than 130,000 current sheet intervals were identified. The occurrence rate decreases with heliocentric distance following an approximate $1/r$ law—from roughly 150 events per day near 1 AU to ~30 per day near 5 AU. Through a geometric model accounting for the expanding spherical shell and the Parker spiral orientation of current sheet normals, we show that this apparent decrease may be largely a geometric effect: when the global occurrence rate is corrected for the fractional cross-sectional coverage of a planar structure at distance $r$, the factor of 5 decrease in local occurrence rate reduces to a factor of only ~1.3, with the residual attributable to physical annihilation processes such as magnetic reconnection. The normalized thickness and current density continue to remain nearly constant from 1 to 5 AU, confirming that current sheets maintain their kinetic character throughout the outer inner heliosphere and that the characteristic velocity $\tilde{v}_B \propto BL$ is approximately radially invariant—with direct implications for particle scattering efficiency. + +**Quantitative modeling of particle scattering and transport.** The second program translates these observational constraints into a physics-based model of energetic particle scattering and transport. Chapter 6 develops an exact Hamiltonian formulation for a force-free, one-dimensional current sheet—retaining the full unexpanded magnetic vector potential rather than relying on Taylor expansions near the sheet center—and uses it to conduct extensive test-particle simulations parameterized by the observed distributions of $\theta$, $\omega_{in}$, and $\tilde{v}_B$ from the 1 AU current sheet catalog of ~100,000 events. + +The scattering outcome of each particle–sheet interaction is captured in a transition matrix (TM) mapping initial to final pitch angle $\alpha_0 \to \alpha_1$, averaged over the gyrophase. A central result is that for certain current sheets and particle energies, the TM combines a bright diagonal (weakly scattered particles) with off-diagonal structure representing large, non-diffusive pitch-angle jumps. The character of the scattering depends strongly on the shear half-angle $\beta$ and the ratio $v_p/v_B$. Large $\beta$ (large rotation angle) and $v_p \sim v_B$ (gyroradius comparable to sheet thickness) produce enhanced diffusion and particle reflection; very high energies ($v_p \gg v_B$) result in weak scattering over a narrow pitch-angle range. The uncertainty curve length $L_\text{uc}/\sqrt{H}$, which governs the probability of separatrix crossing, increases with both $\beta$ and particle energy—confirming that the geometric chaotization mechanism is more pronounced for larger field rotations and higher-energy particles interacting with current sheets of comparable scale. + +A statistically weighted transition matrix (WTM) is constructed by averaging individual TMs over the observed distribution of current sheet properties at 1 AU. For 100 keV protons, the WTM shows a strong diagonal (most encounters produce little scattering) but with significant probability of both diffusive spreading and rare, large-angle jumps—a non-Markovian structure that cannot be reduced to a single diffusion coefficient. The long-term pitch-angle evolution is simulated through the iterative mapping $\alpha_{n+1} = W_\Pi(\alpha_n, \xi_n)$, where $\xi_n$ is a uniformly sampled random variable encoding the gyrophase randomization. This produces pitch-angle histories characterized by infrequent but substantial jumps—exactly the non-Brownian dynamics anticipated from the intermittent distribution of current sheets. An effective pitch-angle diffusion rate $D_{\mu\mu}$ is extracted by fitting the second moment $M_2(n)$ of the pitch-angle distribution to a saturating exponential, and converted to a physical timescale $\mathcal{D}_{\mu\mu}$ using the observed current sheet occurrence rate and mean solar wind speed. + +Chapter 7 extends this framework to spatial transport. Using a Monte Carlo approach in which particles traverse fixed separations $d$ between successive current sheet encounters—drawn from the observed spacing distribution—and undergo scattering according to the WTM, we compute both parallel displacement $\langle \Delta z^2 \rangle$ and perpendicular displacement $\langle \Delta \mathbf{r}_\perp^2 \rangle$ as functions of time for particle energies from 100 keV to 1 MeV. Both exhibit asymptotically diffusive behavior, with coefficients $D_\parallel$ and $D_\perp$ extracted from the linear growth of mean-square displacement at late times. A key finding is that the perpendicular diffusion coefficient $D_\perp$ scales more steeply with particle energy than $D_\parallel$, causing the ratio $D_\perp/D_\parallel$ to increase with energy and to exceed standard predictions from turbulent wave models at high energies. This arises because the perpendicular guiding-center displacement at each current sheet crossing—produced by demagnetization when the gyroradius is comparable to the sheet thickness—grows rapidly with particle energy, while the parallel transport is regulated by the pitch-angle scattering rate in a more gradual fashion. Furthermore, the characteristic velocity $\tilde{v}_B \propto BL$ is nearly radially invariant (confirmed by the observational results of Chapters 4–5), implying that the scattering efficiency—and thus both $D_\parallel$ and $D_\perp$—remains significant throughout the inner heliosphere, with the higher occurrence rates closer to the Sun amplifying the effect further. + +Taken together, these results establish current sheets as systematic, quantifiable agents of energetic particle scattering throughout the inner heliosphere. The observational invariance of normalized current sheet properties across 0.1–5 AU provides the empirical foundation; the Hamiltonian framework and test-particle simulations provide the physical mechanism and transport coefficients; and the non-Brownian, large-jump character of the scattering offers a natural explanation for the anomalous transport behavior—broad particle reservoirs, rapid longitudinal spreading—that has long challenged purely wave-based models of SEP propagation. + +--- + +We have demonstrated—both theoretically and through numerical modeling—that SWDs play a significant role in modulating particle transport, particularly by enhancing pitch-angle scattering beyond quasilinear expectations (see @sec-modeling). Furthermore, we have shown that their internal structure, including multifluid effects and Alfvénicity variations, are essential to understanding their properties and thereby their transport-modifying capacity (see @sec-multifluid). + +## Opportunities for Future Research + +The results of this thesis open several directions for future investigation: + +**Origins and evolution of solar wind current sheets.** A fundamental question left open by this thesis is the origin of the current sheet population we characterize. The observational invariance of normalized current sheet properties across 0.1–5 AU is consistent with two competing pictures: structures generated near the Sun (potentially by turbulence there) and convected outward in a way that evolves consistently with local plasma properties, and structures generated locally and continuously by turbulence throughout the heliosphere. One way to distinguishing is tracking the propagation velocity. @liuCharacteristicsInterplanetaryDiscontinuities2021 showed that outward-propagating rotational discontinuities have velocities scaling as $r^{-1}$—consistent with Alfvénic advection—while a minority propagate inward, suggesting local turbulent generation. A more systematic investigation of propagation velocities as a function of heliocentric distance, spatial scale, and solar wind type and current sheet charater (RD or TD) would help quantify the relative contributions of solar-origin and locally-generated sheets. Closely related is the question of what fraction of the current sheet population arises from the steepening of large-amplitude Alfvén waves versus direct turbulent cascade. A statistical decomposition of the current sheet population by generation mechanism would provide critical insights. + +**Curvature statistics and connections to the broader intermittency picture.** The quasi-adiabatic framework developed here treats scattering in terms of the internal magnetic field structure of individual current sheets. A complementary perspective, developed by @lemoineParticleTransportLocalized2023 and @kempskiCosmicRayTransport2023, frames scattering as a consequence of regions where the magnetic field line curvature radius $\kappa^{-1}$ and perpendicular reversal scale, drops below the particle gyroradius—a condition met whenever the turbulence is sufficiently intermittent on scales $l \sim r_g$. This curvature-based picture is broader than the current sheet picture: sharp bends of the magnetic field line, whether or not they constitute a recognized current sheet, can produce order-unity changes in the magnetic moment. Unifying these two perspectives requires characterizing the curvature statistics of the solar wind directly, and connecting them to the current sheet population identified in this thesis. @huInterplanetaryMagneticField2025 recently took a step in this direction, using combined MMS and Solar Orbiter observations to measure magnetic field curvature PDFs in the solar wind from 0.3 to 1 AU, finding a two-power-law distribution similar to that in the magnetosheath but with distinct exponents, and a U-shaped radial profile of average curvature with a minimum near 0.5 AU. An important next step is to relate these curvature statistics to current sheet occurrence rates and internal field geometries across the same radial range, asking whether the power-law tails of the curvature PDF are dominated by current sheet crossings, by smaller-scale bends within flux tubes, or by some combination. Such a synthesis would allow the scattering model developed here to be embedded within—and validated against—a more general, structure-agnostic turbulent scattering framework. + +- Expanded observation studies utilizing upcoming spacecraft missions designed to better resolve the kinetic structure of solar wind at unprecedented resolution and have some statistical analysis. + +- What is the origin of the solar current sheet? And I would like to point out studying the propagation velocity and the evolution in the heliosphere may be helpful. And I will be the next step to understand what percentage or what subgroup of the current sheet is generated in the alpha turbulence and what percentage may be formed from the steepening of Alfvén waves. Previous studies have explored this parts and a more detailed examination may be useful. + - @liuCharacteristicsInterplanetaryDiscontinuities2021 + - "Figure 6(c) shows the joint distribution of the RD propagation velocity in the plasma rest frame. Positive and negative velocities indicate the outward (antisunward) and inward (sunward) propagation, respectively. As can be seen, the outward-propagating RDs predominate among all RDs. A minority of RDs also exhibit inward propagation, implying potential mechanisms, such as turbulence, are at work locally generating and pitching RDs in all directions. Then we focus on the outward-propagating RDs and calculate their mean propagation velocity as a function of distance, as shown by the plus signs in Figure 6(d). The fitting shows r−1.03 scaling of the RD propagation velocity, consistent with the expected r−1 scaling from the Walén relation. It implies that the RDs at smaller distances always move faster than those at larger distances, which may in turn influence the particles in the solar wind through some mechanisms like the Fermi process (Fu et al. 2011; Guo et al. 2014; Park et al. 2015; Liu et al. 2019)." + +**Curvature statistics and the broader turbulence context.** The quasi-adiabatic framework treats scattering in terms of the internal structure of individual current sheets, but a complementary perspective is offered by @lemoineParticleTransportLocalized2023, who showed that any region where the magnetic field line curvature radius $\kappa^{-1}$ drops below the particle gyroradius can produce an order-unity change in magnetic moment. In MHD turbulence, such regions are rare in a root-mean-square sense but guaranteed to exist in sufficient numbers by the non-Gaussian, power-law tails of the curvature probability distribution—a direct signature of intermittency. This raises the question of how the curvature statistics of the solar wind relate to the current sheet population characterized in this thesis, and whether a unified statistical description can bridge these two perspectives. Measurements of field-line curvature statistics directly in the solar wind, building on recent magnetosheath observations, would be a natural next step. + +It seems that @huInterplanetaryMagneticField2025 just did some estimation of magnetic field curvature using Solar Orbiter. So you need to think about some other future research. + +![](figures/ref/lemoineParticleTransportLocalized2023_fig3.png) + +- **Interplanetary Magnetic Field Curvature and Its Role in Particle Acceleration: Magnetospheric Multiscale and Solar Orbiter Observations** #[[.zotero]] #[[publication]] #journalArticle + - Citation Key:: @huInterplanetaryMagneticField2025 + - Title:: Interplanetary Magnetic Field Curvature and Its Role in Particle Acceleration: Magnetospheric Multiscale and Solar Orbiter Observations + - Author(s):: + - D. K. Hu + - Y. Y. Liu + - J. B. Cao + - Year:: 2025 + - Publication:: Astrophysical Journal + - Abstract:: The curvature of the magnetic field plays a crucial role in magnetic confinement, magnetic reconnection, particle heating, and acceleration. Though the magnetic field curvature has been sufficiently investigated in the terrestrial magnetosheath, there is a lack of research in the more extensive interplanetary space. This study, combining the Magnetospheric Multiscale and Solar Orbiter (SolO) observations, investigates the magnetic field curvature in the near-Earth and interplanetary space. The spatial distributions and probability distribution functions (PDFs) of the curvature are revealed in the solar wind, magnetosheath, and magnetosphere. It shown that the field curvature in the solar wind follows a two-power-law PDF, similar to that in the magnetosheath but with distinct exponents. We then extend the research to a heliocentric distance of [0.3, 1.0] au, with the aid of an indirect estimation method of magnetic field curvature that can be applied to the SolO data. The average curvature, as a function of heliocentric distance, exhibits a U-shape variation with the minimum curvature appearing at 0.5 au. A possible explanation for such a variation is also given and supported by the spacecraft data. This work could improve our understanding of the turbulent environment in the interplanetary space and demonstrate the universality of a recently proposed acceleration scenario of electrons due to the interaction of shocks and curved magnetic field lines. + + +**Modelling framework: statistics of current sheets and non-diffusive transport.** Connecting the microphysics of individual scattering events to macroscopic transport requires a statistical framework that goes beyond the standard Fokker–Planck diffusion equation. When pitch-angle changes are dominated by rare but large jumps—as in geometrical chaotization—the assumptions of continuous, Markovian scattering break down. The intermittent, clustered distribution of current sheets in the solar wind naturally generates the heavy-tailed step-size distributions that underlie anomalous transport: subdiffusive when sheets trap particles, and superdiffusive when intense encounters produce large pitch-angle jumps and correspondingly large spatial displacements. Building a transport model that incorporates the observed statistics of current sheet properties—occurrence rate, thickness distribution, guide field distribution—and propagates these through to a prediction for non-Brownian or Lévy-flight particle statistics represents an important open problem that this thesis takes initial steps toward addressing. + +These considerations motivate the development of transport models that go beyond the classical diffusion framework and explicitly account for the discrete, structured nature of particle scattering by current sheets. The quantitative characterization of pitch-angle scattering at individual current sheets—using the quasi-adiabatic theory and test-particle simulations developed in this thesis—provides the necessary microphysical input for such models. + +Across all major transport models, the presense of current sheets would influence energetic particle dynamics. In the Parker equation framework, current sheets modify the pitch-angle scattering rate and hence directly affect $\kappa_\parallel$. In the context of the focused transport equation, they introduce strong pitch-angle dependencies and rapid scattering events. Moreover, current sheets can induce memory effects that violate the Markov assumption [@zimbardoNonMarkovianPitchangleScattering2020] of classical diffusion models and contribute to anomalous diffusion. These structures also challenge the conventional picture of diffusion. For parallel transport, the intense magnetic shear and sharp field gradients in current sheets can induce nonlinear effects, producing pitch-angle jumps that are too large to be treated as diffusive. For perpendicular transport, it is often assumed that field-line random walk dominates cross-field motion, as the magnetic field is typically smooth on scales comparable to SEP gyro-radii. However, near current sheets, the magnetic field becomes highly inhomogeneous—often varying on scales similar to or smaller than the gyro-radius—thus enabling enhanced particle transfer between field lines and more significant perpendicular diffusion. Because of their coherent, localized nature and their ability to shape both pitch-angle and spatial scattering processes, current sheets play a central role in accurately modeling particle transport in the turbulent heliospheric environment. + +How to properly model this non-diffusive transport is a chanllenge [@dingModellingEnergeticParticle2024]. Derive the effective diffusion coefficient is one way but needs verification. Modelling it as multiphase diffusive media is another interesting approach [@ewartCosmicrayTransportInhomogeneous2025] where current sheets may be statistically different in different patches (in solar wind there are different types of plasma[@xuNewFourplasmaCategorization2015]) and they may be the interfaces between different diffusion regimes. As shown in @ewartCosmicrayTransportInhomogeneous2025, the multiphase medium is only capable of altering the energy dependence of cosmic-ray transport when there is a moderate (but not excessive) level of perpendicular diffusion across magnetic-field lines. + +- **Cosmic-ray transport in inhomogeneous media** + - Citation Key:: @ewartCosmicrayTransportInhomogeneous2025 + - Title:: Cosmic-ray transport in inhomogeneous media + - Author(s):: + - Robert J. Ewart + - Patrick Reichherzer + - Shuzhe Ren + - Year:: 2025 + - Publication:: Monthly Notices of the Royal Astronomical Society + - Abstract:: A theory of cosmic-ray transport in multiphase diffusive media is developed, with the specific application to cases in which the cosmic-ray diffusion coefficient has large spatial fluctuations that may be inherently multiscale. We demonstrate that the resulting transport of cosmic rays is diffusive in the long-time limit, with an average diffusion coefficient equal to the harmonic mean of the spatially varying diffusion coefficient. Thus, cosmic-ray transport is dominated by areas of low diffusion even if these areas occupy a relatively small, but not infinitesimal, fraction of the volume. On intermediate time-scales, the cosmic rays experience transient effective subdiffusion, as a result of low-diffusion regions interrupting long flights through high-diffusion regions. In the simplified case of a two-phase medium, we show that the extent and extremity of the subdiffusivity of cosmic-ray transport is controlled by the spectral exponent of the distribution of patch sizes of each of the phases. We finally show that, despite strongly influencing the confinement times, the multiphase medium is only capable of altering the energy dependence of cosmic-ray transport when there is a moderate (but not excessive) level of perpendicular diffusion across magnetic-field lines. +- **A new four-plasma categorization scheme for the solar wind** #[[.zotero]] #[[publication]] #journalArticle + - Citation Key:: @xuNewFourplasmaCategorization2015 + - Title:: A new four-plasma categorization scheme for the solar wind + - Tags:: solar cycle,coronal holes,solar wind source,streamer belt + - Author(s):: + - Fei Xu + - Joseph E. Borovsky + - Year:: 2015 + - Publication:: Journal of Geophysical Research: Space Physics + - Abstract:: A three-parameter algebraic scheme is developed to categorize the solar wind at 1 AU into four plasma types: coronal-hole-origin plasma, streamer-belt-origin plasma, sector-reversal-region plasma, and ejecta. The three parameters are the proton-specific entropy Sp = Tp/np2/3, the proton Alfvén speed vA, and the proton temperature Tp compared with a velocity-dependent expected temperature. Four measurements are needed to apply the scheme: the proton number density np, the proton temperature Tp, the magnetic field strength B, and the solar wind speed vsw. The scheme is tested and is found to be more accurate than existing categorization schemes. The categorization scheme is applied to the 1963–2013 OMNI2 data set spanning four solar cycles and to the 1998–2008 ACE data set. The statistical properties of the four types of plasma are examined. The sector-reversal-region plasma is found to have statistically low alpha-to-proton density ratios and high Alfvén Mach numbers. The statistical relations between the proton and alpha-particle-specific entropies and oxygen and carbon charge-state-density ratios Sp, Sα, O7+/O6+, and C6+/C5+ from ACE are examined for the four types of plasma: the patterns observed imply a connection between sector-reversal-region plasma and ejecta and a connection between streamer-belt-origin plasma and coronal-hole-origin plasma. Plasma occurrence rates are examined and solar cycle patterns are found for ejecta, for coronal-hole-origin plasma, and for sector-reversal-region plasma. + + +**Other structures** + +- Therefore, understanding SEP transport requires more than bulk statistical descriptions of turbulence; it demands detailed knowledge of its intermittent nature and the embedded coherent structures that mediate particle scattering. Accurately characterizing these features is essential for developing realistic models of SEP propagation throughout the heliosphere. + +Also, we emphasize the magnetic configurations will determine the particle scattering efficiency, and therefore, understanding the varieties of the magnetic configurations in the turbulence and how specific configurations contribute is an important topic. In this thesis, we explore a specific configuration (one dimension force-free current sheet), but there are other structures in the solar wind. How multiple dimensions and more realistic current configurations together with the ambient environment implement particle dynamics with further exploration. + + + +**Role in turbulence dissipation and spectral structure.** Current sheets account for a substantial fraction of the magnetic fluctuation power in the inertial range and may be responsible for the commonly observed Kolmogorov spectral scaling. Their disruption by reconnection at kinetic scales is theorized to mediate the transition from the inertial to the kinetic cascade, but observational confirmation of this picture — particularly the relationship between the tearing disruption scale, the observed spectral break, and local plasma beta — remains incomplete. A systematic characterization of how current sheet properties connect to the local turbulence environment would help test these theoretical predictions. + + +Finally, understanding how these different classes of current sheets influence the **dynamics of the solar wind plasma** remains an important open problem. Their presence may affect processes such as turbulence dissipation, particle scattering, and energy transport. Establishing the statistical properties and evolutionary behavior of these structures is therefore a necessary step toward a more comprehensive understanding of solar wind turbulence and heliospheric plasma dynamics. + +There are contradicting evidence + +- **No evidence for heating of the solar wind at strong current sheets** + - Citation Key:: @borovskyNoEvidenceHeating2011 + - Title:: No evidence for heating of the solar wind at strong current sheets + - Author(s):: + - Joseph E. Borovsky + - Michael H. Denton + - Year:: 2011 + - Publication:: The Astrophysical Journal Letters + - Abstract:: It has been conjectured that strong current sheets are the sites of proton heating in the solar wind. For the present study, a strong current sheet is defined by a >45° rotation of the solar-wind magnetic-field direction in 128 s. A total of 194,070 strong current sheets at 1 AU are analyzed in the 1998–2010 ACE solar-wind data set. The proton temperature, proton specific entropy, and electron temperature at each current sheet are compared with the same quantities in the plasmas adjacent to the current sheet. Statistically, the plasma at the current sheets is not hotter or of higher entropy than the plasmas just outside the current sheets. This is taken as evidence that there is no significant localized heating of the solar-wind protons or electrons at strong current sheets. Current sheets are, however, found to be more prevalent in hotter solar-wind plasma. This is because more current sheets are counted in the fast solar wind than in the slow solar wind, and the fast solar wind is hotter than the slow solar wind. + "No evidence for proton or electron heating is found at the locations of strong current sheets in the solar wind. As summarized in Figure 1 and Table 1, the proton temperature, proton specific entropy, and electron temperature at the sites of the strong current sheets are statistically the same as the proton temperature, proton specific entropy, and electron temperature in the adjacent plasmas. This rebuts the conjecture that strong currents in the solar wind are the sites of solar-wind heating (e.g., Leamon et al. 2000; Matthaeus et al. 2003; Greco et al. 2010; Osman et al. 2011). The statistical results presented in Figure 2 of this Letter are consistent with the statistical results presented in Figure 2 of Osman et al. (2011), with the interpretation here that strong current sheets are more prevalent in the hotter solar wind than in cooler solar wind. Two reasons why strong current sheets should be more prevalent in the fast wind were given in Borovsky (2008): (1) hot solar wind statistically is fast solar wind (Richardson & Cane 1995; Borovsky & Steinberg 2006) and in the fast wind structures are swept past a spacecraft faster, increasing the counting frequency of current sheets, and (2) slow wind (cooler) is expanded more near the Sun than is fast (hot) wind (Wang & Sheeley 1990; Arge & Pizzo 2000; Arge et al. 2003), reducing the spatial density of chromospheric structure seeded into the cooler slow wind. The motivation for the conjecture that solar-wind heating should occur at current sheets came from observations of reconnection and heating at current sheets in incompressible resistive-MHD simulations of turbulence. Perhaps in the solar wind there is heating at weaker current sheets, which are more likely to be part of an MHD turbulence (e.g., Bruno et al. 2004; Neugebauer & Giacalone 2010; Miao et al. 2011). Note, however, that the collisionless solar-wind plasma is not well described by incompressible resistive MHD (Borovsky & Gary 2009). In collisionless plasmas reconnection can only occur when current sheets are thin, of the order of the ion skin depth c/ωpi whereas strong current sheets in the solar wind have thicknesses much greater than c/ωpi (Siscoe et al. 1968; Vasquez et al. 2007): indeed, reconnection of solar-wind strong current sheets is observed to be rare (Gosling 2007, 2010). In incompressible resistive MHD, reconnection tends to be of the diffusive Sweet–Parker type in which Ohmic dissipation is important (Ugai 1995); reconnection in collisionless plasmas is of the Petschek type where Ohmic heating is unimportant and where the plasma specific entropy is conserved (Birn et al. 2006, 2008). Using similar methodology, a recent study (Borovsky & Steinberg 2011) finds no evidence for heating of the solar-wind plasma at regions of strong velocity shear in the solar wind." + + + + +- Exploration of current sheet interactions in other astrophysical environments, such as planetary magnetospheres. diff --git a/docs/others/phd/2026_grad/_transport.qmd b/docs/others/phd/2026_grad/_transport.qmd new file mode 100644 index 0000000..59a5615 --- /dev/null +++ b/docs/others/phd/2026_grad/_transport.qmd @@ -0,0 +1,9 @@ +# Energetic Particle Transport driven by Solar Wind Current Sheets + +Building on previous results, the next phase of research will extend the pitch-angle scattering framework to comprehensively model spatial diffusion processes (both parallel and perpendicular). This extension is crucial for accurately capturing the full scope of SEP transport influenced by current sheets. + +The key parameters—$v_{\parallel,1}$, $T_{cs}$, $\Delta s_\perp$, and $\Delta t$—are directly extracted from test-particle simulations, while quantities such as the current sheet separation distance $s_{fs}$, thickness, shear angle, and normal orientation are treated as system parameters derived from solar wind observations. Together, these inputs enable a systematic and physically grounded estimation of spatial diffusion coefficients under realistic heliospheric conditions. + +![Example trajectory of a particle interacting with a current sheet](figures/dR_perp.png){#fig-dR-perp width=70%} + +To ensure consistency with heliospheric observations, we use realistic solar wind current sheet parameters to derive the diffusion coefficients using multiple spacecraft spanning radial distances from 0.1 to 5 AU. The derived diffusion coefficients will then be incorporated into turbulence-based transport models, providing a current-sheet-informed extension to global energetic particle transport frameworks. \ No newline at end of file diff --git a/docs/others/phd/2026_grad/_transport_v0.qmd b/docs/others/phd/2026_grad/_transport_v0.qmd new file mode 100644 index 0000000..ceb0aea --- /dev/null +++ b/docs/others/phd/2026_grad/_transport_v0.qmd @@ -0,0 +1,7 @@ +Building on previous results, the next phase of research will extend the pitch-angle scattering framework to comprehensively model spatial diffusion processes (both parallel and perpendicular). This extension is crucial for accurately capturing the full scope of SEP transport influenced by current sheets. + +The key parameters—$v_{\parallel,1}$, $T_{cs}$, $\Delta s_\perp$, and $\Delta t$—are directly extracted from test-particle simulations, while quantities such as the current sheet separation distance $s_{fs}$, thickness, shear angle, and normal orientation are treated as system parameters derived from solar wind observations. Together, these inputs enable a systematic and physically grounded estimation of spatial diffusion coefficients under realistic heliospheric conditions. + +![Example trajectory of a particle interacting with a current sheet](figures/dR_perp.png){#fig-dR-perp width=70%} + +To ensure consistency with heliospheric observations, we use realistic solar wind current sheet parameters to derive the diffusion coefficients using multiple spacecraft spanning radial distances from 0.1 to 5 AU. The derived diffusion coefficients will then be incorporated into turbulence-based transport models, providing a current-sheet-informed extension to global energetic particle transport frameworks. \ No newline at end of file diff --git a/docs/others/phd/2026_grad/index.md b/docs/others/phd/2026_grad/index.md new file mode 100644 index 0000000..db22e77 --- /dev/null +++ b/docs/others/phd/2026_grad/index.md @@ -0,0 +1,3 @@ +## Elsewhere + +- [Solar Orbiter SWA-related Publications | Faculty of Mathematical & Physical Sciences](https://www.ucl.ac.uk/mathematical-physical-sciences/mssl/space-research/solar-system/space-plasma-physics/space-plasma-missions/current-plasma-missions/solar-orbiter/solar-orbiter-swa-related-publications) \ No newline at end of file diff --git a/docs/others/phd/2026_grad/index.typ b/docs/others/phd/2026_grad/index.typ new file mode 100644 index 0000000..5e5f03f --- /dev/null +++ b/docs/others/phd/2026_grad/index.typ @@ -0,0 +1,25 @@ +#import "@local/uclathesis:0.1.0": uclathesis +#import "@preview/cmarker:0.1.8" + +#let citet(..citation) = cite(..citation, form: "prose") + +#show: uclathesis.with( + title: [My Dissertation Title], + author: "Your Name", + degree: "Doctor of Philosophy", + major: "Your Major", + year: 2024, + doc-type: "dissertation", // or "thesis" for master's + committee-chair: "Vassilis Angelopoulos", + committee-members: ( + "Member One", + "Member Two", + ), + abstract: [Your abstract text here.], + bibliography: bibliography("../../../../files/bibliography/research.bib"), +) + + +#pagebreak() + +#include "_review_current_sheet.typ" diff --git a/docs/others/phd/2026_grad/justfile b/docs/others/phd/2026_grad/justfile new file mode 100644 index 0000000..b2f0858 --- /dev/null +++ b/docs/others/phd/2026_grad/justfile @@ -0,0 +1,16 @@ +render: + quarto render thesis.qmd --to typst + +link_figures: + ln -s ~/projects/ids_spatial_evolution_juno/overleaf/figures figures/juno + ln -s ~/projects/ion_scattering_by_SWD/overleaf/figures figures/scattering + # ln -s ~/projects/spedas/overleaf/figures figures/spedas + ln -s ~/projects/psp_conjunction/overleaf figures/psp + +convert: + pandoc -t typst _review_current_sheet.qmd -o _review_current_sheet.typ + +sync: + rsync -av --delete thesis.typ uclathesis/ + rsync -av --delete figures/ uclathesis/figures/ + rsync -av ~/projects/share/bibliography/research.bib uclathesis/ diff --git a/docs/others/phd/2026_grad/notes.md b/docs/others/phd/2026_grad/notes.md new file mode 100644 index 0000000..2101085 --- /dev/null +++ b/docs/others/phd/2026_grad/notes.md @@ -0,0 +1,12 @@ +- Why are there no RDs with large normal components? Are they never created or are they unstable? + - What is the statistics bias of multi-spacecraft measurement? + [@wangSolarWindCurrent2024] [@knetterNewPerspectiveSolar2005] + - Figure 6.4: Number of DDs identified by the TS-method. + - Figure 6.5: Relative number of DDs found simultaneously at all four spacecraft as a function of the spacecraft separation. +- Why are Δv and ΔB aligned across tangential discontinuities? + +To Read + +- [ ] vinogradovEmbeddedCoherentStructures2024 +- [ ] artemyevComparativeStudyElectric2021 +- [ ] neishtadtPolymorphismsAdiabaticChaos2011 diff --git a/docs/others/phd/2026_grad/prompt.md b/docs/others/phd/2026_grad/prompt.md new file mode 100644 index 0000000..f3124d6 --- /dev/null +++ b/docs/others/phd/2026_grad/prompt.md @@ -0,0 +1,7 @@ +- I am writing PhD thesis "Kinetic-scale solar wind current sheets: statistical characteristics and their role in energetic particle transport". Please improve the clarity, coherence and connection of my writing, removes redundancy, and strengthens the logical progression: + +- Understand the raw material here and help me write the section with improved clarity, coherence and connection, removed redundancy, and strengthened logical progression: + +- Understand the following references (pdfs) and help me write the introduction for the quasi-adiabatic dynamics of charged particles in the magnetic field, providing the general theory/understanding and focusing on the ion motion in the current sheet. Please cover both the quasi-adiabatic theory and the non-adiabatic effects, and the associated ion scattering. + +- Please help me write a review part for the energetic particles in the review chapter "Energetic Particle Interaction With Solar