pith. sign in

arxiv: 2607.01644 · v1 · pith:IE4V7C4Onew · submitted 2026-07-02 · 🌌 astro-ph.SR

Numerical Investigation of Efficient Electron Acceleration at an Unsteady Solar Flare Loop-Top

Yoshiaki Sato , Takafumi Kaneko , Noriyuki Narukage , Shinsuke Takasao This is my paper

Pith reviewed 2026-07-03 05:31 UTC · model grok-4.3

classification 🌌 astro-ph.SR
keywords solar flareselectron accelerationloop-topbetatron mechanismMHD simulationtest-particleplasmoid
0
0 comments X

The pith

Time-varying loop-top structures in solar flares accelerate electrons more efficiently than steady ones via betatron energization at the edges.

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

The paper investigates how time-dependent dynamics at the top of solar flare loops affect electron acceleration using MHD fields and guiding-center test-particle calculations. It establishes that unsteady velocity fields enhance net acceleration by allowing the betatron mechanism to energize electrons in compressed magnetic regions at the loop-top edge. In contrast, quasi-steady fields lead to suppression because betatron cooling offsets gains. The central argument is that acceleration depends on dynamic events such as plasmoid collisions, not only the static magnetic geometry, and that processes at the exit point where electrons leave the loop-top are important. This matters because prior work has emphasized reconnection outflows like termination shocks, so including loop-top dynamics could change how models account for observed flare particle energies.

Core claim

A time-varying loop-top velocity field enhances acceleration efficiency because the betatron mechanism readily accelerates electrons in the compressed magnetic field at the edge, whereas a quasi-steady field suppresses net acceleration through the decelerating effect of betatron cooling; thus acceleration at the loop-top is driven by dynamic events such as plasmoid collisions in addition to static structure.

What carries the argument

The unsteady MHD velocity field at the loop-top, which modulates guiding-center electron motion to produce net betatron acceleration in time-varying compressed B regions.

If this is right

  • Electrons gain energy specifically at the loop-top edge when the velocity field varies in time.
  • Plasmoid collisions and similar dynamic events contribute to driving the acceleration.
  • Processes at the exit point where electrons escape the loop-top affect the overall energization.
  • Acceleration models limited to reconnection outflows may understate the role of loop-top dynamics.

Where Pith is reading between the lines

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

  • Flare observations with variable loop-top motions may show higher particle energies than steady cases.
  • Simulations adding turbulence or wave effects could test whether the steady-unsteady efficiency gap persists.
  • Similar unsteady structures in other reconnection environments might produce comparable acceleration patterns.

Load-bearing premise

The MHD fields combined with guiding-center test-particle motion capture the main energization without other processes such as wave-particle interactions or non-adiabatic effects contributing substantially.

What would settle it

Direct comparison of electron energy gains or spectra from flares with identified steady versus unsteady loop-tops that shows no difference in acceleration efficiency.

Figures

Figures reproduced from arXiv: 2607.01644 by Noriyuki Narukage, Shinsuke Takasao, Takafumi Kaneko, Yoshiaki Sato.

Figure 1
Figure 1. Figure 1: Time evolution of the number density distribu￾tion in the reconnection region from the MHD simulation. The upper panels show the overall view at t = 100, 150, and 200 s, illustrating the large-scale evolution of the recon￾nection region. The lower panels show zoomed-in views of the loop-top region at t = 130, 134, and 138 s, highlighting the transition from a quasi-steady state (t = 130 s) to an unsteady s… view at source ↗
Figure 2
Figure 2. Figure 2: Electron acceleration in the quasi-steady loop-top configuration at t = 130 s. This figure shows the trajectories and energy evolution for two selected electrons. (a) Overview of electron trajectories with magnetic field lines. (b, f) Trajectories on the magnetic-field strength (B) map. (c, g) Time evolution of kinetic energy (black), cumulative Fermi reflection energy gain (orange), and cumulative betatro… view at source ↗
Figure 3
Figure 3. Figure 3: Spatial distributions of quantities governing acceleration mechanisms at t = 130 s. Panels (a)-(c) show factors for Fermi reflection: (a) vertical plasma flow (uey), (b) magnetic curvature ([(b · ∇)b]y), and (c) the resulting acceleration rate. Panels (d)-(f) show factors for betatron acceleration: (d) transverse plasma flow (uex), (e) magnetic field gradient (∂B/∂x), and (f) the resulting acceleration rat… view at source ↗
Figure 4
Figure 4. Figure 4: Electron acceleration in the unsteady loop-top at t = 138 s, perturbed by a plasmoid collision. The panel layout is the same as in [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Same as [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: A schematic of the electron acceleration mechanism at the flare loop-top. In a quasi-steady state (top right), the inward plasma flow (uE) driven by slow shocks opposes the outward magnetic field gradient (∇B), resulting in betatron cooling. Conversely, in an unsteady, expanding loop-top (bottom right), the outward plasma flow aligns with the magnetic field gradient, leading to positive betatron accelerati… view at source ↗
Figure 7
Figure 7. Figure 7: Spatial regions used for particle selection in the calculation with loop-top-only initial conditions. Panels (a) and (b) show magnetic field lines for the quasi-steady and unsteady snapshots; black contours mark the selected loop-top-edge regions [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
read the original abstract

