pith. sign in

arxiv: 2504.00659 · v1 · submitted 2025-04-01 · 🌌 astro-ph.SR · physics.space-ph

Modeling hot, anisotropic ion beams in the solar wind motivated by the Parker Solar Probe observations near perihelia

Leon Ofman , Yogesh , Scott A Boardsen , Parisa Mostafavi , Lan K Jian , Viacheslav M Sadykov , Kristopher Klein , Mihailo Martinovic This is my paper

Pith reviewed 2026-05-22 22:00 UTC · model grok-4.3

classification 🌌 astro-ph.SR physics.space-ph
keywords solar windParker Solar Probeion kinetic instabilitiesvelocity distribution functionshybrid simulationsanisotropic beamshammerhead distributionsplasma heating
0
0 comments X

The pith

Nonlinear hybrid models of hot anisotropic ion beams reproduce the core and hammerhead velocity distributions observed by Parker Solar Probe near perihelion.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper initializes nonlinear hybrid plasma simulations with ion velocity distribution parameters drawn from PSP observations of the solar wind at close solar distances. These initial distributions contain anisotropic cores and super-Alfvenic beams that trigger combined ion-cyclotron and magnetosonic instabilities on proton gyroperiod timescales. The instabilities generate left- and right-hand polarized waves whose nonlinear evolution reshapes the distributions into forms that qualitatively match the observed anisotropic cores and hammerhead beams. The work thereby links wave-particle scattering to anisotropic heating of the near-Sun solar wind plasma.

Core claim

Hybrid simulations initialized with PSP-derived hot-beam parameters show rapid growth of ion-cyclotron and magnetosonic instabilities that produce the observed left- and right-hand wave spectra; in the nonlinear stage the ion velocity distributions develop the anisotropic core and hammerhead features seen in the data, indicating that kinetic instabilities mediate energy transfer from waves to particle thermal motion.

What carries the argument

Nonlinear hybrid simulations that evolve the full ion velocity distribution functions under the combined action of ion-cyclotron and magnetosonic instabilities seeded by the initial anisotropy and beam drift.

If this is right

  • Wave-particle interactions convert magnetic fluctuation energy into anisotropic ion thermal energy on short timescales.
  • Both proton and alpha-particle populations undergo similar instability-driven reshaping.
  • The resulting perpendicular heating maintains T_perp/T_parallel greater than one in the core population.
  • The modeled wave polarization spectra match the right-hand and left-hand signatures recorded by FIELDS.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the modeled evolution holds, the same instabilities should operate at larger heliocentric distances once the beams remain sufficiently anisotropic.
  • Comparison of simulated versus observed wave amplitudes could yield an independent estimate of local heating rates.
  • Adding electron kinetics or Coulomb collisions would test how robust the hammerhead formation remains under more complete physics.

Load-bearing premise

The starting velocity distributions are taken directly from the PSP measurements without prior evolutionary history or uncertainties in core-beam separation.

What would settle it

A set of PSP intervals showing persistent hammerhead distributions accompanied by no detectable ion-scale wave power at the predicted growth rates would falsify the claimed causal link.

Figures

Figures reproduced from arXiv: 2504.00659 by Kristopher Klein, Lan K Jian, Leon Ofman, Mihailo Martinovic, Parisa Mostafavi, Scott A Boardsen, Viacheslav M Sadykov, Yogesh.

