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arxiv: 2604.17119 · v1 · submitted 2026-04-18 · 🌌 astro-ph.HE · astro-ph.SR· physics.plasm-ph

Recognition: unknown

Strong MHD Turbulence and Coherent Structures as Drivers of Cosmic Particle Acceleration

Loukas Vlahos
Authors on Pith no claims yet

Pith reviewed 2026-05-10 06:12 UTC · model grok-4.3

classification 🌌 astro-ph.HE astro-ph.SRphysics.plasm-ph
keywords MHD turbulenceparticle accelerationcoherent structuresplasma heatingmagnetic reconnectionsolar windcosmic rays
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The pith

Coherent structures are the main drivers of particle acceleration in strong MHD turbulence, not secondary byproducts of cascades.

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

The paper claims that models of energy conversion in astrophysical plasmas must treat the spontaneous formation of coherent structures as central to heating and particle acceleration. These structures concentrate electric fields and localized dissipation, repeatedly energizing particles to high energies. A reader would care because this reframes turbulence away from statistical spectra toward the concrete sites that produce the suprathermal populations observed in space. The review synthesizes how this view connects solar, wind, and shock environments while outlining computational hurdles for future models.

Core claim

Any physically complete picture of turbulent plasma heating and particle acceleration must place the self-consistent emergence of coherent structures at its center. Current sheets, vortical structures, magnetic flux ropes, shocklets, and confined reconnection sites are not secondary by-products of the turbulent cascade; they are its dynamically dominant dissipative and energizing elements where electric fields intensify and particles undergo repeated acceleration.

What carries the argument

Self-consistently emerging coherent structures (current sheets, flux ropes, shocklets, confined reconnection sites) that localize intense electric fields and dissipation within strong MHD turbulence.

Load-bearing premise

Coherent structures are the dynamically dominant sites of dissipation and acceleration rather than secondary features produced by the turbulent cascade.

What would settle it

A high-resolution simulation or in-situ observation demonstrating that particle energization rates remain uniform across the volume and uncorrelated with the locations of current sheets or flux ropes would falsify the central claim.

Figures

Figures reproduced from arXiv: 2604.17119 by Loukas Vlahos.