Using magnetohydrodynamic (MHD) fields and guiding-center test-particle calculations, this study investigates how time-dependent loop-top dynamics modulate the adiabatic energization of electrons in a solar flare. Our results indicate that a time-varying loop-top structure enhances acceleration efficiency compared to a quasi-steady one. In the quasi-steady velocity field, the net acceleration is suppressed due to the decelerating effect of betatron cooling. Conversely, in the unsteady velocity field, the betatron mechanism readily accelerates electrons within the compressed magnetic field at the edge of the loop-top. These findings suggest that the acceleration of electrons at the loop-top is driven not only by the static shape of the magnetic structure but also by dynamic events such as plasmoid collisions. While previous studies have primarily focused on acceleration processes within the reconnection outflow, such as at termination shocks or within plasmoids, our research highlights the importance of the acceleration and deceleration processes at the exit point where electrons escape from the loop-top.

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

1 major / 1 minor

Summary. The manuscript uses MHD fields combined with guiding-center test-particle calculations to investigate adiabatic electron energization at a solar flare loop-top. It claims that time-varying (unsteady) loop-top structures enhance acceleration efficiency relative to quasi-steady cases, where betatron cooling suppresses net gains; in the unsteady case, betatron acceleration occurs readily in compressed fields at the loop-top edge, driven by dynamic events such as plasmoid collisions. The work argues that acceleration at the loop-top exit point is important in addition to processes in reconnection outflows.

Significance. If the central distinction between unsteady and quasi-steady cases holds under the stated approximations, the result would usefully redirect attention to time-dependent loop-top dynamics as a contributor to flare electron acceleration. The direct numerical experiments (MHD + test particles) constitute a concrete strength that allows falsifiable comparison of the two velocity-field regimes.

major comments (1)
  1. [Numerical methods / test-particle section] The central claim rests on the guiding-center test-particle results distinguishing unsteady from quasi-steady acceleration. No checks of the adiabaticity ordering (gyroradius ≪ field scale length; temporal variation slow compared with gyroperiod) or cross-validation against full-orbit integration are reported for the rapid compressions and plasmoid collisions at the loop-top edge. This omission is load-bearing because violation of the ordering would render the reported betatron enhancement an artifact of the approximation rather than a physical effect.
minor comments (1)
  1. [Abstract] Abstract and results sections lack quantitative metrics (e.g., energy spectra, acceleration timescales, efficiency ratios), grid resolution, or error estimates, making it difficult to assess the magnitude and robustness of the claimed enhancement.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thoughtful review and for highlighting the importance of validating the guiding-center approximation. We address the single major comment below and commit to revisions that strengthen the numerical methods section.

read point-by-point responses

  1. Referee: [Numerical methods / test-particle section] The central claim rests on the guiding-center test-particle results distinguishing unsteady from quasi-steady acceleration. No checks of the adiabaticity ordering (gyroradius ≪ field scale length; temporal variation slow compared with gyroperiod) or cross-validation against full-orbit integration are reported for the rapid compressions and plasmoid collisions at the loop-top edge. This omission is load-bearing because violation of the ordering would render the reported betatron enhancement an artifact of the approximation rather than a physical effect.