Wind Current Sheets", with a special attention to solar energetic particles. Raw materials and texts are attached (no need to follow). diff --git a/docs/others/phd/2026_grad/thesis.qmd b/docs/others/phd/2026_grad/thesis.qmd index 68e66b9..ab2ecb5 100644 --- a/docs/others/phd/2026_grad/thesis.qmd +++ b/docs/others/phd/2026_grad/thesis.qmd @@ -1,136 +1,32 @@ --- title: "Kinetic-scale solar wind current sheets: statistical characteristics and their role in energetic particle transport" -subtitle: Thesis Outline +subtitle: Thesis author: Zijin Zhang date: last-modified number-sections: true -format: - html: default -# typst: -# include-in-header: -# - text: | -# #show link: set text(fill: rgb("#239dad")) -# #show cite: set text(fill: rgb("#239dad")) +format: + typst: + keep-typ: true # citeproc: true bibliographystyle: apa --- - +{{< include _acknowledgments.qmd >}} -## Motivation and Significance +{{< include _intro.qmd >}} -Solar energetic particles (SEPs), originating from solar flares and coronal mass ejections, pose significant risks to satellite operations, human spaceflight, and communication systems. Accurate prediction of SEP events and their propagation through the heliosphere requires a detailed understanding of particle transport mechanisms in turbulent solar wind environments. - -Traditionally, theoretical studies and numerical models of particle transport in the solar wind have focused on turbulence characterized by broadband, low-amplitude, random-phase magnetic fluctuations described by power-law spectra [@jokipiiCosmicRayPropagationCharged1966; @jokipiiCosmicRayPropagationIi1967]. However, observations consistently reveal the abundance of intermittent, meso-scale, coherent structures within this turbulent medium, notably current sheets—thin plasma boundaries marked by abrupt magnetic field changes. These current sheets deviate significantly from classical magnetohydrodynamic (MHD) picture due to their kinetic-scale features and strong local magnetic gradients. - -Recent theoretical and numerical studies suggest that these coherent structures play a critical role in particle scattering, potentially surpassing the scattering efficiency predicted by traditional quasilinear theories [@malaraChargedparticleChaoticDynamics2021; @artemyevSuperfastIonScattering2020]. Current sheets, generated naturally through nonlinear turbulence cascade, provide localized regions of intense electromagnetic interactions, leading to enhanced scattering and modification of the particle's spatial distribution. Despite their importance, a quantitative and systematic understanding of how these structures influence SEP transport remains incomplete. - -Addressing this critical gap, this dissertation aims to comprehensively investigate and quantify the impact of solar wind current sheets on SEP transport processes. Specifically, this research seeks to: - -1. Characterize the properties and occurrence of current sheets throughout different regions of the heliosphere. -2. Develop and validate theoretical models that describe particle scattering induced by these coherent structures. Provide a improved, quantitative modeling of SEP interactions with current sheets, thereby enabling predictive capabilities and contributing to space weather modeling. - -The motivation for this research lies in the critical need for improved SEP transport models that accurately reflect real-world solar wind conditions. By integrating observational data and advanced theoretical frameworks, this dissertation will provide novel insights into heliospheric particle dynamics, ultimately enhancing our ability to predict and mitigate the risks associated with SEP events. - -## Research Context and Background - -The study of solar energetic particles [@anastasiadisSolarEnergeticParticles2019; @kleinAccelerationPropagationSolar2017; @desaiLargeGradualSolar2016; @reamesTwoSourcesSolar2013], turbulent magnetic fields [@schekochihinMHDTurbulenceBiased2022; @matthaeusTurbulenceSpacePlasmas2021; @oughtonSolarWindTurbulence2021; @brunoTurbulenceSolarWind2016;@brunoSolarWindTurbulence2013; @verscharenMultiscaleNatureSolar2019; @tuMHDStructuresWaves1995], and charged particle transport [@engelbrechtTheoryCosmicRay2022; @vandenbergPrimerFocusedSolar2020] has produced a vast body of literature spanning decades of theoretical, observational, and numerical research. Within this context, current sheets have increasingly been recognized as key structures. In the following sections, we highlight a selection of foundational observations, models, and theoretical developments that are directly or indirectly connected to the role of current sheets. These include both classical frameworks and recent advances that point to the importance of coherent structures in turbulent plasmas. In @sec-cs, we summarize how a deeper understanding of current sheets can enhance our ability to model energetic particle transport and, more broadly, improve our understanding of heliospheric particle dynamics. - -### Solar Energetic Particles - -Solar energetic particles (SEPs) are high-energy ions and electrons originating at or near the Sun. They span a broad energy spectrum, from Solar energetic particles (SEPs) consist of high-energy ions and electrons originating at or near the Sun. Unlike the solar wind and galactic cosmic rays (GCRs), solar energetic particles (SEPs) manifest as discrete episodic events with intensities that can vary dramatically—by several orders of magnitude—in just minutes. Additionally, SEP events exhibit significant variations in heavy ion composition, spectral shape, and spatial distribution. - -SEPs are primarily accelerated through two distinct mechanisms [@reamesTwoSourcesSolar2013]: (1) shock-wave acceleration associated with fast coronal mass ejections (CMEs), resulting in large gradual SEP events [@desaiLargeGradualSolar2016], and (2) magnetic reconnection-driven processes during solar flares, producing impulsive SEP events. - -Gradual SEP events typically last for several days and are predominantly proton-rich, often associated with fast CMEs driving shocks in the solar corona and interplanetary space. These shocks accelerate particles over extended regions, producing widespread and intense radiation storms. In contrast, impulsive SEP events are related to short duration (less than 1 h) solar flares. These events typically have shorter durations, lasting from minutes to a few hours, and feature characteristically higher electron-to-proton ratios and enrichments of heavy ions (${ }^3 \mathrm{He} /{ }^4 \mathrm{He}$ and $\mathrm{Fe} / \mathrm{O}$ ratios). - -![The two-class picture for SEP events. @desaiLargeGradualSolar2016](figures/desaiLargeGradualSolar2016-fig3.png) - -In the decay phase of large gradual SEP events, a characteristic phenomenon known as the **reservoir effect** frequently occurs, where particle intensities measured by widely separated spacecraft become nearly uniform across large regions and exhibit similar temporal evolutions. One traditional explanation for reservoir formation suggests that particles become trapped behind a CME-driven magnetic structure, resulting in spatially uniform spectra that adiabatically decrease in intensity as the confining magnetic bottle expands. However, high heliolatitude observations from the Ulysses mission revealed the three-dimensional character of the reservoir effect and favor the cross-field diffusion explanation [@larioHeliosphericEnergeticParticle2010; @dallaPropertiesHighHeliolatitude2003]. - -In contrast to these smooth, widespread distributions, certain impulsive SEP events demonstrate remarkably sharp spatial variations (abrupt depletions) in particle intensity, known as dropout events [@tesseinEffectCoherentStructures2015; @neugebauerEnergeticParticlesTangential2015]. Such behavior is attributed to spacecraft traversing alternating particle-filled and particle-empty magnetic flux tubes, suggesting extremely limited lateral transport of particles across magnetic fields [@mazurInterplanetaryMagneticField2000]. This phenomenon is typically interpreted as resulting from particles being effectively confined within distinct magnetic flux tubes, due to minimal cross-field diffusion. The sharply defined spatial gradients scales observed in dropout events, are often comparable to particle gyro-radii. - -Together, these contrasting observations—extensive spatial uniformity in gradual SEP events (reservoir effects) versus sharp intensity variations in impulsive events (dropouts)—underscore the complexity of SEP transport mechanisms, motivating ongoing studies to reconcile these phenomena within comprehensive transport models. - -### Turbulent Magnetic Fluctuations - -Solar wind turbulence spans scales from the large‑scale coherence length (∼0.01 AU) down to kinetic dissipation scales on the order of the thermal ion gyro‑radius (∼100 km). Of particular importance for energetic particle transport is the turbulence at intermediate scales, often referred to as inertial-range turbulence. For a 5 nT magnetic field, this range corresponds to proton gyro-radii from about 1 GeV to 1 keV, encompassing nearly all SEPs, whose gyro-radii lie between these two bounds. - -The transport of SEPs through the heliosphere is shaped by the properties of magnetic turbulence. Key parameters—such as the spatial inhomogeneity, turbulence level ($δB/B₀$), spectral index, and anisotropy of wave vectors [@pucciEnergeticParticleTransport2016] —strongly influence how particles scatter in velocity space. These properties govern both parallel and perpendicular transport through mechanisms including pitch-angle diffusion, magnetic field-line meandering, and gradient or curvature drift. - -Classical scattering theories and numerical models of particle transport [@giacaloneTransportCosmicRays1999] typically model turbulence as a sea of random, phase-uncorrelated fluctuations (common constructions of magnetic fluctuations for the slab component $δ𝐁^s$ and two-dimensional component $δ𝐁^{2D}$ are shown below in @eq-δ𝐁). However, this idealized view neglects the intricate internal nonlinear structures of turbulence. Increasingly, observations and simulations show that solar wind turbulence is highly intermittent and populated with coherent structures—especially current sheets—that arise naturally through nonlinear cascade processes. - -$$ -\begin{aligned} -& δ𝐁^s=\sum_{n=1}^{N_m} A_n\left[\cos α_n\left(\cos \phi_n \hat{x}+\sin \phi_n \hat{y}\right)+i \sin α_n(-\sin \phi_n \hat{x}+\cos \phi_n \hat{y})\right] \times \exp \left(i k_n z+i β_n\right) \\ -& δ𝐁^{2 D}=\sum_{n=1}^{N_m} A_n i\left(-\sin \phi_n \hat{x}+\cos \phi_n \hat{y}\right) \times \exp \left[i k_n\left(\cos \phi_n x+\sin \phi_n y\right)+i β_n\right] -\end{aligned} -$$ {#eq-δ𝐁} - - -$$ -\begin{aligned} -&D_{\|}=\frac{v^2}{8} \int_{-1}^1 \frac{\left(1-\mu^2\right)^2}{D_{\mu \mu}} d \mu\\ -&D_{\mu \mu}=\frac{\pi \omega_{c i} k}{2 B^2 / \mu_0}\left(1-\mu^2\right) \sum_{+,-} I_{ \pm}(k) -\end{aligned} -$$ - -#### Geometrical Chaotization - -A key physical mechanism underlying the strong scattering induced by current sheets is geometrical chaotization—a rapid breakdown of adiabatic invariants caused by separatrix crossings in the particle’s phase space [@tennysonChangeAdiabaticInvariant1986; @zelenyiQuasiadiabaticDynamicsCharged2013]. In such slow-fast Hamiltonian systems, even weak asymmetries in the current sheet configuration can produce large, abrupt pitch-angle changes, leading to fast and efficient chaotization of particle motion [@artemyevSuperfastIonScattering2020; @artemyevRapidGeometricalChaotization2014]. This mechanism departs from the diffusive assumptions of classical quasilinear theory and underscores the importance of kinetic-scale structure in driving non-diffusive scattering behaviors. - -Therefore, understanding SEP transport requires more than bulk statistical descriptions of turbulence; it demands detailed knowledge of its intermittent nature and the embedded coherent structures that mediate particle scattering. Accurately characterizing these features is essential for developing realistic models of SEP propagation throughout the heliosphere. - -### Charged Particle Transport and Turbulence Transport Models - -The large-scale behavior of energetic charged particles in the heliosphere is commonly described using a diffusive approximation, justified when the particle scattering time is short compared to the timescale of interest. Under this assumption, the evolution of an approximately isotropic particle distribution is governed by the Parker transport equation [@parkerPassageEnergeticCharged1965]. This foundational framework captures four main transport processes: spatial diffusion due to particle scattering, advection with the solar wind, drifts (such as gradient and curvature drifts due to variations in the large-scale magnetic field), and adiabatic energy change: - -$$ -\frac{∂ f}{∂ t}=\frac{∂}{∂ x_i}\left[κ_{i j} \frac{∂ f}{∂ x_j}\right]-U_i \frac{∂ f}{∂ x_i}-V_{d, i} \frac{∂ f}{∂ x_i}+\frac{1}{3} \frac{∂ U_i}{∂ x_i}\left[\frac{∂ f}{∂ \ln p}\right]+ \text{Sources} - \text{Losses}, -$$ {#eq-parker} - -where $f$ is the phase-space distribution as a function of the particle momentum, $p$, position, $x_i$, and time, $t$; $κ_{i j}$ is the symmetric part of the diffusion tensor; $U_i$ is the bulk plasma velocity; $V_{d, i}$ is the drift velocity. The drift velocity can be formally derived from the guiding center approximation averaged over a nearly isotropic distribution, and can be included as the antisymmetric part of the diffusion tensor. - -The symmetric diffusion tensor can be decomposed into components parallel and perpendicular to the mean magnetic field using: $κ_{ij}=κ_{\perp} \delta_{ij}-\frac{\left(κ_{\perp}-κ_{\|}\right) B_i B_j}{B^2}$. The parallel diffusion coefficient, $κ_{\|}$, is related to the pitch-angle diffusion coefficient $D_{\mu\mu}$ through the quasilinear theory (QLT) framework [@jokipiiCosmicRayPropagationCharged1966; @jokipiiAddendumErratumCosmicRay1968] as $κ_{\|}=\frac{v^2}{8} \int_{-1}^1 \frac{\left(1-\mu^2\right)^2}{D_{\mu \mu}(\mu)} d \mu$. -While parallel transport is relatively well understood, perpendicular (cross-field) diffusion ($κ_\perp$) remains more elusive due to its nonlinear and non-resonant nature [@shalchiPerpendicularDiffusionEnergetic2021; @costajr.CrossfieldDiffusionEnergetic2013]. A key factor influencing cross-field transport is the dimensionality of the turbulence [@giacaloneChargedParticleMotionMultidimensional1994]: in models with at least one ignorable spatial coordinate (e.g., slab geometry), cross-field diffusion is artificially suppressed, failing to capture essential physics. In general, cross-field transport arises from two distinct mechanisms: (1) particle motion along stochastic, meandering magnetic field lines, which can lead to substantial displacements relative to the mean field direction; and (2) the true decorrelation of particles from their initial field lines, allowing them to effectively jump between neighboring lines. Though often considered a small fraction of $κ_\parallel$ [@giacaloneTransportCosmicRays1999], recent simulations reveal that $κ_\perp$ can be significant and strongly dependent on particle energy and turbulence structure [@dundovicNovelAspectsCosmic2020]. - -Anisotropy in particle distributions is common in SEP events, particularly in the early phases or near upstream regions of interplanetary shocks. One fundamental source of anisotropy is adiabatic focusing in a diverging magnetic field. To account for such effects, the focused transport equation [@roelofPropagationSolarCosmic1969; @earlEffectAdiabaticFocusing1976] extends the Parker equation by explicitly retaining the pitch-angle dependence: - -$$ -\frac{∂ f}{∂ t}+\mu v \frac{∂ f}{∂ z}+\frac{v}{2 L}\left(1-\mu^2\right) \frac{∂ f}{∂ \mu}=\frac{∂}{∂ \mu}\left(D_{\mu \mu} \frac{∂ f}{∂ \mu}\right) -$$ - -where $f= f(z,\mu,t)$ is the phase-space distribution, $\mu$ is the pitch-angle cosine and $L = -B \left(\frac{dB}{dz}\right)^{-1}$ is the focusing length. - -Beyond classical diffusion, observations of SEP events and near interplanetary shocks often reveal anomalous transport behavior [@zimbardoSuperdiffusiveSubdiffusiveTransport2006], characterized by subdiffusive or superdiffusive scaling of particle displacement with time [@zimbardoSuperdiffusiveTransportLaboratory2015] $\left\langle\Delta x^2(t)\right\rangle \propto t^α$. These deviations from normal diffusion are attributed to the intermittent and structured nature of solar wind turbulence, and are better described using generalized frameworks such as fractional diffusion models [@del-castillo-negreteNondiffusiveTransportPlasma2005] or Lévy statistics [@zaburdaevLevyWalks2015]. - -### The Role of Current Sheets in Particle Transport {#sec-cs} - -Across all major transport models, current sheets emerge as a critical feature influencing energetic particle dynamics. In the Parker equation framework, current sheets modify the pitch-angle scattering rate and hence directly affect $\kappa_\parallel$. In the context of the focused transport equation, they introduce strong pitch-angle dependencies and rapid scattering events. Moreover, current sheets can induce memory effects that violate the Markov assumption [@zimbardoNonMarkovianPitchangleScattering2020] of classical diffusion models and contribute to anomalous diffusion. - -These structures also challenge the conventional picture of diffusion. For parallel transport, the intense magnetic shear and sharp field gradients in current sheets can induce nonlinear effects, producing pitch-angle jumps that are too large to be treated as diffusive. For perpendicular transport, it is often assumed that field-line random walk dominates cross-field motion, as the magnetic field is typically smooth on scales comparable to SEP gyro-radii. However, near current sheets, the magnetic field becomes highly inhomogeneous—often varying on scales similar to or smaller than the gyro-radius—thus enabling enhanced particle transfer between field lines and more significant perpendicular diffusion. - -Because of their coherent, localized nature and their ability to shape both pitch-angle and spatial scattering processes, current sheets play a central role in accurately modeling particle transport in the turbulent heliospheric environment. - - +{{< pagebreak >}} -## Objectives and Thesis Plan +{{< include _review_current_sheet.qmd >}} -The overall goal of this thesis is to quantify and model the impact of solar wind current sheets on energetic particle transport. This research is structured around two primary objectives: + -- Observational characterization of solar wind current sheets across the heliosphere +{{< include _review_energetic_particles.qmd >}} -- Development of data-driven theoretical models for current sheet-induced particle scattering and transport +{{< pagebreak >}} -## Work Completed -### Observational Analysis of Current Sheets {#sec-obs} +# Observational Analysis of Current Sheets {#sec-obs} **Context:** A critical first step in understanding the role of current sheets in energetic particle transport is to characterize their statistical properties and quantify the parameters most relevant to particle scattering. Although current sheets have been extensively observed—especially near $1$ AU—our knowledge of how their properties evolve across heliocentric distances, and how key scattering-related parameters vary with radial distance, has remained incomplete. Previous studies [@sodingRadialLatitudinalDependencies2001, @lotekarKineticscaleCurrentSheets2022, @liuCharacteristicsInterplanetaryDiscontinuities2021, @vaskoKineticscaleCurrentSheets2022, @vaskoKineticScaleCurrentSheets2024] often lacked simultaneous, multi-point measurements and did not adequately separate temporal variability from spatial trends, leading to persistent uncertainties regarding their role in particle transport, their origin, and their evolution within the turbulent solar wind. @@ -138,185 +34,53 @@ The overall goal of this thesis is to quantify and model the impact of solar win ![Current sheets detected by PSP, Juno, STEREO and near-Earth ARTEMIS satellite: red, blue, and black lines are $𝐵_𝑙$, $𝐵_𝑚$, and $𝐵$](figures/fig-ids_examples.png) -**Results:** Our analysis reveals that solar wind current sheets maintain kinetic-scale thicknesses throughout the inner heliosphere, with occurrence rates decreasing approximately as $1/r$ with radial distance between 1 and 5 AU. When normalized to the local ion inertial length and Alfvén current, both the current density and thickness of these structures remain nearly constant over the range from 0.1 to 5 AU (see @fig-juno-distribution-r-sw). This suggests that current sheets consistently influence energetic particle transport across heliocentric distances, with their higher occurrence rates closer to the Sun indicating a more pronounced role in shaping particle dynamics in the inner heliosphere. Furthermore, by leveraging simultaneous observations from spacecraft at different radial distances, we demonstrate that the observed radial trends reflect genuine spatial evolution rather than temporal or solar-cycle effects. In particular, we propose that the observed reduction in current sheet occurrence rate at larger heliocentric distances is partly attributable to a geometric effect—namely, the decreasing probability that a spacecraft intersects inclined structures as distance from the Sun increases. This represents an observational bias that must be accounted for when interpreting occurrence statistics. However, even after correcting for this geometric effect, a modest residual decrease remains, which we attribute to possible physical dissipation or annihilation of current sheets as they propagate outward through the solar wind. +**Results:** Our analysis reveals that the majority of solar wind current sheets maintain kinetic-scale thicknesses throughout the inner heliosphere. When normalized to the local ion inertial length and Alfvén current, both the current density and thickness of these structures remain nearly constant over the range from 0.1 to 5 AU (see @fig-juno-distribution-r-sw). This suggests that current sheets consistently influence energetic particle transport across heliocentric distances, with their higher occurrence rates closer to the Sun indicating a more pronounced role in shaping particle dynamics in the inner heliosphere. Furthermore, by leveraging simultaneous observations from spacecraft at different radial distances, we demonstrate that the observed radial trends reflect genuine spatial evolution rather than temporal or solar-cycle effects. -Together, these results provide critical empirical constraints for particle transport modeling and establish a robust observational foundation for the theoretical and numerical components of this thesis. This work is presented in *"Solar wind discontinuities in the outer heliosphere: Spatial distribution between 1 and 5 AU"* (Zhang et al., submitted to JGR Space Physics, 2025, manuscript is available at [10.22541/essoar.174431869.93012071/v1](https://www.authorea.com/users/814634/articles/1283652-solar-wind-discontinuities-in-the-outer-heliosphere-spatial-distribution-between-1-and-5-au)). +Together, these results provide critical empirical constraints for particle transport modeling and establish a robust observational foundation for the theoretical and numerical components of this thesis. -![Waiting time probability density functions $p(\tau)$ for Juno at 1 AU in 2011 (top) and 5 AU in 2016 (bottom). Observed data (black) are fitted with Weibull (blue) and exponential (orange) distributions. Vertical dashed lines denote the mean waiting times for each fitted distribution.](../2025_atc/figures/fig_wt_dist_no_duplicates.pdf) + This work is presented in *"Solar wind discontinuities in the outer heliosphere: Spatial distribution between 1 and 5 AU"* (Zhang et al., submitted to JGR Space Physics, 2025, manuscript is available at [10.22541/essoar.174431869.93012071/v1](https://www.authorea.com/users/814634/articles/1283652-solar-wind-discontinuities-in-the-outer-heliosphere-spatial-distribution-between-1-and-5-au)). -![Distribution of various SWD properties observed by Juno, grouped by radial distance from the Sun (with colors shown at the top). Panel (a) thickness, (b) current density, (c) normalized thickness, (d) normalized current density.](figures/juno_distribution_r_sw.png){#fig-juno-distribution-r-sw} +In particular, we propose that the observed reduction in current sheet occurrence rate at larger heliocentric distances is partly attributable to a geometric effect—namely, the decreasing probability that a spacecraft intersects inclined structures as distance from the Sun increases. This represents an observational bias that must be accounted for when interpreting occurrence statistics. However, even after correcting for this geometric effect, a modest residual decrease remains, which we attribute to possible physical dissipation or annihilation of current sheets as they propagate outward through the solar wind. -![Comparison of solar wind properties (top) and discontinuity properties (bottom) between / using model (x-axis) and JADE observations (y-axis). (a-d) Solar wind velocity, density, temperature, and plasma beta. (e-h) Discontinuity thickness, current density, normalized thickness, and normalized current density. Blue dots indicate values derived using the cross-product normal method, while yellow dots correspond to values obtained using minimum variance analysis.](figures/fig_juno_sw_comparision.png){#fig-juno-sw-comparision} -### Quantitative Modeling of Particle Scattering {#sec-modeling} +# Quantitative Modeling of Particle Scattering {#sec-modeling} **Context:** While it is well established that turbulence governs energetic particle transport in the heliosphere, the specific role of coherent structures—particularly current sheets—in shaping scattering processes remained under-explored. A central objective of this thesis is to develop a physics-based, observation-informed model that directly links solar wind current sheet properties to pitch-angle scattering rates of energetic particles. **Approach:** To this end, we combined statistical measurements of current sheets at 1 AU with a Hamiltonian analytical framework and test particle simulations to investigate how particle scattering efficiency varies with current sheet geometry and particle energy, using a realistic magnetic field configuration: -$$ -\mathbf{B} = B_0 (\cos θ \ \mathbf{e_z} + \sin θ ( \sin φ(z) \ \mathbf{e_x} + \cos φ (z) \ \mathbf{e_y})) -$$ - -where $B_0$ is the magnitude of the magnetic field, $θ$ is the azimuthal angle between the normal and the magnetic field, and $φ(z)$ is the rotation profile of the magnetic field as a function of $z$. - -**Results:** Using a newly formulated Hamiltonian framework (see dimensionless form in @eq-Hamiltonian, below) that incorporates the effects of magnetic field shear angle $\beta$ and particle energy $H$, we demonstrate that scattering rates depend strongly on the current density—which is directly tied to $\beta$—as well as on the ratio of the particle gyroradius to the current sheet thickness. Notably, our results show that current sheets can induce rapid, non-diffusive pitch-angle jumps, particularly for SEPs in the 100 keV to 1 MeV energy ranges (see @fig-example-subset). This behavior deviates significantly from classical quasilinear predictions and highlights the need to account for coherent structures in transport models. To describe long-term pitch-angle evolution, we developed a simplified probabilistic model of pitch-angle scattering due to current sheets and derived an effective pitch-angle diffusion coefficient $D_{\mu\mu}$ (see @fig-mixing-rate). - -These diffusion rate estimates enable direct comparison with other scattering mechanisms, facilitate the incorporation of SWD-induced scattering into global SEP transport models, and directly support the broader goal of this thesis to improve our understanding of how energetic particles interact with turbulence in the solar wind. This work is presented in "Quantification of Ion Scattering by Solar Wind Current Sheets: Pitch-Angle Diffusion Rates" (Zhang et al., submitted to Physical Review E, 2025, manuscript is available at [GitHub](https://github.com/Beforerr/ion_scattering_by_SWD/blob/ec33d3d082bcd463faf7a233ba80138414231b51/files/2024PRE_Scattering_Zijin.pdf)). - -$$ -\begin{aligned} -\tilde{H} &= \frac{1}{2} \left(\left(\tilde{p_x}-f_1(z)\right)^2+\left(\tilde{x} \cot θ + f_2(z)\right)^2+\tilde{p_z}^2\right) -\\ -f_1(z) &=\frac{1}{2} \cos β \ \left(\text{Ci}\left(βs_+(z) \right)-\text{Ci}\left(βs_-(z)\right)\right) + \frac{1}{2} \sin β \ \left(\text{Si}\left(βs_+(z) \right)-\text{Si}\left(βs_-(z)\right)\right), -\\ -f_2(z) &=\frac{1}{2} \sin β \ \left(\text{Ci}\left(β s_+(z) \right)+\text{Ci}\left(β s_-(z)\right)\right) -\frac{1}{2} \cos β \ \left(\text{Si}\left(β s_+(z) \right)+\text{Si}\left(β s_-(z)\right)\right) -\end{aligned} -$$ {#eq-Hamiltonian} - - +**Results:** Using a newly formulated Hamiltonian framework (see dimensionless form in @eq-Hamiltonian, below) that incorporates the effects of magnetic field shear angle $\beta$ and particle energy $H$, we demonstrate that scattering rates depend strongly on the current density—which is directly tied to $\beta$—as well as on the ratio of the particle gyroradius to the current sheet thickness. Notably, our results show that current sheets can induce rapid, non-diffusive pitch-angle jumps, particularly for SEPs in the 100 keV to 1 MeV energy ranges (see @fig-example-subset). This behavior deviates significantly from classical quasilinear predictions and highlights the need to account for coherent structures in transport models. To describe long-term pitch-angle evolution and spatial transport, we developed a simplified probabilistic model of pitch-angle scattering due to current sheets and derived an effective pitch-angle diffusion and spatial transport coefficient $D_{\mu\mu}$. -![Transition matrix for 100 keV protons under four distinct magnetic field configurations: (i) $v_p = 8 v_0$, $θ = 85°$, $β = 50°$; (ii) $v_p = 8 v_0$, $θ = 85°$, $β = 75°$; (iii) $v_p = 8 v_0$, $θ = 60°$, $β = 50°$; and (iv) $v_p = v_0$, $θ = 85°$, $β = 50°$.](figures/example_subset.png){#fig-example-subset width=70%} +These diffusion rate estimates enable direct comparison with other scattering mechanisms, facilitate the incorporation of SWD-induced scattering into global SEP transport models, and directly support the broader goal of this thesis to improve our understanding of how energetic particles interact with turbulence in the solar wind. -![Second moment of the pitch-angle distribution, $M_2(n)$, as a function of interaction number ($n$) for different particle energies (\textasciitilde100 eV, \textasciitilde5 keV, \textasciitilde100 keV, \textasciitilde1 MeV). The estimated mixing rates, $D_{μμ}$, are indicated in the legend.](figures/mixing_rate.png){#fig-mixing-rate width=60%} -### Multifluid Model for Current Sheet Alfvénicity {#sec-multifluid} +This work is presented in "Quantification of Ion Scattering by Solar Wind Current Sheets: Pitch-Angle Diffusion Rates" (Zhang et al., submitted to Physical Review E, 2025, manuscript is available at [GitHub](https://github.com/Beforerr/ion_scattering_by_SWD/blob/ec33d3d082bcd463faf7a233ba80138414231b51/files/2024PRE_Scattering_Zijin.pdf)). -**Context:** Early in this thesis, we identified a consistent radial trend in the Alfvénicity of solar wind current sheets—defined as the ratio of the plasma velocity jump to the Alfvén speed jump. While current sheets near the Sun exhibit high Alfvénicity, this value systematically decreases with increasing heliocentric distance. This raised a fundamental question: why do current sheets appear increasingly non-Alfvénic with distance, despite their force-free magnetic structure? Understanding the internal structure, stability, and evolution of current sheets is crucial, as it directly relates to their role in modulating the transport of energetic particles across the heliosphere. - -![Statistics of the asymptotic velocity ratio from PSP, Wind, and ARTEMIS spacecraft observations during PSP encounter 7 period from 2021-01-14 to 2021-01-21.](figures/vl_ratio.png){width=60%} - -**Approach:** While classical single-fluid MHD models, even when extended to include pressure anisotropy, provide useful modeling of current sheet properties, they fall short of capturing the full complexity encountered in observed current sheets. A natural and necessary extension is to adopt a multifluid framework, which allows for a more realistic treatment of multiple ion populations. - -**Results:** To address this challenge, we developed a multifluid theoretical model that includes both a nonzero normal magnetic field and a guide field, and explicitly accounts for the dynamics of counter-streaming ion populations (see @fig-profiles). The model reveals a clear physical interpretation: close to the Sun, current sheets are dominated by a single ion population, leading to high Alfvénicity, while at larger radial distances, the ion populations become more balanced, resulting in reduced Alfvénicity (see @fig-velocity-profiles). By bridging the gap between overly simplified single-fluid models and fully kinetic treatments, this multifluid model offers a physically consistent and computationally tractable framework. It establishes a critical connection between the macroscopic evolution of current sheets and the microscopic processes relevant to energetic particle scattering—thus advancing the broader thesis goal of modeling SEP transport in a realistic, structure-rich solar wind. - -This work is presented in "On the Alfvénicity of Multifuild Force-Free Current Sheets" (Zhang et al., submitted to Physics of Plasmas, 2025, manuscript is available at [GitHub](https://github.com/Beforerr/cs_theory/blob/3812b6f1e62b4c954b7b1aa7dcbf092df23834bb/files/2025PoP_Model_Zijin.pdf)). - -![Magnetic field, ion density, and ion bulk velocity for the case where $n_1 = 1.5 n_2$ and $L=d_i, B_0 = 2 B_z$. Here, $z$ is the distance from the center of the 1-D current sheet, $n_α$ denotes the number density of ion species $α$, $d_i$ is the asymptotic ion inertial length, and $B_0$ is the in-plane magnetic field strength.](figures/profiles_n1Inf=0.6.svg){#fig-profiles width=60%} - -![Ion bulk velocity in the $x$ direction (maximum variance direction) $U_x$ profiles normalized by local Alfvén velocity $v_{A,x}(z) = B_x(z) / \sqrt{μ_0 m_p n(z)}$ for different $n_1(∞)$](figures/UxNormBx.svg){#fig-velocity-profiles width=60%} - -### Software Development - -**Context**: A central requirement for this thesis is the ability to perform high-performance, interactive, and reproducible analysis of space plasma data and particle dynamics. While the established SPEDAS framework—originally developed in IDL and later ported to Python—remains widely used in the community, its design limitations hinder modern scientific workflows (big data, parallel/distributed computing, etc.). - -**Approach**: To address this, we developed a suite of Julia-based software tools that combine the flexibility and speed of a modern language with the functionality of legacy systems. - -**Results**: The core of this framework is `SPEDAS.jl`, which has interfaces directly with [`pyspedas`](https://github.com/spedas/pyspedas), [`speasy`](https://github.com/SciQLop/speasy), and [`HAPI`](https://hapi-server.org/) while introducing new routines with significantly improved performance. To enable efficient test-particle tracing in both analytic presets and numerical derived electromagnetic fields, we developed `TestParticle.jl`, a lightweight tool for rapid particle trajectory simulations. Additionally, we created [`SpaceDataModel.jl`](https://github.com/beforerr/SpaceDataModel.jl) to implement flexible, standards-compliant data structures aligned with SPASE and HAPI specifications, and contributed physics utilities through [`ChargedParticles.jl`](https://github.com/JuliaPlasma/ChargedParticles.jl) and [`PlasmaFormulary.jl`](https://github.com/JuliaPlasma/PlasmaFormulary.jl). These tools have been integral to the data analysis (e.g., @fig-sp), modeling, and simulation components of this thesis, enabling scalable and transparent research workflows essential for studying