Figure 1
Figure 1. Figure 1: Energy introduced at large scales is gradually transferred to progressively smaller scales until it reaches the dissipation range, where it is converted into heat. is τℓ ∼ ℓ/uℓ, while the energy transfer rate is Πℓ ∼ u 2 ℓ τℓ ∼ u 3 ℓ ℓ . (2) By setting Πℓ ∼ ϵ, one recovers the Kolmogorov scaling uℓ ∼ (ϵℓ) 1/3 , (3) which implies the familiar energy spectrum E(k) ∝ ϵ 2/3 k −5/3 . (4) Although idealized, thi… view at source ↗
Figure 2
Figure 2. Figure 2: Visualization of the turbulent magnetic field strength |B| (left) and electric field strength |E| (right) across the simulation domain. Regions of high intensity are shown in yellow (light), whereas low-intensity regions are shown in blue (dark) (P. Dmitruk et al. 2003). After several Alfv´en crossing times, τA (τA = L/vA, with vA denoting the Alfv´en speed (F. F. Chen 2016)), the system reaches a fully de… view at source ↗
Figure 1
Figure 1. Figure 1: The distribution (volume rendering) of the magnitude of the current density in the reconnection region at ωpet = 797 for the standard run 3D-1. Under the influence of injected turbulence, the current sheet breaks into a turbulent reconnection region filled with many structures such as flux ropes and current sheets. 10 https://youtu.be/-2EsinquZjA. 11 https://youtu.be/5-eL9oXXCLs 3 The Astrophysical Journal… view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17 [PITH_FULL_IMAGE:figures/full_fig_p005_17.png] view at source ↗
Figure 8
Figure 8. Figure 8: Within the convection zone, turbulent motions generate organized magnetic structures known as flux tubes, which rise toward the solar surface and eventually form active regions in the solar atmosphere (Y. Fan 2009). Differential rotation, together with dynamo action, generates large-scale toroidal magnetic configurations, some of which become buoyantly unstable and reorga￾nize into coherent magnetic flux t… view at source ↗
Figure 9
Figure 9. Figure 9 [PITH_FULL_IMAGE:figures/full_fig_p006_9.png] view at source ↗
Figure 11
Figure 11. Figure 11: Diagram illustrating turbulent regions in near-Earth space together with the locations of large-scale reconnection events across the system. Turbulence can both interact with these global reconnection processes and gener￾ate additional, smaller-scale reconnection events within the turbulent regions (J. E. Stawarz et al. 2024). Earth’s magnetotail. —Earth’s magnetotail is a driven and highly dynamic plasma… view at source ↗
Figure 10
Figure 10. Figure 10: (Left) Turbulence in the solar wind. (Right) Power spectral density of magnetic field fluctuations in the solar wind. The black lines show power-law fits with slopes of −1, −5/3, and −2.8, corresponding to the injection, iner￾tial, and dissipation ranges, respectively (D. Verscharen et al. 2019). With increasing heliocentric distance, however, they develop stronger anisotropy and intermittency, accom￾pani… view at source ↗
Figure 2
Figure 2. Figure 2: AcompilationofB-fieldmorphologiestracedindifferentregionsoftheISM.Theseregionsinclude:(a)Taurus cores (Eswaraiah et al.2021), (b) cometary globule (CG) Gal 110–13 (Neha et al.2016), (c) filament G34.43+0.24 (Soam et al.2019), and (d) bright-rimmed cloud (BRC) L1172/1174 cloud complex (Saha et al.2021). Many detailed reviews, such as by Pattle et al.(2022), have attempted to put together all the observation… view at source ↗
Figure 14
Figure 14. Figure 14: Artist’s illustration of an astrophysical jet, with an inset displaying the jet’s internal electromagnetic structure. The presence of irregular electric fields, local￾ized current filaments, and a twisted (helical) magnetic field configuration indicates that relativistic jets provide favor￾able conditions for magnetic-energy dissipation, turbulent energy transport, and the acceleration of non-thermal part… view at source ↗
Figure 13
Figure 13. Figure 13: Schematic depiction of a magnetized accretion disk that feeds a compact central object and launches bipolar jets. The disk shows differential rotation, turbulent motion, and outward transport of angular momentum, while large-s￾cale magnetic fields collimate and power outflows along the polar axis. Together, these interconnected multiscale plasma processes regulate energy dissipation and non-thermal parti￾… view at source ↗
Figure 15
Figure 15. Figure 15: (Left) Young supernova remnant displaying narrow striped patterns and filamentary structures associ￾ated with turbulence driven by shocks and amplified mag￾netic fields. (Right) Supernova remnant showing an intri￾cate network of luminous filaments formed by the nonlinear interaction between the outward-moving shock front and the surrounding plasma. These structures highlight the inter￾mittent character of… view at source ↗
Figure 16
Figure 16. Figure 16: Artist’s illustration of a pair of black holes in orbit, surrounded by bright accretion flows. Gas orbiting each compact object forms hot, chaotic accretion disks, while part of the infalling plasma is ejected as narrow relativis￾tic jets aligned with the spin axes. These systems provide an extreme setting for angular-momentum transport, mag￾netic-energy dissipation, and the acceleration of high-energy pa… view at source ↗
Figure 17
Figure 17. Figure 17: Measured all-particle cosmic-ray energy spec￾trum, highlighting the solar contribution at low energies and the principal spectral features of the Galactic and extragalac￾tic components, in particular the knee at about 3 × 1015 eV and the ankle near 3 × 1018 eV (P. D. Group 2024). and explosive plasma flows transfer and redistribute en￾ergy over a broad range of scales [PITH_FULL_IMAGE:figures/full_fig_p0… view at source ↗
Figure 18
Figure 18. Figure 18: (Left) Broadband photon spectrum of a major solar flare, showing the transition from thermal soft X-ray emission to non-thermal hard X-rays and gamma rays pro￾duced by accelerated electrons and ions, as well as by nuclear interactions and pion decay (R. P. Lin et al. 2003). (Right) Energy spectrum of heliospheric ions extending from the so￾lar-wind peak through the suprathermal tail into the ener￾getic-pa… view at source ↗
Figure 20
Figure 20. Figure 20: (a) Evolving turbulent current sheet. (b)–(d) Spatial profiles of (b) ER, (c) Vey, and (d) J⃗·E⃗ along the most active x-line, with the dashed curve indicating the global reconnection rate. (e) Temporal evolution of ER (Y.-N. Liu et al. 2024). Particle motion in the evolving turbulent electric fields leads to heating and acceleration within the current sheet. the fluctuating velocity field naturally produ… view at source ↗
Figure 21
Figure 21. Figure 21: Representative results from test-particle calcula￾tions in strongly turbulent electromagnetic fields, illustrating particle transport and energization in an environment dom￾inated by spontaneously emerging coherent structures. At the ensemble level, the particle distribution func￾tion is often described by a Fokker–Planck equation, ∂P(W, t) ∂t = − ∂[F(W)P(W, t)] ∂W + ∂ 2 [D(W)P(W, t)] ∂W2 − P(W, t) tesc(W… view at source ↗
Figure 23
Figure 23. Figure 23: (Left) A range of coherent structures within a three-dimensional turbulent plasma (T. Richard et al. 2022). (Right) Schematic depiction of coherent current structures in a magnified portion of the turbulent region. As particles move through this volume, they interact with these struc￾tures in both stochastic and systematic ways (N. Sioulas et al. 2022). N. Sioulas et al. (2022) investigated this scenario … view at source ↗
Figure 22
Figure 22. Figure 22: Particle energy spectra at different times during a representative PIC simulation. At late times, the spectrum develops a clear non-thermal tail with a spectral index of p = 1.8 F. Guo et al. (2021). MHD and test-particle approaches indicate where en￾ergization regions are expected to appear and character￾ize how particles move through a turbulent multiscale environment. PIC simulations, in turn, reveal t… view at source ↗
Figure 24
Figure 24. Figure 24: Steady-state kinetic-energy distribution show￾ing the coexistence of a heated Maxwellian-like core and a power-law tail (N. Sioulas et al. 2022). strategies is not to replace first-principles plasma physics, but to be guided by MHD and test-particle results while tracking the effective statistical behav￾ior of energization events, waiting times, acceleration timescales, and escape times. In this way, one … view at source ↗
read the original abstract