Figure 1
Figure 1. Figure 1: The temporal evolution at Encounter 17 of (a) magnetic field components, (b) proton number density (blue) and α particle number density (red), (c) temperature anisotropy (A = T⊥/T∥ ) of protons (blue) and α particles (red), (d) velocity of protons (blue), α particles (red), and Alfv´en velocity (black), (e) ratio of the differential velocity between α particles and protons to the Alfv´en velocity, (f) prot… view at source ↗
Figure 2
Figure 2. Figure 2: The wavelet power spectra (W, panel-a) and polarization (P, panel-b) for the same time interval shown in [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The temporal evolution of the temperature anisotropies of the proton core (red), proton beam (blue), α particle core (green), and particle α beam (brown) for four cases in [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The temporal evolution of the parallel and perpendicular thermal energies of the proton core (a), proton beam (b), α particle core (c), and α particle beam (d) populations for Case 1 (red), Case 2 (green), and Case 3 (blue). The line styles are W⊥ solid, W∥ dots. In [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The temporal evolution of the proton (solid) and α particle (dashed) beam-core drift velocities (Vd) for Case 1 (red), Case 2 (green), Case 3 (blue). The VDFs of protons and α particles in Vx-Vz phase-space plane obtained from the 2.5D hybrid model at the end of the run at t = 3000Ω−1 p are shown in [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The initial state of the VDFs and the results of the 2.5 D hybrid model at the end of the run for Case 2 (a) protons at t = 0, (b) α particles at t = 0, (c) protons at t = 3000Ω−1 p , (d) α particles at t = 3000. Same for Case 3 (e) protons at t = 0, (f) α particles at t = 0, (g) protons at t = 3000, (h) α particles at t = 3000. The corresponding line plots show the 1D VDFs dependence on Vx at Vz = 0 [PIT… view at source ↗
Figure 7
Figure 7. Figure 7: (a) The initial state of the proton core and beam VDFs in the 2.5D hybrid model for Case 4 (no α particles). (b)The evolved state of the proton VDFs at t = 600τA. The corresponding line plots show the 1D VDFs dependence on Vx at Vz = 0 (lower panels). In [PITH_FULL_IMAGE:figures/full_fig_p018_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The dispersion relation obtained from the 2.5D hybrid modeling results for Case 2. The frequency ω is in the unit of Ωp, while the k-vector is in the unit of inverse proton inertial length ωpp/c = Ωp/VA, where VA is the Alfv´en speed, and ωpp is the proton plasma frequency. Comprehensive evidence of complex anisotropic non-Maxwellian proton velocity distributions com￾prising core, beam, and ‘hammerhead’ po… view at source ↗
Figure 9
Figure 9. Figure 9: The kx − ky power spectrum of the perpendicular magnetic field component (By) obtained from the 2.5D hybrid modeling at t = 100, 400, 800, 1500 Ω−1 p for Case 2. The k-vector is in units of inverse proton inertial length ωpp/c = Ωp/CA. Motivated by recent PSP observations at perihelia, we use linear stability analysis, followed by nonlinear hybrid models to study the evolution of beam and ion temperature a… view at source ↗
Figure 10
Figure 10. Figure 10: Linear growth rates as a function of wavevector γmax(k)/Ωp for the seven cases described in [PITH_FULL_IMAGE:figures/full_fig_p024_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Linear frequencies and damping rates for the parallel forward fast (left column), parallel forward Alfv´en/ICW (center) and oblique Alfv´en solutions. (Top row) Real frequency ωr/Ωp for the designated solution for three selected cases. (Second, Third and Fourth Rows) Power absorption and emission per wave period from population j, |γj |/|ωr|, as well as the total absorption or emission. Solid and dashed l… view at source ↗
read the original abstract

Recent observations of the solar wind ions by the SPAN-I instruments on board the Parker Solar Probe (PSP) spacecraft at solar perihelia (Encounters) 4 and closer find ample evidence of complex anisotropic non-Maxwellian velocity distributions that consist of core, beam, and `hammerhead' (i.e., anisotropic beam) populations. The proton core populations are anisotropic, with T_perp/T||>1, and the beams have super-Alfvenic speed relative to the core (we provide an example from Encounter 17). The alpha-particle population show similar features as the protons. These unstable VDFs are associated with enhanced, right-hand (RH) and left-hand (LH) polarized ion-scale kinetic wave activity, detected by the FIELDS instrument. Motivated by PSP observations, we employ nonlinear hybrid models to investigate the evolution of the anisotropic hot-beam VDFs and model the growth and the nonlinear stage of ion kinetic instabilities in several linearly unstable cases. The models are initialized with ion VDFs motivated by the observational parameters. We find rapidly growing (in terms of proton gyroperiods) combined ion-cyclotron (IC) and magnetosonic (MS) instabilities, which produce LH and RH ion-scale wave spectra, respectively. The modeled ion VDFs in the nonlinear stage of the evolution are qualitatively in agreement with PSP observations of the anisotropic core and `hammerhead' velocity distributions, quantifying the effect of the ion kinetic instabilities on wind plasma heating close to the Sun. We conclude that the wave-particle interactions play an important role in the energy transfer between the magnetic energy (waves) and random particle motion leading to anisotropic solar wind plasma heating.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The paper claims that nonlinear hybrid simulations of the solar wind, initialized with anisotropic hot-beam ion VDFs drawn from PSP SPAN-I observations (e.g., Encounter 17 parameters for core-beam density ratio, super-Alfvénic beam speed, and T_perp/T_parallel ratios), exhibit rapid growth of combined ion-cyclotron (LH) and magnetosonic (RH) instabilities. The nonlinear stage produces ion VDFs that are qualitatively similar to the observed anisotropic core plus hammerhead distributions, thereby quantifying the contribution of wave-particle interactions to anisotropic heating near the Sun.

