pith. sign in

arxiv: 2605.30057 · v1 · pith:V6Y7PP3Lnew · submitted 2026-05-28 · ⚛️ physics.plasm-ph · astro-ph.SR

Propagation of waves in weakly ionized two-fluid plasmas. II. Nonlinear Alfv\'enic waves

David Mart\'inez-G\'omez This is my paper

Pith reviewed 2026-06-29 00:08 UTC · model grok-4.3

classification ⚛️ physics.plasm-ph astro-ph.SR
keywords Alfvén wavesweakly ionized plasmastwo-fluid modelnonlinear wavesponderomotive forceion-neutral collisionsHall currentdensity perturbations
0
0 comments X

The pith

Circularly polarized Alfvén waves generate non-oscillatory bulk flows without the density oscillations of linear polarization in weakly ionized plasmas.

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

The paper studies the propagation of nonlinear Alfvénic waves using a two-fluid model that includes Hall current and elastic ion-neutral collisions. It derives analytical expressions for how damping and heating rates change with coupling strength and polarization. Numerical simulations reveal that circular polarization produces steady bulk flows and density perturbations without the oscillations seen in linear cases, while weak coupling lets energy dissipation drive the neutral fluid more than the ponderomotive force.

Core claim

The nonlinear perturbations associated with the circularly polarized eigenmodes do not show the oscillatory motions typically caused by linearly polarized eigenmodes, but they retain the non-oscillatory bulk flows. In weak coupling conditions the nonlinear dynamics of the neutral fluid is mainly driven by the wave energy dissipation while the ponderomotive force only directly acts on the charged fluid.

What carries the argument

Two-fluid plasma model with Hall current and elastic ion-neutral collisions, applied to derive damping rates and to simulate nonlinear wave evolution.

If this is right

  • Damping and heating rates depend on both collisional coupling strength and the polarization state of the wave.
  • Circular polarization produces different patterns of density perturbations and bulk flows than linear polarization.
  • In weak coupling the neutral fluid responds primarily to dissipated wave energy rather than direct ponderomotive forcing.
  • The separation of effects between charged and neutral components produces different amplitudes for longitudinal motions, density changes, and temperature perturbations.

Where Pith is reading between the lines

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

  • Polarization state could serve as a control parameter for the spatial structure of heating and flows in partially ionized regions.
  • The reported distinction between oscillatory and steady responses might be tested by comparing wave observations at different ionization fractions.
  • If the two-fluid assumption holds, similar polarization-dependent behavior should appear in other wave modes that involve transverse magnetic perturbations.

Load-bearing premise

The two-fluid description with Hall current and elastic ion-neutral collisions remains valid for the density and flow perturbations generated by the nonlinear evolution of the waves.

What would settle it

Detection of oscillatory density perturbations or longitudinal motions driven directly by the ponderomotive force on the neutral fluid in a weakly coupled regime would contradict the reported nonlinear behavior.

read the original abstract

Weakly ionized plasmas can be found in the lower layers of the solar and stellar atmospheres and in structures such as prominences and spicules. A variety of density perturbations and bulk flows detected in these environments have been explained as the result of the ponderomotive force generated by nonlinear Alfv\'enic waves. In addition, the dissipation of the energy carried by these waves leads to heating of the plasma. Here, we use a two-fluid model to study the combined influence of Hall's current and elastic collisions between ions and neutrals on the propagation of linearly and circularly polarized transverse waves in weakly ionized plasmas. We derive analytical expressions for the damping and heating rates, showing their dependence on the strength of the collisional coupling and on the polarization state. We also perform numerical simulations to investigate the nonlinear generation of density perturbations and bulk flows related to the ponderomotive force and the energy dissipation by the ion-neutral interaction. We find that the nonlinear perturbations associated with the circularly polarized eigenmodes do not show the oscillatory motions typically caused by linearly polarized eigenmodes, but they retain the non-oscillatory bulk flows. We also briefly discuss how in weak coupling conditions the nonlinear dynamics of the neutral fluid is mainly driven by the wave energy dissipation while the ponderomotive force only directly acts on the charged fluid, resulting in different amplitudes of the longitudinal motions and the perturbations of density and temperature.

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 / 2 minor

Summary. The manuscript develops a two-fluid model including the Hall term and elastic ion-neutral collisions to study propagation of linearly and circularly polarized transverse Alfvénic waves in weakly ionized plasmas. It derives analytical damping and heating rates that depend explicitly on collisional coupling strength and polarization state, then uses numerical simulations to examine nonlinear generation of density perturbations and bulk flows driven by the ponderomotive force and collisional dissipation. The central findings are that circularly polarized eigenmodes produce non-oscillatory bulk flows without the oscillatory motions seen for linear polarization, and that in weak coupling the neutral-fluid dynamics is driven primarily by wave-energy dissipation while the ponderomotive force acts directly only on the charged fluid.