Magnetohydrodynamic (MHD) turbulence is a ubiquitous dynamical state of astrophysical plasmas and a primary agent in the redistribution, dissipation, and conversion of energy into particle populations. Yet turbulence is still most often described in terms of cascades, spectra, and scale-to-scale transfer, while its role in producing localized sites of intense energization remains comparatively underemphasized. In this forward-looking review, aimed at a broad astrophysical readership, I argue that any physically complete picture of turbulent plasma heating and particle acceleration must place the self-consistent emergence of coherent structures at its center. Current sheets, vortical structures, magnetic flux ropes, shocklets, and confined reconnection sites are not secondary by-products of the turbulent cascade; they are its dynamically dominant dissipative and energizing elements, where electric fields intensify, dissipation becomes highly localized, and particles undergo repeated acceleration. Viewed in this way, strong turbulence provides a unifying framework that links large-scale plasma dynamics to the generation of suprathermal particles and non-thermal energy distributions in the solar atmosphere, the solar wind, shock environments, and a wide range of other cosmic plasmas. Rather than attempting an exhaustive survey of the literature, this article offers a selective and physically organized synthesis of the field, emphasizing the mechanisms, regimes, and open problems most relevant to the development of predictive theories of particle acceleration in turbulent plasmas. It also identifies the principal conceptual and computational challenges that must be overcome if the next generation of models is to connect multiscale plasma dynamics with observable energetic-particle signatures.

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

0 major / 2 minor

Summary. The manuscript is a selective, forward-looking review arguing that any complete description of MHD turbulent plasma heating and particle acceleration in astrophysical settings must center on the self-consistent emergence of coherent structures (current sheets, flux ropes, vortical structures, shocklets, and localized reconnection sites) as the dominant dissipative and energizing elements, rather than treating them as secondary by-products of cascades. It synthesizes selected literature to link large-scale dynamics to suprathermal particle generation across solar, heliospheric, and other cosmic plasmas, while identifying conceptual and computational challenges for predictive models.