    Authors: We agree that explicit verification of the adiabaticity ordering is necessary to support the central claim. In the revised manuscript we will add a dedicated subsection (and associated figure) that quantifies, for representative electron energies and locations including the loop-top edge during plasmoid collisions, (i) the ratio of gyroradius to local magnetic-field gradient scale length and (ii) the ratio of the MHD temporal variation timescale to the gyroperiod. We will also integrate a statistically meaningful subset of trajectories with the full Lorentz-force equations and directly compare the resulting energy spectra and spatial distributions with the guiding-center results; any discrepancies will be reported. These additions will be placed in the numerical-methods section so that readers can assess the validity of the approximation under the specific conditions of the unsteady runs. revision: yes

Circularity Check

0 steps flagged

No circularity; results from direct numerical experiments

full rationale

The paper performs MHD simulations of quasi-steady versus unsteady loop-top flows, then integrates guiding-center test particles in the resulting time-dependent fields. The reported efficiency difference follows directly from comparing the computed particle energy gains in the two cases; no parameters are fitted to observations and re-used as predictions, no equations are defined in terms of their own outputs, and no load-bearing claims rest on self-citations. The derivation chain is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no specific free parameters, axioms, or invented entities can be extracted from the full text.

pith-pipeline@v0.9.1-grok · 5712 in / 1082 out tokens · 38814 ms · 2026-07-03T05:31:26.894478+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

74 extracted references · 74 canonical work pages

  1. [1]

    2025, Publications of the Astronomical Society of Japan, doi: 10.1093/pasj/psaf123

    Akutagawa, K., Imada, S., & Shoda, M. 2025, Publications of the Astronomical Society of Japan, doi: 10.1093/pasj/psaf123

    work page doi:10.1093/pasj/psaf123 2025

  2. [2]

    2021, Physical Review Letters, 126, 135101, doi: 10.1103/PhysRevLett.126.135101

    Arnold, H., Drake, J., Swisdak, M., et al. 2021, Physical Review Letters, 126, 135101, doi: 10.1103/PhysRevLett.126.135101

    work page doi:10.1103/physrevlett.126.135101 2021

  3. [3]

    2004, The Astrophysical Journal, 605, L77, doi: 10.1086/420768

    Asai, A., Yokoyama, T., Shimojo, M., & Shibata, K. 2004, The Astrophysical Journal, 605, L77, doi: 10.1086/420768

    work page doi:10.1086/420768 2004

  4. [4]

    Aschwanden, M. J. 2002, Space Science Reviews, 101, 1, doi: 10.1023/A:1019712124366

    work page doi:10.1023/a:1019712124366 2002

  5. [5]

    Ashfield, W., Polito, V., Yu, S., Collier, H., & Hayes, L. A. 2024, The Astrophysical Journal, 973, 96, doi: 10.3847/1538-4357/ad64ca

    work page doi:10.3847/1538-4357/ad64ca 2024

  6. [6]

    2002, Astronomy and Astrophysics, 384, 273, doi: 10.1051/0004-6361:20011735

    Aurass, H., Vrˇ snak, B., & Mann, G. 2002, Astronomy and Astrophysics, 384, 273, doi: 10.1051/0004-6361:20011735

    work page doi:10.1051/0004-6361:20011735 2002

  7. [7]

    2024, MNRAS, 529, 2399, doi: 10.1093/mnras/stae723

    Bacchini, F., Ruan, W., & Keppens, R. 2024, MNRAS, 529, 2399, doi: 10.1093/mnras/stae723

    work page doi:10.1093/mnras/stae723 2024

  8. [8]

    Benz, A. O. 2017, Living Reviews in Solar Physics, 14, 2, doi: 10.1007/s41116-016-0004-3

    work page doi:10.1007/s41116-016-0004-3 2017

  9. [9]

    V., Baker, D

    Birn, J., Artemyev, A. V., Baker, D. N., et al. 2012, Space Science Reviews, 173, 49, doi: 10.1007/s11214-012-9874-4

    work page doi:10.1007/s11214-012-9874-4 2012

  10. [10]