Significance. If the qualitative agreement can be placed on a quantitative footing, the work would provide a useful bridge between PSP observations of unstable VDFs and the kinetic processes that regulate inner-heliosphere heating. The hybrid approach is a standard tool for ion-scale dynamics, and the initialization from independent measurements avoids direct fitting of the final state. However, the current reliance on visual comparison alone limits the strength of the heating-quantification claim.

major comments (2)
  1. [Abstract] Abstract: the central claim that the nonlinear-stage VDFs are 'qualitatively in agreement' with PSP observations and thereby 'quantify' the effect of instabilities on heating rests entirely on visual inspection. No quantitative metrics (e.g., Wasserstein distance between VDFs, relative errors in core anisotropy or beam speed, or resolution-convergence tests) are reported, which is load-bearing for the assertion that the simulations demonstrate a causal pathway to the observed features.
  2. [Model initialization paragraph] Model initialization paragraph: while the initial core-beam parameters are taken from PSP data without post-hoc fitting of the saturated state, the manuscript does not describe control runs initialized with stable or sub-critical anisotropy ratios. Such runs would strengthen the demonstration that the hammerhead morphology emerges specifically from the instability evolution rather than being inherited from the chosen initial conditions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments that highlight opportunities to strengthen the quantitative support for our claims. We address each major comment below and will incorporate revisions to improve the manuscript.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that the nonlinear-stage VDFs are 'qualitatively in agreement' with PSP observations and thereby 'quantify' the effect of instabilities on heating rests entirely on visual inspection. No quantitative metrics (e.g., Wasserstein distance between VDFs, relative errors in core anisotropy or beam speed, or resolution-convergence tests) are reported, which is load-bearing for the assertion that the simulations demonstrate a causal pathway to the observed features.

    Authors: We agree that the current reliance on visual comparison alone limits the strength of the heating-quantification claim. In the revised manuscript we will add quantitative metrics, including Wasserstein distances between the simulated nonlinear-stage VDFs and the PSP observations, relative errors on core anisotropy, beam speed, and density ratio, plus resolution-convergence tests for the primary runs. These additions will place the qualitative agreement on a firmer footing while preserving the paper's focus on the instability-driven evolution. revision: yes

  2. Referee: [Model initialization paragraph] Model initialization paragraph: while the initial core-beam parameters are taken from PSP data without post-hoc fitting of the saturated state, the manuscript does not describe control runs initialized with stable or sub-critical anisotropy ratios. Such runs would strengthen the demonstration that the hammerhead morphology emerges specifically from the instability evolution rather than being inherited from the chosen initial conditions.

    Authors: We acknowledge that explicit control runs would strengthen the demonstration of causality. In the revised manuscript we will include additional hybrid simulations initialized with stable (sub-critical) anisotropy ratios drawn from the same observational parameter space. These runs will show that the hammerhead features do not develop in the absence of the instability, confirming that the morphology arises from the nonlinear wave-particle interactions rather than from the initial conditions alone. revision: yes

Circularity Check

0 steps flagged

No significant circularity: forward hybrid evolution from independent observational initial conditions

full rationale

The paper initializes nonlinear hybrid simulations using ion VDF parameters taken directly from PSP observations and evolves them under the standard hybrid plasma equations. The reported qualitative agreement in the nonlinear stage arises from the dynamical action of the model (IC and MS instabilities, wave-particle interactions) rather than any redefinition, fitting of final states, or self-citation chain that reduces the result to the inputs. The central claim therefore remains a genuine prediction from the physics, not a tautology.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The model depends on observational parameters extracted from PSP data as initial conditions and on the standard hybrid plasma approximation; no new entities are postulated.

free parameters (1)
  • initial core-beam density ratio, beam speed, and T_perp/T_parallel ratios
    Chosen to match PSP SPAN-I measurements for the unstable cases; these set the linear growth rates and saturation levels.
axioms (1)
  • domain assumption Hybrid approximation (kinetic ions, fluid electrons) captures the dominant ion-scale instabilities
    Invoked when the authors state they employ nonlinear hybrid models to investigate ion kinetic instabilities.

pith-pipeline@v0.9.0 · 5875 in / 1362 out tokens · 40980 ms · 2026-05-22T22:00:48.619381+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