Significance. If the perspective holds, the review supplies a unifying interpretive framework that reframes turbulence as a structure-driven process connecting multiscale plasma dynamics to non-thermal energy distributions and observable energetic-particle signatures. The selective synthesis of mechanisms, regimes, and open problems provides a useful organizing principle for the field without attempting exhaustive coverage.

minor comments (2)
  1. [Abstract] Abstract: the phrase 'any physically complete picture' is repeated in spirit across the opening and closing paragraphs; a single consolidated statement would improve conciseness.
  2. The synthesis sections would benefit from explicit cross-references to the specific open problems listed later, to tighten the logical flow between literature summary and forward-looking challenges.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive and accurate summary of the manuscript, which correctly identifies it as a selective, forward-looking review that places self-consistently emerging coherent structures at the center of turbulent plasma heating and particle acceleration. The significance assessment is also appreciated, as it aligns with our goal of providing an organizing framework for the field. The recommendation for minor revision is noted, and we are prepared to address any such issues. Since the report contains no enumerated major comments, we have no specific points to rebut or revise at this time.

Circularity Check

0 steps flagged

No circularity: conceptual review with no derivations or self-referential reductions

full rationale

The manuscript is a selective forward-looking review and conceptual synthesis. It advances an interpretive thesis that coherent structures must be placed at the center of turbulent energization models, but contains no equations, no fitted parameters, no quantitative predictions, and no derivation chain. All claims are supported by reference to external literature rather than internal self-citation loops or ansatzes. The argument structure is self-contained as an organizing perspective and does not reduce any result to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No new free parameters, axioms, or invented entities are introduced; the paper synthesizes existing concepts from the literature on MHD turbulence and particle acceleration.

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Works this paper leans on

61 extracted references · 48 canonical work pages

  1. [1]

    P., Zrake, J., & Fiuza, F

    Alves, E. P., Zrake, J., & Fiuza, F. 2018, PhRvL, 121, 245101, doi: 10.1103/PhysRevLett.121.245101

    work page doi:10.1103/physrevlett.121.245101 2018

  2. [2]

    Archontis, V., & Hood, A. W. 2010, Astronomy & Astrophysics, 514, A56, doi: 10.1051/0004-6361/200913495 13

    work page doi:10.1051/0004-6361/200913495 2010

  3. [3]

    2006, Astrophys

    Vlahos, L. 2006, Astrophys. J., 637, 322, doi: 10.1086/498341

    work page doi:10.1086/498341 2006

  4. [4]

    K., & Langdon, A

    Birdsall, C. K., & Langdon, A. B. 1991, Plasma Physics via Computer Simulation (IOP Publishing)

    1991

  5. [5]

    Boyd, T. J. M., & Sanderson, J. J. 1969, Plasma dynamics

    1969

  6. [6]

    Caprioli \ and\ author A

    Caprioli, D., & Spitkovsky, A. 2014, Astrophys. J, 783, 91, doi: 10.1088/0004-637X/783/2/91

    work page doi:10.1088/0004-637x/783/2/91 2014

  7. [7]

    Chen, F. F. 2016, Introduction to Plasma Physics and Controlled Fusion, doi: 10.1007/978-3-319-22309-4

    work page doi:10.1007/978-3-319-22309-4 2016

  8. [8]

    2019, ApJ, 886, 122, doi: 10.3847/1538-4357/ab4c33 —

    Comisso, L., & Sironi, L. 2019, Astrophysical Journal, 886, 122, doi: 10.3847/1538-4357/ab4c33

    work page doi:10.3847/1538-4357/ab4c33 2019

  9. [9]

    Dawson, J. M. 1983, Reviews of Modern Physics, 55, 403, doi: 10.1103/RevModPhys.55.403 de Jager, C. 1986, Space Science Reviews, 44, 43

    work page doi:10.1103/revmodphys.55.403 1983

  10. [10]

    Dmitruk, P., Matthaeus, W., Seenu, N., & Brown, M. R. 2003, Astrophys. J. Lett., 597, L81

    2003

  11. [11]

    H., & Seenu, N

    Dmitruk, P., Matthaeus, W. H., & Seenu, N. 2004, Astrophys. J., 617, 667, doi: 10.1086/425301

    work page doi:10.1086/425301 2004

  12. [12]

    2022, Science Advances, 8, eabn7627, doi: 10.1126/sciadv.abn7627

    Dong, C., Wang, L., Huang, Y.-M., et al. 2022, Science Advances, 8, eabn7627, doi: 10.1126/sciadv.abn7627

    work page doi:10.1126/sciadv.abn7627 2022

  13. [13]