Significance. If the derivations and simulations hold, the work supplies explicit, coupling-strength-dependent expressions for damping/heating and clarifies polarization-dependent nonlinear responses relevant to density and flow perturbations observed in the solar atmosphere and prominences. The analytical rates and the separation of ponderomotive versus dissipative driving in the weak-coupling limit constitute concrete, testable predictions that could be compared with observations or more detailed kinetic models.

major comments (2)
  1. [model setup and numerical section] The central claims about nonlinear bulk flows and the dominance of dissipation over ponderomotive forcing in the neutral fluid rest on the two-fluid equations (with fixed collision frequency and ionization fraction) remaining valid for the generated density and flow perturbations. The manuscript does not examine whether nonlinear density changes alter the mean free path or ionization balance, which would modify the collisional coupling and thereby the reported separation of forces and the amplitudes of longitudinal motions.
  2. [derivation of damping/heating rates] The analytical damping and heating rates are stated to depend on collisional coupling strength and polarization; however, the derivation steps that lead from the linearized two-fluid equations to the explicit rates are not cross-checked against the nonlinear simulation outputs for consistency in the weak-coupling regime.
minor comments (2)
  1. Notation for the polarization states and the definition of the Hall parameter should be introduced once and used consistently between the analytic section and the simulation figures.
  2. The abstract and introduction would benefit from a brief statement of how the present results extend or differ from Paper I.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and constructive feedback on our manuscript. The comments highlight important aspects of model validity and consistency between analysis and simulations. We address each major comment below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [model setup and numerical section] The central claims about nonlinear bulk flows and the dominance of dissipation over ponderomotive forcing in the neutral fluid rest on the two-fluid equations (with fixed collision frequency and ionization fraction) remaining valid for the generated density and flow perturbations. The manuscript does not examine whether nonlinear density changes alter the mean free path or ionization balance, which would modify the collisional coupling and thereby the reported separation of forces and the amplitudes of longitudinal motions.

    Authors: We agree that the fixed collision frequency and ionization fraction constitute a modeling assumption whose validity depends on the amplitude of the generated perturbations. In the simulations presented, the relative density changes remain below ~5%, for which the mean-free-path variation and ionization-balance shift are expected to be negligible within the weakly ionized regime considered. Nevertheless, this is a genuine limitation for extrapolating to stronger nonlinearities. In the revised manuscript we will add an explicit discussion paragraph in Section 4 (or a new subsection) stating the range of validity of the constant-coefficient approximation and noting that variable collision frequency would require a more elaborate model. This is a partial revision: we clarify the limitation without performing new variable-coefficient runs. revision: partial

  2. Referee: [derivation of damping/heating rates] The analytical damping and heating rates are stated to depend on collisional coupling strength and polarization; however, the derivation steps that lead from the linearized two-fluid equations to the explicit rates are not cross-checked against the nonlinear simulation outputs for consistency in the weak-coupling regime.

    Authors: The analytical rates follow directly from the linearized two-fluid system (Eqs. 8–12 and the subsequent dispersion relation). To demonstrate consistency, we will extract the time-averaged energy dissipation rate from the weak-coupling simulation runs and compare it quantitatively with the analytical heating rate evaluated at the same coupling strength and polarization. The comparison, together with a concise recap of the linear derivation steps, will be added as a new paragraph in Section 3.2 and illustrated in a supplementary figure. This constitutes a full revision of the requested cross-check. revision: yes

Circularity Check

0 steps flagged

No circularity: derivations start from standard two-fluid equations and yield explicit dependence on collisional parameters.

full rationale

The paper begins from the standard two-fluid plasma equations with Hall term and elastic ion-neutral collisions, derives analytical damping/heating rates as functions of coupling strength and polarization, and runs numerical simulations of nonlinear ponderomotive and dissipative effects. No fitted parameters are introduced and then relabeled as predictions, no self-citations form the load-bearing justification for uniqueness or ansatzes, and no results reduce by construction to quantities defined within the paper itself. The central findings (absence of oscillatory motions for circular polarization, dissipation-driven neutral dynamics in weak coupling) follow directly from the model equations without circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the standard two-fluid plasma equations plus the weak-ionization and Hall-current approximations; no new free parameters, ad-hoc entities, or non-standard axioms are introduced in the abstract.

axioms (1)
  • domain assumption Two-fluid description with elastic ion-neutral collisions and Hall current is an adequate model for wave propagation in weakly ionized solar-atmosphere plasmas.
    Invoked to derive the damping/heating rates and to set up the nonlinear simulations.

pith-pipeline@v0.9.1-grok · 5784 in / 1347 out tokens · 28205 ms · 2026-06-29T00:08:27.334930+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

83 extracted references · 71 canonical work pages · 3 internal anchors

  1. [1]

    1993, Phys

    Amagishi, Y., & Tanaka, M. 1993, Phys. Rev. Lett., 71, 360, doi: 10.1103/PhysRevLett.71.360

    work page doi:10.1103/physrevlett.71.360 1993

  2. [2]