78 extracted references · 78 canonical work pages

  1. [1]

    K., Hellinger, P., et al

    Alexandrova, O., Jagarlamudi, V. K., Hellinger, P., et al. 2021, PhRvE, 103, 063202, doi: 10.1103/PhysRevE.103.063202

    work page doi:10.1103/physreve.103.063202 2021

  2. [2]

    A., Marsch, E., & Vi˜ nAs, A

    Araneda, J. A., Marsch, E., & Vi˜ nAs, A. F. 2007, Journal of Geophysical Research (Space Physics), 112, A04104, doi: 10.1029/2006JA011999

    work page doi:10.1029/2006ja011999 2007

  3. [3]

    D., Kasper, J

    Bale, S. D., Kasper, J. C., Howes, G. G., et al. 2009, PhRvL, 103, 211101, doi: 10.1103/PhysRevLett.103.211101

    work page doi:10.1103/physrevlett.103.211101 2009

  4. [4]

    D., Goetz, K., Harvey, P

    Bale, S. D., Goetz, K., Harvey, P. R., et al. 2016, SSRv, 204, 49, doi: 10.1007/s11214-016-0244-5

    work page doi:10.1007/s11214-016-0244-5 2016

  5. [5]

    J., et al

    McComas, D. J., et al. 2022, ApJL, 926, L1, doi: 10.3847/2041-8213/ac4a5c

    work page doi:10.3847/2041-8213/ac4a5c 2022

  6. [6]

    A., Hospodarsky, G

    Boardsen, S. A., Hospodarsky, G. B., Kletzing, C. A., et al. 2016, Journal of Geophysical Research (Space Physics), 121, 2902, doi: 10.1002/2015JA021844

    work page doi:10.1002/2015ja021844 2016

  7. [7]

    Bourouaine, S., & Chandran, B. D. G. 2013, The Astrophysical Journal, 774, 96, doi: 10.1088/0004-637X/774/2/96

    work page doi:10.1088/0004-637x/774/2/96 2013

  8. [8]

    Bourouaine, S., Verscharen, D., Chandran, B. D. G., Maruca, B. A., & Kasper, J. C. 2013, ApJL, 777, L3, doi: 10.1088/2041-8205/777/1/L3

    work page doi:10.1088/2041-8205/777/1/l3 2013

  9. [9]

    A., Mallet, A., Huang, J., et al

    Bowen, T. A., Mallet, A., Huang, J., et al. 2020, ApJS, 246, 66, doi: 10.3847/1538-4365/ab6c65

    work page doi:10.3847/1538-4365/ab6c65 2020

  10. [10]

    A., Vasko, I

    Bowen, T. A., Vasko, I. Y., Bale, S. D., et al. 2024a, ApJL, 972, L8, doi: 10.3847/2041-8213/ad6b2e —. 2024b, ApJL, 972, L8, doi: 10.3847/2041-8213/ad6b2e

    work page doi:10.3847/2041-8213/ad6b2e 2041

  11. [11]

    A., Bale, S

    Bowen, T. A., Bale, S. D., Chandran, B. D. G., et al. 2024c, Nature Astronomy, 8, 482, doi: 10.1038/s41550-023-02186-4

    work page doi:10.1038/s41550-023-02186-4

  12. [12]

    Chen, C. H. K., Leung, L., Boldyrev, S., Maruca, B. A., & Bale, S. D. 2014, Geophys. Res. Lett., 41, 8081, doi: 10.1002/2014GL062009

    work page doi:10.1002/2014gl062009 2014

  13. [13]

    Chen, C. H. K., Matteini, L., Schekochihin, A. A., et al. 2016, ApJL, 825, L26, doi: 10.3847/2041-8205/825/2/L26

    work page doi:10.3847/2041-8205/825/2/l26 2016

  14. [14]

    J., Velli, M

    Fox, N. J., Velli, M. C., Bale, S. D., et al. 2016, SSRv, 204, 7, doi: 10.1007/s11214-015-0211-6

    work page doi:10.1007/s11214-015-0211-6 2016

  15. [15]

    P., Yin, L., & Winske, D

    Gary, S. P., Yin, L., & Winske, D. 2006, Journal of Geophysical Research (Space Physics), 111, A06105, doi: 10.1029/2005JA011552

    work page doi:10.1029/2005ja011552 2006

  16. [16]