    2017, The Astrophysical Journal, 848, 116, doi: 10.3847/1538-4357/aa8ad4

    Birn, J., Battaglia, M., Fletcher, L., Hesse, M., & Neukirch, T. 2017, The Astrophysical Journal, 848, 116, doi: 10.3847/1538-4357/aa8ad4

    work page doi:10.3847/1538-4357/aa8ad4 2017

  11. [11]

    F., & Hesse, M

    Birn, J., Thomsen, M. F., & Hesse, M. 2004, Physics of Plasmas, 11, 1825, doi: 10.1063/1.1704641

    work page doi:10.1063/1.1704641 2004

  12. [12]

    2016, Solar Physics, 291, 1385, doi: 10.1007/s11207-016-0915-0

    Borissov, A., Neukirch, T., & Threlfall, J. 2016, Solar Physics, 291, 1385, doi: 10.1007/s11207-016-0915-0

    work page doi:10.1007/s11207-016-0915-0 2016

  13. [13]

    C., et al

    Cai, Q., Shen, C., Raymond, J. C., et al. 2019, Monthly Notices of the Royal Astronomical Society, 489, 3183, doi: 10.1093/mnras/stz2167

    work page doi:10.1093/mnras/stz2167 2019

  14. [14]

    Che, H., & Zank, G. P. 2019, Journal of Physics: Conference Series, 1332, 012003, doi: 10.1088/1742-6596/1332/1/012003

    work page doi:10.1088/1742-6596/1332/1/012003 2019

  15. [15]

    Che, H., & Zank, G. P. 2020, The Astrophysical Journal, 889, 11, doi: 10.3847/1538-4357/ab5d3b

    work page doi:10.3847/1538-4357/ab5d3b 2020

  16. [16]

    S., Shen, C., et al

    Chen, B., Bastian, T. S., Shen, C., et al. 2015, Science, doi: 10.1126/science.aac8467

    work page doi:10.1126/science.aac8467 2015

  17. [17]

    2024, The Astrophysical Journal, 971, 85, doi: 10.3847/1538-4357/ad531a

    Chen, B., Kong, X., Yu, S., et al. 2024, The Astrophysical Journal, 971, 85, doi: 10.3847/1538-4357/ad531a

    work page doi:10.3847/1538-4357/ad531a 2024

  18. [18]

    T., Drake, J

    Dahlin, J. T., Drake, J. F., & Swisdak, M. 2014, Physics of Plasmas, 21, 092304, doi: 10.1063/1.4894484

    work page doi:10.1063/1.4894484 2014

  19. [19]

    T., Drake, J

    Dahlin, J. T., Drake, J. F., & Swisdak, M. 2017, Physics of Plasmas, 24, 092110, doi: 10.1063/1.4986211

    work page doi:10.1063/1.4986211 2017

  20. [20]

    2002, Journal of Computational Physics, 175, 645, doi: 10.1006/jcph.2001.6961

    Dedner, A., Kemm, F., Kr¨ oner, D., et al. 2002, Journal of Computational Physics, 175, 645, doi: 10.1006/jcph.2001.6961

    work page doi:10.1006/jcph.2001.6961 2002

  21. [21]

    F., Swisdak, M., Che, H., & Shay, M

    Drake, J. F., Swisdak, M., Che, H., & Shay, M. A. 2006, Nature, 443, 553, doi: 10.1038/nature05116

    work page doi:10.1038/nature05116 2006

  22. [22]

    2024, Astronomy and Astrophysics, 684, A171, doi: 10.1051/0004-6361/202347600 Eradat Oskoui, S., & Neukirch, T

    Druett, M., Ruan, W., & Keppens, R. 2024, Astronomy and Astrophysics, 684, A171, doi: 10.1051/0004-6361/202347600 Eradat Oskoui, S., & Neukirch, T. 2014, Astronomy and Astrophysics, 567, A131, doi: 10.1051/0004-6361/201423886 Eradat Oskoui, S., Neukirch, T., & Grady, K. J. 2014, Astronomy and Astrophysics, 563, A73, doi: 10.1051/0004-6361/201322519

    work page doi:10.1051/0004-6361/202347600 2024

  23. [23]

    1949, Physical Review, 75, 1169, doi: 10.1103/PhysRev.75.1169

    Fermi, E. 1949, Physical Review, 75, 1169, doi: 10.1103/PhysRev.75.1169

    work page doi:10.1103/physrev.75.1169 1949

  24. [24]