particle transport in the heliosphere. - -```julia -f = Figure() -tvars1 = ["cda/OMNI_HRO_1MIN/flow_speed", "cda/OMNI_HRO_1MIN/E", "cda/OMNI_HRO_1MIN/Pressure"] -tvars2 = ["cda/THA_L2_FGM/tha_fgs_gse"] -tvars3 = ["cda/OMNI_HRO_1MIN/BX_GSE", "cda/OMNI_HRO_1MIN/BY_GSE"] -t0,t1 = "2008-09-05T10:00:00", "2008-09-05T22:00:00" -tplot(f[1, 1], tvars1, t0, t1) -tplot(f[1, 2], tvars2, t0, t1) -tplot(f[2, 1:2], tvars3, t0, t1) -f -``` - -![Example code snippet and resulting output from the Julia implementation of the widely used tplot function.](figures/spedas_jl.png){#fig-sp} - -{{< pagebreak >}} - -## Proposed Research Direction - -### Spatial Diffusion Model Refinement - -The work completed in this thesis has established that solar wind current sheets are persistent, kinetic-scale structures whose statistical properties evolve predictably with heliocentric distance (see @sec-obs). We have demonstrated—both theoretically and through numerical modeling—that SWDs play a significant role in modulating particle transport, particularly by enhancing pitch-angle scattering beyond quasilinear expectations (see @sec-modeling). Furthermore, we have shown that their internal structure, including multifluid effects and Alfvénicity variations, are essential to understanding their properties and thereby their transport-modifying capacity (see @sec-multifluid). - -Building on previous results, the next phase of research will extend the pitch-angle scattering framework to comprehensively model spatial diffusion processes (both parallel and perpendicular). This extension is crucial for accurately capturing the full scope of SEP transport influenced by current sheets. - -### Methodology - -**Analysis:** To estimate the spatial diffusion coefficients parallel and perpendicular to the mean magnetic field, we must quantify the net displacement a particle experiences due to multiple, consecutive interactions with realistic current sheets. This includes estimating the parallel and perpendicular displacements, $\Delta s_\parallel$ and $\Delta s_\perp$, over the duration of one interaction cycle. - -The total time between two consecutive current sheet encounters is modeled as the sum of the time spent inside the current sheet $T_{cs}$, and the time spent free-streaming between sheets $T_{fs}$, given by: - -$$ -T = T_{cs} + T_{fs}, \quad T_{fs} = \frac{s_{fs}}{|v_{\parallel,1}|} -$$ - -where $v_{\parallel,0}$, $v_{\parallel,1}$ are the particle's changed parallel velocity before and after interacting with the current sheet, respectively. - -In the absence of scattering, the particle would follow the field line and travel a distance: - -$$ -s_0 = v_{\parallel,0} \cdot T = v_{\parallel,0} \left( T_{cs} + \frac{s_{fs}}{|v_{\parallel,1}|} \right). -$$ - -However, when scattering occurs, the total distance traveled becomes: $$ -s = s_{cs}^* + \text{sign}\left(\frac{v_{\parallel,1}}{v_{\parallel,0}}\right) s_{fs} -$$ - -where $s_{cs}^*$ is the effective parallel distance the particle travels within the current sheet. The net displacement compared to the unperturbed case is then: - -$$ -\Delta s_\parallel = s - s_0 = (s_{cs}^* - v_{\parallel,0} T_{cs}) + s_{fs} \left(1 - \frac{v_{\parallel,0}}{v_{\parallel,1}} \right). -$$ - -Under the approximation that $s_{cs}^* << s_{fs}$, we obtain: - -$$ -\Delta s_\parallel \approx s_{fs} \left(1 - \frac{v_{\parallel,0}}{v_{\parallel,1}} - \frac{v_{\parallel,0} T_{cs}}{s_{fs}} \right). -$$ - -The parallel spatial diffusion coefficient is then expressed as: - -$$ -\kappa_\parallel = \frac{(\Delta s_\parallel)^2}{\Delta t} = \frac{\left[s_{fs} \left(1 - \frac{v_{\parallel,0}}{v_{\parallel,1}} - \frac{v_{\parallel,0} T_{cs}}{s_{fs}} \right) \right]^2}{T_{cs} + \frac{s_{fs}}{v_{\parallel,1}}}. -$$ - -Similarly, for the perpendicular direction: - -$$ -\kappa_\perp = \frac{(\Delta s_\perp)^2}{T_{cs} + \frac{s_{fs}}{v_{\parallel,1}}}. +\mathbf{B} = B_0 (\cos θ \ \mathbf{e_z} + \sin θ ( \sin φ(z) \ \mathbf{e_x} + \cos φ (z) \ \mathbf{e_y})) $$ -The key parameters—$v_{\parallel,1}$, $T_{cs}$, $\Delta s_\perp$, and $\Delta t$—are directly extracted from test-particle simulations, while quantities such as the current sheet separation distance $s_{fs}$, thickness, shear angle, and normal orientation are treated as system parameters derived from solar wind observations. Together, these inputs enable a systematic and physically grounded estimation of spatial diffusion coefficients under realistic heliospheric conditions. - -![Example trajectory of a particle interacting with a current sheet](figures/dR_perp.png){#fig-dR-perp width=70%} - -**Data-Integrated Analytical Modeling:** To ensure consistency with heliospheric observations, we will use realistic solar wind current sheet parameters to derive the diffusion coefficients using multiple spacecraft spanning radial distances from 0.1 to 5 AU. The derived diffusion coefficients will then be incorporated into turbulence-based transport models, providing a current-sheet-informed extension to global energetic particle transport frameworks. - -### Timeline - -**Months 1–4:** - - - Refine the pitch-angle scattering model to incorporate both parallel and perpendicular spatial diffusion effects. - - - Conduct detailed test-particle simulations using solar wind parameters derived from multi-spacecraft observations (PSP, Wind, Juno, etc.). - -**Months 5–7:** - - - Identify observational signatures supporting the proposed scattering model. - - - Examine how current sheet properties influence SEP scattering across different heliocentric distances. +where $B_0$ is the magnitude of the magnetic field, $θ$ is the azimuthal angle between the normal and the magnetic field, and $φ(z)$ is the rotation profile of the magnetic field as a function of $z$. -**Months 8–10:** +{{< pagebreak >}} - - Finalize the spatial diffusion model and assess its implications for large-scale SEP propagation. + - - Synthesize simulation results and observational insights into dissertation chapters. Integrate observational and theoretical findings into comprehensive thesis documentation. +{{< pagebreak >}} -### Relevance and Broader Implications + -This thesis substantially advances our understanding of particle transport mechanisms within turbulent space plasmas, offering significant enhancements to SEP prediction models. By accurately quantifying the influence of coherent structures such as current sheets, the research outcomes have direct applications to improving space weather forecasting, enhancing spacecraft operational safety, and contributing to the broader understanding of energetic particles transport and acceleration in the solar wind. +{{< pagebreak >}} -### Opportunities for Future Research + -Completion of this thesis opens several avenues for future investigations: -- Exploration of current sheet interactions in other astrophysical environments, such as planetary magnetospheres. + -- Advanced integration of mapping techniques with numerical simulations to further refine SEP transport models. +- @podestaMostIntenseCurrent2017 + - Abstract: Electric currents in the solar wind plasma are investigated using 92 ms fluxgate magnetometer data acquired in a high-speed stream near 1 AU. The minimum resolvable scale is roughly 0.18 s in the spacecraft frame or, using Taylor's “frozen turbulence” approximation, one proton inertial length di in the plasma frame. A new way of identifying current sheets is developed that utilizes a proxy for the current density J obtained from the derivatives of the three orthogonal components of the observed magnetic field B. The most intense currents are identified as 5σ events, where σ is the standard deviation of the current density. The observed 5σ events are characterized by an average scale size of approximately 3di along the flow direction of the solar wind, a median separation of around 50di or 100di along the flow direction of the solar wind, and a peak current density on the order of 0.5 pA/cm2. The associated current-carrying structures are consistent with current sheets; however, the planar geometry of these structures cannot be confirmed using single-point, single-spacecraft measurements. If Taylor's hypothesis continues to hold for the energetically dominant fluctuations at kinetic scales , then the results suggest that the most intense current-carrying structures in high-speed wind occur at electron scales, although the peak current densities at kinetic and electron scales are predicted to be nearly the same as those found in this study. -- Expanded observational campaigns utilizing upcoming spacecraft missions designed to probe heliospheric turbulence and particle dynamics at unprecedented resolution. {{< pagebreak >}} - -## References diff --git a/docs/others/phd/2026_grad/thesis.typ b/docs/others/phd/2026_grad/thesis.typ index 2a7d295..07455ef 100644 --- a/docs/others/phd/2026_grad/thesis.typ +++ b/docs/others/phd/2026_grad/thesis.typ @@ -1,19 +1,59 @@ +// Simple numbering for non-book documents +#let equation-numbering = "(1)" +#let callout-numbering = "1" +#let subfloat-numbering(n-super, subfloat-idx) = { + numbering("1a", n-super, subfloat-idx) +} + +// Theorem configuration for theorion +// Simple numbering for non-book documents (no heading inheritance) +#let theorem-inherited-levels = 0 + +// Theorem numbering format (can be overridden by extensions for appendix support) +// This function returns the numbering pattern to use +#let theorem-numbering(loc) = "1.1" + +// Default theorem render function +#let theorem-render(prefix: none, title: "", full-title: auto, body) = { + if full-title != "" and full-title != auto and full-title != none { + strong[#full-title.] + h(0.5em) + } + body +} // Some definitions presupposed by pandoc's typst output. +#let content-to-string(content) = { + if content.has("text") { + content.text + } else if content.has("children") { + content.children.map(content-to-string).join("") + } else if content.has("body") { + content-to-string(content.body) + } else if content == [ ] { + " " + } +} + #let horizontalrule = line(start: (25%,0%), end: (75%,0%)) #let endnote(num, contents) = [ #stack(dir: ltr, spacing: 3pt, super[#num], contents) ] +// Use nested show rule to preserve list structure for PDF/UA-1 accessibility +// See: https://github.com/quarto-dev/quarto-cli/pull/13249#discussion_r2678934509 #show terms: it => { - it.children - .map(child => [ - #strong[#child.term] - #block(inset: (left: 1.5em, top: -0.4em))[#child.description] - ]) - .join() + show terms.item: item => { + set text(weight: "bold") + item.term + block(inset: (left: 1.5em, top: -0.4em))[#item.description] + } + it } +// Prevent breaking inside definition items, i.e., keep term and description together. +#show terms.item: set block(breakable: false) + // Some quarto-specific definitions. #show raw.where(block: true): set block( @@ -64,7 +104,6 @@ label: none, supplement: str, position: none, - subrefnumbering: "1a", subcapnumbering: "(a)", body, ) = { @@ -77,7 +116,10 @@ supplement: supplement, caption: caption, { - show figure.where(kind: kind): set figure(numbering: _ => numbering(subrefnumbering, n-super, quartosubfloatcounter.get().first() + 1)) + show figure.where(kind: kind): set figure(numbering: _ => { + let subfloat-idx = quartosubfloatcounter.get().first() + 1 + subfloat-numbering(n-super, subfloat-idx) + }) show figure.where(kind: kind): set figure.caption(position: position) show figure: it => { @@ -125,10 +167,13 @@ } // TODO use custom separator if available + // Use the figure's counter display which handles chapter-based numbering + // (when numbering is a function that includes the heading counter) + let callout_num = it.counter.display(it.numbering) let new_title = if empty(old_title) { - [#kind #it.counter.display()] + [#kind #callout_num] } else { - [#kind #it.counter.display(): #old_title] + [#kind #callout_num: #old_title] } let new_title_block = block_with_new_content( @@ -173,86 +218,124 @@ + #let article( title: none, subtitle: none, authors: none, + keywords: (), date: none, - abstract: none, abstract-title: none, + abstract: none, + thanks: none, cols: 1, lang: "en", region: "US", - font: "libertinus serif", + font: none, fontsize: 11pt, title-size: 1.5em, subtitle-size: 1.25em, - heading-family: "libertinus serif", + heading-family: none, heading-weight: "bold", heading-style: "normal", heading-color: black, heading-line-height: 0.65em, + mathfont: none, + codefont: none, + linestretch: 1, sectionnumbering: none, + linkcolor: none, + citecolor: none, + filecolor: none, toc: false, toc_title: none, toc_depth: none, toc_indent: 1.5em, doc, ) = { - set par(justify: true) + // Set document metadata for PDF accessibility + set document(title: title, keywords: keywords) + set document( + author: authors.map(author => content-to-string(author.name)).join(", ", last: " & "), + ) if authors != none and authors != () + set par( + justify: true, + leading: linestretch * 0.65em + ) set text(lang: lang, region: region, - font: font, size: fontsize) + set text(font: font) if font != none + show math.equation: set text(font: mathfont) if mathfont != none + show raw: set text(font: codefont) if codefont != none + set heading(numbering: sectionnumbering) - if title != none { - align(center)[#block(inset: 2em)[ - #set par(leading: heading-line-height) - #if (heading-family != none or heading-weight != "bold" or heading-style != "normal" - or heading-color != black) { - set text(font: heading-family, weight: heading-weight, style: heading-style, fill: heading-color) - text(size: title-size)[#title] - if subtitle != none { - parbreak() - text(size: subtitle-size)[#subtitle] - } - } else { - text(weight: "bold", size: title-size)[#title] - if subtitle != none { - parbreak() - text(weight: "bold", size: subtitle-size)[#subtitle] - } + + show link: set text(fill: rgb(content-to-string(linkcolor))) if linkcolor != none + show ref: set text(fill: rgb(content-to-string(citecolor))) if citecolor != none + show link: this => { + if filecolor != none and type(this.dest) == label { + text(this, fill: rgb(content-to-string(filecolor))) + } else { + text(this) + } + } + + place( + top, + float: true, + scope: "parent", + clearance: 4mm, + block(below: 1em, width: 100%)[ + + #if title != none { + align(center, block(inset: 2em)[ + #set par(leading: heading-line-height) if heading-line-height != none + #set text(font: heading-family) if heading-family != none + #set text(weight: heading-weight) + #set text(style: heading-style) if heading-style != "normal" + #set text(fill: heading-color) if heading-color != black + + #text(size: title-size)[#title #if thanks != none { + footnote(thanks, numbering: "*") + counter(footnote).update(n => n - 1) + }] + #(if subtitle != none { + parbreak() + text(size: subtitle-size)[#subtitle] + }) + ]) } - ]] - } - if authors != none { - let count = authors.len() - let ncols = calc.min(count, 3) - grid( - columns: (1fr,) * ncols, - row-gutter: 1.5em, - ..authors.map(author => - align(center)[ - #author.name \ - #author.affiliation \ - #author.email - ] - ) - ) - } + #if authors != none and authors != () { + let count = authors.len() + let ncols = calc.min(count, 3) + grid( + columns: (1fr,) * ncols, + row-gutter: 1.5em, + ..authors.map(author => + align(center)[ + #author.name \ + #author.affiliation \ + #author.email + ] + ) + ) + } - if date != none { - align(center)[#block(inset: 1em)[ - #date - ]] - } + #if date != none { + align(center)[#block(inset: 1em)[ + #date + ]] + } - if abstract != none { - block(inset: 2em)[ - #text(weight: "semibold")[#abstract-title] #h(1em) #abstract + #if abstract != none { + block(inset: 2em)[ + #text(weight: "semibold")[#abstract-title] #h(1em) #abstract + ] + } ] - } + ) if toc { let title = if toc_title == none { @@ -269,431 +352,772 @@ ] } - if cols == 1 { - doc - } else { - columns(cols, doc) - } + doc } #set table( inset: 6pt, stroke: none ) -#show link: set text(fill: rgb("#239dad")) +#let brand-color = (:) +#let brand-color-background = (:) +#let brand-logo = (:) #set page( paper: "us-letter", margin: (x: 1.25in, y: 1.25in), numbering: "1", + columns: 1, ) #show: doc => article( title: [Kinetic-scale solar wind current sheets: statistical characteristics and their role in energetic particle transport], - subtitle: [Thesis Outline], + subtitle: [Thesis], authors: ( ( name: [Zijin Zhang], affiliation: [], email: [] ), ), - date: [2026-01-06], + date: [2026-03-13], sectionnumbering: "1.1.a", toc_title: [Table of contents], toc_depth: 3, - cols: 1, doc, ) -= Motivation and Significance - -Solar energetic particles (SEPs), originating from solar flares and coronal mass ejections, pose significant risks to satellite operations, human spaceflight, and communication systems. Accurate prediction of SEP events and their propagation through the heliosphere requires a detailed understanding of particle transport mechanisms in turbulent solar wind environments. += Acknowledgments + += Introduction + +The solar wind is a hot, magnetized plasma that continuously expands from the solar corona into interplanetary space, driven by the pressure gradient between the dense coronal plasma and the surrounding interstellar medium @parkerDynamicsInterplanetaryGas1958@parkerDynamicalTheorySolar1965@velliSupersonicWindsAccretion1994@velliHydrodynamicsSolarWind2001. As it propagates outward, the solar wind accelerates to super-Alfvénic speeds, transporting mass, momentum, and energy throughout the heliosphere. + +Because the solar wind is highly electrically conductive, magnetic field lines are frozen into the plasma flow to a good approximation. The resulting large-scale interplanetary magnetic field (IMF) is shaped by the combination of radial expansion and solar rotation, producing the Archimedean spiral geometry described by the Parker model---a configuration well supported by observations when averaged over sufficiently long intervals @zhaoStatisticsInterplanetaryMagnetic2025@zhaoStatisticsInterplanetaryMagnetic2025a. + +Superposed on this ordered background, however, are substantial fluctuations spanning a broad range of scales. These fluctuations are a manifestation of MHD turbulence @schekochihinMHDTurbulenceBiased2022@matthaeusTurbulenceSpacePlasmas2021@brunoTurbulenceSolarWind2016@brunoSolarWindTurbulence2013@tuMHDStructuresWaves1995@goldsteinMagnetohydrodynamicTurbulenceSolar1995, which is multiscale, intermittent, and strongly non-Gaussian @chandranIntermittentReflectiondrivenStrong2025@brunoIntermittencySolarWind2019@verscharenMultiscaleNatureSolar2019@matthaeusIntermittencyNonlinearDynamics2015. A key feature of this turbulence is that energy does not distribute uniformly across space as it cascades from large to small scales. Instead, it concentrates into localized, coherent structures---magnetic flux ropes, magnetic holes, plasma waves, and, most prominently, current sheets @perroneCoherentEventsIon2020@khabarovaCurrentSheetsPlasmoids2021@pezziCurrentSheetsPlasmoids2021. These structures are not passive byproducts of the cascade; they feed back into the ambient plasma and magnetic environment, contributing significantly to plasma heating, particle acceleration, and departures from classical MHD behavior @degiorgioCoherentStructureFormation2017@borovskyContributionStrongDiscontinuities2010@liEffectCurrentSheets2011@grecoPartialVarianceIncrements2017. + +The heliospheric magnetic field thus exhibits a dual character: while the Parker spiral describes the mean magnetic topology of the heliosphere, a complete physical picture of the solar wind requires explicit consideration of turbulence and embedded coherent structures. This multiscale structure has important consequences for a range of heliophysical processes, particularly the transport and acceleration of energetic particles. + +Solar energetic particles (SEPs) are high-energy ions and electrons episodically accelerated in the solar and interplanetary environment @anastasiadisSolarEnergeticParticles2019@kleinAccelerationPropagationSolar2017@desaiLargeGradualSolar2016@reamesTwoSourcesSolar2013. They originate either in the low solar corona during eruptive events, or at interplanetary shock fronts driven by fast coronal mass ejections, where diffusive shock acceleration can energize particles over extended spatial and temporal scales. SEP events pose radiation hazards to spacecraft electronics, astronauts, and high-altitude aviation, and they provide a natural laboratory for studying particle acceleration and transport in magnetized plasmas---with implications extending to astrophysical systems more broadly. + +Early theoretical descriptions of SEP propagation treated the interplanetary medium as a smooth background threaded by broadband turbulent waves, within which particles scatter quasi-linearly. However, growing observational and numerical evidence demonstrates that the fine-scale magnetic structure of the solar wind can exert a decisive influence on particle dynamics @malaraEnergeticParticleDynamics2023@malaraChargedparticleChaoticDynamics2021@artemyevSuperfastIonScattering2020. When a particle's gyroradius becomes comparable to the thickness of a current sheet, classical guiding-center theory breaks down: the particle undergoes a non-adiabatic interaction that can produce large, abrupt changes in pitch angle. Because current sheets are abundant throughout the heliosphere and their thickness overlaps with the gyroradii of suprathermal and energetic ions, such interactions are not rare events but a systematic feature of SEP propagation. They can enhance pitch-angle scattering, induce trapping or reflection, enable localized acceleration, and produce non-diffusive transport that cannot be captured by models based solely on homogeneous wave turbulence. + +Despite their importance, a quantitative understanding of how current sheets influence SEP transport has remained incomplete, for two reasons. First, the statistical properties of kinetic-scale current sheets---their occurrence rate, thickness distribution, and internal magnetic configuration---have not been systematically characterized across different regions of the heliosphere. Second, the connection between these microphysical structural properties and macroscopic transport coefficients has not been rigorously established. This dissertation addresses both gaps. + +Specifically, this dissertation (1) develops a coherent observational framework to characterize the properties and occurrence rates of current sheets from kinetic to large scales across different heliospheric regions, and (2) constructs a quantitative, statistics-based model of SEP interactions with current sheets that describes pitch-angle scattering induced by these structures and estimates the resulting spatial transport coefficients. Together, these contributions provide a bridge between the microphysics of individual current sheet encounters and the macroscopic diffusion of energetic particles through the heliosphere, offering new insight into heliospheric particle dynamics and a foundation for improved space weather prediction. -Traditionally, theoretical studies and numerical models of particle transport in the solar wind have focused on turbulence characterized by broadband, low-amplitude, random-phase magnetic fluctuations described by power-law spectra @jokipiiCosmicRayPropagationCharged1966@jokipiiCosmicRayPropagationIi1967. However, observations consistently reveal the abundance of intermittent, meso-scale, coherent structures within this turbulent medium, notably current sheets---thin plasma boundaries marked by abrupt magnetic field changes. These current sheets deviate significantly from classical magnetohydrodynamic (MHD) picture due to their kinetic-scale features and strong local magnetic gradients. +== Thesis Organization + +This thesis is organized into four parts, progressing from background reviews through observational characterization of current sheets, to their impact on energetic particle transport, and finally to the methodological developments that support and extend the primary scientific results. -Recent theoretical and numerical studies suggest that these coherent structures play a critical role in particle scattering, potentially surpassing the scattering efficiency predicted by traditional quasilinear theories @malaraChargedparticleChaoticDynamics2021@artemyevSuperfastIonScattering2020. Current sheets, generated naturally through nonlinear turbulence cascade, provide localized regions of intense electromagnetic interactions, leading to enhanced scattering and modification of the particle's spatial distribution. Despite their importance, a quantitative and systematic understanding of how these structures influence SEP transport remains incomplete. +#strong[Part I] (Chapters 2--3) provides the observational and theoretical background. #emph[Chapter 2] reviews the observational properties of solar wind current sheets, including the identification and characterization methods, their kinetic nature, and the statistical characteristics reported in the literature. #emph[Chapter 3] introduces the solar energetic particle context: the sources and classification of SEP events, the transport frameworks within which particle scattering is parameterized, and the motivation---rooted in solar wind intermittency---for going beyond classical quasilinear theory. This chapter develops the quasi-adiabatic theory of particle motion near magnetic field reversals and the mechanisms by which the quasi-adiabatic invariant is destroyed at separatrix crossings. -Addressing this critical gap, this dissertation aims to comprehensively investigate and quantify the impact of solar wind current sheets on SEP transport processes. Specifically, this research seeks to: +#strong[Part II] (Chapters 4--5) presents the observational characterization of kinetic-scale current sheets across the inner heliosphere from a coordinated set of spacecraft: Parker Solar Probe (PSP) at distances as close as 0.1 AU, Wind, ARTEMIS, and STEREO near 1 AU, and Juno during its cruise phase out to 5 AU. These chapters systematically quantify the statistical distributions of current sheet thickness and current density---two parameters that directly determine the magnetic field configurations. Because the magnetic field geometry directly influences particle motion, these structural properties serve as essential inputs for understanding particle scattering and transport processes. The observational results therefore provide the empirical foundation for the particle-transport studies that follow. -+ Characterize the properties and occurrence of current sheets throughout different regions of the heliosphere. -+ Develop and validate theoretical models that describe particle scattering induced by these coherent structures. Provide a improved, quantitative modeling of SEP interactions with current sheets, thereby enabling predictive capabilities and contributing to space weather modeling. +#strong[Part III] (Chapters 6--7) investigates how kinetic-scale current sheets scatter and transport energetic particles. #emph[Chapter 6] quantifies pitch-angle scattering through test-particle simulations and statistical analysis, establishing how specific current sheet configurations produce non-adiabatic behavior and how an ensemble of current sheets contributes to long-term pitch-angle diffusion. #emph[Chapter 7] extends this analysis to spatial transport, deriving parallel and perpendicular diffusion coefficients from ensembles of particles interacting with statistically representative current sheet populations. Together, these chapters bridge microscopic scattering physics between local structure physics and macroscopic heliospheric transport. -The motivation for this research lies in the critical need for improved SEP transport models that accurately reflect real-world solar wind conditions. By integrating observational data and advanced theoretical frameworks, this dissertation will provide novel insights into heliospheric particle dynamics, ultimately enhancing our ability to predict and mitigate the risks associated with SEP events. +#strong[Part IV] (Chapters 8--9) presents two complementary developments. #emph[Chapter 8] introduces a multi-fluid current sheet model that provides a self-consistent description of magnetic field and plasma velocity structure within a current sheet, extending beyond simplified one-dimensional configurations. #emph[Chapter 9] describes the Julia-based computational framework developed to support the large-scale statistical analyses and test-particle simulations performed throughout this work. -= Research Context and Background - -The study of solar energetic particles @anastasiadisSolarEnergeticParticles2019@kleinAccelerationPropagationSolar2017@desaiLargeGradualSolar2016@reamesTwoSourcesSolar2013, turbulent magnetic fields @schekochihinMHDTurbulenceBiased2022@matthaeusTurbulenceSpacePlasmas2021@oughtonSolarWindTurbulence2021@brunoTurbulenceSolarWind2016@brunoSolarWindTurbulence2013@verscharenMultiscaleNatureSolar2019@tuMHDStructuresWaves1995, and charged particle transport @engelbrechtTheoryCosmicRay2022@vandenbergPrimerFocusedSolar2020 has produced a vast body of literature spanning decades of theoretical, observational, and numerical research. Within this context, current sheets have increasingly been recognized as key structures. In the following sections, we highlight a selection of foundational observations, models, and theoretical developments that are directly or indirectly connected to the role of current sheets. These include both classical frameworks and recent advances that point to the importance of coherent structures in turbulent plasmas. In #ref(, supplement: [Section]), we summarize how a deeper understanding of current sheets can enhance our ability to model energetic particle transport and, more broadly, improve our understanding of heliospheric particle dynamics. +#emph[Chapter 10] summarizes the principal findings and discusses open questions and future directions, with emphasis on the origins of solar wind current sheets and on strategies for incorporating intermittency effects into operational SEP transport models. -== Solar Energetic Particles - -Solar energetic particles (SEPs) are high-energy ions and electrons originating at or near the Sun. They span a broad energy spectrum, from Solar energetic particles (SEPs) consist of high-energy ions and electrons originating at or near the Sun. Unlike the solar wind and galactic cosmic rays (GCRs), solar energetic particles (SEPs) manifest as discrete episodic events with intensities that can vary dramatically---by several orders of magnitude---in just minutes. Additionally, SEP events exhibit significant variations in heavy ion composition, spectral shape, and spatial distribution. +#pagebreak() += Observations of Solar Wind Current Sheets + +This chapter reviews the observational landscape of solar wind current sheets @tsurutaniReviewDiscontinuitiesAlfven1999@neugebauerProgressStudyInterplanetary2010@khabarovaCurrentSheetsPlasmoids2021, tracing their study from the earliest spacecraft measurements to the high-cadence, multi-point observations available today. The review is organized as follows. We begin by recounting how abrupt magnetic field rotations were first recognized as magnetohydrodynamic discontinuities and how subsequent high-resolution measurements revealed their fundamentally kinetic character. We then discuss the analysis methods used to determine current sheet orientation and thickness, with particular attention to the strengths and limitations of single-spacecraft and multi-spacecraft techniques. The bulk of the chapter is devoted to the statistical properties of current sheets --- their identification, magnetic field configuration, spatial scales, current densities, occurrence rates, Alfvénicity, and orientation --- as established by surveys spanning from the inner heliosphere to beyond 5 AU. Throughout, we emphasize the extent to which reported statistics depend on the identification method employed, a recurring theme that must be kept in mind when comparing results across studies. The chapter concludes by identifying several open questions that motivate the work presented in subsequent chapters. -SEPs are primarily accelerated through two distinct mechanisms @reamesTwoSourcesSolar2013: (1) shock-wave acceleration associated with fast coronal mass ejections (CMEs), resulting in large gradual SEP events @desaiLargeGradualSolar2016, and (2) magnetic reconnection-driven processes during solar flares, producing impulsive SEP events. +== From MHD Discontinuities to Kinetic-Scale Current Sheets + +Early observations from the Pioneer 6 mission revealed that the direction of the IMF is highly variable, an unexpected finding at the time @nessPreliminaryResultsPioneer1966. As shown in #ref(, supplement: [Figure]), on hour-long timescales, these abrupt directional changes are clearly distinguishable from the comparatively quiet background in which the magnetic field evolves slowly. Such rapid variations in the magneto-plasma parameters were recognized as fundamental solar wind features @colburnDiscontinuitiesSolarWind1966 and were identified as magnetohydrodynamic discontinuities---spatial boundaries separating two distinct plasma regions. -Gradual SEP events typically last for several days and are predominantly proton-rich, often associated with fast CMEs driving shocks in the solar corona and interplanetary space. These shocks accelerate particles over extended regions, producing widespread and intense radiation storms. In contrast, impulsive SEP events are related to short duration (less than 1 h) solar flares. These events typically have shorter durations, lasting from minutes to a few hours, and feature characteristically higher electron-to-proton ratios and enrichments of heavy ions ($zws^3 upright(H e) \/ zws^4 upright(H e)$ and $upright(F e) \/ upright(O)$ ratios). +MHD theory permits such discontinuities but constrains the changes allowed across them through the Rankine--Hugoniot jump conditions. The early solar wind measurements spurred the development of theory for anisotropic plasmas @hudsonDiscontinuitiesAnisotropicPlasma1970. Five distinct types are possible, the most relevant here being tangential discontinuities (TDs), rotational discontinuities (RDs), and shocks (relatively rare in the solar wind). Classifying observed discontinuities as RDs or TDs attracted considerable early research interest because the distinction carries physical implications for the topology of the IMFs @knetterNewPerspectiveSolar2005. A TD separates two topologically distinct plasma regions with no field-normal component, whereas an RD is a propagating structure that connects magnetically linked regions. This distinction has consequences for energetic particle diffusion coefficients and bears on possible generation mechanisms operating in the solar corona. The relative abundance of RDs and TDs in the solar wind has been the subject of longstanding debate @smithIdentificationInterplanetaryTangential1973@neugebauerReexaminationRotationalTangential1984@neugebauerCommentAbundancesRotational2006. #figure([ -#box(image("figures/desaiLargeGradualSolar2016-fig3.png")) +#box(image("figures/ref/nessPreliminaryResultsPioneer1966-fig6.png")) ], caption: figure.caption( position: bottom, [ -The two-class picture for SEP events. #cite(, form: "prose") +Two-hour example of 1-minute averages of the interplanetary magnetic field for which the magnitude is average, but the direction is highly variable and principally inclined at large angles ($theta approx 90^compose$) to the ecliptic plane @nessPreliminaryResultsPioneer1966 ]), kind: "quarto-float-fig", supplement: "Figure", ) + -In the decay phase of large gradual SEP events, a characteristic phenomenon known as the #strong[reservoir effect] frequently occurs, where particle intensities measured by widely separated spacecraft become nearly uniform across large regions and exhibit similar temporal evolutions. One traditional explanation for reservoir formation suggests that particles become trapped behind a CME-driven magnetic structure, resulting in spatially uniform spectra that adiabatically decrease in intensity as the confining magnetic bottle expands. However, high heliolatitude observations from the Ulysses mission revealed the three-dimensional character of the reservoir effect and favor the cross-field diffusion