    P., Yin, L., Winske, D., & Ofman, L

    Gary, S. P., Yin, L., Winske, D., & Ofman, L. 2001, J. Geophys. Res., 106, 10715, doi: 10.1029/2000JA000406

    work page doi:10.1029/2000ja000406 2001

  17. [17]

    P., Yin, L., Winske, D., et al

    Gary, S. P., Yin, L., Winske, D., et al. 2003, Journal of Geophysical Research (Space Physics), 108, 1068, doi: 10.1029/2002JA009654

    work page doi:10.1029/2002ja009654 2003

  18. [18]

    P., Yin, L., Winske, D., & Reisenfeld, D

    Gary, S. P., Yin, L., Winske, D., & Reisenfeld, D. B. 2000, Geophys. Res. Lett., 27, 1355, doi: 10.1029/2000GL000019

    work page doi:10.1029/2000gl000019 2000

  19. [19]

    2006, Journal of Geophysical Research (Space Physics), 111, A01107, doi: 10.1029/2005JA011318

    Hellinger, P., & Tr´ avn´ ıˇ cek, P. 2006, Journal of Geophysical Research (Space Physics), 111, A01107, doi: 10.1029/2005JA011318

    work page doi:10.1029/2005ja011318 2006

  20. [20]

    C., Lazarus, A

    Kasper, J. C., Lazarus, A. J., & Gary, S. P. 2008, Physical Review Letters, 101, 261103, doi: 10.1103/PhysRevLett.101.261103 Modeling hot, anisotropic ion beams 29

    work page doi:10.1103/physrevlett.101.261103 2008

  21. [21]

    C., Abiad, R., Austin, G., et al

    Kasper, J. C., Abiad, R., Austin, G., et al. 2016, SSRv, 204, 131, doi: 10.1007/s11214-015-0206-3

    work page doi:10.1007/s11214-015-0206-3 2016

  22. [22]

    C., Klein, K

    Kasper, J. C., Klein, K. G., Lichko, E., et al. 2021, PhRvL, 127, 255101, doi: 10.1103/PhysRevLett.127.255101

    work page doi:10.1103/physrevlett.127.255101 2021

  23. [23]

    Vech, D., & Kasper, J. C. 2018, PhRvL, 120, 205102, doi: 10.1103/PhysRevLett.120.205102

    work page doi:10.1103/physrevlett.120.205102 2018

  24. [24]

    G., & Howes, G

    Klein, K. G., & Howes, G. G. 2015, Phys. Plasma, 22, 032903, doi: 10.1063/1.4914933

    work page doi:10.1063/1.4914933 2015

  25. [25]

    Valentini, F. 2020, J. Phys. (Paris), 86, 905860402, doi: 10.1017/S0022377820000689

    work page doi:10.1017/s0022377820000689 2020

  26. [26]

    Stevens, M. L. 2017, J. Geophys. Res., 122, 9815, doi: 10.1002/2017JA024486

    work page doi:10.1002/2017ja024486 2017

  27. [27]

    G., Verniero, J

    Klein, K. G., Verniero, J. L., Alterman, B., et al. 2021, ApJ, 909, 7, doi: 10.3847/1538-4357/abd7a0

    work page doi:10.3847/1538-4357/abd7a0 2021

  28. [28]

    E., Kasper, J

    Livi, R., Larson, D. E., Kasper, J. C., et al. 2021, Earth and Space Science Open Archive, 20, doi: 10.1002/essoar.10508651.1

    work page doi:10.1002/essoar.10508651.1 2021

  29. [29]

    E., Kasper, J

    Livi, R., Larson, D. E., Kasper, J. C., et al. 2022, ApJ, 938, 138, doi: 10.3847/1538-4357/ac93f5

    work page doi:10.3847/1538-4357/ac93f5 2022

  30. [30]

    2021, A&A, 656, A36, doi: 10.1051/0004-6361/202141095

    Louarn, P., Fedorov, A., Prech, L., et al. 2021, A&A, 656, A36, doi: 10.1051/0004-6361/202141095

    work page doi:10.1051/0004-6361/202141095 2021

  31. [31]

    G., Ofman, L., & Vi˜ nas, A

    Maneva, Y. G., Ofman, L., & Vi˜ nas, A. 2015, A&A, 578, A85, doi: 10.1051/0004-6361/201424401

    work page doi:10.1051/0004-6361/201424401 2015

  32. [32]

    G., Vi˜ nAs, A

    Maneva, Y. G., Vi˜ nAs, A. F., & Ofman, L. 2013, Journal of Geophysical Research (Space Physics), 118, 2842, doi: 10.1002/jgra.50363

    work page doi:10.1002/jgra.50363 2013

  33. [33]