    2024, Annual Review of Astronomy and Astrophysics, 62, 437, doi: 10.1146/annurev-astro-052920-010547

    Fletcher, L. 2024, Annual Review of Astronomy and Astrophysics, 62, 437, doi: 10.1146/annurev-astro-052920-010547

    work page doi:10.1146/annurev-astro-052920-010547 2024

  25. [25]

    J., Hayes, L

    French, R. J., Hayes, L. A., Kazachenko, M. D., et al. 2024, The Astrophysical Journal, 977, 207, doi: 10.3847/1538-4357/ad8ed1

    work page doi:10.3847/1538-4357/ad8ed1 2024

  26. [26]

    2005, The Astrophysical Journal, 635, 636, doi: 10.1086/497366

    Giuliani, P., Neukirch, T., & Wood, P. 2005, The Astrophysical Journal, 635, 636, doi: 10.1086/497366

    work page doi:10.1086/497366 2005

  27. [27]

    Gordovskyy, M., Browning, P., & Pinto, R. F. 2019, Advances in Space Research, 63, 1453, doi: 10.1016/j.asr.2018.09.024

    work page doi:10.1016/j.asr.2018.09.024 2019

  28. [28]

    J., & Neukirch, T

    Grady, K. J., & Neukirch, T. 2009, Astronomy and Astrophysics, 508, 1461, doi: 10.1051/0004-6361/200913230

    work page doi:10.1051/0004-6361/200913230 2009

  29. [29]

    J., Neukirch, T., & Giuliani, P

    Grady, K. J., Neukirch, T., & Giuliani, P. 2012, Astronomy and Astrophysics, 546, A85, doi: 10.1051/0004-6361/201218914

    work page doi:10.1051/0004-6361/201218914 2012

  30. [30]

    V., & Nordlund, ˚A

    Haahr, M., Gudiksen, B. V., & Nordlund, ˚A. 2025, Astronomy and Astrophysics, 696, A191, doi: 10.1051/0004-6361/202452117

    work page doi:10.1051/0004-6361/202452117 2025

  31. [31]

    2017, AIAA Journal, 55, 1487, doi: 10.2514/1.J055493 Jel´ ınek, P., Karlick´ y, M., Van Doorsselaere, T., & B´ arta, M

    Jameson, A. 2017, AIAA Journal, 55, 1487, doi: 10.2514/1.J055493 Jel´ ınek, P., Karlick´ y, M., Van Doorsselaere, T., & B´ arta, M. 2017, ApJ, 847, 98, doi: 10.3847/1538-4357/aa88a6

    work page doi:10.2514/1.j055493 2017

  32. [32]

    2017, ApJ, 845, 12, doi: 10.3847/1538-4357/aa7d59 Karlick´ y, M., & B´ arta, M

    Kaneko, T., & Yokoyama, T. 2017, ApJ, 845, 12, doi: 10.3847/1538-4357/aa7d59 Karlick´ y, M., & B´ arta, M. 2006, The Astrophysical Journal, 647, 1472, doi: 10.1086/505460

    work page doi:10.3847/1538-4357/aa7d59 2017

  33. [33]

    2020, The Astrophysical Journal, 905, L16, doi: 10.3847/2041-8213/abcbf5

    Kong, X., Guo, F., Shen, C., et al. 2020, The Astrophysical Journal, 905, L16, doi: 10.3847/2041-8213/abcbf5

    work page doi:10.3847/2041-8213/abcbf5 2020

  34. [34]

    2025, Frontiers in Astronomy and Space Sciences, 11, 1510579, doi: 10.3389/fspas.2024.1510579

    Kong, X., Ning, H., & Chen, Y. 2025, Frontiers in Astronomy and Space Sciences, 11, 1510579, doi: 10.3389/fspas.2024.1510579

    work page doi:10.3389/fspas.2024.1510579 2025

  35. [35]

    2019, The Astrophysical Journal, 887, L37, doi: 10.3847/2041-8213/ab5f67

    Kong, X., Guo, F., Shen, C., et al. 2019, The Astrophysical Journal, 887, L37, doi: 10.3847/2041-8213/ab5f67

    work page doi:10.3847/2041-8213/ab5f67 2019

  36. [36]