    2025, SSRv, 221, 75, doi: 10.1007/s11214-025-01203-4

    Effenberger, F., Walter, D., Fichtner, H., et al. 2025, SSRv, 221, 75, doi: 10.1007/s11214-025-01203-4

    work page doi:10.1007/s11214-025-01203-4 2025

  14. [14]

    2009, Living Reviews in Solar Physics, 6, 4, doi: 10.12942/lrsp-2009-4

    Fan, Y. 2009, Living Reviews in Solar Physics, 6, 4, doi: 10.12942/lrsp-2009-4

    work page doi:10.12942/lrsp-2009-4 2009

  15. [15]

    1949, Physical Review, 75, 1169, doi: 10.1103/PhysRev.75.1169 21

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

    work page doi:10.1103/physrev.75.1169 1949

  16. [17]

    1954, Ap

    Fermi, E. 1954, Astrophys. J., 119, 1, doi: 10.1086/145789

    work page doi:10.1086/145789 1954

  17. [18]

    Group, P. D. 2024, Prog. Theor. Exp. Phys. https: //pdg.lbl.gov/2025/reviews/rpp2024-rev-cosmic-rays.pdf

    2024

  18. [19]

    2021, ApJ, 919, 111, doi: 10.3847/1538-4357/ac0918

    Guo, F., Li, X., Daughton, W., et al. 2021, The Astrophysical Journal, 919, 111, doi: 10.3847/1538-4357/ac0918

    work page doi:10.3847/1538-4357/ac0918 2021

  19. [20]

    Hillas, A. M. 2005, Journal of Physics G: Nuclear and Particle Physics, 31, R95, doi: 10.1088/0954-3899/31/5/R02

    work page doi:10.1088/0954-3899/31/5/r02 2005

  20. [21]

    W., & Eastwood, J

    Hockney, R. W., & Eastwood, J. W. 1988, Computer Simulation Using Particles (CRC Press)

    1988

  21. [22]

    2019, The Astrophysical Journal, 882, 57, doi: 10.3847/1538-4357/ab30c6

    Isliker, H., Archontis, V., & Vlahos, L. 2019, The Astrophysical Journal, 882, 57, doi: 10.3847/1538-4357/ab30c6

    work page doi:10.3847/1538-4357/ab30c6 2019

  22. [23]

    2022, Physics of Plasmas, 29, 112306, doi: 10.1063/5.0115754

    Isliker, H., Cathey, A., Hoelzl, M., Pamela, S., & Vlahos, L. 2022, Physics of Plasmas, 29, 112306, doi: 10.1063/5.0115754

    work page doi:10.1063/5.0115754 2022

  23. [24]

    2025, Physics of Plasmas, 32, 090501, doi: 10.1063/5.0285560

    Isliker, H., & Vlahos, L. 2025, Physics of Plasmas, 32, 090501, doi: 10.1063/5.0285560

    work page doi:10.1063/5.0285560 2025

  24. [25]

    Isliker, L

    Isliker, H., Vlahos, L., & Constantinescu, D. 2017, Physical Review Letters, 119, 045101, doi: 10.1103/PhysRevLett.119.045101

    work page doi:10.1103/physrevlett.119.045101 2017

  25. [26]

    A., & Trivelpiece, A

    Krall, N. A., & Trivelpiece, A. W. 1973, Principles of plasma physics

    1973

  26. [27]

    2021, Physical Review D, 104, doi: 10.1103/physrevd.104.063020

    Lemoine, M. 2021, Physical Review D, 104, 063020, doi: 10.1103/PhysRevD.104.063020

    work page doi:10.1103/physrevd.104.063020 2021

  27. [28]

    Particle transport through localized interactions with sharp magnetic field bends in MHD turbulence.Journal of Plasma Physics2023,89, 175890501, [arXiv:physics.plasm-ph/2304.03023]

    Lemoine, M. 2023, Journal of Plasma Physics, 89, 175890501, doi: 10.1017/S0022377823000946

    work page doi:10.1017/s0022377823000946 2023

  28. [29]