explanation @larioHeliosphericEnergeticParticle2010@dallaPropertiesHighHeliolatitude2003. +At smaller scales (on the order of seconds), many discontinuities retain their sharp character, though some reveal resolvable internal structure even in early measurements @siscoePowerSpectraDiscontinuities1968@burlagaDirectionalDiscontinuitiesInterplanetary1969. At these scales, the term "discontinuity" becomes inappropriate: the structure width is comparable to the ion inertial length, and MHD theory may no longer be applicable. The term #emph[current sheet] is adopted instead, emphasizing the current layer that maintains the field rotation. In this dissertation, we use "current sheet" throughout to emphasize the kinetic nature of these structures. The kinetic scale generally refers to structures with widths comparable to the ion inertial length, corresponding roughly to temporal scales below 30 seconds at 1 AU. (Four scale regimes were introduced by #cite(, form: "prose"): macro-scale (\> 100 h), meso-scale (1--100 h), micro-scale (30 s -- 1 h), and kinetic-scale (\< 30 s).) -In contrast to these smooth, widespread distributions, certain impulsive SEP events demonstrate remarkably sharp spatial variations (abrupt depletions) in particle intensity, known as dropout events @tesseinEffectCoherentStructures2015@neugebauerEnergeticParticlesTangential2015a. Such behavior is attributed to spacecraft traversing alternating particle-filled and particle-empty magnetic flux tubes, suggesting extremely limited lateral transport of particles across magnetic fields @mazurInterplanetaryMagneticField2000. This phenomenon is typically interpreted as resulting from particles being effectively confined within distinct magnetic flux tubes, due to minimal cross-field diffusion. The sharply defined spatial gradients scales observed in dropout events, are often comparable to particle gyro-radii. +Recent observations from multiple missions have confirmed that many solar wind current sheets possess fundamentally kinetic properties that defy a purely MHD description (see #ref(, supplement: [Figure]) for an example). #cite(, form: "prose") analyzed discontinuities observed by the ARTEMIS and MMS missions and found that these structures exhibit characteristics of both TDs and RDs simultaneously: tangential velocity jumps correlate well with Alfvén speed jumps (an RD signature), yet electron density and temperature vary significantly across them (a TD signature). -Together, these contrasting observations---extensive spatial uniformity in gradual SEP events (reservoir effects) versus sharp intensity variations in impulsive events (dropouts)---underscore the complexity of SEP transport mechanisms, motivating ongoing studies to reconcile these phenomena within comprehensive transport models. - -== Turbulent Magnetic Fluctuations - -Solar wind turbulence spans scales from the large‑scale coherence length (∼0.01 AU) down to kinetic dissipation scales on the order of the thermal ion gyro‑radius (∼100 km). Of particular importance for energetic particle transport is the turbulence at intermediate scales, often referred to as inertial-range turbulence. For a 5 nT magnetic field, this range corresponds to proton gyro-radii from about 1 GeV to 1 keV, encompassing nearly all SEPs, whose gyro-radii lie between these two bounds. +#figure([ +#box(image("figures/ref/artemyevKineticNatureSolar2019-fig3.jpg")) +], caption: figure.caption( +position: bottom, +[ +Discontinuity observations by two ARTEMIS (at ∼ 07:38:00 and ∼ 07:38:30) and the MMS 1 (at ∼ 07:53:00) spacecraft. (a) The magnetic field $B_l$ (left axis) and plasma velocity $v_l$ (right axis). (b1 and b2) The electron density $n_e$ (left axis) and temperature $T_e$ (right axis). (c1, c2, d1, and d2) The electron pitch angle distributions for two energy ranges. (e1 and e2) The electron flux anisotropy. +]), +kind: "quarto-float-fig", +supplement: "Figure", +) + -The transport of SEPs through the heliosphere is shaped by the properties of magnetic turbulence. Key parameters---such as the spatial inhomogeneity, turbulence level ($delta B \/ B ₀$), spectral index, and anisotropy of wave vectors @pucciEnergeticParticleTransport2016 ---strongly influence how particles scatter in velocity space. These properties govern both parallel and perpendicular transport through mechanisms including pitch-angle diffusion, magnetic field-line meandering, and gradient or curvature drift. -Classical scattering theories and numerical models of particle transport @giacaloneTransportCosmicRays1999 typically model turbulence as a sea of random, phase-uncorrelated fluctuations (common constructions of magnetic fluctuations for the slab component $delta 𝐁^s$ and two-dimensional component $delta 𝐁^(2 D)$ are shown below in #ref(, supplement: [Equation])). However, this idealized view neglects the intricate internal nonlinear structures of turbulence. Increasingly, observations and simulations show that solar wind turbulence is highly intermittent and populated with coherent structures---especially current sheets---that arise naturally through nonlinear cascade processes. +Complementary statistical work by #cite(, form: "prose") using ARTEMIS data revealed that ion-scale discontinuities are accompanied by density and temperature variations extending over tens of ion inertial lengths. The inversely correlated density and temperature variations suggest a nearly force-free configuration. These structures exhibit two characteristic spatial scales: an intense inner current layer (\> 1 nA/m²) enveloped by a broader outer structure. The magnetic field rotation occurs at the outer scale, while plasma pressure gradients provide pressure balance at the inner scale. The electron kinetic behavior around these structures is strongly energy-dependent: near-thermal electrons (10--30 eV) exhibit significant pitch-angle changes across the discontinuity, whereas hot electrons (100--1000 eV) retain their distribution properties on both sides, suggesting they can cross the structure freely. -#math.equation(block: true, numbering: "(1)", [ $ & delta 𝐁^s = sum_(n = 1)^(N_m) A_n [cos alpha_n (cos phi.alt_n hat(x) + sin phi.alt_n hat(y)) + i sin alpha_n (- sin phi.alt_n hat(x) + cos phi.alt_n hat(y))] times exp (i k_n z + i beta_n)\ - & delta 𝐁^(2 D) = sum_(n = 1)^(N_m) A_n i (- sin phi.alt_n hat(x) + cos phi.alt_n hat(y)) times exp [i k_n (cos phi.alt_n x + sin phi.alt_n y) + i beta_n] $ ]) +Further insight into the kinetic nature of current sheets was provided by #cite(, form: "prose"). Rotational discontinuities are typically accompanied by velocity jumps $Delta v_l$ that are systematically smaller than the corresponding Alfvén speed jumps $Delta v_A$, contrary to the stationary MHD prediction of equality. Previous explanations invoked either pressure anisotropy reducing $Delta v_A$ or non-stationarity from residual magnetic energy. #cite(, form: "prose") proposed an alternative: nonadiabatic ion interactions with intense thin discontinuities produce nongyrotropic ion distributions with a nondiagonal pressure tensor component whose cross-discontinuity gradient reduces $Delta v_A$. ARTEMIS observations confirmed the existence of such an ion population with sufficient amplitude and spatial profile to account for the discrepancy, demonstrating that ion kinetic effects fundamentally shape the internal structure of solar wind current sheets. -$ & D_parallel = v^2 / 8 integral_(- 1)^1 (1 - mu^2)^2 / D_(mu mu) d mu\ - & D_(mu mu) = frac(pi omega_(c i) k, 2 B^2 \/ mu_0) (1 - mu^2) sum_(+ \, -) I_plus.minus (k) $ +Consequently, investigating the formation, evolution, and particle interactions of current sheets requires moving beyond MHD theory and into the realm of ion and electron kinetics. -=== Geometrical Chaotization - -A key physical mechanism underlying the strong scattering induced by current sheets is geometrical chaotization---a rapid breakdown of adiabatic invariants caused by separatrix crossings in the particle's phase space @tennysonChangeAdiabaticInvariant1986@zelenyiQuasiadiabaticDynamicsCharged2013. In such slow-fast Hamiltonian systems, even weak asymmetries in the current sheet configuration can produce large, abrupt pitch-angle changes, leading to fast and efficient chaotization of particle motion @artemyevSuperfastIonScattering2020@artemyevRapidGeometricalChaotization2014. This mechanism departs from the diffusive assumptions of classical quasilinear theory and underscores the importance of kinetic-scale structure in driving non-diffusive scattering behaviors. +== Analysis Methods: Normal Determination and the Planar Approximation + +Solar wind current sheets are commonly approximated as one-dimensional structures in which the dominant variation occurs along the direction normal to the sheet. Under this approximation, the most important spatial parameter is the thickness---the scale over which the magnetic field rotates across the structure. Estimating the thickness requires knowledge of the current sheet normal, and the accuracy of this estimate depends critically on the method used to determine it. +A detailed treatment and comprehensive review of both single-spacecraft and multi-spacecraft analysis techniques can be found in #cite(, form: "prose"). In this section, we do not attempt to reproduce that earlier review. Instead, we focus on developments that have emerged since then. -Therefore, understanding SEP transport requires more than bulk statistical descriptions of turbulence; it demands detailed knowledge of its intermittent nature and the embedded coherent structures that mediate particle scattering. Accurately characterizing these features is essential for developing realistic models of SEP propagation throughout the heliosphere. +=== Single-Spacecraft Methods + +For single-spacecraft observations, two methods are most widely used. The first is minimum variance analysis (MVA) of the magnetic field, based on the continuity of the normal component of the magnetic field. This method identifies the direction of minimum magnetic field variance in the transition layer and interprets it as the sheet normal @sonnerupMagnetopauseStructureAttitude1967@sonnerupMinimumMaximumVariance1998. The second is the cross-product method, which estimates the normal from the cross product of the upstream and downstream magnetic field vectors @burlagaTangentialDiscontinuitiesSolar1969@wangSolarWindCurrent2024. -== Charged Particle Transport and Turbulence Transport Models - -The large-scale behavior of energetic charged particles in the heliosphere is commonly described using a diffusive approximation, justified when the particle scattering time is short compared to the timescale of interest. Under this assumption, the evolution of an approximately isotropic particle distribution is governed by the Parker transport equation @parkerPassageEnergeticCharged1965. This foundational framework captures four main transport processes: spatial diffusion due to particle scattering, advection with the solar wind, drifts (such as gradient and curvature drifts due to variations in the large-scale magnetic field), and adiabatic energy change: +MVA has been the dominant technique in the literature, and much of the global statistical picture of solar wind discontinuities has been built upon it. A key advantage of MVA is that it makes no a priori assumption about the magnetic field geometry---it can in principle recover the normal direction for both TDs and RDs. The method is designed to handle realistic deviations from an ideal one-dimensional layer---including 2D or 3D internal structure, temporal fluctuations in normal orientation, and measurement uncertainties---provided there is no systematic change in the normal direction during the spacecraft traversal @sonnerupMinimumMaximumVariance1998. In practice, however, these non-ideal effects can be severe enough to cause MVA to fail. Systematic comparisons with multi-spacecraft determinations have revealed that MVA can be inaccurate for solar wind current sheets. #cite(, form: "prose"), using three-spacecraft timing with Geotail, Wind, and IMP-8, found that many MVA normal estimates lie far from the timing-derived normals. In their dataset, 77% of events were likely TDs based on timing, yet single-spacecraft MVA would have classified a much larger fraction as RDs---a discrepancy they attributed partly to surface waves on the discontinuities (which may cause the minimum variance directions approximately perpendicular to discontinuity normals). -#math.equation(block: true, numbering: "(1)", [ $ frac(partial f, partial t) = frac(partial, partial x_i) [kappa_(i j) frac(partial f, partial x_j)] - U_i frac(partial f, partial x_i) - V_(d \, i) frac(partial f, partial x_i) + 1 / 3 frac(partial U_i, partial x_i) [frac(partial f, partial ln p)] + upright("Sources") - upright("Losses") \, $ ]) +The nature of these MVA failures has been clarified by subsequent work. #cite(, form: "prose"), applying Grad--Shafranov reconstruction (based on the ideal 2D MHD equations in steady state) to Cluster observations, showed that internal structures such as magnetic islands (flux ropes) within the transition layer can cause MVA to fail as a predictor of the normal direction. Since MVA assumes a planar, one-dimensional geometry, any two-dimensional substructure may shift the minimum variance direction away from the true normal. (Good agreement with timing methods can be achieved by imposing the additional constraint $chevron.l B_N chevron.r = 0$ on MVA.) #cite(, form: "prose") conducted a comprehensive statistical assessment using 6,752 Cluster discontinuities with timing-derived normals as a benchmark. They found that while increasing the eigenvalue ratio $lambda_2 \/ lambda_3$ and narrowing the analysis window can reduce scatter, MVA suffers from an inherent geometric defect: discontinuities with small normal magnetic field component ($\| B_N \| \/ \| B \| < 0.2$) and small magnitude change ($Delta \| B \| \/ \| B \| < 0.05$) are systematically misidentified. In these cases, MVA confuses the dominant in-plane magnetic field component with the normal component, producing a spurious large $\| B_N \|$ and an apparent in-plane rotation angle of $tilde.op 180 degree$. This mechanism causes genuine TDs and EDs to be misclassified as RDs with dominant normal fields, explaining the false RD predominance reported in many earlier MVA-based studies. Despite these limitations, #cite(, form: "prose") showed that MVA can achieve acceptable accuracy under favorable geometric conditions. Since the magnitude change $Delta \| B \| \/ \| B \|$ and the rotation angle $omega$ do not depend on knowledge of the normal and are known a priori, they can serve as pre-selection criteria: MVA errors remain generally below $30 degree$ when either $Delta \| B \| \/ \| B \| > 0.05$ or $omega > 60 degree$. The problematic population is therefore confined to small-angle, nearly constant-magnitude current sheets. -where $f$ is the phase-space distribution as a function of the particle momentum, $p$, position, $x_i$, and time, $t$; $kappa_(i j)$ is the symmetric part of the diffusion tensor; $U_i$ is the bulk plasma velocity; $V_(d \, i)$ is the drift velocity. The drift velocity can be formally derived from the guiding center approximation averaged over a nearly isotropic distribution, and can be included as the antisymmetric part of the diffusion tensor. +The cross-product method, by contrast, has been shown to perform substantially better in accuracy. #cite(, form: "prose"), comparing single-spacecraft estimates against four-spacecraft Cluster timing for 1,831 current sheets, demonstrated that the cross-product normal agrees with the timing normal to within $15 degree$ at the 90% confidence level, whereas MVA normals frequently deviate by more than $60 degree$---likely due to contamination by #emph[anisotropic] turbulent fluctuations. Combined with the Taylor frozen-in hypothesis (validated by the finding that current sheet propagation velocities agree with local ion flow velocities within \~20%), the cross-product method delivers current sheet thickness and current density amplitude within 20% of their multi-spacecraft values at the 90% confidence level. However, the cross-product method has an important limitation: it assumes a vanishing normal magnetic field component ($B_N = 0$), which is strictly valid only for tangential discontinuities. For structures with a significant normal field component, the cross product of the boundary fields does not coincide with the true normal. In practice, this limitation is mitigated by the observational finding that the vast majority of kinetic-scale current sheets have very small $\| B_N \| \/ \| B \|$ @erdosDensityDiscontinuitiesHeliosphere2008@wangSolarWindCurrent2024, making the assumption a good approximation for the bulk of the population. This single-spacecraft methodology has been widely adopted in recent statistical studies of kinetic-scale current sheets @vaskoKineticscaleCurrentSheets2021@vaskoKineticscaleCurrentSheets2022@lotekarKineticscaleCurrentSheets2022@vaskoKineticScaleCurrentSheets2024. -The symmetric diffusion tensor can be decomposed into components parallel and perpendicular to the mean magnetic field using: $kappa_(i j) = kappa_perp delta_(i j) - frac((kappa_perp - kappa_parallel) B_i B_j, B^2)$. The parallel diffusion coefficient, $kappa_parallel$, is related to the pitch-angle diffusion coefficient $D_(mu mu)$ through the quasilinear theory (QLT) framework @jokipiiCosmicRayPropagationCharged1966@jokipiiAddendumErratumCosmicRay1968 as $kappa_parallel = v^2 / 8 integral_(- 1)^1 frac((1 - mu^2)^2, D_(mu mu) (mu)) d mu$. -While parallel transport is relatively well understood, perpendicular (cross-field) diffusion ($kappa_perp$) remains more elusive due to its nonlinear and non-resonant nature @shalchiPerpendicularDiffusionEnergetic2021@costajr.CrossfieldDiffusionEnergetic2013. A key factor influencing cross-field transport is the dimensionality of the turbulence @giacaloneChargedParticleMotionMultidimensional1994: in models with at least one ignorable spatial coordinate (e.g., slab geometry), cross-field diffusion is artificially suppressed, failing to capture essential physics. In general, cross-field transport arises from two distinct mechanisms: (1) particle motion along stochastic, meandering magnetic field lines, which can lead to substantial displacements relative to the mean field direction; and (2) the true decorrelation of particles from their initial field lines, allowing them to effectively jump between neighboring lines. Though often considered a small fraction of $kappa_parallel$ @giacaloneTransportCosmicRays1999, recent simulations reveal that $kappa_perp$ can be significant and strongly dependent on particle energy and turbulence structure @dundovicNovelAspectsCosmic2020. +#cite(, form: "prose") reached a partially reconciling conclusion from the Ulysses dataset: for the subset of well-defined discontinuities with reliable normals, MVA and the cross-product technique yield consistent results. They advocated retaining only such well-defined events for further analysis, noting that the vast majority of these have a small magnetic field component parallel to the normal. The practical implication is that MVA can be used reliably when applied with appropriate quality filters, but uncritical application to the full population of discontinuities introduces systematic biases that have historically distorted the statistical picture---particularly the RD/TD classification. -Anisotropy in particle distributions is common in SEP events, particularly in the early phases or near upstream regions of interplanetary shocks. One fundamental source of anisotropy is adiabatic focusing in a diverging magnetic field. To account for such effects, the focused transport equation @roelofPropagationSolarCosmic1969@earlEffectAdiabaticFocusing1976 extends the Parker equation by explicitly retaining the pitch-angle dependence: +==== Multi-Spacecraft Methods and Higher-Dimensional Structure + +Multi-spacecraft observations have played a crucial role in advancing our understanding of current sheets, as they allow improved determination of the normal direction and direct separation of spatial structure from temporal evolution @knetterNewPerspectiveSolar2005@knetterFourpointDiscontinuityObservations2004@knetterDiscontinuityObservationsCluster2003@horburyThreeSpacecraftObservations2001@nessSimultaneousMeasurementsInterplanetary1966. The timing method compares the arrival times of the same structure at several spacecraft with known relative positions, enabling geometric inference of the sheet orientation and propagation velocity @vogtAnalysisDataMultisatellite2020@paschmannMultispacecraftAnalysisMethods2008@knetterNewPerspectiveSolar2005@paschmannAnalysisMethodsMultispacecraft2000. Missions such as Cluster and MMS have greatly expanded the use of these techniques. -$ frac(partial f, partial t) + mu v frac(partial f, partial z) + frac(v, 2 L) (1 - mu^2) frac(partial f, partial mu) = frac(partial, partial mu) (D_(mu mu) frac(partial f, partial mu)) $ +Beyond providing more accurate normals, multi-spacecraft observations have revealed that many solar wind current sheets are not perfectly planar. #cite(, form: "prose"), using Wind and IMP-8 data for 134 large-angle ($omega > 90 degree$) discontinuities, estimated a weighted-average radius of curvature of $tilde.op 380 thin R_E$ (with a most probable value of $tilde.op 290 thin R_E$\; however, their analysis is unable to distinguish real curvature from shorter-scale surface variations using only two-spacecraft data sets), and an average thickness of $tilde.op 14 thin R_E$ (most probable $tilde.op 6 thin R_E$). These results caution against the simplistic use of the planar assumption when projecting a distantly observed discontinuity to predict its arrival characteristics at a downstream location. #cite(, form: "prose"), exploiting the variable separation of the twin STEREO spacecraft, studied tens of thousands of discontinuities and found that the distributions of thickness, normal orientation, shear angle, and waiting times differ systematically between discontinuities observed by both spacecraft and those seen by only one. The population observed by both spacecraft---those with sufficient lateral extent to be intercepted at two separated points---was most consistently interpreted as the walls of solar wind flux tubes. -where $f = f (z \, mu \, t)$ is the phase-space distribution, $mu$ is the pitch-angle cosine and $L = - B (frac(d B, d z))^(- 1)$ is the focusing length. +#cite(, form: "prose") proposed a method to determine the orientation and propagation velocity of two-dimensional structures using two-spacecraft data under the assumption of a steady-state, divergence-free magnetic field. While no clear 2D structures were identified on the \~10-hour scales examined with Wind and IMP-8, the method established a framework for probing departures from planarity at smaller scales. #cite(, form: "prose") subsequently demonstrated with Grad--Shafranov reconstruction that directional discontinuities can contain internal magnetic islands, making them irreducible to simple TD or RD classifications and underscoring the importance of accounting for multidimensional geometry when interpreting spacecraft crossings. -Beyond classical diffusion, observations of SEP events and near interplanetary shocks often reveal anomalous transport behavior @zimbardoSuperdiffusiveSubdiffusiveTransport2006, characterized by subdiffusive or superdiffusive scaling of particle displacement with time @zimbardoSuperdiffusiveTransportLaboratory2015 $⟨Delta x^2 (t)⟩ prop t^alpha$. These deviations from normal diffusion are attributed to the intermittent and structured nature of solar wind turbulence, and are better described using generalized frameworks such as fractional diffusion models @del-castillo-negreteNondiffusiveTransportPlasma2005 or Lévy statistics @zaburdaevLevyWalks2015. +== Statistical surveys and identification methods + +Following the initial discovery of solar wind current sheets, research shifted toward systematic statistical surveys @tsurutaniReviewDiscontinuitiesAlfven1999@neugebauerProgressStudyInterplanetary2010. This section reviews general statistical properties and the methods used to identify current sheets. Parameters central to understanding both their physical nature and their dynamical influence on energetic particles --- magnetic field configuration, spatial scale #ref(, supplement: [Section]), and occurrence rate #ref(, supplement: [Section]) --- are discussed in the subsequent sections. -== The Role of Current Sheets in Particle Transport - -Across all major transport models, current sheets emerge as a critical feature influencing energetic particle dynamics. In the Parker equation framework, current sheets modify the pitch-angle scattering rate and hence directly affect $kappa_parallel$. In the context of the focused transport equation, they introduce strong pitch-angle dependencies and rapid scattering events. Moreover, current sheets can induce memory effects that violate the Markov assumption @zimbardoNonMarkovianPitchangleScattering2020 of classical diffusion models and contribute to anomalous diffusion. +A key point that must be emphasized at the outset is that statistical properties depend critically on the identification method. Different selection criteria---thresholds on magnetic field rotation angle, magnetic field increments, partial variance of increments (PVI), or relative standard deviation---introduce systematic biases into the sampled population. Methods optimized for large-amplitude, well-defined discontinuities preferentially select broader, MHD-scale structures, whereas gradient-based or increment-based approaches are more sensitive to thinner, kinetic-scale current sheets. Reported distributions of thickness, current density, and occurrence rate are therefore inherently method-dependent, and care must be taken when comparing results across studies. -These structures also challenge the conventional picture of diffusion. For parallel transport, the intense magnetic shear and sharp field gradients in current sheets can induce nonlinear effects, producing pitch-angle jumps that are too large to be treated as diffusive. For perpendicular transport, it is often assumed that field-line random walk dominates cross-field motion, as the magnetic field is typically smooth on scales comparable to SEP gyro-radii. However, near current sheets, the magnetic field becomes highly inhomogeneous---often varying on scales similar to or smaller than the gyro-radius---thus enabling enhanced particle transfer between field lines and more significant perpendicular diffusion. +To process the vast amounts of spacecraft data, various automated identification algorithms have been developed. #ref(, supplement: [Table]) summarizes the primary quantitative criteria utilized in the literature. -Because of their coherent, localized nature and their ability to shape both pitch-angle and spatial scattering processes, current sheets play a central role in accurately modeling particle transport in the turbulent heliospheric environment. +#figure([ +#table( + columns: (15.69%, 23.53%, 33.33%, 27.45%), + align: (auto,auto,auto,auto,), + table.header([Method], [Description], [Method Reference], [Applications],), + table.hline(), + [Directional change], [Change in the direction of #strong[B]], [#cite(, form: "prose")], [#cite(, form: "prose")], + [Relative field change], [Relative change in magnetic field #strong[B]], [#cite(, form: "prose")], [#cite(, form: "prose")], + [Correlation / angle distribution], [Two-time correlation functions and distribution of angle change over a time lag], [#cite(, form: "prose")], [#cite(, form: "prose")], + [PVI], [Partial Variance of Increments], [#cite(, form: "prose")], [#cite(, form: "prose")\; #cite(, form: "prose")\; #cite(, form: "prose")], + [Relative standard deviation], [Relative standard deviation of #strong[B]], [#cite(, form: "prose")], [#cite(, form: "prose")], +) +], caption: figure.caption( +position: top, +[ +Summary of current sheet identification methods used in the literature. +]), +kind: "quarto-float-tbl", +supplement: "Table", +) + -= Objectives and Thesis Plan - -The overall goal of this thesis is to quantify and model the impact of solar wind current sheets on energetic particle transport. This research is structured around two primary objectives: -- Observational characterization of solar wind current sheets across the heliosphere +Most early statistical studies focused on large-scale or mesoscale current sheets that are readily identifiable in lower-cadence data. While these works provide essential context, their results cannot be directly compared with statistics derived from high-cadence measurements targeting kinetic-scale structures. Differences in scale selection, detection thresholds, and instrumental resolution introduce subtleties that must be carefully navigated. For instance, as demonstrated by #cite(, form: "prose"), current sheets become exponentially more numerous at smaller spread angles, a population often missed by earlier methods which focus on isolated, large-angle events and exclude structures in close proximity to one another. Therefore, while prior studies inform the broader landscape, the results presented in this thesis pertain specifically to the kinetic-scale population and should be interpreted within the framework of the identification methodology employed herein. -- Development of data-driven theoretical models for current sheet-induced particle scattering and transport +=== Small Intensity jump, Δ|B| + +While it is possible for a current sheet to exhibit a large jump in magnetic field magnitude, the vast majority are predominantly characterized by a rotation of the magnetic field across the sheet, with the magnitude remaining nearly constant. This was first recognized by #cite(, form: "prose"), who observed that most sharp changes in the IMF are primarily directional and introduced the term #emph[directional discontinuity] (DD). Quantitatively, the change in $\| B \|$ is less than 20% for approximately 75% of the discontinuities in their study. -= Work Completed - -== Observational Analysis of Current Sheets - -#strong[Context:] A critical first step in understanding the role of current sheets in energetic particle transport is to characterize their statistical properties and quantify the parameters most relevant to particle scattering. Although current sheets have been extensively observed---especially near $1$ AU---our knowledge of how their properties evolve across heliocentric distances, and how key scattering-related parameters vary with radial distance, has remained incomplete. Previous studies @sodingRadialLatitudinalDependencies2001[#cite(, form: "prose");, #cite(, form: "prose");, #cite(, form: "prose");, #cite(, form: "prose");] often lacked simultaneous, multi-point measurements and did not adequately separate temporal variability from spatial trends, leading to persistent uncertainties regarding their role in particle transport, their origin, and their evolution within the turbulent solar wind. +More recent high-resolution analyses confirm this near-constant magnitude at smaller scales. Using 1/3-second resolution ACE data, #cite(, form: "prose") found that most discontinuities have ramp-like internal profiles---the field varies nearly monotonically within the layer, with no evidence of rapid compression or dissipation @tsurutaniRapidEvolutionMagnetic2005. Furthermore, the intensity jump distribution for solar wind discontinuities is best fit with a lognormal function and is narrowly confined about unity, in contrast to the much broader distribution found in phase-randomized surrogate fields. This difference implies that the layer is regulated by specific physical processes (e.g., nonlinear wave magnetic pressure and Landau damping) rather than random superposition of fluctuations. -#strong[Approach:] To bridge this observational gap, we conducted a detailed statistical analysis using continuous solar wind data from multiple spacecraft: Parker Solar Probe (PSP) at distances down to 0.1 AU, Wind, ARTEMIS, and STEREO at 1 AU, and Juno during its cruise phase out to 5 AU near Jupiter. This combination allowed us to track the evolution of current sheet properties across a wide radial distance, from near Alfvénic critical surface to the outer inner heliosphere. +This conclusion is further reinforced by #cite(, form: "prose") and #cite(, form: "prose"), who analyzed 11200 proton kinetic-scale current sheets near the sun By Parker Solar Probe and 16903 current sheets at 5 AU observed by Ulysses. The magnetic field rotates through a shear angle with only weak magnitude variation. The maximum variation of $\| B \|$ within a current sheet is statistically larger than the variation between its boundaries, and larger magnitude variations are typical at higher plasma $beta$ (#ref(, supplement: [Figure])). #figure([ -#box(image("figures/fig-ids_examples.png")) +#box(image("figures/ref/vaskoKineticScaleCurrentSheets2024-fig4.png")) ], caption: figure.caption( position: bottom, [ -Current sheets detected by PSP, Juno, STEREO and near-Earth ARTEMIS satellite: red, blue, and black lines are $𝐵_𝑙$, $𝐵_𝑚$, and $𝐵$ +Probability and cumulative distributions of parameters $Delta B \/ 〈 B 〉$, $Delta B_max \/ 〈 B 〉$ and $Delta B \/ 〈 B 〉 Delta theta$ for subsets of the current sheets (CSs) observed at different plasma betas, β \< 1 and β \> 3. The bottom panels also present the cumulative distributions corresponding to all the CSs in our data set. Note that parameter $Delta B \/ 〈 B 〉 Delta theta$ quantifies the ratio between average perpendicular and parallel current densities within CS. ]), kind: "quarto-float-fig", supplement: "Figure", ) + -#strong[Results:] Our analysis reveals that solar wind current sheets maintain kinetic-scale thicknesses throughout the inner heliosphere, with occurrence rates decreasing approximately as $1 \/ r$ with radial distance between 1 and 5 AU. When normalized to the local ion inertial length and Alfvén current, both the current density and thickness of these structures remain nearly constant over the range from 0.1 to 5 AU (see #ref(, supplement: [Figure])). This suggests that current sheets consistently influence energetic particle transport across heliocentric distances, with their higher occurrence rates closer to the Sun indicating a more pronounced role in shaping particle dynamics in the inner heliosphere. Furthermore, by leveraging simultaneous observations from spacecraft at different radial distances, we demonstrate that the observed radial trends reflect genuine spatial evolution rather than temporal or solar-cycle effects. In particular, we propose that the observed reduction in current sheet occurrence rate at larger heliocentric distances is partly attributable to a geometric effect---namely, the decreasing probability that a spacecraft intersects inclined structures as distance from the Sun increases. This represents an observational bias that must be accounted for when interpreting occurrence statistics. However, even after correcting for this geometric effect, a modest residual decrease remains, which we attribute to possible physical dissipation or annihilation of current sheets as they propagate outward through the solar wind. +=== Field Rotation Angle + +The field rotation angle $omega$ (also referred to as the spread angle, shear angle, or directional change) is one of the most fundamental parameters characterizing current sheets. Its distribution depends on the statistical ensemble (i.e., the identification criteria), solar activity level, radial distance from the Sun, and the type of discontinuity. A robust finding across all studies is that discontinuities become more abundant at smaller spread angles. -Together, these results provide critical empirical constraints for particle transport modeling and establish a robust observational foundation for the theoretical and numerical components of this thesis. This work is presented in #emph["Solar wind discontinuities in the outer heliosphere: Spatial distribution between 1 and 5 AU"] (Zhang et al., submitted to JGR Space Physics, 2025, manuscript is available at #link("https://www.authorea.com/users/814634/articles/1283652-solar-wind-discontinuities-in-the-outer-heliosphere-spatial-distribution-between-1-and-5-au")[10.22541/essoar.174431869.93012071/v1];). +#cite(, form: "prose") first showed that the number of discontinuities falls off rapidly with increasing $omega$. Subsequent studies confirmed this behavior and characterized the distribution quantitatively \[#cite(, form: "prose")\; #cite(, form: "prose") #cite(, form: "prose")\;\]. For $omega gt.eq 30 degree$, the distribution is well described by $N \( omega \) prop exp [- (omega / omega_s)^2]$, where the scale parameter $omega_s$ encodes the characteristic width of the distribution. #cite(, form: "prose") found $omega_s = 75 degree$ during a solar minimum period (December 1965--January 1966), while #cite(, form: "prose") obtained a significantly smaller $omega_s = 44 degree$ during a period of higher solar activity. This difference likely reflects a dependence on solar cycle phase @marianiVariationsOccurrenceRate1973. #cite(, form: "prose"), using our coordinated spacecraft (Cluster) for four different periods between 2001 and 2003, further demonstrated that $omega$ depends on solar wind type: the spread angle tends to be smaller in slow solar wind from active regions and larger in fast solar wind originating from coronal holes. -#figure([ -#box(image("../2025_atc/figures/fig_wt_dist_no_duplicates.pdf")) -], caption: figure.caption( -position: bottom, -[ -Waiting time probability density functions $p (tau)$ for Juno at 1 AU in 2011 (top) and 5 AU in 2016 (bottom). Observed data (black) are fitted with Weibull (blue) and exponential (orange) distributions. Vertical dashed lines denote the mean waiting times for each fitted distribution. -]), -kind: "quarto-float-fig", -supplement: "Figure", -) +Extending the analysis to small rotation angles, #cite(, form: "prose") showed that the small-spread-angle population forms a smooth continuation of the larger-angle distribution, which is best fit by a #emph[lognormal] function. For most discontinuities, the maximum spread angle within the layer is nearly equal to the net edge-to-edge value, confirming that the field rotation is approximately monotonic across the structure. This finding was subsequently cited by #cite(, form: "prose") as evidence that the method of #cite(, form: "prose") captures a previously unexamined but physically continuous population. +A radial dependence of $omega_s$ was established by #cite(, form: "prose"), who found that the distribution steepens with increasing heliocentric distance: $omega_s$ decreases from \~82° to \~50° between the inner heliosphere and several AU (see #ref(, supplement: [Figure])). Fewer events with $omega > 60 degree$ are observed at larger distances, indicating that discontinuities evolve during their outward propagation. Whether this evolution leads to eventual annihilation of current sheets remains unclear. Notably, the radial dependence of $omega_s$ differs depending on the identification criterion: it is present for the Tsurutani--Smith (TS) criterion but absent for the Burlaga (B) criterion inside 2.3 AU, suggesting that the evolution is driven by the additionally identified population of anisotropic RDs. For TDs alone, the mean rotation angle $chevron.l omega chevron.r approx 78 degree$ shows no radial dependence, whereas for RDs, smaller $omega$ is more probable and fewer large-$omega$ events survive at greater distances. #figure([ -#box(image("figures/juno_distribution_r_sw.png")) +#box(image("figures/ref/sodingRadialLatitudinalDependencies2001-fig11.png")) ], caption: figure.caption( position: bottom, [ -Distribution of various SWD properties observed by Juno, grouped by radial distance from the Sun (with colors shown at the top). Panel (a) thickness, (b) current density, (c) normalized thickness, (d) normalized current density. +Relative frequency of $omega$ for Helios 2 (top) and Voyager 2 (bottom) as a histogram; thin solid line is a fit to the distribution proportional to $exp [- (omega / omega_s)^2]$ ]), kind: "quarto-float-fig", supplement: "Figure", ) - + + + +=== Occurrence rate + +The occurrence rate of current sheets is important from both plasma-physics and particle-transport perspectives. From the standpoint of solar wind physics, occurrence statistics constrain the generation mechanisms of current sheets---including their relation to turbulence intermittency, flux-tube boundaries, and large-scale solar wind structuring---and provide information about their stability and evolution during outward propagation. From the standpoint of energetic particle transport, the frequency with which particles encounter current sheets determines the cumulative scattering rate and therefore influences large-scale diffusion properties in both momentum and configuration space. + +A central question is whether current sheets are formed close to the Sun and subsequently convected outward by the solar wind, or whether they are generated in situ at all heliocentric distances, for example in colliding solar wind streams. As summarized by #cite(, form: "prose"), early radial surveys spanning approximately 0.3 to 10 AU consistently reported a decrease in occurrence rate with increasing radial distance. However, interpreting this trend is not straightforward. The observed decrease may indicate genuine disintegration of current sheets during their outward propagation. Alternatively, it could reflect a changing balance between local generation and destruction processes. It may also arise from observational effects: as structures evolve, their orientation relative to the radial direction may change, reducing the locally detected occurrence rate. In addition, current sheets may thicken with increasing distance, causing them to fall below instrumental detection thresholds and thereby introducing an observational bias @leppingMagneticFieldDirectional1986. + +The earliest radial studies established the basic phenomenology. #cite(, form: "prose"), using Pioneer 6 data, found that the occurrence rate at 0.82 AU is only slightly lower than at 1 AU ("this may be due to the lower data quality and increase in the number of data gaps when the spacecraft is far from the earth"), and that the distributions of rotation angle and discontinuity normals are very similar across the range 0.8--1.0 AU. This suggested that most discontinuities originate inside 0.8 AU and do not evolve appreciably over this distance range. Importantly, the occurrence rate in regions of increasing bulk speed was only slightly higher than elsewhere, arguing against stream collision as the primary generation mechanism. #cite(, form: "prose"), analyzing over 16,000 events from Pioneer 8, reported an average occurrence rate of approximately 3.6 per hour near 1 AU (with \~1.6 per hour identified as TD-like) and found a correlation with the directional change $omega$ and a decrease with increasing heliocentric distance. However, they also noted a possible dependence on heliographic latitude. + +#cite(, form: "prose") made a major contribution by using simultaneous Pioneer 10 and 11 data to separate spatial from temporal variations. This distinction was essential, as occurrence rates display substantial day-to-day and solar-rotation-scale fluctuations well outside #emph[Poisson] expectations. They found that the rates averaged over Bartels rotations were strongly correlated between the two spacecraft despite their \~2 AU separation, and that the statistical properties of discontinuities at 1 and 5 AU were remarkably similar. Both findings support a scenario in which discontinuities originate within 1 AU and are subsequently convected outward by the solar wind. The radial dependence of the occurrence rate follows $rho = 50 thin e^(- \( R - 1 \) \/ 4)$ per day, corresponding to an apparent decrease of about 25% per AU. However, they argued that this radial gradient may not represent true physical decay: it could arise from progressive thickening of current sheets such that they no longer satisfy identification criteria. Finally, they demonstrated that temporal variations, persisting over several months, had likely been misinterpreted as latitudinal gradients in earlier Pioneer 8 results @marianiVariationsOccurrenceRate1973. + +The picture was enriched by Ulysses observations at high heliographic latitudes. #cite(, form: "prose") found a radial decrease from 1 to 5 AU following $e^(- \( r - 1 \) \/ 5)$. More strikingly, the occurrence rate increased by a factor of \~5 as Ulysses moved from Jupiter at 5 AU to 2.5 AU over the south pole (from the ecliptic plane to −80° heliographic latitudes), with a one-to-one correspondence between high occurrence rates and high-speed streams from coronal holes. In these streams, nonlinear outward-propagating Alfvén waves with large transverse fluctuations are ubiquitous, and rotational discontinuities frequently form the edges of phase-steepened Alfvén waves---offering a natural explanation for the elevated occurrence rates. +#cite(, form: "prose") synthesized data from five missions spanning 0.3--19 AU and $- 80 degree$ to $+ 10 degree$ latitude during solar minimum. They found that the occurrence rate depends linearly on solar wind velocity (a geometric effect: faster wind sweeps more plasma volume past the spacecraft per unit time) and decreases radially as $r^(- 0.78)$ (TS criterion) or $r^(- 1.28)$ (B criterion). After normalizing to 400 km/s and 1 AU, approximately 64 discontinuities per day were identified with both criteria, and no residual dependence on heliographic latitude or solar wind structure type was observed---indicating that current sheets are uniformly distributed on spherical shells. Nonetheless, large day-to-day variations persisted even after normalization. The RD-to-TD ratio depended on solar wind structure, with relatively more RDs in high-speed streams, but showed no radial or latitudinal dependence in the inner heliosphere ($r < 10$ AU). + +#cite(, form: "prose"), using their method sensitive to small-spread-angle events, found dramatically higher rates than earlier surveys: an average exceeding 243 per day for all discontinuities, 117 per day above $15 degree$, and 52 per day above $30 degree$ (comparable to the \~30 per day above $30 degree$ reported by classical methods). These rates exhibit pronounced temporal variability on both daily and hourly timescales, and discontinuities occur in spatial groupings with #emph[lognormally] distributed separations. Even excluding active periods (interplanetary shocks, solar ejections), the rates were only weakly correlated with solar wind speed. + +A methodological subtlety that pervades all occurrence rate studies was highlighted by #cite(, form: "prose"), who used the extensive Ulysses magnetometer dataset to critically examine the role of the identification method. They showed in #ref(, supplement: [Figure]) that occurrence rates differ dramatically depending on whether events are selected by their temporal rate of change (in the spacecraft frame) or by their spatial gradient (transformed into the solar wind frame): the temporal criterion systematically overestimates the number of discontinuities in fast solar wind, because structures convect more rapidly past the observer. After correcting for this bias, they confirmed the radial decrease in spatial density with increasing distance from the Sun. And surprisingly, they found that at a given radial distance, periods with slower solar wind tended to contain more discontinuities. #figure([ -#box(image("figures/fig_juno_sw_comparision.png")) +#box(image("figures/ref/erdosDensityDiscontinuitiesHeliosphere2008-fig4.png")) ], caption: figure.caption( position: bottom, [ -Comparison of solar wind properties (top) and discontinuity properties (bottom) between / using model (x-axis) and JADE observations (y-axis). (a-d) Solar wind velocity, density, temperature, and plasma beta. (e-h) Discontinuity thickness, current density, normalized thickness, and normalized current density. Blue dots indicate values derived using the cross-product normal method, while yellow dots correspond to values obtained using minimum variance analysis. +The number of discontinuities as a function of the distance from the Sun (horizontal scale) and the velocity of solar wind (color coded). Upper panel: selection of events by time rate of change. Lower panel: selection of events by spatial gradients. ]), kind: "quarto-float-fig", supplement: "Figure", ) - + -== Quantitative Modeling of Particle Scattering - -#strong[Context:] While it is well established that turbulence governs energetic particle transport in the heliosphere, the specific role of coherent structures---particularly current sheets---in shaping scattering processes remained under-explored. A central objective of this thesis is to develop a physics-based, observation-informed model that directly links solar wind current sheet properties to pitch-angle scattering rates of energetic particles. +Recent inner-heliosphere measurements from Parker Solar Probe and Solar Orbiter have extended these statistics closer to the Sun than previously possible. #cite(, form: "prose") analyzed 3,948 discontinuities between 0.13 and 0.9 AU and found a steep radial decrease following $r^(- 2.00)$. A particularly interesting finding was that the RD occurrence rate decreases more steeply ($r^(- 2.17)$) than the TD rate, so that the RD-to-TD ratio drops sharply from \~8 at $r < 0.3$ AU to \~1 at $r > 0.4$ AU, exhibiting distinct evolution with distance. -#strong[Approach:] To this end, we combined statistical measurements of current sheets at 1 AU with a Hamiltonian analytical framework and test particle simulations to investigate how particle scattering efficiency varies with current sheet geometry and particle energy, using a realistic magnetic field configuration: +#cite(, form: "prose"), combining Solar Orbiter and Parker Solar Probe data, identified over 140,000 discontinuities between 0.06 and 1.01 AU and confirmed the power-law decrease in spatial density, fitting an exponent of $- 0.93$---somewhat shallower than the $r^(- 2.00)$ of #cite(, form: "prose"), likely reflecting differences in identification criteria and the correction for solar wind velocity effects. They identified several competing mechanisms that shape the radial profile: the increasing Parker spiral angle with distance affects how many TD-like flux-tube boundaries are swept past the spacecraft; the smaller cross-section of flux tubes near the Sun makes boundary crossings more probable; and any radial evolution of current sheet thickness introduces selection biases for gradient-based detection methods. + +Taken together, these studies establish that current sheet occurrence rates decrease with heliocentric distance, but the precise radial scaling depends sensitively on identification criteria and the population of discontinuity types sampled with corrections for solar wind velocity effects. The much higher rates found by methods sensitive to small rotation angles @vasquezNumerousSmallMagnetic2007 underscore that the total population of current sheets is substantially larger than suggested by classical surveys restricted to large-angle events. The differential radial evolution of RDs and TDs points to fundamentally different origins and lifetimes for these two populations---a distinction with direct implications for understanding how they are generated and sustained by solar wind turbulence. + +=== Spatial Scale and Current Density + +In the classical MHD framework, current sheets are treated as infinitely thin discontinuities---mathematical step functions across which plasma parameters change instantaneously. In reality, however, the transition between upstream and downstream regions occurs over a finite thickness, requiring a treatment beyond the MHD approximation. The impact of a current sheet on the plasma --- and specifically on the dynamical process of particles within the sheet @yamadaMagneticReconnection2010 --- is fundamentally governed by this spatial scale @sergeevCurrentSheetThickness1990 and the associated internal current density (the ratio between gyroradius and the characteristic scale of magnetic inhomogeneity). These two closely coupled parameters (usually compared to local proton inertial length and Alfvén current density) dictate the transition from fluid-like MHD behavior to kinetic physics and are therefore central to understanding the role of current sheets in both turbulence dissipation and particle transport. -$ upright(bold(B)) = B_0 (cos theta med upright(bold(e_z)) + sin theta (sin phi (z) med upright(bold(e_x)) + cos phi (z) med upright(bold(e_y)))) $ +The thickness of a current sheet determines the physical regime in which the structure operates. The critical threshold occurs when the thickness approaches fundamental kinetic length scales: the ion inertial length $d_i = c \/ omega_(p i)$ or the thermal ion gyroradius $rho_i$. When $lambda tilde.op rho_i$, the assumptions of ideal MHD break down: ions become demagnetized within the sheet while electrons---with their much smaller gyroradius---remain magnetized. Test particle simulations reveal the kinetic consequences of this intermittent structure: #cite(, form: "prose") found that the current sheets spontaneously formed by MHD turbulence produce differential energization, with electrons developing large parallel velocities within current sheets while protons are energized preferentially in the perpendicular direction by nonuniform electric fields varying on proton kinetic scales. This differential response generates Hall electric fields and enables the onset of collisionless magnetic reconnection, which requires current sheet thicknesses comparable to $d_i$ to proceed at sufficiently fast rates. #cite(, form: "prose") showed that as a Sweet--Parker dissipation region dynamically thins during reconnection, a critical transition occurs when its width drops below $d_i$: the Hall effect becomes dominant, the resistive MHD solution ceases to exist, and the system transitions abruptly to fast collisionless reconnection with rates orders of magnitude higher. #cite(, form: "prose") extended this picture by investigating the tearing instability in both the MHD and Hall-MHD regimes. They showed that when a current sheet achieves a sufficiently small aspect ratio ($a \/ L tilde.op S^(- 1 \/ 3)$ for Lundquist number $S gt.double 1$), reconnection proceeds on ideal Alfvén timescales independent of $S$. In the nonlinear phase, secondary current sheets spontaneously form and, at high $S$, naturally adjust to this critical aspect ratio, driving very rapid reconnection. When the Hall term is included---appropriate once the resistive layer width $delta$ becomes comparable to $d_i$---the secondary reconnection rate is enhanced by up to a factor of two relative to the pure MHD case and up to ten times faster than the linear phase, leading to explosive energy release on super-Alfvénic timescales. -where $B_0$ is the magnitude of the magnetic field, $theta$ is the azimuthal angle between the normal and the magnetic field, and $phi (z)$ is the rotation profile of the magnetic field as a function of $z$. +The interplay between reconnection and the turbulent cascade is now recognized as fundamental @boldyrevTearingInstabilityAlfven2020@boldyrevRoleReconnectionInertial2019. In dynamically aligned Alfvénic turbulence, magnetic fluctuations naturally form progressively thinner sheet-like structures at smaller scales @boldyrevSpectrumMagnetohydrodynamicTurbulence2006. Analytic theories predict that below a critical thickness these sheets become tearing-unstable, disrupting the classical cascade. #cite(, form: "prose") calculated the disruption scale $lambda_D$ at which this onset occurs in a low-$beta$ collisionless plasma, showing that $lambda_D$ can exceed the ion sound scale $rho_s$ and produce a spectral break at $lambda_D$ rather than at $rho_s$, with a steepened "transition range" between them---a feature sometimes observed in solar wind turbulence intervals. #cite(, form: "prose") proposed a complementary picture in which the tearing instability modifies the effective alignment of field lines, balancing the eddy turnover rate at all scales below the critical threshold and yielding a reconnection-mediated energy spectrum steeper than the classical prediction. These theoretical expectations have been confirmed numerically. #cite(, form: "prose"), in high-resolution 2D MHD simulations at magnetic Reynolds number $R_m = 10^6$, showed that the combined effects of dynamic alignment and turbulent intermittency produce copious plasmoid formation in intense current sheets; the resulting disruption steepens the energy spectrum toward a spectral index near $- 2.2$, consistent with the analytic predictions. #cite(, form: "prose") extended this to three dimensions, demonstrating that rapid reconnection breaks elongated current sheets into chains of plasmoids and opens a previously unrecognized range of energy cascade in which the transfer rate is controlled by plasmoid growth, again producing a $- 2.2$ spectral index accompanied by modified turbulence anisotropy. #cite(, form: "prose"), using high-resolution hybrid-kinetic simulations that retain ion kinetic effects, provided further confirmation: reconnection at current sheets with $a tilde.eq d_i$ actively drives the sub-ion-scale cascade, generating a stable power-law spectrum below the ion break as soon as the first reconnection events occur---regardless of the state of the large-scale cascade. Taken together, these results establish that current sheets at kinetic scales actively shape the turbulence spectrum, mediate the energy cascade across the ion break, and control the pathway by which magnetic energy is ultimately converted into particle heating. -#strong[Results:] Using a newly formulated Hamiltonian framework (see dimensionless form in #ref(, supplement: [Equation]), below) that incorporates the effects of magnetic field shear angle $beta$ and particle energy $H$, we demonstrate that scattering rates depend strongly on the current density---which is directly tied to $beta$---as well as on the ratio of the particle gyroradius to the current sheet thickness. Notably, our results show that current sheets can induce rapid, non-diffusive pitch-angle jumps, particularly for SEPs in the 100 keV to 1 MeV energy ranges (see #ref(, supplement: [Figure])). This behavior deviates significantly from classical quasilinear predictions and highlights the need to account for coherent structures in transport models. To describe long-term pitch-angle evolution, we developed a simplified probabilistic model of pitch-angle scattering due to current sheets and derived an effective pitch-angle diffusion coefficient $D_(mu mu)$ (see #ref(, supplement: [Figure])). +For energetic particle transport, the spatial scale is equally decisive. When particles encounter a broad MHD-scale structure ($lambda gt.double rho_(upright(S E P))$), their motion remains adiabatic and they smoothly follow guiding-center trajectories. When $lambda tilde.op rho_(upright(S E P))$, however, the magnetic field changes too abruptly for adiabaticity to be maintained, leading to strong pitch-angle scattering, temporary trapping, or reflection. The implications of this resonance condition for SEP transport will be examined in detail in the following sections. -These diffusion rate estimates enable direct comparison with other scattering mechanisms, facilitate the incorporation of SWD-induced scattering into global SEP transport models, and directly support the broader goal of this thesis to improve our understanding of how energetic particles interact with turbulence in the solar wind. This work is presented in "Quantification of Ion Scattering by Solar Wind Current Sheets: Pitch-Angle Diffusion Rates" (Zhang et al., submitted to Physical Review E, 2025, manuscript is available at #link("https://github.com/Beforerr/ion_scattering_by_SWD/blob/ec33d3d082bcd463faf7a233ba80138414231b51/files/2024PRE_Scattering_Zijin.pdf")[GitHub];). +Current density is inextricably linked to spatial scale through Ampère's law: $upright(bold(J)) = nabla times upright(bold(B)) \/ mu_0$. A thin magnetic field rotation necessarily implies an intense current layer. In the context of magnetic turbulence, the energy cascading from large to small scales is not dissipated uniformly but is concentrated within coherent structures---predominantly thin, high-current-density sheets. Numerical simulations have quantified this intermittency in detail. #cite(, form: "prose"), analyzing current sheets in driven reduced-MHD turbulence, found that structures with peak current density exceeding eight times the rms value occupy less than 1% of the simulation volume yet account for more than 25% of all Ohmic dissipation. They also showed that while not all intense current sheets contain magnetic X-points (about 55% do not), the most intense structures are preferentially reconnecting ones, with the probability of containing an X-point rising to \~90% for the strongest events. -#math.equation(block: true, numbering: "(1)", [ $ tilde(H) & = 1 / 2 ((tilde(p_x) - f_1 (z))^2 + (tilde(x) cot theta + f_2 (z))^2 + tilde(p_z)^2)\ -f_1 (z) & = 1 / 2 cos beta med (upright("Ci") (beta s_(+) (z)) - upright("Ci") (beta s_(-) (z))) + 1 / 2 sin beta med (upright("Si") (beta s_(+) (z)) - upright("Si") (beta s_(-) (z))) \,\ -f_2 (z) & = 1 / 2 sin beta med (upright("Ci") (beta s_(+) (z)) + upright("Ci") (beta s_(-) (z))) - 1 / 2 cos beta med (upright("Si") (beta s_(+) (z)) + upright("Si") (beta s_(-) (z))) $ ]) +Spacecraft observations corroborate this picture. Current sheets are correlated with enhanced electron and ion temperatures @osmanEvidenceInhomogeneousHeating2011, and these structures, while constituting only \~19% of the data, can account for \~50% of the total plasma internal energy @osmanIntermittencyLocalHeating2012. If reconnection is triggered within an intense current sheet, the contracting magnetic islands can further accelerate particles to high energies. Recent Parker Solar Probe observations have provided direct evidence of proton acceleration up to \~400 keV within the reconnection exhaust of the heliospheric current sheet at \~16 $R_dot.circle$ @desaiMagneticReconnectionDriven2025. + +Understanding how thickness and current density of current sheets evolve with heliocentric distance is therefore crucial for revealing their role in the thermodynamics of solar wind turbulence and dynamics of energetic paticles: whether these structures maintain their kinetic-scale character and whether their current density weakens in tandem with the radial drop in magnetic field constrains theories of their local generation, evolution, and overall contribution to energy dissipation and particle transport throughout the heliosphere. + +==== Thickness + +The earliest thickness estimates, necessarily limited by instrumental resolution, revealed structures with spatial scales of thousands of kilometers or tens of proton inertial lengths @burlagaDirectionalDiscontinuitiesInterplanetary1969. + +The radial evolution of current sheet thickness was first systematically studied by #cite(, form: "prose"), who analyzed discontinuities from five missions spanning 0.3--19 AU. They found that the mean thickness in kilometers increases with heliocentric distance, as expected from the radial decrease in magnetic field strength and the associated expansion of kinetic length scales. However, when normalized to the local proton gyroradius, the thickness decreases dramatically---by a factor of \~50, from $chevron.l d_(rho_g) chevron.r approx 201 thin rho_g$ at 0.3 AU down to $chevron.l d_(rho_g) chevron.r approx 4.3 thin rho_g$ at 19 AU. A similar decrease, from $127 thin d_i$ to $2.6 thin d_i$, was found when normalizing to the ion inertial length. RDs were consistently thicker than TDs by a factor of \~1.5. This strong radial thinning in normalized units indicates that current sheets do not simply expand passively with the solar wind but evolve dynamically, progressively approaching kinetic scales at larger distances. + +Parker Solar Probe has extended these measurements into the pristine inner heliosphere. #cite(, form: "prose"), analyzing discontinuities between 0.13 and 0.9 AU, found distinct behavior for the two types: TD thicknesses normalized by $d_i$ show no clear spatial scaling and range broadly from 5--35 $d_i$, whereas RD thicknesses decrease as $r^(- 1.09)$ in normalized units. In absolute terms, the average RD thickness of \~574 km changes little with distance, implying that the normalized thinning reflects the radial increase of $d_i$ itself. #cite(, form: "prose"), using combined Solar Orbiter and PSP data from 0.06 to 1.01 AU, reported a more nuanced picture: RD thickness first #emph[decreases] between 0.06 and 0.30 AU, then increases beyond 0.30 AU in proportion to the local ion inertial length. TD thickness, by contrast, scales with $d_i$ throughout the entire distance range. The authors interpreted these trends as evidence for different physical origins of the two types of discontinuities. RDs are thought to arise from the nonlinear steepening of Alfvén waves, a process that tends to generate structures with a significant magnetic-field component normal to the discontinuity surface. As the steepening progresses, the characteristic thickness decreases until it approaches ion kinetic scales, where dispersive and kinetic effects limit further steepening. TDs, on the other hand, are more likely associated with boundaries between magnetic flux tubes, whose widths tracks the local kinetic scale. + +High-cadence measurements have made it possible to resolve the kinetic-scale population directly. #cite(, form: "prose"), using Wind data at 11 samples/s, characterized 17,043 current sheets at 1 AU with thicknesses from a few tens to \~1,000 km, corresponding to \$$0.1 dash.en 10$,\_p\$ with typical values around 100 km (\~a few $lambda_p$). Near the Sun, #cite(, form: "prose") analyzed 11,200 current sheets around PSP's first perihelion, finding thicknesses from a few to \~200 km (typical value \~30 km), or \$\$0.1--10 $lambda_p$ with a typical value of \~2 $lambda_p$. At 5 AU, #cite(, form: "prose") found half-thicknesses of 200--2,000 km for non-bifurcated current sheets and 500--5,000 km for bifurcated ones, corresponding to 0.5--5 $lambda_p$ and 0.7--15 $lambda_p$ respectively. Despite the enormous difference in absolute scale across these distances, the remarkable consistency in normalized thickness---typically a few proton inertial lengths---indicates that current sheets at all heliocentric distances are predominantly kinetic-scale structures whose width is set by the local plasma conditions. + +==== Current Density and Scale-Dependent Properties + +The current density within kinetic-scale current sheets is not independent of their spatial scale. #cite(, form: "prose") found at 1 AU that the current density increases systematically for thinner structures, following $J_0 approx 6 #h(0em) upright(n A \/ m^2) dot.op \( lambda \/ 100 #h(0em) upright(k m) \)^(- 0.56)$, but does not statistically exceed a critical value $J_A$ corresponding to an ion-electron drift at the local Alfvén speed. In normalized units, this becomes $J_0 \/ J_A approx 0.17 dot.op \( lambda \/ lambda_p \)^(- 0.51)$. A corresponding power-law correlation was observed near the Sun by #cite(, form: "prose"), who found $J_0 approx 0.15 #h(0em) mu upright(A \/ m^2) dot.op \( lambda \/ 100 #h(0em) upright(k m) \)^(- 0.76)$ with current densities in the range 0.1--10 $mu$A/m², and at 5 AU by #cite(, form: "prose"), who reported $J_0 \/ J_A approx 0.14 dot.op \( lambda \/ lambda_p \)^(- 0.66)$ with typical current densities of 0.05--0.5 nA/m². + +These current sheets are statistically force-free: the current density is dominated by its magnetic-field-aligned component, consistent with the observation that $\| B \|$ does not vary substantially across them. The magnetic shear angle is also correlated with spatial scale: #cite(, form: "prose") found $Delta theta approx 19 degree dot.op \( lambda \/ lambda_p \)^0.5$ at 1 AU, while #cite(, form: "prose") found $Delta theta approx 16.6 degree dot.op \( lambda \/ lambda_p \)^0.34$ at 5 AU. These scale-dependent correlations---thinner sheets carrying proportionally stronger currents with smaller shear angles---are a natural consequence of the turbulent cascade, in which magnetic field gradients steepen as energy is transferred to smaller scales. The approximate scale-invariance of these relationships across heliocentric distances (from 0.17 AU to 5 AU), together with the matching of magnetic field rotation and compressibility between current sheets and the ambient turbulence, provides strong evidence that the majority of kinetic-scale current sheets are produced by the turbulent cascade. + +The observation that current density does not exceed the Alfvén current density $J_A$ is physically significant but not yet fully understood. From the standpoint of reconnection, #cite(, form: "prose") showed that essentially all 18,785 kinetic-scale current sheets in their dataset satisfy the necessary condition for reconnection not to be suppressed by diamagnetic drift of the X-line. This condition, $Delta beta lt.tilde 2 \( L \/ lambda_p \) tan \( Delta theta \/ 2 \)$, is automatically met due to the geometry of the current sheets as dictated by the turbulent cascade, rather than being a coincidence of local plasma parameters. The same conclusion was reached near the Sun @lotekarKineticscaleCurrentSheets2022 and at 5 AU @vaskoKineticScaleCurrentSheets2024. + +=== Alfvénicity, Walén Relation and Propagation Direction + +The relationship between velocity and magnetic field variations across current sheets---their degree of Alfvénicity---has been a central and persistently debated topic, bearing directly on the classification of discontinuities as RDs or TDs and on their dynamical role in the solar wind. + +The pioneering observation by #cite(, form: "prose"), using IMP 8 and Voyager 2 data, revealed that velocity and magnetic field jumps ($Delta upright(bold(v))$ and $Delta upright(bold(B)) \/ sqrt(rho)$) across tangential discontinuities (a large change in magnetic field strength) are closely aligned---either parallel or antiparallel---in the sense associated with outward-propagating Alfvén waves. This alignment was found to be independent of solar wind stream structure and heliocentric distance between 1 and 2.2 AU. The result was unexpected for structures classified as TDs, and several explanations were proposed, including interplanetary turbulence, large-amplitude Alfvénic fluctuations propagating independently on both sides of the discontinuity, and surface waves on TDs. + +#cite(, form: "prose") confirmed using Helios data that this alignment is already well established inside 0.4 AU, suggesting it is not a product of in situ evolution but may instead reflect a selection effect: TDs for which $Delta upright(bold(v))$ and $Delta upright(bold(B))$ are not aligned are destroyed by the Kelvin--Helmholtz instability, so that only Alfvénically aligned TDs survive. The observed decrease in the total number of discontinuities with increasing heliocentric distance may be associated with the growth of this instability as the Alfvén speed declines. + +An earlier comprehensive study by #cite(, form: "prose"), using ISEE 3 data, showed that the relative directions of velocity and field changes across all three discontinuity types (RD, TD, and the intermediate "either" category, ED) are consistent with outward propagation. The magnitude of the velocity change at RDs was found to be systematically smaller than the MHD prediction --- a discrepancy only partially reduced by using a two-stream proton fit --- foreshadowing the broader $R < 1$ puzzle discussed below. Further, the plasma jump conditions at EDs showed closer resemblance to RDs than to TDs. + +Quantitative assessment of Alfvénicity relies on the Walén relation, which states that the velocity jump across an RD should equal the corresponding Alfvén velocity jump. Two complementary approaches have been developed (see the bottom panels of #ref(, supplement: [Figure])). The first evaluates the Walén relation as a jump condition by comparing velocity and Alfvén velocity changes between two carefully chosen measurement times on opposite sides of the