    2022, The Astrophysical Journal Letters, 941, L22, doi: 10.3847/2041-8213/aca65c 16

    Kong, X., Chen, B., Guo, F., et al. 2022, The Astrophysical Journal Letters, 941, L22, doi: 10.3847/2041-8213/aca65c 16

    work page doi:10.3847/2041-8213/aca65c 2022

  37. [37]

    T., & Dahlin, J

    Kumar, P., Karpen, J. T., & Dahlin, J. T. 2025, The Astrophysical Journal, 980, 158, doi: 10.3847/1538-4357/ada293

    work page doi:10.3847/1538-4357/ada293 2025

  38. [38]

    2021, Physics of Plasmas, 28, 052905, doi: 10.1063/5.0047644

    Li, X., Guo, F., & Liu, Y.-H. 2021, Physics of Plasmas, 28, 052905, doi: 10.1063/5.0047644

    work page doi:10.1063/5.0047644 2021

  39. [39]

    2013, The Astrophysical Journal, 767, 168, doi: 10.1088/0004-637X/767/2/168

    Liu, W., Chen, Q., & Petrosian, V. 2013, The Astrophysical Journal, 767, 168, doi: 10.1088/0004-637X/767/2/168

    work page doi:10.1088/0004-637x/767/2/168 2013

  40. [40]

    1994, Nature, 371, 495, doi: 10.1038/371495a0

    Masuda, S., Kosugi, T., Hara, H., Tsuneta, S., & Ogawara, Y. 1994, Nature, 371, 495, doi: 10.1038/371495a0

    work page doi:10.1038/371495a0 1994

  41. [41]

    A., Nakariakov, V

    McLaughlin, J. A., Nakariakov, V. M., Dominique, M., Jel´ ınek, P., & Takasao, S. 2018, Space Science Reviews, 214, 45, doi: 10.1007/s11214-018-0478-5

    work page doi:10.1007/s11214-018-0478-5 2018

  42. [42]

    J., Hilker M., Baumgardt H., C \^o t \'e P., Grebel E

    Meyer, C. D., Balsara, D. S., & Aslam, T. D. 2012, MNRAS, 422, 2102, doi: 10.1111/j.1365-2966.2012.20744.x

    work page doi:10.1111/j.1365-2966.2012.20744.x 2012

  43. [43]

    D., Balsara, D

    Meyer, C. D., Balsara, D. S., & Aslam, T. D. 2014, Journal of Computational Physics, 257, 594, doi: 10.1016/j.jcp.2013.08.021

    work page doi:10.1016/j.jcp.2013.08.021 2014

  44. [44]

    2025, Monthly Notices of the Royal Astronomical Society, 536, 609, doi: 10.1093/mnras/stae2668

    Mowbray, K., Neukirch, T., & Threlfall, J. 2025, Monthly Notices of the Royal Astronomical Society, 536, 609, doi: 10.1093/mnras/stae2668

    work page doi:10.1093/mnras/stae2668 2025

  45. [45]

    M., Foullon, C., Verwichte, E., & Young, N

    Nakariakov, V. M., Foullon, C., Verwichte, E., & Young, N. P. 2006, Astronomy and Astrophysics, 452, 343, doi: 10.1051/0004-6361:20054608

    work page doi:10.1051/0004-6361:20054608 2006

  46. [46]

    2014, The Astrophysical Journal, 787, 125, doi: 10.1088/0004-637X/787/2/125

    Narukage, N., Shimojo, M., & Sakao, T. 2014, The Astrophysical Journal, 787, 125, doi: 10.1088/0004-637X/787/2/125

    work page doi:10.1088/0004-637x/787/2/125 2014

  47. [47]

    2015, The Astrophysical Journal, 799, 126, doi: 10.1088/0004-637X/799/2/126

    Nishizuka, N., Karlick´ y, M., Janvier, M., & B´ arta, M. 2015, The Astrophysical Journal, 799, 126, doi: 10.1088/0004-637X/799/2/126

    work page doi:10.1088/0004-637x/799/2/126 2015

  48. [48]