    2025, PhRvE, 112, 015205, doi: 10.1103/3xxg-x5dg

    Lemoine, M. 2025, PhRvE, 112, 015205, doi: 10.1103/3xxg-x5dg

    work page doi:10.1103/3xxg-x5dg 2025

  29. [30]

    P., Krucker, S., Hurford, G

    Lin, R. P., Krucker, S., Hurford, G. J., et al. 2003, Astrophysical Journal Letters, 595, L69, doi: 10.1086/378932

    work page doi:10.1086/378932 2003

  30. [31]

    Litvinenko, Y. E. 1996, Astrophysical Journal, 462, 997, doi: 10.1086/177213

    work page doi:10.1086/177213 1996

  31. [32]

    E., & Craig, I

    Litvinenko, Y. E., & Craig, I. J. D. 1996, The Astrophysical Journal, 462, 997

    1996

  32. [33]

    E., & Somov, B

    Litvinenko, Y. E., & Somov, B. V. 1993, Solar Physics, 146, 127

    1993

  33. [34]

    2024, Geophys

    Liu, Y.-N., Fujimoto, K., & Cao, J.-B. 2024, Geophys. Res. Lett., 51, e2023GL106466, doi: 10.1029/2023GL106466

    work page doi:10.1029/2023gl106466 2024

  34. [35]

    Longair, M. S. 2011, High Energy Astrophysics (Cambridge University Press)

    2011

  35. [36]

    N., & Hoshino, M

    Matsumoto, Y., Amano, T., Kato, T. N., & Hoshino, M. 2015, Science, 347, 974, doi: 10.1126/science.1260168

    work page doi:10.1126/science.1260168 2015

  36. [37]

    Matthaeus, W. H. 2021, Physics of Plasmas, 28, 032306, doi: 10.1063/5.0041540

    work page doi:10.1063/5.0041540 2021

  37. [38]

    Space Science Reviews 160, 145–168

    Matthaeus, W. H., & Velli, M. 2011, Space Science Reviews, 160, 145, doi: 10.1007/s11214-011-9793-9

    work page doi:10.1007/s11214-011-9793-9 2011

  38. [39]

    A., et al

    Mewaldt, R. A., et al. 2001, in AIP Conference Proceedings, Vol. 598, Solar and Galactic Composition (AIP), 165–176

    2001

  39. [40]

    2006, Physical Review Letters, 96, 151102, doi: 10.1103/PhysRevLett.96.151102

    Onofri, M., Isliker, H., & Vlahos, L. 2006, Physical Review Letters, 96, 151102, doi: 10.1103/PhysRevLett.96.151102

    work page doi:10.1103/physrevlett.96.151102 2006

  40. [41]

    Pezzi, O., Blasi, P., & Matthaeus, W. H. 2022, ApJ, 928, 25, doi: 10.3847/1538-4357/ac5332

    work page doi:10.3847/1538-4357/ac5332 2022

  41. [42]

    D., Verniero, J

    Phan, T. D., Verniero, J. L., Larson, D., et al. 2022, Geophysical Research Letters, 49, e2021GL096986, doi: 10.1029/2021GL096986 Princeton Plasma Physics Laboratory. 2005, Annual Report 2004,

    work page doi:10.1029/2021gl096986 2022

  42. [43]

    Pritchett, P. L. 2003, in Space Plasma Simulation (Springer), 1–30

    2003

  43. [44]

    F., Velli, M., Einaudi, G., & Dahlburg, R

    Rappazzo, A. F., Velli, M., Einaudi, G., & Dahlburg, R. B. 2008, Astroph. J., 677, 1348, doi: 10.1086/528786 14

    work page doi:10.1086/528786 2008

  44. [45]

    Reames, D. V. 1999, Space Science Reviews, 90, 413, doi: 10.1023/A:1005105831781

    work page doi:10.1023/a:1005105831781 1999

  45. [46]

    2022, Astronomy and Astrophysics, 664, A193, doi: 10.1051/0004-6361/202142531

    Richard, T., Lesaffre, P., Falgarone, E., & Lehmann, A. 2022, Astronomy and Astrophysics, 664, A193, doi: 10.1051/0004-6361/202142531

    work page doi:10.1051/0004-6361/202142531 2022

  46. [47]