discontinuity. The second checks the level of Alfvénicity continuously for all measurements between those two points: plasma velocity components, after transformation into the de Hoffmann--Teller (HT) frame, are plotted against the corresponding Alfvén velocity components, and the slope of the regression line, $W_(upright(s l))$, serves as the quality index, with $W_(upright(s l)) = plus.minus 1$ indicating perfect Alfvénic agreement. #figure([ -#box(image("figures/example_subset.png", width: 70.0%)) +#box(image("figures/ref/paschmannDiscontinuitiesAlfvenicFluctuations2013-fig2case1.png")) ], caption: figure.caption( position: bottom, [ -Transition matrix for 100 keV protons under four distinct magnetic field configurations: (i) $v_p = 8 v_0$, $theta = 85 degree$, $beta = 50 degree$; (ii) $v_p = 8 v_0$, $theta = 85 degree$, $beta = 75 degree$; (iii) $v_p = 8 v_0$, $theta = 60 degree$, $beta = 50 degree$; and (iv) $v_p = v_0$, $theta = 85 degree$, $beta = 50 degree$. +Overview plots for DD crossings. For each case, the five panels at the top show the magnetic field magnitude, the plasma density, followed by a comparison between the three components of $upright(bold(v))' = \( upright(bold(v)) - upright(bold(V))_(upright(H T)) \)$ (in black) and (in red) the three components of $- upright(bold(V))_A$ or $upright(bold(V))_A$ (depending on the sign of the Walén slope), all from Cluster C1, with the DD at the center of the time series. The panels along the bottom show the HT scatterplot for the 10 min interval, and the Walén scatterplots for the full 10 min and for the 1 min interval centered on the DD. In these scatterplots the vector components are distinguished by their color (black for x, red for y, and green for z). ]), kind: "quarto-float-fig", supplement: "Figure", ) - + + + +#cite(, form: "prose") reported that the magnitude ratio $R = \| Delta upright(bold(v)) \| \/ \| Delta upright(bold(v))_A \|$ from the jump approach is commonly around 0.6---systematically less than unity. #cite(, form: "prose"), using Cluster data, performed a comprehensive Walén analysis on 188 directional discontinuities and found that a substantial fraction (77 out of 127 with a good de Hoffmann--Teller frame) exhibited plasma flow speeds exceeding 80% of the Alfvén speed, with 33 cases exceeding 90%. Their analysis also established that the degree of Alfvénicity of the coherent current sheets is nearly the same as that of the fluctuations in which they are embedded, suggesting that whatever process causes deviations from ideal Alfvénicity operates equally on both. This result places current sheets on a continuum with the ambient Alfvénic turbulence rather than as dynamically distinct structures. + +A critical complication in using the Walén test for RD/TD classification was identified by #cite(, form: "prose"), who analyzed over 140,000 directional discontinuities between 0.06 and 1.01 AU using Parker Solar Probe and Solar Orbiter data. They showed that Alfvén waves propagating along the surface of TDs can produce positive Walén test results, mimicking RD signatures. To disentangle the two populations, they examined the velocity in the HT frame: for surface waves on TDs, the residual velocity $\( upright(bold(V)) - upright(bold(V))_(upright(H T)) \)$ lies close to the discontinuity plane, whereas for genuine RDs it is quasi-perpendicular to the surface. A scatter plot of $B_n \/ B_max$ against $\( upright(bold(V)) - upright(bold(V))_(upright(H T)) \) dot.op hat(n) \/ \| upright(bold(V)) - upright(bold(V))_(upright(H T)) \|$ revealed two clearly distinct populations, confirming that many apparent RD candidates are in fact TDs with surface Alfvén waves. After this reclassification, they found that most discontinuities with small normal magnetic field components are TDs, regardless of the jump in field magnitude. + +The systematic shortfall $R < 1$ has remained a longstanding puzzle. As noted by #cite(, form: "prose"), this deficiency mirrors the behavior of Alfvénic fluctuations more broadly: the Alfvén ratio $r_A = delta v^2 \/ delta v_A^2$ is known to decrease systematically with heliocentric distance, reaching approximately 0.5 at 1 AU @belcherLargeamplitudeAlfvenWaves1971@borovskyVelocityMagneticField2012. + +Refined scalar measures for evaluating Alfvénicity have been developed to disentangle directional and magnitude deviations. #cite(, form: "prose") introduced a quality index $Q$ that incorporates both the angular deviation and the magnitude ratio between $Delta upright(bold(v))$ and $Delta upright(bold(v))_A$, with $Q = plus.minus 1$ indicating perfect agreement. #cite(, form: "prose") systematically compared the jump-based index $Q$ with the regression slope $W_(upright(s l))$ across nearly 1,000 magnetopause crossings, finding that a substantially higher threshold is needed for $\| Q \|$ than for $\| W_(upright(s l)) \|$ to yield comparable numbers of RD candidates, and that the events selected by the two methods are not identical. They concluded that a complete evaluation of Alfvénicity requires two scalar quality measures: the magnitude ratio $R = \| Delta upright(bold(v)) \| \/ \| Delta upright(bold(v))_A \|$ and the angle $Theta$ between $Delta upright(bold(v))$ and $Delta upright(bold(v))_A$ @paschmannLargeScaleSurveyStructure2018. + +=== Additional Statistical Properties + +This subsection briefly summarizes several additional properties of solar wind current sheets that, while not the primary focus of this thesis, form an important part of the broader observational picture and provide useful context for interpreting the current sheet population. + +==== Orientation + +The orientation of current sheet normals relative to the local magnetic field depends on both the type of discontinuity and heliocentric distance. In the inner heliosphere, #cite(, form: "prose") analyzed Pioneer 8 data (10 s cadence) and found that TD normals are predominantly perpendicular to the local Parker Archimedean spiral field, consistent with TDs separating adjacent flux tubes. #cite(, form: "prose") corroborated this and highlighted a contrasting behavior for rotational discontinuities (RDs): inner heliospheric RD normals are primarily parallel to the mean field, $upright(bold(B))_0$ (i.e., the normal-to-field angle $gamma approx 0^compose$). This aligns with Alfvén waves propagating along the field, although obliquely propagating RDs are also present. Conversely, in the middle heliosphere (10--40 AU), the distribution of $gamma$ becomes nearly uniform across all directions, with a notable depletion of RDs propagating parallel to $upright(bold(B))_0$. To explain this radial evolution, #cite(, form: "prose") suggested that field-aligned RDs are unstable over large spatial and temporal scales so the effects of this instability are absent in the inner heliosphere (0.3--2.3 AU) but become pronounced at greater distances. + +==== Plasma Beta Dependence + +The properties of current sheets depend on the local plasma $beta$. #cite(, form: "prose"), using interplanetary coronal mass ejections as a natural laboratory spanning a broad range of beta ($10^(- 2) lt.tilde beta_e lt.tilde 10$, $10^(- 3) lt.tilde beta_p lt.tilde 10$), showed that both the shear angle $Delta theta$ and the normalized thickness $lambda \/ lambda_p$ of current sheets depend on electron and proton beta. They argued that the beta dependence of the shear angle is an intrinsic feature of solar wind turbulence arising from the natural correlation between turbulence intensity and plasma beta. According to recent theory, current sheets formed in turbulence are disrupted by the electron tearing instability once their thickness falls below a critical scale, mediating the transition from the inertial to the kinetic cascade @malletDisruptionSheetlikeStructures2017. #cite(, form: "prose") demonstrated that normalizing current sheet thickness by this critical scale eliminates the beta dependence across the entire measured range. + +==== Bifurcated Current Sheets and Reconnection Exhausts + +A distinctive subclass of solar wind current sheets exhibits bifurcated structure---a double-step magnetic field rotation rather than as a single monotonic rotation @goslingMagneticReconnectionSolar2012@neugebauerProgressStudyInterplanetary2010. As reviewed by #cite(, form: "prose"), these bifurcated current sheets are the observational signature of magnetic reconnection in the solar wind. When reconnection occurs, the reconnecting current sheet splits into a pair of back-to-back rotational discontinuities that bound a wedge of accelerated plasma---the reconnection exhaust. The exhaust plasma flows at roughly the local Alfvén speed, with correlated changes in $upright(bold(V))$ and $upright(bold(B))$ at one boundary and anti-correlated changes at the other, reflecting the oppositely propagating Alfvénic disturbances generated by the reconnection process. + +The observational picture of reconnection in the solar wind has developed rapidly since the first unambiguous identification of reconnection exhausts by #cite(, form: "prose"). Using Wind 3-second data, #cite(, form: "prose") reported typical occurrence rates of 40--80 exhausts per month at 1 AU near solar minimum, with most events having temporal widths of tens of seconds (local widths of order $10^4$ km). Reconnection occurs most frequently at current sheets with field shear angles below $90 degree$---simply because such current sheets are the dominant type in the solar wind---and has been observed at shear angles as small as $11 degree$ @goslingBifurcatedCurrentSheets2008. The narrowest exhaust identified had a local width of $tilde.op 10^3$ km ($tilde.op 18 thin lambda_i$), and current sheets thinner than $tilde.op 3 thin lambda_i$ were absent from the dataset, suggesting that such ultrathin structures are quickly disrupted by reconnection. + +Multi-spacecraft observations have revealed that reconnection in large-scale current sheets typically occurs in a quasi-stationary fashion at a single dominant, extended X-line. The most extensive event documented involved five spacecraft and demonstrated continuous reconnection persisting for over 5 hours along an X-line extending at least $4.26 times 10^6$ km @goslingMagneticReconnectionSolar2012. Exhaust boundaries are roughly planar on large scales, though finer-scale corrugations are sometimes observed. The occurrence of reconnection depends on a combination of magnetic shear angle and the plasma $beta$ difference across the current sheet: for low $beta$, reconnection occurs at essentially all shear angles, whereas for high $beta$ it is restricted to large shear angles---consistent with the theoretical prediction that diamagnetic drift of the X-line suppresses reconnection at low-shear, high-$beta$ current sheets. + +#cite(, form: "prose"), examining an interval containing 11 reconnection exhausts and 27 thin current sheets within a magnetic cloud and its trailing high-speed stream, found that at least three of the thin sheets also exhibited bifurcated structure, indicating that they had been disrupted by reconnection. The relative absence of ultrathin current sheets was interpreted as evidence that such structures are rapidly disrupted once they form. More recently, #cite(, form: "prose") showed through particle-in-cell simulations that bifurcated current sheets can also arise naturally from the collisionless equilibration of a disequilibrated current sheet, through transitions among single-particle orbit classes, without requiring active reconnection. This suggests that not all observed bifurcated structures necessarily indicate ongoing or recent reconnection, and that collisionless relaxation may be an additional pathway to bifurcation. + +It should be noted that the reconnection exhausts described above are observed at relatively large-scale current sheets resolvable with 3-second plasma cadence ($gt.tilde 10^3$ km). Whether reconnection at kinetic-scale current sheets---the focus of this thesis---produces qualitatively similar or distinct signatures remains an active area of investigation, as discussed in the context of reconnection onset conditions earlier in this chapter. + +==== Contribution to the Magnetic Fluctuation Spectrum + +Current sheets contribute significantly to the power of magnetic field fluctuations in the solar wind. #cite(, form: "prose"), analyzing 8.5 years of ACE magnetometer data, constructed an artificial time series preserving only the timing and amplitudes of strong (large-rotation-angle) discontinuities. The power spectrum of this discontinuity series follows a power law in the inertial subrange with spectral index near the Kolmogorov $- 5 \/ 3$ value, and accounts for approximately half of the total spectral power of the solar wind magnetic field over this range. This result warns that the measured spectral properties of solar wind turbulence are heavily influenced by the discrete contribution of current sheets, complicating the interpretation of spectral indices. +#cite(, form: "prose") provided complementary evidence from three years of Ulysses data in which over 28,000 current sheets were identified. They showed that during the longest current-sheet-free intervals, the magnetic field power spectra are consistently described by the Iroshnikov--Kraichnan $k^(- 3 \/ 2)$ scaling, whereas during the most current-sheet-abundant intervals, the spectra exhibit Kolmogorov $k^(- 5 \/ 3)$ scaling. This finding implies that the commonly observed Kolmogorov scaling in the solar wind may be a consequence of the ubiquitous presence of current sheets rather than an intrinsic property of the underlying turbulent cascade, and that a proper analysis of solar wind power spectra must account for the contribution of intermittent structures. + +==== Other Plasma Jump Conditions + +Beyond the magnetic field signatures discussed above, the behavior of plasma parameters across current sheets provides additional constraints on their nature and on the validity of the RD/TD classification. + +#cite(, form: "prose") conducted a comprehensive examination of plasma jump conditions across 221 discontinuities using ISEE 3 magnetic field and proton data. They found that the first and second adiabatic invariants ($T_(p perp) \/ B$ and $T_(p parallel) B^2 \/ n^2$) are approximately conserved across RDs but not across TDs, confirming that the MVA-based classification into these two types captures a genuine physical distinction. The product of plasma density and the anisotropy factor, $rho A$, tends to be conserved across all three discontinuity types (RDs, TDs, and EDs)---a result expected for RDs but not required by MHD theory for TDs. The helium abundance $n_alpha \/ n_p$ is generally conserved across RDs but can change substantially across TDs; however, a broad tail in the $n_alpha \/ n_p$ distribution for RDs indicates that the helium abundance does change at a small fraction of them. + +The multi-species dynamics at RDs proved particularly revealing. #cite(, form: "prose") showed that the primary and secondary proton beams flow through RDs in opposite directions with oppositely directed velocity changes, while alpha particles interact much more weakly---with velocity changes clustered near zero and no preferred direction. They proposed a simple model in which the RD moves through the primary proton fluid at slightly less than the local Alfvén speed; because the alpha particles drift relative to the primary protons at approximately this same speed, they effectively co-move with the RD and do not cross it. Under these conditions, the jump conditions for alphas resemble those across a contact discontinuity, explaining how the helium abundance can change even across a genuine RD. When this multi-stream model was used to recalculate the Walén ratio $R_(V B)$, it increased from $0.59 plus.minus 0.03$ (single-stream proton moments) to $0.74 plus.minus 0.02$, and further inclusion of alpha particle anisotropy and estimated electron contributions raised it to $0.77 plus.minus 0.03$---diminishing but not eliminating the longstanding $R_(V B) < 1$ discrepancy. + +The properties of the magnetically ambiguous EDs were found to resemble those of RDs much more closely than those of TDs across essentially all parameters: adiabatic invariants, density conservation, helium abundance, Walén ratio, and mean solar wind speed. This led #cite(, form: "prose") to conclude that the ED population consists predominantly of obliquely propagating RDs with small but finite $B_n$, rather than TDs with coincidentally small $Delta \| B \|$. + +== Summary and Open Questions + +The body of work reviewed in this chapter establishes a rich but incomplete picture of solar wind current sheets. Their basic phenomenology is well characterized: the magnetic field rotates through a shear angle with only weak magnitude variation; the thickness of kinetic-scale current sheets is typically a few proton inertial lengths regardless of heliocentric distance; current density scales inversely with thickness in a manner consistent with the turbulent cascade; and occurrence rates decrease with radial distance from the Sun, though the precise scaling depends sensitively on identification criteria and corrections for solar wind speed and current sheet orientation. The majority of these structures are statistically force-free, satisfy the necessary conditions for magnetic reconnection onset, and share the Alfvénic character of the ambient turbulence in which they are embedded. + +Despite this progress, several important questions remain open and motivate the investigations presented in subsequent chapters of this dissertation. + +#strong[Unified method across scales and distances.] Previous studies have typically examined either large-scale discontinuities or kinetic-scale current sheets, often with different identification methods, instrumental cadences, and spacecraft datasets. As a result, a coherent picture of how the current sheet population evolves across spatial scales and heliocentric distances has not yet emerged. Developing a detection framework capable of capturing current sheets over a broad range of scales --- from the kinetic regime to the MHD regime --- and applying it consistently across multiple radial distances would therefore be highly valuable. + +#strong[Separation of temporal variability from radial evolution.] A persistent difficulty in interpreting radial trends is the contamination by temporal variability. Occurrence rates fluctuate substantially on timescales from hours to solar rotations, and single-spacecraft surveys at different heliocentric distances inevitably sample different solar wind conditions at different times. Multi-spacecraft observations and extended orbital coverage offer opportunities to separate genuine spatial evolution from temporal modulation, but this separation has rarely been achieved systematically for the kinetic-scale population. + +#strong[Multi-scale nature, distinct sub-populations, and their origins.] Current sheets span a vast range of spatial scales, and it is plausible that structures at different scales originate from different physical mechanisms --- for example, coronal flux-tube boundaries, nonlinear Alfvén wave steepening, or turbulence-driven intermittency. Furthermore, these distinct populations may follow different evolutionary paths as they propagate outward with the solar wind. Observational evidence already hints at such differentiation: the RD-to-TD ratio declines steeply inside 0.5 AU, pointing to fundamentally different generation mechanisms and lifetimes for the two classes. Yet at kinetic scales, the classical TD/RD distinction becomes blurred, as individual structures frequently exhibit signatures of both types simultaneously. Identifying physically meaningful sub-populations within the current sheet ensemble, characterizing their properties, and tracking their evolution with heliocentric distance remains an important challenge. + +#strong[Interaction with energetic particles.] The spatial scales of kinetic-scale current sheets --- comparable to the gyroradii of energetic particles in the keV to MeV range --- place them in a regime where adiabatic particle motion breaks down. The cumulative effect of encounters with such structures on pitch-angle scattering, cross-field transport, and particle energization is not yet quantitatively understood from an observational standpoint. Establishing the statistical framework of current sheet properties is a necessary prerequisite for addressing these transport questions, which will be taken up in later chapters. + +These questions collectively define the motivation for the analyses presented in the following chapters, where we develop and apply a consistent methodology for identifying and characterizing current sheets across heliocentric distances, and examine their implications for both solar wind turbulence and energetic particle dynamics. + += Energetic Particle Interaction With Solar Wind Current Sheets + +The transport of energetic particles through the heliosphere is governed not only by the large-scale structure of the interplanetary magnetic field, but also by the small-scale, intermittent structures embedded within it @ewartCosmicrayTransportInhomogeneous2025@engelbrechtTheoryCosmicRay2022@oughtonSolarWindTurbulence2021@vandenbergPrimerFocusedSolar2020. Chief among these are current sheets---thin layers of intense current and rapid magnetic field rotation that occupy a small fraction of the heliospheric volume yet exert a disproportionate influence on particle dynamics. This chapter reviews the theoretical and observational foundations necessary to understand how energetic particles interact with these structures, with particular focus on the mechanisms by which current sheets scatter particles in pitch angle and modulate their transport through the heliosphere. + +We begin with the heliospheric context: the sources and classification of solar energetic particles, and the observational phenomena---reservoir effects and dropout events---that reveal the dual role of current sheets as both barriers and facilitators of transport. We then describe the standard transport frameworks within which pitch-angle scattering is parameterized, and discuss how solar wind turbulence intermittency motivates going beyond classical wave-based scattering theories. The core of the chapter develops the quasi-adiabatic theory of charged particle dynamics near magnetic field reversals, tracing how the adiabatic invariant breaks down at separatrix crossings through geometrical and dynamical jumps---a scattering mechanism qualitatively distinct from, and potentially more efficient than, wave--particle resonance @artemyevElectronPitchangleDiffusion2013. The chapter closes with a summary of the key results and a discussion of the open questions that motivate the quantitative program developed in subsequent chapters. + +== Energetic Particles in the Heliosphere + +=== Sources of Energetic Particles + +The heliosphere hosts several distinct populations of energetic particles spanning a vast range of energies and origins @simnettEnergeticParticlesHeliosphere2017. #emph[Galactic cosmic rays] (GCRs) are high-energy particles originating from outside the solar system---primarily accelerated by supernova shocks---that propagate inward through the heliosphere and are modulated by the solar wind and heliospheric magnetic field @prantzosOriginCompositionGalactic2012@moraalCosmicRayModulationEquations2013. #emph[Anomalous cosmic rays] (ACRs) represent a population of singly charged ions that originate as interstellar neutral atoms, become ionized upon entering the heliosphere (forming pickup ions), and are subsequently accelerated to energies of tens of MeV at the heliospheric termination shock @giacaloneAnomalousCosmicRays2022. #emph[Solar energetic particles] (SEPs), the population most relevant to this thesis, are accelerated at or near the Sun during transient eruptive events and span energies from suprathermal (a few keV) to relativistic ($tilde.op$few GeV) @reamesTwoSourcesSolar2013@desaiLargeGradualSolar2016@kleinAccelerationPropagationSolar2017@anastasiadisSolarEnergeticParticles2019. + +Unlike the quasi-steady GCR background, SEP events are episodic and highly variable. Intensities can increase by several orders of magnitude within minutes and exhibit dramatic event-to-event variations in spectral shape, heavy-ion composition, charge states, and spatial distribution @anastasiadisSolarEnergeticParticles2019. This variability reflects both the diverse acceleration mechanisms at the Sun and the complex transport processes that particles undergo as they propagate through the structured, turbulent interplanetary medium @kleinAccelerationPropagationSolar2017. + +=== Two Classes of Solar Energetic Particle Events + +SEP events are broadly classified into two categories based on their acceleration mechanism and observational characteristics @reamesTwoSourcesSolar2013@desaiLargeGradualSolar2016@vlahosSourcesSolarEnergetic2019: #figure([ -#box(image("figures/mixing_rate.png", width: 60.0%)) +#box(image("figures/ref/desaiLargeGradualSolar2016-fig3.png", alt: "The two-class picture for SEP events. @desaiLargeGradualSolar2016")) ], caption: figure.caption( position: bottom, [ -Second moment of the pitch-angle distribution, $M_2 (n)$, as a function of interaction number ($n$) for different particle energies (100 eV, 5 keV, 100 keV, 1 MeV). The estimated mixing rates, $D_(mu mu)$, are indicated in the legend. +The two-class picture for SEP events. #cite(, form: "prose") ]), kind: "quarto-float-fig", supplement: "Figure", ) - -== Multifluid Model for Current Sheet Alfvénicity - -#strong[Context:] Early in this thesis, we identified a consistent radial trend in the Alfvénicity of solar wind current sheets---defined as the ratio of the plasma velocity jump to the Alfvén speed jump. While current sheets near the Sun exhibit high Alfvénicity, this value systematically decreases with increasing heliocentric distance. This raised a fundamental question: why do current sheets appear increasingly non-Alfvénic with distance, despite their force-free magnetic structure? Understanding the internal structure, stability, and evolution of current sheets is crucial, as it directly relates to their role in modulating the transport of energetic particles across the heliosphere. +#emph[Impulsive SEP events] are produced by magnetic reconnection-driven processes during solar flares. In the standard picture, reconnection in the solar corona---occurring along open magnetic field lines---accelerates particles through stochastic processes related to turbulence and fragmented current sheets in the reconnection region @vlahosSourcesSolarEnergetic2019. Impulsive events are typically short-lived (minutes to hours) and are characterized by several distinctive compositional signatures: large enhancements of $""^3 upright("He") \/^4 upright("He")$ (by factors up to $tilde.op$ 1000 relative to solar wind values), elevated heavy-ion ratios such as Fe/O, high charge states indicating source temperatures of $tilde.op 3 times 10^7$ K, and relatively high electron-to-proton ratios @reamesTwoSourcesSolar2013. Because the accelerated particles are injected onto a narrow range of magnetic flux tubes rooted in the compact flare site, impulsive events are observed only by spacecraft that are magnetically well-connected to the source---typically spanning only a few tens of degrees in solar longitude. + +#emph[Gradual SEP events] are driven by shock waves propagating through the corona and interplanetary space, typically associated with fast coronal mass ejections (CMEs) @leeShockAccelerationIons2012. The CME-driven shock provides a spatially extended acceleration region that can fill a broad range of magnetic field lines as it expands outward, beginning when the shock reaches approximately 2 solar radii @desaiLargeGradualSolar2016. Particles are accelerated through diffusive shock acceleration (DSA), in which ions are scattered back and forth across the shock front by self-generated Alfvén waves amplified by the streaming accelerated protons themselves @kleinAccelerationPropagationSolar2017. Gradual events typically last for several days, are proton-dominant, and exhibit elemental abundances similar to the solar corona and charge states corresponding to source temperatures of $tilde.op 3 times 10^6$ K. These events produce by far the highest SEP intensities observed near Earth and represent the primary source of radiation hazard to astronauts, spacecraft electronics, and passengers on high-altitude polar flight routes @anastasiadisSolarEnergeticParticles2019. + +While this two-class paradigm provides a useful organizational framework, it is important to note that the boundary between the two classes is not always sharp @kleinOriginSolarEnergetic2001. Many large gradual events show elevated $""^3$He and Fe/O ratios at high energies, suggesting that residual suprathermal ions from previous impulsive events can serve as a seed population for subsequent shock acceleration @desaiLargeGradualSolar2016. Furthermore, recent multi-spacecraft observations from widely distributed vantage points (STEREO, ACE, SOHO, and more recently Parker Solar Probe and Solar Orbiter) have revealed that SEPs can fill remarkably broad regions of the heliosphere---often extending over more than 180° in longitude---challenging simple models of localized injection and raising fundamental questions about the relative roles of extended shock geometry, coronal transport, and interplanetary cross-field diffusion in distributing particles throughout the inner heliosphere @anastasiadisSolarEnergeticParticles2019@kleinAccelerationPropagationSolar2017. + +=== Reservoir and Dropout Phenomena + +Two contrasting observational signatures of SEP transport provide particularly compelling evidence for the role of magnetic structure in controlling particle propagation: + +The #strong[reservoir effect] (or invariant spectra) is frequently observed during the decay phase of large gradual SEP events. Particle intensities measured by widely separated spacecraft (sometimes at different heliocentric distances and heliolatitudes) become nearly equal and exhibit similar temporal decay profiles, with energy spectra that maintain an invariant shape as overall intensities decrease @reamesTwoSourcesSolar2013@desaiLargeGradualSolar2016. The three-dimensional character of the reservoir, revealed by high-heliolatitude observations from the Ulysses mission @dallaPropertiesHighHeliolatitude2003, favors explanations involving substantial cross-field transport that distributes particles throughout a large volume of the inner heliosphere, rather than simple trapping behind an expanding magnetic-bottle-like structure. The velocity-dependent migration of particles through the tangled interplanetary field---potentially mediated by interactions with current sheets and other coherent structures---is a natural candidate mechanism for establishing such spatially uniform distributions. + +In stark contrast, #strong[dropout events] are characterized by abrupt, sharp variations (depletions) in energetic particle intensity, often observed during impulsive SEP events @mazurInterplanetaryMagneticField2000@tesseinEffectCoherentStructures2015. Spacecraft crossing from one magnetic flux tube into an adjacent one observe sudden drops (or increases) in particle count rates, with the intensity boundaries occurring on spatial scales comparable to particle gyroradii @neugebauerEnergeticParticlesTangential2015. This behavior indicates that particles are effectively confined within distinct flux tubes with minimal lateral transport. The sharp boundaries between particle-filled and particle-empty flux tubes are frequently identified with tangential discontinuities in the solar wind magnetic field, which act as barriers to cross-field propagation because no magnetic field component threads through their surface ($B_n = 0$). + +Together, these observations highlight a fundamental duality: current sheets can both #emph[impede] particle transport (when acting as flux tube boundaries at tangential discontinuities) and #emph[facilitate] it (through scattering that decouples particles from their field lines at rotational discontinuities). The relative importance of these two effects depends on the internal structure of the current sheets and the particle energy---a theme developed quantitatively in this thesis. + +=== Turbulence Transport Frameworks + +The large-scale transport of energetic particles through the heliosphere is governed by a competition between parallel streaming along magnetic field lines, scattering by magnetic fluctuations, adiabatic focusing in the diverging interplanetary magnetic field, convection with the solar wind, and gradient-curvature drifts @dallaSolarEnergeticParticle2013. The foundational framework is the Parker transport equation @parkerPassageEnergeticCharged1965, which describes the evolution of the nearly isotropic part of the particle distribution using a diffusive approximation (justified when the particle scattering time is short compared to the timescale of interest): + +$ frac(partial f, partial t) = frac(partial, partial x_i) [kappa_(i j) frac(partial f, partial x_j)] - U_i frac(partial f, partial x_i) - V_(d \, i) frac(partial f, partial x_i) + 1 / 3 frac(partial U_i, partial x_i) frac(partial f, partial ln p) + upright("Sources") - upright("Losses") \, $ + +where $f$ is the omnidirectional distribution function, $kappa_(i j)$ is the symmetric part of the diffusion tensor, $U_i$ is the solar wind velocity, $V_(d \, i)$ is the gradient-curvature drift velocity, and $p$ is the particle momentum. The drift velocity can be formally derived from the guiding center approximation averaged over a nearly isotropic distribution, and can be included as the antisymmetric part of the diffusion tensor. The diffusion tensor can be decomposed into components parallel ($kappa_parallel$) and perpendicular ($kappa_perp$) to the mean magnetic field: $kappa_(i j) = kappa_perp delta_(i j) + \( kappa_parallel - kappa_perp \) B_i B_j \/ B^2$. + +The parallel diffusion coefficient is related to the pitch-angle diffusion coefficient $D_(mu mu)$ through the quasilinear theory (QLT) framework @jokipiiCosmicRayPropagationCharged1966: + +$ kappa_parallel = v^2 / 8 integral_(- 1)^1 frac(\( 1 - mu^2 \)^2, D_(mu mu) \( mu \)) thin d mu \, $ + +where $mu = cos alpha$ is the pitch-angle cosine and $v$ is the particle speed. This relation makes $D_(mu mu)$ the central microphysical quantity controlling parallel transport: all the physics of particle--field interaction is encoded in this single function of pitch angle. + +When significant anisotropy is present---as during the early phases of SEP events, near interplanetary shocks, or when focusing effects are important---the focused transport equation @roelofPropagationSolarCosmic1969@earlEffectAdiabaticFocusing1976 retains the explicit pitch-angle dependence: + +$ frac(partial f, partial t) + mu v frac(partial f, partial z) + frac(v, 2 L) \( 1 - mu^2 \) frac(partial f, partial mu) = frac(partial, partial mu) (D_(mu mu) frac(partial f, partial mu)) \, $ + +where $L = - B \( d B \/ d z \)^(- 1)$ is the focusing length characterizing the divergence of the magnetic field. + +In both frameworks, the pitch-angle diffusion coefficient $D_(mu mu)$ is the quantity most directly shaped by magnetic turbulence. Its value is governed by several key turbulence parameters --- spatial inhomogeneity, fluctuation level ($delta B \/ B_0$), spectral index, and wave-vector anisotropy @pucciEnergeticParticleTransport2016@chandranScatteringEnergeticParticles2000. Solar wind turbulence spans an enormous range of scales, from the large-scale coherence length (\~0.01 AU) down to kinetic dissipation scales near the thermal ion gyroradius (\~100 km). The most relevant portion for energetic particle transport is the inertial range: for a 5 nT background field, this range corresponds to proton gyroradii from roughly 1 keV to 1 GeV, a window encompassing nearly all SEPs. + +Classical scattering theory evaluates $D_(mu mu)$ by modeling the turbulence as a superposition of random, low-amplitude, phase-uncorrelated waves with power-law spectra, in which particles scatter via cyclotron resonance with fluctuations at scales matching their gyroradius @jokipiiCosmicRayPropagationCharged1966@jokipiiCosmicRayPropagationIi1967. Many numerical models adopt the same idealization, constructing magnetic fluctuations --- such as the slab component $delta upright(bold(B))^s$ and two-dimensional component $delta upright(bold(B))^(2 D)$ (#ref(, supplement: [Equation])) --- without accounting for the intermittent, structured nature of real solar wind turbulence. This approach has well-known limitations: it fails to scatter particles through $mu = 0$ (the 90° pitch-angle problem) and produces mean free paths that disagree with observational inferences. + +#math.equation(block: true, numbering: equation-numbering, [ $ & delta 𝐁^s = sum_(n = 1)^(N_m) A_n [cos alpha_n (cos phi.alt_n hat(x) + sin phi.alt_n hat(y)) + i sin alpha_n \( - sin phi.alt_n hat(x) + cos phi.alt_n hat(y) \)] times exp (i k_n z + i beta_n)\ + & delta 𝐁^(2 D) = sum_(n = 1)^(N_m) A_n i (- sin phi.alt_n hat(x) + cos phi.alt_n hat(y)) times exp [i k_n (cos phi.alt_n x + sin phi.alt_n y) + i beta_n] $ ]) + +=== Pitch-Angle Scattering by Current Sheets + +Solar wind turbulence is highly intermittent: magnetic energy and current density concentrate into thin, intense current sheets that occupy a small fraction of the volume yet account for a disproportionate share of the dissipation @borovskyContributionStrongDiscontinuities2010. The scattering produced by these coherent structures is fundamentally different from wave--particle resonance. When a particle's gyroradius is comparable to the sheet thickness ($rho tilde.op L$), the interaction involves separatrix crossings in phase space and the associated destruction of the quasi-adiabatic invariant $I_z$ (discussed in detail in the following sections). The resulting pitch-angle changes are large---$Delta alpha tilde.op cal(O) \( 1 \)$ in the case of geometrical chaotization---and occur on timescales comparable to a single gyroperiod, far faster than the gradual diffusion assumed in quasilinear theory. Because kinetic-scale current sheets are abundant in the solar wind @vaskoKineticscaleCurrentSheets2022@vasquezNumerousSmallMagnetic2007, and their thickness overlaps with the gyroradii of energetic particles at typical field strengths, a propagating particle will encounter many such structures en route from the Sun to Earth. The cumulative pitch-angle scattering from these encounters contributes to an effective $D_(mu mu)$ through a mechanism that is qualitatively distinct from---and potentially more efficient than---classical wave--particle resonance. + +A complementary perspective on structure-mediated scattering comes from #cite(, form: "prose"), who studied the role of sharp magnetic field line bends---characterized by the local curvature $kappa equiv B^(- 1) \| upright(bold(b)) times \( upright(bold(b)) dot.op nabla \) upright(bold(B)) \|$---in MHD turbulence more broadly. When a particle with gyroradius $r_g$ crosses a region where $kappa > 1 \/ r_g$ on a scale $l tilde.op r_g$, the interaction becomes non-adiabatic and the magnetic moment changes by order unity---precisely the same physics as in the quasi-adiabatic framework, but framed through field-line geometry rather than the Hamiltonian structure of a specific current sheet model. The key insight of that work is that, while such regions are rare in a root-mean-square sense (the r.m.s. curvature on scale $l$ is too weak to scatter), the non-Gaussian, power-law tails of the curvature p.d.f.