    A., & Vasquez, B

    Markovskii, S. A., & Vasquez, B. J. 2024, ApJ, 966, 1, doi: 10.3847/1538-4357/ad3727

    work page doi:10.3847/1538-4357/ad3727 2024

  34. [34]

    A., Vasquez, B

    Markovskii, S. A., Vasquez, B. J., & Chandran, B. D. G. 2020, ApJ, 889, 7, doi: 10.3847/1538-4357/ab5af3

    work page doi:10.3847/1538-4357/ab5af3 2020

  35. [35]

    Marsch, Living Rev

    Marsch, E. 2006, Living Reviews in Solar Physics, 3, 1, doi: 10.12942/lrsp-2006-1 —. 2012, SSRv, 172, 23, doi: 10.1007/s11214-010-9734-z

    work page doi:10.12942/lrsp-2006-1 2006

  36. [37]

    H., & Neubauer, F

    Muehlhaeuser, K. H., & Neubauer, F. M. 1982b, J. Geophys. Res., 87, 35, doi: 10.1029/JA087iA01p00035 Martinovi´ c, M. M., & Klein, K. G. 2023, ApJ, 952, 14, doi: 10.3847/1538-4357/acdb79 Martinovi´ c, M. M., Klein, K. G., De Marco, R., et al. 2024, arXiv e-prints, arXiv:2412.04885, doi: 10.48550/arXiv.2412.04885 Martinovi´ c, M. M., Klein, K. G.,ˇDurovcov...

    work page doi:10.1029/ja087ia01p00035 2023

  37. [38]

    A., Kasper, J

    Maruca, B. A., Kasper, J. C., & Gary, S. P. 2012, ApJ, 748, 137, doi: 10.1088/0004-637X/748/2/137

    work page doi:10.1088/0004-637x/748/2/137 2012

  38. [39]

    D., Verniero, J., Bale, S

    McManus, M. D., Verniero, J., Bale, S. D., et al. 2022, ApJ, 933, 43, doi: 10.3847/1538-4357/ac6ba3

    work page doi:10.3847/1538-4357/ac6ba3 2022

  39. [40]

    D., Klein, K

    McManus, M. D., Klein, K. G., Bale, S. D., et al. 2024, ApJ, 961, 142, doi: 10.3847/1538-4357/ad05ba

    work page doi:10.3847/1538-4357/ad05ba 2024

  40. [41]

    C., McManus, M

    Mostafavi, P., Allen, R. C., McManus, M. D., et al. 2022, The Astrophysical Journal Letters, 926, L38, doi: 10.3847/2041-8213/ac51e1

    work page doi:10.3847/2041-8213/ac51e1 2022

  41. [42]

    C., Jagarlamudi, V

    Mostafavi, P., Allen, R. C., Jagarlamudi, V. K., et al. 2024, A&A, 682, A152, doi: 10.1051/0004-6361/202347134 M¨ uller, D., St. Cyr, O. C., Zouganelis, I., et al. 2020, A&A, 642, A1, doi: 10.1051/0004-6361/202038467

    work page doi:10.1051/0004-6361/202347134 2024

  42. [43]

    Neugebauer, M. 1976, J. Geophys. Res., 81, 78, doi: 10.1029/JA081i001p00078

    work page doi:10.1029/ja081i001p00078 1976

  43. [44]

    2010a, J

    Ofman, L. 2010a, J. Geophys. Res., 115, 4108, doi: 10.1029/2009JA015094 —. 2010b, Journal of Geophysical Research (Space Physics), 115, A04108, doi: 10.1029/2009JA015094 —. 2019, SoPh, 294, 51, doi: 10.1007/s11207-019-1440-8

    work page doi:10.1029/2009ja015094 2019

  44. [45]

    A., Jian, L

    Ofman, L., Boardsen, S. A., Jian, L. K., Verniero, J. L., & Larson, D. 2022, ApJ, 926, 185, doi: 10.3847/1538-4357/ac402c

    work page doi:10.3847/1538-4357/ac402c 2022

  45. [46]

    Ofman, L., & Vi˜ nas, A. F. 2007, J. Geophys. Res., 112, 6104, doi: 10.1029/2006JA012187

    work page doi:10.1029/2006ja012187 2007

  46. [47]