    Northrop, T. G. 1963, Reviews of Geophysics and Space Physics, 1, 283, doi: 10.1029/RG001i003p00283

    work page doi:10.1029/rg001i003p00283 1963

  49. [49]

    2010, The Astrophysical Journal, 714, 915, doi: 10.1088/0004-637X/714/1/915

    Shinohara, I. 2010, The Astrophysical Journal, 714, 915, doi: 10.1088/0004-637X/714/1/915

    work page doi:10.1088/0004-637x/714/1/915 2010

  50. [50]

    2023, Space Science Reviews, 219, 75, doi: 10.1007/s11214-023-01011-8

    Oka, M., Birn, J., Egedal, J., et al. 2023, Space Science Reviews, 219, 75, doi: 10.1007/s11214-023-01011-8

    work page doi:10.1007/s11214-023-01011-8 2023

  51. [51]

    2012, Space Science Reviews, 173, 535, doi: 10.1007/s11214-012-9900-6

    Petrosian, V. 2012, Space Science Reviews, 173, 535, doi: 10.1007/s11214-012-9900-6

    work page doi:10.1007/s11214-012-9900-6 2012

  52. [52]

    Q., & McTiernan, J

    Petrosian, V., Donaghy, T. Q., & McTiernan, J. M. 2002, The Astrophysical Journal, 569, 459, doi: 10.1086/339240

    work page doi:10.1086/339240 2002

  53. [53]

    R., & Forbes, T

    Priest, E. R., & Forbes, T. G. 2002, Astronomy and Astrophysics Review, 10, 313, doi: 10.1007/s001590100013

    work page doi:10.1007/s001590100013 2002

  54. [54]

    K., Polito, V., Chen, B., et al

    Reeves, K. K., Polito, V., Chen, B., et al. 2020, The Astrophysical Journal, 905, 165, doi: 10.3847/1538-4357/abc4e0

    work page doi:10.3847/1538-4357/abc4e0 2020

  55. [55]

    2014, ApJ, 789, 132, doi: 10.1088/0004-637X/789/2/132

    Rempel, M. 2014, ApJ, 789, 132, doi: 10.1088/0004-637X/789/2/132

    work page doi:10.1088/0004-637x/789/2/132 2014

  56. [56]

    2018, ApJS, 235, 21, doi: 10.3847/1538-4365/aab114

    Ripperda, B., Bacchini, F., Teunissen, J., et al. 2018, ApJS, 235, 21, doi: 10.3847/1538-4365/aab114

    work page doi:10.3847/1538-4365/aab114 2018

  57. [57]

    2023, ApJ, 947, 67, doi: 10.3847/1538-4357/ac9b4e

    Ruan, W., Yan, L., & Keppens, R. 2023, ApJ, 947, 67, doi: 10.3847/1538-4357/ac9b4e

    work page doi:10.3847/1538-4357/ac9b4e 2023

  58. [58]

    Sharma, P., & Hammett, G. W. 2007, Journal of Computational Physics, 227, 123, doi: 10.1016/j.jcp.2007.07.026

    work page doi:10.1016/j.jcp.2007.07.026 2007

  59. [59]

    K., et al

    Shen, C., Chen, B., Reeves, K. K., et al. 2022, Nature Astronomy, 6, 317, doi: 10.1038/s41550-021-01570-2

    work page doi:10.1038/s41550-021-01570-2 2022

  60. [60]

    1995, The Astrophysical Journal, 451, L83, doi: 10.1086/309688

    Shibata, K., Masuda, S., Shimojo, M., et al. 1995, The Astrophysical Journal, 451, L83, doi: 10.1086/309688

    work page doi:10.1086/309688 1995

  61. [61]

    Shibata, K., Takasao, S., & Reeves, K. K. 2023, The Astrophysical Journal, 943, 106, doi: 10.3847/1538-4357/acaa9c

    work page doi:10.3847/1538-4357/acaa9c 2023

  62. [62]

    V., & Kosugi, T

    Somov, B. V., & Kosugi, T. 1997, The Astrophysical Journal, 485, 859, doi: 10.1086/304449

    work page doi:10.1086/304449 1997

  63. [63]

    1968, SIAM Journal on Numerical Analysis, 5, 506, doi: 10.1137/0705041

    Strang, G. 1968, SIAM Journal on Numerical Analysis, 5, 506, doi: 10.1137/0705041

    work page doi:10.1137/0705041 1968

  64. [64]