---with $p_(hat(kappa)_l l) prop \( hat(kappa)_l l \)^(- 2)$ as measured in a direct numerical simulation of incompressible MHD---guarantee that sufficiently many scattering-strength bends exist on all scales $l < ell_c$ to sustain transport. This yields a mean free path $lambda_s tilde.op ell_c^0.7 macron(r)_g^0.3$, a scaling that is distinct from quasilinear predictions and arises from the intermittent structure of the turbulence. Importantly, particle tracking in that simulation confirmed that magnetic moment diffusion proceeds through localized, violent interactions rather than continuous weak scattering, producing non-Brownian transport on scales $lt.tilde ell_c$. + +=== Perpendicular Transport + +While parallel transport is relatively well understood, perpendicular (cross-field) transport remains more elusive due to its nonlinear and non-resonant nature @shalchiPerpendicularDiffusionEnergetic2021@costajr.CrossfieldDiffusionEnergetic2013. In the classical picture, cross-field diffusion arises from two mechanisms: the random walk of magnetic field lines, which carries particles across the mean field, and the decorrelation of particles from their original field lines through scattering. The perpendicular diffusion coefficient $kappa_perp$ is typically assumed to be a small fraction of $kappa_parallel$ @giacaloneTransportCosmicRays1999, but this assumption is challenged on two fronts: observations---such as the reservoir effect and the broad longitudinal spread of SEP events---demand significant cross-field transport, and recent simulations show that $kappa_perp$ can be substantial and strongly dependent on particle energy and turbulence structure @dundovicNovelAspectsCosmic2020. The dimensionality of the turbulence also matters @giacaloneChargedParticleMotionMultidimensional1994: in models with at least one ignorable spatial coordinate (e.g., slab geometry), cross-field diffusion is artificially suppressed, omitting essential physics. + +The role of current sheets in perpendicular transport is underexplored, yet potentially important @lemoineParticleTransportLocalized2023. It is commonly assumed that field-line random walk dominates cross-field motion, since the magnetic field is typically smooth on scales comparable to SEP gyro-radii. Near current sheets, however, the field becomes highly inhomogeneous---varying on scales comparable to or smaller than the gyro-radius---enabling enhanced transfer of particles between field lines. This current-sheet-driven mechanism is distinct from field-line random walk and may constitute a genuinely non-diffusive contribution to cross-field transport. + +=== Evidence for Current Sheet Modulation of SEP Intensity + +Direct observational evidence for the influence of current sheets on SEP transport comes from studies at both large and small scales. At the large scale of the heliospheric current sheet (HCS), #cite(, form: "prose") performed a superposed epoch analysis of 319 HCS crossings observed by the Wind spacecraft, finding a systematic drop in 2--10 MeV/nucleon helium flux at the HCS that was strongest at low energies and diminished at higher energies. They identified 15 individual SEP flux dropout events coinciding with HCS crossings, all originating from western-hemisphere sources at longitudes far from the crossing location---indicating that the HCS severed the magnetic connection between the particle source and the observer. The energy dependence of the dropout fraction is consistent with more effective scattering or blocking of lower-energy particles whose gyroradii are comparable to the current sheet thickness. The transport and drift effects of the HCS have also been characterized through simulation. #cite(, form: "prose") integrated fully three-dimensional proton trajectories near an analytically defined flat HCS in the 1--800 MeV range, finding that gradient and curvature drifts along the sheet can carry protons to longitudes far from their injection site---producing multi-component intensity profiles that could be misinterpreted as evidence for multiple injection events---and confirming that the HCS acts as an effective barrier to cross-hemisphere transport. Extending this work to a more realistic geometry, #cite(, form: "prose") modeled SEP propagation near a wavy HCS whose position was constrained by fits to magnetic source surface maps. They found that the waviness of the sheet introduces longitudinally periodic enhancements in particle fluence and enables efficient longitudinal transport along the sheet, with the magnitude and spatial distribution of these effects depending sensitively on the IMF polarity configuration (A+ vs.~A−) and the HCS tilt angle. + +At smaller scales, #cite(, form: "prose") analyzed over 12 years of ACE observations and found a strong statistical correlation between coherent structures---identified using the partial variance of increments (PVI) method, which detects current sheets and sharp magnetic field gradients---and energetic particle intensity variations in the 0.047--4.78 MeV range. Local PVI maxima frequently coincided with regions of rising or falling particle intensity, suggesting that magnetic discontinuities act as local barriers or modulators of transport. This correlation persisted after removing shock-associated intervals, confirming that the effect is intrinsic to the current sheets rather than a byproduct of shock-related enhancements. + +=== Non-Diffusive Transport Effects + +Beyond classical diffusion, SEP observations and measurements near interplanetary shocks frequently reveal anomalous transport behavior, characterized by subdiffusive or superdiffusive scaling of mean-square displacement with time, $chevron.l Delta x^2 \( t \) chevron.r prop t^alpha$ with $alpha eq.not 1$ @zimbardoSuperdiffusiveSubdiffusiveTransport2006@zimbardoSuperdiffusiveTransportLaboratory2015. These deviations from normal diffusion ($alpha = 1$) are attributed to the intermittent, structured nature of solar wind turbulence and are better captured by generalized frameworks such as fractional diffusion models @del-castillo-negreteNondiffusiveTransportPlasma2005 or Lévy statistics @zaburdaevLevyWalks2015. + +The interactions with current sheets have implications that extend beyond modifications to the diffusion coefficients. When pitch-angle changes are dominated by rare but intense jumps rather than continuous weak scattering, the statistical assumptions underlying the diffusion approximation can break down @lemoineParticleTransportLocalized2023. Current sheets can induce non-Markovian memory effects @zimbardoNonMarkovianPitchangleScattering2020: correlations in the spatial arrangement and properties of current sheets mean that scattering at one structure influences both the likelihood and character of scattering at the next. The intermittent, clustered distribution of current sheets in the solar wind naturally generates the heavy-tailed step-size distributions that underlie anomalous transport in both regimes---subdiffusive ($alpha < 1$) when sheets trap particles, and superdiffusive ($alpha > 1$) when intense encounters produce large pitch-angle jumps and correspondingly large spatial displacements. + +== Quasi-Adiabatic Dynamics of Charged Particles and Its Destruction in Current Sheets + +=== Adiabatic Motion and Its Breakdown + +A charged particle moving in a magnetic field that varies slowly in space and time possesses an approximate conservation law. The particle gyrates around the magnetic field line with a characteristic frequency---the cyclotron frequency $Omega_c = q B \/ m c$---and a characteristic radius---the gyroradius (or Larmor radius) $rho = v_perp \/ Omega_c$, where $v_perp$ is the velocity component perpendicular to the magnetic field. If the magnetic field changes only gradually over the spatial scale of the gyro-orbit, the particle's magnetic moment + +$ mu = frac(m v_perp^2, 2 B) $ + +is approximately conserved. This quantity, the first adiabatic invariant, is the foundation of guiding-center theory @northropAdiabaticChargedparticleMotion1963, in which the rapid gyromotion is averaged out and the particle is described by the drift of its guiding center along and across the magnetic field. The conservation of $mu$ underlies many fundamental phenomena in space physics, including magnetic mirroring, radiation belt trapping, and the confinement of particles in magnetic bottles. + +The key assumption of guiding-center theory is that the magnetic field varies on spatial scales much larger than the particle's gyroradius: $L_B gt.double rho$, where $L_B$ is the characteristic length scale of the field gradient. When a particle encounters a magnetic structure whose thickness $lambda$ is comparable to or smaller than its gyroradius---that is, when $lambda tilde.op rho$---this assumption breaks down. The particle can no longer complete a full gyration within a region of approximately uniform field. The magnetic moment ceases to be conserved, and the guiding-center description fails @escandeBreakdownAdiabaticInvariance2021. This is precisely the situation that arises when energetic particles interact with kinetic-scale current sheets in the solar wind, where the structure thickness is on the order of the ion inertial length $d_i$ or a few ion gyroradii. + +A different theoretical framework is needed to describe particle dynamics in this regime. Rather than averaging over the gyromotion (as in guiding-center theory), one must identify the appropriate fast periodic motion for the specific field geometry and construct the corresponding adiabatic invariant. This is the domain of the quasi-adiabatic theory @zelenyiQuasiadiabaticDynamicsCharged2013@whippleAdiabaticTheoryRegions1986, which we now describe. + +=== Adiabatic Invariants in Slow--Fast Hamiltonian Systems + +The mathematical foundation for understanding particle motion in current sheets lies in the general theory of slow--fast Hamiltonian systems and their adiabatic invariants. We follow the review by #cite(, form: "prose"), which provides a comprehensive treatment of the mechanisms by which adiabatic invariance can be destroyed. + +Consider a Hamiltonian system in which the variables naturally separate into fast and slow degrees of freedom. The fast variables $\( p \, q \)$ oscillate rapidly, while the slow variables $\( y \, x \)$ evolve on a much longer timescale, controlled by a small parameter $epsilon lt.double 1$. The equations of motion take the form + +$ dot(p) = - frac(partial E, partial q) \, quad dot(q) = frac(partial E, partial p) \, quad dot(y) = - epsilon frac(partial E, partial x) \, quad dot(x) = epsilon frac(partial E, partial y) . $ + +For frozen values of the slow variables, the fast system describes periodic motion. One can then introduce #emph[action-angle variables] $\( I \, phi \)$ for this periodic motion, where the action + +$ I = frac(1, 2 pi) integral.cont p thin d q $ + +is the area enclosed by the orbit in the fast phase plane, divided by $2 pi$. Averaging the equations of motion over the fast phase $phi$ yields the #emph[adiabatic approximation]: + +$ I = upright("const") \, quad dot(y) = - epsilon frac(partial H_0, partial x) \, quad dot(x) = epsilon frac(partial H_0, partial y) \, $ + +where $H_0 \( I \, y \, x \)$ is the Hamiltonian expressed in terms of the action and slow variables. The action $I$ is an #emph[adiabatic invariant]: it is conserved with accuracy $O \( epsilon \)$ over time intervals of order $1 \/ epsilon$. + +Under favorable conditions, conservation can be much better than this. When the system has two degrees of freedom (one fast, one slow) and the phase portrait of the fast system is everywhere filled by closed trajectories (no separatrices), Arnold showed that invariant tori of the exact system fill the energy surface up to a residue of small measure, and the adiabatic invariant is conserved perpetually: $\| I \( t \) - I \( 0 \) \| = O \( epsilon \)$ for all time @arnoldSmallDenominatorsProblems1963. This result has direct physical consequences: it implies, for example, that charged particles can be confined indefinitely in axisymmetric magnetic traps. + +=== Separatrix Crossing and Destruction of Adiabatic Invariance + +The situation changes fundamentally when the phase portrait of the fast system contains a #emph[separatrix]---a trajectory that separates topologically distinct regions of phase space. #ref(, supplement: [Figure]) illustrates the generic structure: a saddle point $C$ in the fast phase plane generates separatrices $l_1$ and $l_2$ that divide the portrait into three domains $G_1$, $G_2$, and $G_3$, each corresponding to a different type of periodic motion. For example, $G_1$ and $G_2$ might correspond to oscillations in two separate potential wells, while $G_3$ corresponds to oscillations spanning both wells. #figure([ -#box(image("figures/vl_ratio.png", width: 60.0%)) +#box(image("figures/ref/neishtadtMechanismsDestructionAdiabatic2019-fig3.png")) ], caption: figure.caption( position: bottom, [ -Statistics of the asymptotic velocity ratio from PSP, Wind, and ARTEMIS spacecraft observations during PSP encounter 7 period from 2021-01-14 to 2021-01-21. +Phase portrait of the fast system @neishtadtMechanismsDestructionAdiabatic2019. ]), kind: "quarto-float-fig", supplement: "Figure", ) + + + +As the slow variables evolve, the phase portrait of the fast system changes shape, and a trajectory that was initially in one domain can be driven toward the separatrix and cross into another domain. This crossing has profound consequences for the adiabatic invariant @tennysonChangeAdiabaticInvariant1986. There are two distinct contributions to its change: + ++ #strong[Geometrical jump.] When a trajectory crosses from, say, domain $G_3$ (the large outer domain) to $G_1$ (one of the inner domains), the area enclosed by the orbit changes discontinuously---from $S_3 = S_1 + S_2$ to $S_1$. This purely geometric change $Delta I^(upright("geom"))$ is independent of $epsilon$ and is determined entirely by the relative areas of the separatrix loops at the moment of crossing. + ++ #strong[Dynamical jump.] In addition to the geometric change, the logarithmic divergence of the oscillation period $T$ near the separatrix produces a correction $Delta I^(upright("dyn")) tilde.op epsilon ln epsilon$. The precise value of this correction depends sensitively on the phase of the fast motion at the moment of crossing, parameterized by a variable $xi$. This quantity varies rapidly ($tilde.op 1 \/ kappa$ times faster than the slow evolution) and can be treated as a uniformly distributed random variable $xi in \( 0 \, 1 \)$. + +Because the dynamical jump depends sensitively on initial conditions ($O \( epsilon \)$ changes in initial data produce $O \( 1 \)$ relative changes in the jump), the outcome of separatrix crossing is effectively probabilistic. The probability of entering domain $G_i$ after crossing from $G_3$ is $P_i = Theta_i \/ Theta_3$, where $Theta_i = { S_i \, h_c }$ is the Poisson bracket of the area of domain $G_i$ with the value of the Hamiltonian $E$ (energy) at the saddle point $C$. The accumulation of geometrical and dynamical jumps over multiple separatrix crossings destroys the adiabatic invariance and leads to chaotic dynamics. + +=== Application to Current Sheets: the Quasi-Adiabatic Invariant + +The motion of charged ions in a magnetic field reversal---the configuration of a current sheet---provides the most physically important application of this theory. Consider a current sheet with a reversing field $B_x \( z \) = B_0 \( z \/ L \)$ (where $L$ is the half-thickness), a small normal component $B_z$, and an optional guide field $B_y$. In normalized variables, the Hamiltonian for ion motion takes the form @artemyevIonMotionCurrent2013 + +$ H = 1 / 2 p_z^2 + 1 / 2 \( p_x - s z \)^2 + 1 / 2 (kappa x - 1 / 2 z^2)^2 \, $ + +where the two key dimensionless parameters are + +$ kappa = B_z / B_0 sqrt(L / rho_0) \, #h(2em) s = B_y / B_0 sqrt(L / rho_0) \, $ + +with $rho_0 = sqrt(2 h m) c \/ \( q B_0 \)$ being the characteristic Larmor radius for a particle of energy $h$. The parameter $kappa$ controls the separation of timescales: $kappa lt.double 1$ means the coordinate $z$ (normal to the current sheet) oscillates rapidly while $kappa x$ (along the reversing field direction) evolves slowly. For typical ions in the Earth's magnetotail current sheet, $kappa in \[ 0.01 \, 0.1 \]$\; for energetic ions interacting with solar wind current sheets, $kappa$ can range from $tilde.op 0.01$ to $tilde.op 0.3$ depending on the particle energy and current sheet thickness. + +The structure of the fast motion in the $\( z \, p_z \)$ plane depends on the instantaneous values of the slow variables $\( kappa x \, p_x \)$. The effective potential energy is +$ U \( z \) = 1 / 2 \( p_x - s z \)^2 + 1 / 2 (kappa x - 1 / 2 z^2)^2 . $ -#strong[Approach:] While classical single-fluid MHD models, even when extended to include pressure anisotropy, provide useful modeling of current sheet properties, they fall short of capturing the full complexity encountered in observed current sheets. A natural and necessary extension is to adopt a multifluid framework, which allows for a more realistic treatment of multiple ion populations. +This potential can have either two local minima separated by a maximum (forming two potential wells) or a single minimum (a single well), depending on $\( kappa x \, p_x \)$. The two-well structure corresponds physically to the particle oscillating about a magnetic field line on one side of the current sheet, while the single-well structure corresponds to the particle oscillating across the neutral plane $z = 0$, executing figure-eight-like orbits that span both sides of the sheet. -#strong[Results:] To address this challenge, we developed a multifluid theoretical model that includes both a nonzero normal magnetic field and a guide field, and explicitly accounts for the dynamics of counter-streaming ion populations (see #ref(, supplement: [Figure])). The model reveals a clear physical interpretation: close to the Sun, current sheets are dominated by a single ion population, leading to high Alfvénicity, while at larger radial distances, the ion populations become more balanced, resulting in reduced Alfvénicity (see #ref(, supplement: [Figure])). By bridging the gap between overly simplified single-fluid models and fully kinetic treatments, this multifluid model offers a physically consistent and computationally tractable framework. It establishes a critical connection between the macroscopic evolution of current sheets and the microscopic processes relevant to energetic particle scattering---thus advancing the broader thesis goal of modeling SEP transport in a realistic, structure-rich solar wind. +Since the fast motion is periodic at any frozen value of the slow variables, one can define the quasi-adiabatic invariant -This work is presented in "On the Alfvénicity of Multifuild Force-Free Current Sheets" (Zhang et al., submitted to Physics of Plasmas, 2025, manuscript is available at #link("https://github.com/Beforerr/cs_theory/blob/3812b6f1e62b4c954b7b1aa7dcbf092df23834bb/files/2025PoP_Model_Zijin.pdf")[GitHub];). +$ I_z = frac(1, 2 pi) integral.cont p_z thin d z \, $ + +which is the action variable of the fast oscillation. This quantity was first introduced in this context by Büchner and Zelenyi (1986, 1989) and plays a role analogous to the magnetic moment $mu$ of guiding-center theory, but is appropriate for the regime $lambda tilde.op rho$ where guiding-center theory fails. In the adiabatic approximation, $I_z = upright("const")$, and the equation $I_z \( kappa x \, p_x \) = upright("const")$ (at fixed energy $H = h$) determines closed trajectories in the slow-variable plane $\( kappa x \, p_x \)$. These trajectories describe the slow drift of the particle as it oscillates back and forth in $z$. + +The boundary between regions of the slow-variable plane corresponding to the two types of fast motion (two wells vs.~one well) is called the #emph[uncertainty curve]. When a particle's trajectory in the $\( kappa x \, p_x \)$ plane reaches the uncertainty curve, the corresponding trajectory in the $\( z \, p_z \)$ plane crosses the separatrix---the particle switches from oscillating on one side of the current sheet to oscillating across both sides, or vice versa. It is at these crossings that the quasi-adiabatic invariant $I_z$ undergoes jumps and the adiabatic approximation breaks down. + +=== Symmetric Current Sheets: Slow Diffusion + +In the simplest and most-studied case---a current sheet without a guide field ($s = 0$)---the phase portrait in the $\( z \, p_z \)$ plane is symmetric about $z = 0$. The two separatrix loops enclose equal areas, $S_l = S_r$, and they grow and shrink in synchrony as the slow variables evolve. As a consequence, the geometrical jumps at two successive separatrix crossings (one entering the single-well regime, one returning to a double-well regime) exactly cancel and do not contribute to stochastization (in other words, geometrical jumps can be absorbed into a factor-of-two renormalization of $I_z$). Only the dynamical jumps remain, and for the symmetric system these take the well-known form @caryAdiabaticinvariantChangeDue1986@buchnerRegularChaoticCharged1989 + +$ Delta I_z^(upright("dyn")) = - 2 / pi kappa p_x ln \( 2 sin pi xi \) \, $ + +where $xi in \( 0 \, 1 \)$ is the quasi-random phase variable. Since the average over $xi$ vanishes---$chevron.l Delta I_z^(upright("dyn")) chevron.r_xi = 0$---there is no directed drift in the invariant space. The destruction of $I_z$ proceeds as a diffusive random walk: each jump has amplitude $tilde.op kappa$, jumps occur at intervals $tilde.op 1 \/ kappa$ (one period of slow motion), and the variance accumulates as $chevron.l \( Delta I_z \)^2 chevron.r tilde.op kappa^2 dot.op \( kappa t \)$. A substantial (order-unity) change in $I_z$ therefore requires a time $tilde.op kappa^(- 3)$. For typical magnetotail parameters ($kappa tilde.op 0.05$--0.1), this corresponds to stochastization timescales of tens of minutes---slow, but physically relevant for the lifetime of thin current sheet configurations. + +An important phenomenon in the symmetric system is #emph[resonant interaction]. If the phase accumulated between two successive separatrix crossings satisfies a specific condition ($W = pi$, where $W$ is determined by the integral of the fast frequency $Omega_z$ along the adiabatic trajectory), then the two dynamical jumps can exactly compensate: $sum Delta I_z^(upright("dyn")) = 0$. This condition depends on $kappa$ but not on the random variable $xi$, so it can be simultaneously satisfied for an entire population of particles with the appropriate energy. The resulting coherent particle beams ("beamlets") have been observed in the Earth's magnetotail and studied extensively @buchnerRegularChaoticCharged1989. + +=== Guide Field Effects on the Quasi-Adiabatic Theory + +The introduction of a guide field ($B_y eq.not 0$, corresponding to $s eq.not 0$ in the Hamiltonian) breaks the symmetry of the system and profoundly alters the quasi-adiabatic dynamics. #cite(, form: "prose") developed the complete quasi-adiabatic theory for arbitrary values of $s$, identifying new types of particle trajectories and new mechanisms of invariant destruction that have no counterpart in the symmetric ($s = 0$) case. We summarize the key results here; the reader is referred to that paper for full details. + +==== Modified phase-space structure + +In the symmetric system ($s = 0$), the slow-variable plane $\( kappa x \, p_x \)$ is divided into two domains by the uncertainty curve (a half-circle $\( kappa x \)^2 + p_x^2 = 1$, $kappa x > 0$): a domain (t1) where the particle oscillates in one of two symmetric potential wells (motion along field lines on one side of the sheet), and a domain (t2) where the particle oscillates in a single well spanning both sides of the sheet (motion across the neutral plane). The separatrix in the $\( z \, p_z \)$ plane demarcates these two types of motion, and the uncertainty curve is its projection onto the slow-variable plane. + +When $s eq.not 0$, the potential $U \( z \)$ loses its symmetry about $z = 0$: the saddle point shifts to $z = z_c eq.not 0$, and the two separatrix loops enclose unequal areas, $S_l eq.not S_r$ (see #ref(, supplement: [Figure]), schematic). Correspondingly, the slow-variable plane acquires a richer structure. Two new domains appear in addition to (t1) and (t2): domain (t2r), where the particle oscillates in a single well #emph[above] the neutral plane ($z > 0$, both solutions of $U = H$ are positive), and domain (t2l), where it oscillates #emph[below] the neutral plane ($z < 0$). In neither of these new domains does the particle cross $z = 0$. Meanwhile, the (t1) domain---the region with two potential wells and a separatrix---shrinks, and the uncertainty curve contracts from a half-circle to a shorter arc. For $s gt.eq 1$, the uncertainty curve (and with it the separatrix) vanishes entirely. #figure([ -#box(image("figures/profiles_n1Inf=0.6.svg", width: 60.0%)) +#box(image("figures/ref/artemyevIonMotionCurrent2013-fig3.png")) ], caption: figure.caption( position: bottom, [ -Magnetic field, ion density, and ion bulk velocity for the case where $n_1 = 1.5 n_2$ and $L = d_i \, B_0 = 2 B_z$. Here, $z$ is the distance from the center of the 1-D current sheet, $n_alpha$ denotes the number density of ion species $alpha$, $d_i$ is the asymptotic ion inertial length, and $B_0$ is the in-plane magnetic field strength. +The phase plane of slow variables $\( kappa x \, p_x \)$ is shown for two values of the parameter $s$. Various colours are used for domains with different types of particle motion. Dotted grey lines show the position of energy level $U = 1 \/ 2$. ]), kind: "quarto-float-fig", supplement: "Figure", ) - + + + +==== Trajectory types in the current sheet with $B_y$ + +The trajectories of trapped particles in the current sheet without a guide field are simple: the particle oscillates in one potential well, crosses the separatrix into the single well (performing a half-rotation around $B_z$ in the neutral plane), then crosses back into a potential well. In the $\( kappa x \, p_x \)$ plane this appears as a single closed curve, determined by $I_z = upright("const")$ with a factor-of-two renormalization at the separatrix to account for the doubling of enclosed area ($S_l = S_r$, so $S = 2 S_l$). The motion is simple because the equal areas and synchronous evolution of the two loops ensure that geometrical jumps cancel exactly. +In the system with $B_y eq.not 0$, particle trajectories become qualitatively more complex because the areas $S_l$ and $S_r$ are no longer equal and their rates of change $Theta_l$ and $Theta_r$ are no longer identical. #cite(, form: "prose") showed that, as a consequence, the trajectory in the $\( kappa x \, p_x \)$ plane is no longer a single closed curve but a composite object: within each domain, the particle follows a segment of an adiabatic trajectory $I_z \( kappa x \, p_x \) = upright("const")$, but the value of $I_z$ differs from one domain to the next because of the geometrical jumps $Delta I_z^(upright("geom"))$ that occur at each separatrix crossing. The full trajectory is constructed by matching these segments at the uncertainty curve. + +A further complication arises from the asynchronous evolution of $S_l$ and $S_r$. At certain points along the uncertainty curve, one loop area is increasing ($Theta_l > 0$) while the other is decreasing ($Theta_r < 0$). At such points, a particle arriving from the (t1) domain has #emph[two possible continuations]: it can cross into the single-well domain (t2), or it can remain in the (t1) domain but switch from the right potential well to the left one (or vice versa). This phenomenon---called #emph[trajectory splitting]---is unique to the asymmetric system. The choice between the two continuations depends on the precise phase of the fast motion at the separatrix crossing, which is effectively random. The probabilities of the two outcomes, $P_l$, $P_r$, and $P = 1 - P_l - P_r$ (capture into the left well, right well, or single well respectively), can be calculated analytically from the rates $Theta_l$ and $Theta_r$ at the crossing point @artemyevIonMotionCurrent2013. + +Additionally, when $s eq.not 0$, a new type of transition appears that has no counterpart in the symmetric system: the particle can switch between the left and right potential wells #emph[without crossing the separatrix at all]. This occurs when one well disappears as the slow variables evolve, the particle is carried smoothly to the position of the other well, and a new well reappears. In the $\( kappa x \, p_x \)$ plane, this corresponds to the trajectory going around the end of the uncertainty curve rather than crossing it. #figure([ -#box(image("figures/UxNormBx.svg", width: 60.0%)) +#box(image("figures/ref/artemyevIonMotionCurrent2013-fig4.png", alt: "Trajectory splitting in the (\\kappa x, p_x) plane for s=0.5, showing two possible continuations at the uncertainty curve. Schemes of particle trajectories in systems with s = 0 and with s = 0.5 are shown in the phase plane (\\kappa x, p_x). Fragment of (\\kappa x, p_x) plane with trajectory splitting is shown in separated panel. Bottom schemes (C1, C2, C3) show particle trajectories before (dotted curves) and after (solid curves) separatrix crossings in the plane (z, p_z).")) ], caption: figure.caption( position: bottom, [ -Ion bulk velocity in the $x$ direction (maximum variance direction) $U_x$ profiles normalized by local Alfvén velocity $v_(A \, x) (z) = B_x (z) \/ sqrt(mu_0 m_p n (z))$ for different $n_1 (oo)$ +Trajectory splitting in the $\( kappa x \, p_x \)$ plane for $s = 0.5$, showing two possible continuations at the uncertainty curve. Schemes of particle trajectories in systems with $s = 0$ and with $s = 0.5$ are shown in the phase plane $\( kappa x \, p_x \)$. Fragment of $\( kappa x \, p_x \)$ plane with trajectory splitting is shown in separated panel. Bottom schemes (C1, C2, C3) show particle trajectories before (dotted curves) and after (solid curves) separatrix crossings in the plane $\( z \, p_z \)$. ]), kind: "quarto-float-fig", supplement: "Figure", ) - -== Software Development - -#strong[Context];: A central requirement for this thesis is the ability to perform high-performance, interactive, and reproducible analysis of space plasma data and particle dynamics. While the established SPEDAS framework---originally developed in IDL and later ported to Python---remains widely used in the community, its design limitations hinder modern scientific workflows (big data, parallel/distributed computing, etc.). +==== Four regimes of particle motion + +#cite(, form: "prose") identified four distinct regimes of particle dynamics, controlled by the value of $s$: + ++ #strong[$0 < s < s_(upright("bif")) approx 0.25$]: Only one type of trajectory exists, analogous to (but more complex than) the trajectories of the symmetric system. Particles cross the uncertainty curve multiple times per period of slow motion, with the number of crossings increasing as $s arrow.r 0$. At each crossing, geometrical jumps modify $I_z$, and the trajectory splits into segments matched at the uncertainty curve. Despite this splitting, the total number of crossings is finite and well-prescribed for a given $s$. -#strong[Approach];: To address this, we developed a suite of Julia-based software tools that combine the flexibility and speed of a modern language with the functionality of legacy systems. ++ #strong[$s_(upright("bif")) < s < macron(s) approx 0.35$]: A second type of trajectory appears in addition to the first. Trajectories of this new type cross the uncertainty curve only twice (once for $p_x > 0$ and once for $p_x < 0$), and the transition between wells occurs without separatrix crossing. Both trajectory types coexist in this range. -#strong[Results];: The core of this framework is `SPEDAS.jl`, which has interfaces directly with #link("https://github.com/spedas/pyspedas")[`pyspedas`];, #link("https://github.com/SciQLop/speasy")[`speasy`];, and #link("https://hapi-server.org/")[`HAPI`] while introducing new routines with significantly improved performance. To enable efficient test-particle tracing in both analytic presets and numerical derived electromagnetic fields, we developed `TestParticle.jl`, a lightweight tool for rapid particle trajectory simulations. Additionally, we created #link("https://github.com/beforerr/SpaceDataModel.jl")[`SpaceDataModel.jl`] to implement flexible, standards-compliant data structures aligned with SPASE and HAPI specifications, and contributed physics utilities through #link("https://github.com/JuliaPlasma/ChargedParticles.jl")[`ChargedParticles.jl`] and #link("https://github.com/JuliaPlasma/PlasmaFormulary.jl")[`PlasmaFormulary.jl`];. These tools have been integral to the data analysis (e.g., #ref(, supplement: [Figure])), modeling, and