    F., & Maneva, Y

    Ofman, L., Vi˜ nas, A. F., & Maneva, Y. 2014, Journal of Geophysical Research (Space Physics), 119, 4223, doi: 10.1002/2013JA019590

    work page doi:10.1002/2013ja019590 2014

  47. [48]

    F., & Moya, P

    Ofman, L., Vi˜ nas, A. F., & Moya, P. S. 2011, Annales Geophysicae, 29, 1071, doi: 10.5194/angeo-29-1071-2011

    work page doi:10.5194/angeo-29-1071-2011 2011

  48. [49]

    F., & Roberts, D

    Ofman, L., Vi˜ nas, A. F., & Roberts, D. A. 2017, Journal of Geophysical Research (Space Physics), 122, 5839, doi: 10.1002/2016JA023705

    work page doi:10.1002/2016ja023705 2017

  49. [50]

    A., Jian, L

    Ofman, L., Boardsen, S. A., Jian, L. K., et al. 2023, ApJ, 954, 109, doi: 10.3847/1538-4357/acea7e

    work page doi:10.3847/1538-4357/acea7e 2023

  50. [51]

    Ozak, N., Ofman, L., & Vi˜ nas, A. F. 2015, ApJ, 799, 77, doi: 10.1088/0004-637X/799/1/77

    work page doi:10.1088/0004-637x/799/1/77 2015

  51. [52]

    2024, ApJ, 977, 27, doi: 10.3847/1538-4357/ad79fa

    Peng, J., He, J., Duan, D., & Verscharen, D. 2024, ApJ, 977, 27, doi: 10.3847/1538-4357/ad79fa

    work page doi:10.3847/1538-4357/ad79fa 2024

  52. [53]

    N., Bacchini, F., et al

    Pezzini, L., Zhukov, A. N., Bacchini, F., et al. 2024, ApJ, 975, 37, doi: 10.3847/1538-4357/ad7465

    work page doi:10.3847/1538-4357/ad7465 2024

  53. [54]

    1998, ApJ, 500, 978, doi: 10.1086/305770

    Quataert, E. 1998, ApJ, 500, 978, doi: 10.1086/305770

    work page doi:10.1086/305770 1998

  54. [55]

    E., Matteini, L., Squire, J., et al

    Raouafi, N. E., Matteini, L., Squire, J., et al. 2023, SSRv, 219, 8, doi: 10.1007/s11214-023-00952-4

    work page doi:10.1007/s11214-023-00952-4 2023

  55. [56]

    A., & Yogesh

    Sadykov, V., Ofman, L., Boardsen, S. A., & Yogesh. 2025, New Astronomy, in prep

    work page 2025

  56. [57]

    M., Lazar, M., L´ opez, R

    Shaaban, S. M., Lazar, M., L´ opez, R. A., Yoon, P. H., & Poedts, S. 2024, A&A, 692, L6, doi: 10.1051/0004-6361/202452205

    work page doi:10.1051/0004-6361/202452205 2024

  57. [58]

    G., & Martinovi´ c, M

    Shankarappa, N., Klein, K. G., & Martinovi´ c, M. M. 2023, ApJ, 946, 85, doi: 10.3847/1538-4357/acb542 Modeling hot, anisotropic ion beams 31

    work page doi:10.3847/1538-4357/acb542 2023

  58. [59]

    G., Martinovi´ c, M

    Shankarappa, N., Klein, K. G., Martinovi´ c, M. M., & Bowen, T. A. 2024, ApJ, 973, 20, doi: 10.3847/1538-4357/ad5f2a

    work page doi:10.3847/1538-4357/ad5f2a 2024

  59. [60]

    2024, ApJL, 971, L41, doi: 10.3847/2041-8213/ad68fb

    Shi, C., Zhao, J., Liu, S., et al. 2024, ApJL, 971, L41, doi: 10.3847/2041-8213/ad68fb

    work page doi:10.3847/2041-8213/ad68fb 2024

  60. [61]

    Squire, J., Meyrand, R., & Kunz, M. W. 2023, ApJL, 957, L30, doi: 10.3847/2041-8213/ad0779

    work page doi:10.3847/2041-8213/ad0779 2023

  61. [62]

    W., et al

    Squire, J., Meyrand, R., Kunz, M. W., et al. 2022, Nature Astronomy, 6, 715, doi: 10.1038/s41550-022-01624-z

    work page doi:10.1038/s41550-022-01624-z 2022

  62. [63]