    2016, The Astrophysical Journal, 828, 103, doi: 10.3847/0004-637X/828/2/103

    Takasao, S., Asai, A., Isobe, H., & Shibata, K. 2016, The Astrophysical Journal, 828, 103, doi: 10.3847/0004-637X/828/2/103

    work page doi:10.3847/0004-637x/828/2/103 2016

  65. [65]

    2015, The Astrophysical Journal, 805, 135, doi: 10.1088/0004-637X/805/2/135

    Takasao, S., Matsumoto, T., Nakamura, N., & Shibata, K. 2015, The Astrophysical Journal, 805, 135, doi: 10.1088/0004-637X/805/2/135

    work page doi:10.1088/0004-637x/805/2/135 2015

  66. [66]

    2016, The Astrophysical Journal, 823, 150, doi: 10.3847/0004-637X/823/2/150 V¨ ogler, A., Shelyag, S., Sch¨ ussler, M., et al

    Takasao, S., & Shibata, K. 2016, The Astrophysical Journal, 823, 150, doi: 10.3847/0004-637X/823/2/150 V¨ ogler, A., Shelyag, S., Sch¨ ussler, M., et al. 2005, A&A, 429, 335, doi: 10.1051/0004-6361:20041507

    work page doi:10.3847/0004-637x/823/2/150 2016

  67. [67]

    2022, ApJL, 931, L32, doi: 10.3847/2041-8213/ac715a

    Wang, Y., Cheng, X., Ren, Z., & Ding, M. 2022, ApJL, 931, L32, doi: 10.3847/2041-8213/ac715a

    work page doi:10.3847/2041-8213/ac715a 2022

  68. [68]

    2022, Monthly Notices of the Royal Astronomical Society, 509, 406, doi: 10.1093/mnras/stab2954

    Xie, X., Mei, Z., Shen, C., et al. 2022, Monthly Notices of the Royal Astronomical Society, 509, 406, doi: 10.1093/mnras/stab2954

    work page doi:10.1093/mnras/stab2954 2022

  69. [69]

    K., et al

    Xie, X., Shen, C., Reeves, K. K., et al. 2025, The Astrophysical Journal, 984, L27, doi: 10.3847/2041-8213/adc91b

    work page doi:10.3847/2041-8213/adc91b 2025

  70. [70]

    2020, The Astrophysical Journal, 897, 64, doi: 10.3847/1538-4357/ab93b5

    Ye, J., Cai, Q., Shen, C., et al. 2020, The Astrophysical Journal, 897, 64, doi: 10.3847/1538-4357/ab93b5

    work page doi:10.3847/1538-4357/ab93b5 2020

  71. [71]

    K., et al

    Yu, S., Chen, B., Reeves, K. K., et al. 2020, The Astrophysical Journal, 900, 17, doi: 10.3847/1538-4357/aba8a6

    work page doi:10.3847/1538-4357/aba8a6 2020

  72. [72]

    2014, The Astrophysical Journal, 797, 28, doi: 10.1088/0004-637X/797/1/28

    Khabarova, O. 2014, The Astrophysical Journal, 797, 28, doi: 10.1088/0004-637X/797/1/28

    work page doi:10.1088/0004-637x/797/1/28 2014

  73. [73]

    V., Arzner, K., Benz, A

    Zharkova, V. V., Arzner, K., Benz, A. O., et al. 2011, Space Science Reviews, 159, 357, doi: 10.1007/s11214-011-9803-y

    work page doi:10.1007/s11214-011-9803-y 2011

  74. [74]

    2015, The Astrophysical Journal, 815, 6, doi: 10.1088/0004-637X/815/1/6 17 Øyre, E

    Zhou, X., B¨ uchner, J., B´ arta, M., Gan, W., & Liu, S. 2015, The Astrophysical Journal, 815, 6, doi: 10.1088/0004-637X/815/1/6 17 Øyre, E. S., Gudiksen, B. V., & Fletcher, L. 2025, Astronomy and Astrophysics, 703, A97, doi: 10.1051/0004-6361/202555424

    work page doi:10.1088/0004-637x/815/1/6 2015