simulation components of this thesis, enabling scalable and transparent research workflows essential for studying particle transport in the heliosphere. ++ #strong[$macron(s) < s < 1$]: Only the second type of trajectory survives. The rate $Theta_r$ is negative everywhere along the uncertainty curve, meaning that capture into the right well at the separatrix is impossible. All well-switching occurs without separatrix crossing. -```julia -f = Figure() -tvars1 = ["cda/OMNI_HRO_1MIN/flow_speed", "cda/OMNI_HRO_1MIN/E", "cda/OMNI_HRO_1MIN/Pressure"] -tvars2 = ["cda/THA_L2_FGM/tha_fgs_gse"] -tvars3 = ["cda/OMNI_HRO_1MIN/BX_GSE", "cda/OMNI_HRO_1MIN/BY_GSE"] -t0,t1 = "2008-09-05T10:00:00", "2008-09-05T22:00:00" -tplot(f[1, 1], tvars1, t0, t1) -tplot(f[1, 2], tvars2, t0, t1) -tplot(f[2, 1:2], tvars3, t0, t1) -f -``` ++ #strong[$s gt.eq 1$]: The separatrix vanishes entirely ($ell \( s \) = 0$), and with it all geometrical and dynamical jumps. The quasi-adiabatic invariant is exactly conserved in the adiabatic approximation, and the motion is regular. This critical value corresponds to $B_y > B_0 sqrt(L \/ rho_0)$, the same magnetization criterion derived independently by Galeev and Zelenyi (1978). For $s gt.double 1$, the quasi-adiabatic invariant $I_z$ reduces to the ordinary magnetic moment, and the guiding-center theory becomes applicable. + +These four regimes are summarized in Table 1 of #cite(, form: "prose"). The threshold values $s_(upright("bif")) approx 0.25$ and $macron(s) approx 0.35$ are determined by the geometry of the separatrix loop areas and their evolution rates along the uncertainty curve, while $s = 1$ corresponds to the vanishing of the separatrix itself (the uncertainty curve length $ell \( s \) = 2 arctan sqrt(s^(- 4 \/ 3) - 1) - s^(1 \/ 3) sqrt(s^(2 \/ 3) - s^2)$ reaches zero). + +==== Importance of geometrical jumps + +A key result of #cite(, form: "prose") is the demonstration that, for $s > \( 2 \/ pi \) kappa ln 2$ (i.e., $B_y > 0.44 B_z$), the geometrical jumps dominate over the dynamical jumps in shaping particle trajectories. In this regime, the adiabatic trajectories in the $\( kappa x \, p_x \)$ plane are determined primarily by the geometrical jumps---which are of order unity and independent of $kappa$---rather than by the small dynamical jumps $tilde.op kappa ln kappa$. The dynamical jumps produce only a slow diffusion across these geometrically determined trajectories. For $s < \( 2 \/ pi \) kappa ln 2$, the geometrical jumps become smaller than the dynamical ones and the system behaves effectively like the $s = 0$ case. This separation is the gross features of particle dynamics in a current sheet, with geometrical jumps defining the trajectory structure and dynamical jumps providing the stochastic spreading. + +It is important to note that, however, the remaining $z arrow.r - z$, $p_x arrow.r - p_x$ symmetry constrains the possible values of $I_z$ to a finite set: the particle's invariant cycles through a finite number of discrete values, and the trajectories remain closed (though more complex than in the $s = 0$ case). The number of distinct $I_z$ values depends on $s$ and corresponds to the number of uncertainty curve crossings per period of slow motion (which can be counted using the graphical construction in Appendix C of #cite(, form: "prose")). + +=== Non-adiabatic Effects: Destruction of the Quasi-adiabatic Invariant + +The quasi-adiabatic theory developed in the previous subsections treats the invariant $I_z$ as exactly conserved between separatrix crossings and accounts only for the geometrical jumps at crossings. In reality, $I_z$ is only an approximate invariant: it oscillates with amplitude $tilde.op kappa$ about its mean value even far from the separatrix, and it undergoes both geometrical and dynamical jumps at each separatrix crossing. #cite(, form: "prose") provided a comprehensive analysis of these non-adiabatic effects for the current sheet with sheared magnetic field, deriving analytical expressions for the jumps and quantifying their consequences for particle stochastization. We summarize the key results here. + +Understanding the nature, magnitude, and statistics of these jumps is essential for this thesis, because the destruction of $I_z$ is physically equivalent to pitch-angle scattering: $I_z$ determines the particle's oscillation amplitude in the current sheet, which in turn determines its pitch angle relative to the local magnetic field. A jump in $I_z$ therefore corresponds directly to a change in pitch angle, and the rate of $I_z$ destruction determines the pitch-angle diffusion rate. + +==== The improved quasi-adiabatic invariant and its jumps + +Far from the separatrix, one can construct an #emph[improved quasi-adiabatic invariant] $J = I_z + kappa u \( z \, p_z \, kappa x \, p_x \)$, where $u$ is a correction function defined at each point of the four-dimensional phase space (except on the separatrix itself). The improved invariant $J$ is conserved with accuracy $tilde.op kappa^2$, compared to the accuracy $tilde.op kappa$ of the original $I_z$. This improvement is important because it isolates the true non-adiabatic behavior---the jumps at separatrix crossings $Delta J = Delta J^(upright("geom")) + Delta J^(upright("dyn"))$---from the oscillatory variations that occur during regular motion. It is through the accumulation of these jumps over multiple separatrix crossings that the quasi-adiabatic invariant is destroyed and particle motion becomes chaotic. + +==== Nonzero mean dynamical jump and accelerated stochastization + +For the symmetric system ($s = 0$), the well-known expression for the dynamical jump is $Delta J^(upright("dyn")) = - 2 / pi kappa p_x ln \( 2 sin pi xi \) .$ For the asymmetric system ($s eq.not 0$), the expressions for $Delta J^(upright("dyn"))$ are considerably more complex. #cite(, form: "prose") derived the full expressions for the dynamical jumps using the general formulas of #cite(, form: "prose") (the derivation is detailed in Appendix A of that paper). The key difference from the symmetric case is that the expressions involve the #emph[individual] rates of evolution $Theta_l$ and $Theta_r$ (rather than their sum) and the ratio $tilde(theta) = Theta_r \/ Theta_l$, which parameterizes the asymmetry. All these parameters are functions of the crossing point along the uncertainty curve, and their asymmetry produces a nonzero $chevron.l Delta J^(upright("dyn")) chevron.r_xi$ at each crossing. + +The physical consequence of $chevron.l Delta J^(upright("dyn")) chevron.r_xi eq.not 0$ is profound: it introduces a #emph[directed drift] in the invariant space, superimposed on the diffusive spreading from the random component. The stochastization timescale is then set by the drift rather than the diffusion. Each separatrix crossing changes $I_z$ by $tilde.op kappa$ on average (rather than zero on average), and crossings occur at intervals $tilde.op 1 \/ kappa$ (one period of slow motion in the $\( kappa x \, p_x \)$ plane). An order-unity change in $I_z$ therefore requires only $tilde.op 1 \/ kappa$ crossings, taking a total time $tilde.op kappa^(- 2)$---compared to $tilde.op kappa^(- 3)$ for the symmetric system where only diffusion operates. For the Earth's magnetotail with $kappa tilde.op 0.05$--0.1, this reduces the stochastization timescale from tens of minutes to a few minutes. It should be noted that the conservation of phase-space volume (Liouville's theorem) imposes a constraint: the double average $chevron.l chevron.l Delta J^(upright("dyn")) chevron.r chevron.r_(I_z)$ over the entire particle population must vanish. That is, while individual values of $I_z$ experience a nonzero mean drift, the net effect over the full distribution is a #emph[redistribution] of invariants without the appearance of net fluxes in phase space. The presence of current sheet boundaries at $\| z \| = lambda$ can, however, break this constraint and produce a net drift in the invariant space @zelenyiSplittingThinCurrent2003. + +==== Geometrical chaotization: rapid destruction independent of $kappa$ + +While the nonzero mean dynamical jump accelerates stochastization from $kappa^(- 3)$ to $kappa^(- 2)$, an even faster mechanism exists when the system possesses a fully broken symmetry. As discussed in the quasi-adiabatic theory, the Hamiltonian with $s eq.not 0$ retains a residual symmetry: invariance under the combined transformation $z arrow.r - z$, $p_x arrow.r - p_x$. As a result, the net change in $I_z$ after one full period of slow motion is determined by the dynamical jumps alone. + +#cite(, form: "prose") showed that when this residual symmetry is also broken---for example, by a cross-sheet electric field $epsilon$ that adds a term $tilde.op - s z u_d$ to the Hamiltonian (where $u_d = epsilon \/ kappa$ is the drift velocity)---the trajectories in the $\( kappa x \, p_x \)$ plane become asymmetric about $p_x = 0$. The surfaces of constant energy in the four-dimensional phase space that correspond to the two separatrix loops now project onto #emph[different] (shifted) domains in the $\( kappa x \, p_x \)$ plane. Consequently, the two successive geometrical jumps no longer compensate: $sum Delta I_z^(upright("geom")) eq.not 0$. + +The critical feature of this mechanism is that each geometrical jump $Delta I_z^(upright("geom"))$ is of order unity---it equals the difference of the areas of the two separatrix loops, which is a geometric property of the phase portrait and does not depend on $kappa$. This means that even for arbitrarily small $kappa$ (arbitrarily thin current sheets or arbitrarily energetic particles), a #emph[single pair] of separatrix crossings produces an $O \( 1 \)$ change in $I_z$. The particle trajectory in the slow-variable plane jumps between widely separated adiabatic curves, and after only a few separatrix crossings the trajectory fills a large domain, with the adiabatic invariant completely destroyed. + +#cite(, form: "prose") termed this mechanism #emph[geometrical chaotization] (an immediate, large-scale redistribution of $I_z$ values) to distinguish it from the much slower #emph[diffusive chaotization] (a gradual Gaussian spreading of $I_z$ distributions with width growing as $sqrt(t)$) produced by dynamical jumps. Numerical integration of $10^5$ test particle trajectories confirmed that the width of the $I_z$ distribution after ten separatrix crossings is dramatically broader in the fully asymmetric system (3) than in either the symmetric case (1) or the partially symmetric case (2; $s eq.not 0$ but no electric field). As shown in #ref(, supplement: [Figure]), system (3) has a broad, nearly uniform distribution over a wide range of $I_z$ values (complete invariant destruction) while system (1) peaks narrowly at $I_z \/ I_(z \, upright("init")) = 1$ (slow diffusion) and system (2) splits into a few discrete peaks (the finite number of possible $I_z$ values due to geometrical jumps with compensating sums). #figure([ -#box(image("figures/spedas_jl.png")) +#box(image("figures/ref/artemyevRapidGeometricalChaotization2014-fig3.png")) ], caption: figure.caption( position: bottom, [ -Example code snippet and resulting output from the Julia implementation of the widely used tplot function. +Distribution of $I_z$ values for an ensemble of $10^5$ trajectories after 10 separatrix crossings: (a) $s = 0$, $epsilon = 0$\; (b) $s = 0.15$, $epsilon = 0$\; (c) $s = 0.15$, $epsilon = 0.003$. All with $kappa = 0.01$. From #cite(, form: "prose"). ]), kind: "quarto-float-fig", supplement: "Figure", ) - + -#pagebreak() -= Proposed Research Direction - -== Spatial Diffusion Model Refinement - -The work completed in this thesis has established that solar wind current sheets are persistent, kinetic-scale structures whose statistical properties evolve predictably with heliocentric distance (see #ref(, supplement: [Section])). We have demonstrated---both theoretically and through numerical modeling---that SWDs play a significant role in modulating particle transport, particularly by enhancing pitch-angle scattering beyond quasilinear expectations (see #ref(, supplement: [Section])). Furthermore, we have shown that their internal structure, including multifluid effects and Alfvénicity variations, are essential to understanding their properties and thereby their transport-modifying capacity (see #ref(, supplement: [Section])). +==== Current sheet dynamics: particle reflection, transmission and resonance + +These non-adiabatic effects also have several physically important consequences for current sheet dynamics. The guide field introduces an asymmetry in particle reflection and transmission at the current sheet boundaries. #cite(, form: "prose") showed that for $B_y > 0$, particles approaching from the Southern Hemisphere ($z > 0$) are increasingly likely to transit through the sheet rather than be reflected, with transition becoming certain for $s > 0.35$ (i.e., $B_y > 0.35 B_0 sqrt(L \/ rho_0)$). Particles from the Northern Hemisphere ($z < 0$), by contrast, cross the sheet without any scattering for $s > \( pi^(- 1) ln 2 \) kappa$, because the uncertainty curve shrinks and these particles never encounter the separatrix. This directional asymmetry of current sheet interaction has implications for asymmetric auroral precipitation and for the self-consistent equilibrium of current sheets with finite guide fields. -Building on previous results, the next phase of research will extend the pitch-angle scattering framework to comprehensively model spatial diffusion processes (both parallel and perpendicular). This extension is crucial for accurately capturing the full scope of SEP transport influenced by current sheets. +The finite guide field also destroys the resonant condition under which two successive dynamical jumps can exactly compensate---a condition that, in the symmetric case, produces coherent beamlets of accelerated particles. For $s > \( pi^(- 1) ln 2 \) kappa$, the resonance condition, $sum Delta I_z^(upright("dyn")) = 0$, cannot be simultaneously satisfied for a large particle population. Moreover, the dependence of the integral $W$ on $I_z$ steepens with increasing $s$, so that even if the resonance condition is satisfied at one value of $I_z$, particles with slightly different invariants are far from resonance. The coherent beamlet structures predicted for the symmetric system are thus disrupted. -== Methodology - -#strong[Analysis:] To estimate the spatial diffusion coefficients parallel and perpendicular to the mean magnetic field, we must quantify the net displacement a particle experiences due to multiple, consecutive interactions with realistic current sheets. This includes estimating the parallel and perpendicular displacements, $Delta s_parallel$ and $Delta s_perp$, over the duration of one interaction cycle. +=== Application to Force-Free Solar Wind Current Sheets: Superfast Scattering + +The theoretical framework described above was applied to energetic ion scattering by solar wind discontinuities by #cite(, form: "prose"). The key observation motivating that work is that the internal magnetic field structure of observed solar wind current sheets differs in an important way from the idealized models previously considered. In observed compressionless (force-free) discontinuities, the reversal of the maximum-variance component $B_l$ is accompanied by a peak in the intermediate-variance component $B_m$, such that $\| B \| approx upright("const")$ across the structure (as shown earlier in #ref(, supplement: [Figure]) and confirmed statistically by #cite(, form: "prose")). This $B_m$ peak was absent from earlier Hamiltonian models of ion--current sheet interaction, which assumed either $B_m = 0$ (pure field reversal) or $B_m = upright("const")$ (uniform guide field). -The total time between two consecutive current sheet encounters is modeled as the sum of the time spent inside the current sheet $T_(c s)$, and the time spent free-streaming between sheets $T_(f s)$, given by: +To account for this observed field configuration, #cite(, form: "prose") modeled the discontinuity magnetic field as $B_l approx B_0 \( r_n \/ L \)$, $B_n = upright("const")$, $B_m = sqrt(B_0^2 - B_l^2) approx B_0 \( 1 - r_n^2 \/ 2 L^2 \)$, which in the normalized Hamiltonian introduces a cubic term: -$ T = T_(c s) + T_(f s) \, quad T_(f s) = s_(f s) / lr(|v_(parallel \, 1)|) $ +$ H = 1 / 2 p_z^2 + 1 / 2 (p_x - z + z^3 / 6)^2 + 1 / 2 (kappa x - z^2 / 2)^2 . $ -where $v_(parallel \, 0)$, $v_(parallel \, 1)$ are the particle's changed parallel velocity before and after interacting with the current sheet, respectively. +The $z^3 \/ 6$ term, which encodes the $B_m$ peak, breaks the symmetry of the phase portrait in the $\( z \, p_z \)$ plane. This is the essential difference from the $p_x^2 \/ 2$ term (no $B_m$\; pure compressional discontinuity) or the $\( p_x - z \)^2 \/ 2$ term (constant $B_m$\; uniform guide field) used in previous studies. As demonstrated by the theory of geometrical chaotization @artemyevRapidGeometricalChaotization2014, this asymmetry generates large geometrical jumps $Delta I_z tilde.op 1$ at each separatrix crossing, producing very fast destruction of the adiabatic invariant. -In the absence of scattering, the particle would follow the field line and travel a distance: +This result has several important implications for energetic particle transport in the solar wind: -$ s_0 = v_(parallel \, 0) dot.op T = v_(parallel \, 0) (T_(c s) + s_(f s) / lr(|v_(parallel \, 1)|)) . $ +First, the scattering rate (measured in the slow time $kappa t$) is independent of $kappa = \( B_n \/ B_0 \) sqrt(L \/ rho)$. Since $kappa$ determines whether a discontinuity is classified as rotational (finite $B_n$) or tangential ($B_n arrow.r 0$), this means that the distinction between RDs and nearly-TDs is irrelevant for the efficiency of ion scattering, provided the $B_m$ peak is present. In a solar wind containing multiple discontinuities, the timescale between successive scattering events is determined by the occurrence rate of discontinuities rather than by $kappa$, further reducing the importance of $B_n$. -However, when scattering occurs, the total distance traveled becomes: +Second, the condition for strong scattering is $L tilde.op rho$, which can be rewritten in terms of particle energy as $h \/ T_i tilde.op beta_i \( L \/ d_p \)^2$. For the most intense kinetic-scale discontinuities observed in the solar wind ($L \/ d_p in \[ 1 \, 10 \]$) and typical plasma conditions ($beta_i in \[ 0.1 \, 10 \]$), this condition is satisfied for essentially the entire suprathermal ion population. The mechanism is therefore expected to operate broadly. -$ s = s_(c s)^(\*) + upright("sign") (v_(parallel \, 1) / v_(parallel \, 0)) s_(f s) $ +Third, test particle simulations confirmed that in the presence of the $B_m$ peak, initially narrow distributions in both $I_z$ and pitch angle broaden rapidly---within a few separatrix crossings---to fill a broad range. In contrast, when the $B_m$ peak is absent ($B_m = 0$), the distributions remain narrowly peaked around their initial values, consistent with the much slower diffusive destruction of $I_z$ in the symmetric system. The effect is robust for $B_m \/ B_0 > 0.75$, which encompasses the majority of observed force-free discontinuities. -where $s_(c s)^(\*)$ is the effective parallel distance the particle travels within the current sheet. The net displacement compared to the unperturbed case is then: +The identification of this "superfast" scattering mechanism establishes that kinetic-scale current sheets in the solar wind are not merely passive structures but active agents of energetic particle scattering. This mechanism is expected to shape the observed low-anisotropy ion distributions at 1 AU and to contribute significantly to cross-field transport of energetic particle populations. The quantitative characterization of this scattering---including the derivation of analytical pitch-angle diffusion coefficients informed by observed current sheet parameters---forms one of the central contributions of this thesis and is developed in the following chapters. -$ Delta s_parallel = s - s_0 = (s_(c s)^(\*) - v_(parallel \, 0) T_(c s)) + s_(f s) (1 - v_(parallel \, 0) / v_(parallel \, 1)) . $ +== Summary and Open Questions + +This chapter has established the theoretical and observational foundation for understanding energetic particle scattering by solar wind current sheets. We summarize the key points and identify the open questions that motivate the work presented in this thesis. -Under the approximation that $s_(c s)^(\*) < < s_(f s)$, we obtain: +Solar wind turbulence is highly intermittent, with magnetic energy concentrating into current sheets whose thickness overlaps with the gyroradii of suprathermal and energetic ions. When a particle's gyroradius is comparable to the sheet thickness, guiding-center theory fails and the quasi-adiabatic invariant $I_z$ governs the dynamics instead of the magnetic moment $mu$. This invariant is destroyed at separatrix crossings through two mechanisms: dynamical jumps, which drive slow diffusion on a timescale $tilde.op kappa^(- 3)$ (or $tilde.op kappa^(- 2)$ with a guide field), and geometrical jumps, which in asymmetric field configurations---including the force-free current sheets most commonly observed in the solar wind---produce order-unity changes in $I_z$ within a single gyration. This #emph[geometrical chaotization] constitutes a superfast scattering mechanism for energetic particles. -$ Delta s_parallel approx s_(f s) (1 - v_(parallel \, 0) / v_(parallel \, 1) - frac(v_(parallel \, 0) T_(c s), s_(f s))) . $ +Observationally, current sheets modulate SEP intensities at both large and small scales. The heliospheric current sheet acts as a barrier to cross-hemisphere transport and drives systematic flux dropouts whose energy dependence is consistent with gyroradius-scale interactions. At kinetic scales, coherent magnetic structures are statistically correlated with sharp variations in energetic particle intensity. Together, these observations confirm that current sheets are not passive features of the background medium but active agents of particle scattering and transport. -The parallel spatial diffusion coefficient is then expressed as: +Several important questions remain open and motivate the work in this thesis. -$ kappa_parallel = frac((Delta s_parallel)^2, Delta t) = frac([s_(f s) (1 - v_(parallel \, 0) / v_(parallel \, 1) - frac(v_(parallel \, 0) T_(c s), s_(f s)))]^2, T_(c s) + s_(f s) / v_(parallel \, 1)) . $ +#strong[Pitch-angle diffusion coefficients from current sheet parameters.] While the superfast scattering mechanism has been demonstrated theoretically and confirmed in test-particle simulations, a quantitative analytical expression for $D_(mu mu)$ as a function of particle energy and current sheet parameters has not yet been derived. Such an expression---informed by the observed statistical distributions of magnetic field configurations---is needed to incorporate current-sheet scattering into heliospheric transport models and constitutes a central goal of this thesis. -Similarly, for the perpendicular direction: +#strong[Perpendicular transport and cross-field diffusion.] The role of current sheets in cross-field transport remains largely unexplored. Near current sheets, the magnetic field varies on scales comparable to or smaller than the gyroradius, enabling enhanced transfer of particles between field lines through a mechanism distinct from field-line random walk. The anti-correlation between large curvature and weak magnetic field strength @yangRoleMagneticField2019@kempskiCosmicRayTransport2023 further suggests that particles undergoing magnetic moment-violating interactions are simultaneously more likely to hop between field lines---a potentially important source of perpendicular transport that deserves systematic study in the solar wind context. -$ kappa_perp = frac((Delta s_perp)^2, T_(c s) + s_(f s) / v_(parallel \, 1)) . $ +#pagebreak() += Observational Analysis of Current Sheets + +#strong[Context:] A critical first step in understanding the role of current sheets in energetic particle transport is to characterize their statistical properties and quantify the parameters most relevant to particle scattering. Although current sheets have been extensively observed---especially near $1$ AU---our knowledge of how their properties evolve across heliocentric distances, and how key scattering-related parameters vary with radial distance, has remained incomplete. Previous studies @sodingRadialLatitudinalDependencies2001[#cite(, form: "prose"), #cite(, form: "prose"), #cite(, form: "prose"), #cite(, form: "prose")] often lacked simultaneous, multi-point measurements and did not adequately separate temporal variability from spatial trends, leading to persistent uncertainties regarding their role in particle transport, their origin, and their evolution within the turbulent solar wind. -The key parameters---$v_(parallel \, 1)$, $T_(c s)$, $Delta s_perp$, and $Delta t$---are directly extracted from test-particle simulations, while quantities such as the current sheet separation distance $s_(f s)$, thickness, shear angle, and normal orientation are treated as system parameters derived from solar wind observations. Together, these inputs enable a systematic and physically grounded estimation of spatial diffusion coefficients under realistic heliospheric conditions. +#strong[Approach:] To bridge this observational gap, we conducted a detailed statistical analysis using continuous solar wind data from multiple spacecraft: Parker Solar Probe (PSP) at distances down to 0.1 AU, Wind, ARTEMIS, and STEREO at 1 AU, and Juno during its cruise phase out to 5 AU near Jupiter. This combination allowed us to track the evolution of current sheet properties across a wide radial distance, from near Alfvénic critical surface to the outer inner heliosphere. #figure([ -#box(image("figures/dR_perp.png", width: 70.0%)) +#box(image("figures/fig-ids_examples.png", alt: "Current sheets detected by PSP, Juno, STEREO and near-Earth ARTEMIS satellite: red, blue, and black lines are 𝐵_𝑙, 𝐵_𝑚, and 𝐵")) ], caption: figure.caption( position: bottom, [ -Example trajectory of a particle interacting with a current sheet +Current sheets detected by PSP, Juno, STEREO and near-Earth ARTEMIS satellite: red, blue, and black lines are $𝐵_𝑙$, $𝐵_𝑚$, and $𝐵$ ]), kind: "quarto-float-fig", supplement: "Figure", ) - - -#strong[Data-Integrated Analytical Modeling:] To ensure consistency with heliospheric observations, we will use realistic solar wind current sheet parameters to derive the diffusion coefficients using multiple spacecraft spanning radial distances from 0.1 to 5 AU. The derived diffusion coefficients will then be incorporated into turbulence-based transport models, providing a current-sheet-informed extension to global energetic particle transport frameworks. -== Timeline - -#strong[Months 1--4:] +#strong[Results:] Our analysis reveals that the majority of solar wind current sheets maintain kinetic-scale thicknesses throughout the inner heliosphere. When normalized to the local ion inertial length and Alfvén current, both the current density and thickness of these structures remain nearly constant over the range from 0.1 to 5 AU (see #strong[?\@fig-juno-distribution-r-sw]). This suggests that current sheets consistently influence energetic particle transport across heliocentric distances, with their higher occurrence rates closer to the Sun indicating a more pronounced role in shaping particle dynamics in the inner heliosphere. Furthermore, by leveraging simultaneous observations from spacecraft at different radial distances, we demonstrate that the observed radial trends reflect genuine spatial evolution rather than temporal or solar-cycle effects. -- Refine the pitch-angle scattering model to incorporate both parallel and perpendicular spatial diffusion effects. +Together, these results provide critical empirical constraints for particle transport modeling and establish a robust observational foundation for the theoretical and numerical components of this thesis. -- Conduct detailed test-particle simulations using solar wind parameters derived from multi-spacecraft observations (PSP, Wind, Juno, etc.). +This work is presented in #emph["Solar wind discontinuities in the outer heliosphere: Spatial distribution between 1 and 5 AU"] (Zhang et al., submitted to JGR Space Physics, 2025, manuscript is available at #link("https://www.authorea.com/users/814634/articles/1283652-solar-wind-discontinuities-in-the-outer-heliosphere-spatial-distribution-between-1-and-5-au")[10.22541/essoar.174431869.93012071/v1]). -#strong[Months 5--7:] +In particular, we propose that the observed reduction in current sheet occurrence rate at larger heliocentric distances is partly attributable to a geometric effect---namely, the decreasing probability that a spacecraft intersects inclined structures as distance from the Sun increases. This represents an observational bias that must be accounted for when interpreting occurrence statistics. However, even after correcting for this geometric effect, a modest residual decrease remains, which we attribute to possible physical dissipation or annihilation of current sheets as they propagate outward through the solar wind. -- Identify observational signatures supporting the proposed scattering model. - -- Examine how current sheet properties influence SEP scattering across different heliocentric distances. - -#strong[Months 8--10:] += Quantitative Modeling of Particle Scattering + +#strong[Context:] While it is well established that turbulence governs energetic particle transport in the heliosphere, the specific role of coherent structures---particularly current sheets---in shaping scattering processes remained under-explored. A central objective of this thesis is to develop a physics-based, observation-informed model that directly links solar wind current sheet properties to pitch-angle scattering rates of energetic particles. -- Finalize the spatial diffusion model and assess its implications for large-scale SEP propagation. +#strong[Approach:] To this end, we combined statistical measurements of current sheets at 1 AU with a Hamiltonian analytical framework and test particle simulations to investigate how particle scattering efficiency varies with current sheet geometry and particle energy, using a realistic magnetic field configuration: -- Synthesize simulation results and observational insights into dissertation chapters. Integrate observational and theoretical findings into comprehensive thesis documentation. +#strong[Results:] Using a newly formulated Hamiltonian framework (see dimensionless form in #strong[?\@eq-Hamiltonian], below) that incorporates the effects of magnetic field shear angle $beta$ and particle energy $H$, we demonstrate that scattering rates depend strongly on the current density---which is directly tied to $beta$---as well as on the ratio of the particle gyroradius to the current sheet thickness. Notably, our results show that current sheets can induce rapid, non-diffusive pitch-angle jumps, particularly for SEPs in the 100 keV to 1 MeV energy ranges (see #strong[?\@fig-example-subset]). This behavior deviates significantly from classical quasilinear predictions and highlights the need to account for coherent structures in transport models. To describe long-term pitch-angle evolution and spatial transport, we developed a simplified probabilistic model of pitch-angle scattering due to current sheets and derived an effective pitch-angle diffusion and spatial transport coefficient $D_(mu mu)$. -== Relevance and Broader Implications - -This thesis substantially advances our understanding of particle transport mechanisms within turbulent space plasmas, offering significant enhancements to SEP prediction models. By accurately quantifying the influence of coherent structures such as current sheets, the research outcomes have direct applications to improving space weather forecasting, enhancing spacecraft operational safety, and contributing to the broader understanding of energetic particles transport and acceleration in the solar wind. +These diffusion rate estimates enable direct comparison with other scattering mechanisms, facilitate the incorporation of SWD-induced scattering into global SEP transport models, and directly support the broader goal of this thesis to improve our understanding of how energetic particles interact with turbulence in the solar wind. -== Opportunities for Future Research - -Completion of this thesis opens several avenues for future investigations: +This work is presented in "Quantification of Ion Scattering by Solar Wind Current Sheets: Pitch-Angle Diffusion Rates" (Zhang et al., submitted to Physical Review E, 2025, manuscript is available at #link("https://github.com/Beforerr/ion_scattering_by_SWD/blob/ec33d3d082bcd463faf7a233ba80138414231b51/files/2024PRE_Scattering_Zijin.pdf")[GitHub]). -- Exploration of current sheet interactions in other astrophysical environments, such as planetary magnetospheres. +$ upright(bold(B)) = B_0 \( cos theta med upright(bold(e_z)) + sin theta \( sin phi \( z \) med upright(bold(e_x)) + cos phi \( z \) med upright(bold(e_y)) \) \) $ -- Advanced integration of mapping techniques with numerical simulations to further refine SEP transport models. +where $B_0$ is the magnitude of the magnetic field, $theta$ is the azimuthal angle between the normal and the magnetic field, and $phi \( z \)$ is the rotation profile of the magnetic field as a function of $z$. -- Expanded observational campaigns utilizing upcoming spacecraft missions designed to probe heliospheric turbulence and particle dynamics at unprecedented resolution. +#pagebreak() +#pagebreak() +#pagebreak() +- #cite(, form: "prose") + - Abstract: Electric currents in the solar wind plasma are investigated using 92 ms fluxgate magnetometer data acquired in a high-speed stream near 1 AU. The minimum resolvable scale is roughly 0.18 s in the spacecraft frame or, using Taylor's “frozen turbulence” approximation, one proton inertial length di in the plasma frame. A new way of identifying current sheets is developed that utilizes a proxy for the current density J obtained from the derivatives of the three orthogonal components of the observed magnetic field B. The most intense currents are identified as 5σ events, where σ is the standard deviation of the current density. The observed 5σ events are characterized by an average scale size of approximately 3di along the flow direction of the solar wind, a median separation of around 50di or 100di along the flow direction of the solar wind, and a peak current density on the order of 0.5 pA/cm2. The associated current-carrying structures are consistent with current sheets; however, the planar geometry of these structures cannot be confirmed using single-point, single-spacecraft measurements. If Taylor's hypothesis continues to hold for the energetically dominant fluctuations at kinetic scales , then the results suggest that the most intense current-carrying structures in high-speed wind occur at electron scales, although the peak current densities at kinetic and electron scales are predicted to be nearly the same as those found in this study. #pagebreak() -= References - - #set bibliography(style: "apa") - #bibliography(("../../../../files/bibliography/research.bib")) diff --git a/files/bibliography/full.bib b/files/bibliography/full.bib new file mode 120000 index 0000000..c871568 --- /dev/null +++ b/files/bibliography/full.bib @@ -0,0 +1 @@ +/Users/zijin/projects/ion_scattering_by_SWD/overleaf/files/bibliography/full.bib \ No newline at end of file diff --git a/justfile b/justfile index 6574320..4f43c93 100644 --- a/justfile +++ b/justfile @@ -16,6 +16,10 @@ install-julia-deps: ]) Pkg.instantiate() +install-tools: + chmod +x src/latex2qmd.py + cp src/latex2qmd.py ~/.local/bin/latex2qmd + publish: quarto publish gh-pages --no-prompt --no-render @@ -26,6 +30,7 @@ preview: ln-bib: mkdir -p files/bibliography [ -e files/bibliography/research.bib ] || ln -s ~/projects/share/bibliography/research.bib files/bibliography/research.bib + ln -s ~/projects/ion_scattering_by_SWD/overleaf/files/bibliography/full.bib files/bibliography/full.bib cv: rendercv render docs/others/cv_Zijin.yaml \ No newline at end of file