    G., & Kasper, J

    Vech, D., Klein, K. G., & Kasper, J. C. 2017, The Astrophysical Journal Letters, 850, L11, doi: 10.3847/2041-8213/aa9887

    work page doi:10.3847/2041-8213/aa9887 2017

  63. [64]

    G., & Kasper, J

    Vech, D., Mallet, A., Klein, K. G., & Kasper, J. C. 2018, ApJL, 855, L27, doi: 10.3847/2041-8213/aab351

    work page doi:10.3847/2041-8213/aab351 2018

  64. [65]

    M., Klein, K

    Vech, D., Martinovi´ c, M. M., Klein, K. G., et al. 2021, A&A, 650, A10, doi: 10.1051/0004-6361/202039296

    work page doi:10.1051/0004-6361/202039296 2021

  65. [66]

    L., Larson, D

    Verniero, J. L., Larson, D. E., Livi, R., et al. 2020, ApJS, 248, 5, doi: 10.3847/1538-4365/ab86af

    work page doi:10.3847/1538-4365/ab86af 2020

  66. [67]

    L., Chandran, B

    Verniero, J. L., Chandran, B. D. G., Larson, D. E., et al. 2022, ApJ, 924, 112, doi: 10.3847/1538-4357/ac36d5

    work page doi:10.3847/1538-4357/ac36d5 2022

  67. [68]

    Verscharen, D., Bourouaine, S., & Chandran, B. D. G. 2013, ApJ, 773, 163, doi: 10.1088/0004-637X/773/2/163

    work page doi:10.1088/0004-637x/773/2/163 2013

  68. [69]

    G., & Maruca, B

    Verscharen, D., Klein, K. G., & Maruca, B. A. 2019, Living Reviews in Solar Physics, 16, 5, doi: 10.1007/s41116-019-0021-0

    work page doi:10.1007/s41116-019-0021-0 2019

  69. [70]

    2011, Journal of Plasma Physics, 77, 693, doi: 10.1017/S0022377811000080

    Verscharen, D., & Marsch, E. 2011, Journal of Plasma Physics, 77, 693, doi: 10.1017/S0022377811000080

    work page doi:10.1017/s0022377811000080 2011

  70. [71]

    2012, PhRvE, 86, 027401, doi: 10.1103/PhysRevE.86.027401

    Verscharen, D., Marsch, E., Motschmann, U., & M¨ uller, J. 2012, PhRvE, 86, 027401, doi: 10.1103/PhysRevE.86.027401

    work page doi:10.1103/physreve.86.027401 2012

  71. [72]

    G., Lichko, E., et al

    Walters, J., Klein, K. G., Lichko, E., et al. 2023, ApJ, 955, 97, doi: 10.3847/1538-4357/acf1fa

    work page doi:10.3847/1538-4357/acf1fa 2023

  72. [73]

    1978, Computing, 20, 333

    Wambecq, A. 1978, Computing, 20, 333

    work page 1978

  73. [74]

    1993, Computer Space Plasma Physics: Simulation Techniques and

    Winske, D., & Omidi, N. 1993, Computer Space Plasma Physics: Simulation Techniques and

    work page 1993

  74. [75]

    D., Horbury, T

    Woodham, L. D., Horbury, T. S., Matteini, L., et al. 2021, A&A, 650, L1, doi: 10.1051/0004-6361/202039415

    work page doi:10.1051/0004-6361/202039415 2021

  75. [76]

    2004, Journal of Geophysical Research (Space Physics), 109, A08103, doi: 10.1029/2004JA010501

    Xie, H., Ofman, L., & Vi˜ nAs, A. 2004, Journal of Geophysical Research (Space Physics), 109, A08103, doi: 10.1029/2004JA010501

    work page doi:10.1029/2004ja010501 2004

  76. [77]

    Y., Parashar, T

    Xiong, Q. Y., Parashar, T. N., Huang, S. Y., et al. 2024, ApJ, 968, 93, doi: 10.3847/1538-4357/ad4643

    work page doi:10.3847/1538-4357/ad4643 2024

  77. [78]

    Yoon, P. H. 2017, Reviews of Modern Plasma Physics, 1, 4, doi: 10.1007/s41614-017-0006-1

    work page doi:10.1007/s41614-017-0006-1 2017

  78. [79]

    H., Lazar, M., Salem, C., et al

    Yoon, P. H., Lazar, M., Salem, C., et al. 2024, ApJ, 969, 77, doi: 10.3847/1538-4357/ad47f1

    work page doi:10.3847/1538-4357/ad47f1 2024