    2003, Particle Acceleration Processes in Cosmic Plasmas, ed

    Schlickeiser, R. 2003, Particle Acceleration Processes in Cosmic Plasmas, ed. L. Klein, Vol. 612, 230–260

    2003

  47. [48]

    M., Bell, A

    Schure, K. M., Bell, A. R., Drury, L. O., & Bykov, A. M. 2012, Space Science Reviews, 173, 491, doi: 10.1007/s11214-012-9871-7

    work page doi:10.1007/s11214-012-9871-7 2012

  48. [49]

    10.1051/0004-6361/202141361

    Sioulas, N., Isliker, H., & Vlahos, L. 2022, Astronomy and Astrophysics, 657, A8, doi: 10.1051/0004-6361/202141361

    work page doi:10.1051/0004-6361/202141361 2022

  49. [50]

    2024, Journal of Astrophysics and Astronomy, 45, 17, doi: 10.1007/s12036-024-10005-z

    Soam, A., Eswaraiah, C., & Maheswar, G. 2024, Journal of Astrophysics and Astronomy, 45, 17, doi: 10.1007/s12036-024-10005-z

    work page doi:10.1007/s12036-024-10005-z 2024

  50. [51]

    E., Mu˜ noz, P

    Stawarz, J. E., Mu˜ noz, P. A., Bessho, N., et al. 2024, SSRv, 220, 90, doi: 10.1007/s11214-024-01124-8

    work page doi:10.1007/s11214-024-01124-8 2024

  51. [52]

    Sturrock, P. A. 1994, Plasma Physics, An Introduction to the Theory of Astrophysical, Geophysical and Laboratory Plasmas, doi: 10.1017/CBO9781139170598

    work page doi:10.1017/cbo9781139170598 1994

  52. [53]

    K., Kandaswamy, S., Kahniashvili, T., & Brandenburg, A

    Teaca, B., Verma, M. K., Kandaswamy, S., Kahniashvili, T., & Brandenburg, A. 2014, Physical Review Letters, 112, 145001, doi: 10.1103/PhysRevLett.112.145001

    work page doi:10.1103/physrevlett.112.145001 2014

  53. [54]

    Y., & Marsch, E

    Tu, C. Y., & Marsch, E. 1995, Space Science Reviews, 73, 1, doi: 10.1007/BF00748891

    work page doi:10.1007/bf00748891 1995

  54. [55]

    2005, The Astrophysical Journal Letters, 620, L59

    Isliker, H. 2005, The Astrophysical Journal Letters, 620, L59

    2005

  55. [56]

    M., Pouquet, A., Rosenberg, D., Mininni, P

    Uritsky, V. M., Pouquet, A., Rosenberg, D., Mininni, P. D., & Donovan, E. F. 2010, Phys. Rev. E, 82, 056326, doi: 10.1103/PhysRevE.82.056326

    work page doi:10.1103/physreve.82.056326 2010

  56. [57]

    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

  57. [58]

    2019, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 377, 20180095, doi: 10.1098/rsta.2018.0095

    Kouloumvakos, A., & Isliker, H. 2019, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 377, 20180095, doi: 10.1098/rsta.2018.0095

    work page doi:10.1098/rsta.2018.0095 2019

  58. [59]

    2023, Physics of Plasmas, 30, 040502, doi: 10.1063/5.0141512

    Vlahos, L., & Isliker, H. 2023, Physics of Plasmas, 30, 040502, doi: 10.1063/5.0141512

    work page doi:10.1063/5.0141512 2023

  59. [60]

    2004, Astrophys

    Vlahos, L., Isliker, H., & Lepreti, F. 2004, Astrophys. J., 608, 540, doi: 10.1086/386364

    work page doi:10.1086/386364 2004

  60. [61]

    2025, ApJ, 985, 43, doi: 10.3847/1538-4357/adca3d

    Wang, Y., Cheng, X., & Ding, M. 2025, ApJ, 985, 43, doi: 10.3847/1538-4357/adca3d

    work page doi:10.3847/1538-4357/adca3d 2025

  61. [62]

    A., Perez, J

    Zhdankin, V., Uzdensky, D. A., Perez, J. C., & Boldyrev, S. 2013, Astrophs. J., 771, 124, doi: 10.1088/0004-637X/771/2/124

    work page doi:10.1088/0004-637x/771/2/124 2013