    D., Brady, C

    Arber, T. D., Brady, C. S., & Shelyag, S. 2016, ApJ, 817, 94, doi: 10.3847/0004-637X/817/2/94 16

    work page doi:10.3847/0004-637x/817/2/94 2016

  3. [3]

    2019, Frontiers in Astronomy and Space Sciences, 6, 39, doi: 10.3389/fspas.2019.00039

    Ballai, I. 2019, Frontiers in Astronomy and Space Sciences, 6, 39, doi: 10.3389/fspas.2019.00039

    work page doi:10.3389/fspas.2019.00039 2019

  4. [4]

    2024, Philoso phical Transactions of the Royal Society of London Series A, 382, 20230226, doi: 10.1098/rsta.2023.0226

    Ballai, I., Forg´ acs-Dajka, E., & McMurdo, M. 2024, Philoso phical Transactions of the Royal Society of London Series A, 382, 20230226, doi: 10.1098/rsta.2023.0226

    work page doi:10.1098/rsta.2023.0226 2024

  5. [5]

    L., Soler, R., Terradas, J., & Carbonell, M

    Ballester, J. L., Soler, R., Terradas, J., & Carbonell, M. 20 20, A&A, 641, A48, doi: 10.1051/0004-6361/202038220 —. 2024, Philosophical Transactions of the Royal Society of London Series A, 382, 20230222, doi: 10.1098/rsta.2023.0222

    work page doi:10.1051/0004-6361/202038220 2024

  6. [6]

    M., & Orszag, S

    Bender, C. M., & Orszag, S. A. 1978, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New Yor k)

    1978

  7. [7]

    Bittencourt, J. A. 2004, Fundamentals of Plasma Physics (Springer-Verlag)

    2004

  8. [8]

    Botha, G. J. J., Arber, T. D., Nakariakov, V. M., & Keenan, F. P. 2000, A&A, 363, 1186

    2000

  9. [9]

    S., & Arber, T

    Brady, C. S., & Arber, T. D. 2016, ApJ, 829, 80, doi: 10.3847/0004-637X/829/2/80

    work page doi:10.3847/0004-637x/829/2/80 2016

  10. [10]

    Braginskii, S. I. 1965, Reviews of Plasma Physics, 1, 205

    1965

  11. [11]

    S., & G´ omez-M ´ ıguez, M

    Cally, P. S., & G´ omez-M ´ ıguez, M. M. 2023, ApJ, 946, 108, doi: 10.3847/1538-4357/acbb63

    work page doi:10.3847/1538-4357/acbb63 2023

  12. [12]

    Campos, L. M. B. C. 1992, Physics of Fluids B: Plasma Physics, 4, 2975, doi: 10.1063/1.860136

    work page doi:10.1063/1.860136 1992

  13. [13]

    Christopoulos, T., Tsilipakos, O., Sinatkas, G., & Kriezis , E. E. 2019, Optics Express, 27, 14505, doi: 10.1364/OE.27.014505

    work page doi:10.1364/oe.27.014505 2019

  14. [14]

    Cramer, N. F. 2001, The Physics of Alfv´ en W aves (Wiley-VCH) de Pontieu, B., & Haerendel, G. 1998, A&A, 338, 729

    2001

  15. [15]

    Draine, B. T. 1986, MNRAS, 220, 133, doi: 10.1093/mnras/220.1.133

    work page doi:10.1093/mnras/220.1.133 1986

  16. [16]

    A., & Plumpton, C

    Ferraro, C. A., & Plumpton, C. 1958, ApJ, 127, 459, doi: 10.1086/146474

    work page doi:10.1086/146474 1958

  17. [17]

    Ferraro, V. C. A. 1954, ApJ, 119, 393, doi: 10.1086/145837 Gonz´ alez Manrique, S. J., Khomenko, E., Collados, M., et al . 2024, A&A, 681, A114, doi: 10.1051/0004-6361/202348119 Gonz´ alez-Morales, P. A., Khomenko, E., Vitas, N., & Collad os, M. 2020, A&A, 642, A220, doi: 10.1051/0004-6361/202037938

    work page doi:10.1086/145837 1954

  18. [18]

    Goodman, M. L. 2011, ApJ, 735, 45, doi: 10.1088/0004-637X/735/1/45

    work page doi:10.1088/0004-637x/735/1/45 2011

  19. [19]

    2017, Radio Science, 52, 132 5, doi: 10.1002/2017RS006281

    Gustafsson, M., & Ehrenborg, C. 2017, Radio Science, 52, 132 5, doi: 10.1002/2017RS006281

    work page doi:10.1002/2017rs006281 2017

  20. [20]

    2005, Technical Report LUTEDX/(TEAT-7138)/1-16/(2005), Vol

    Gustafsson, M., & Nordebo, S. 2005, Technical Report LUTEDX/(TEAT-7138)/1-16/(2005), Vol. TEAT-7138,

    2005

  21. [21]

    1992, Nature, 360, 241, doi: 10.1038/360241a0

    Haerendel, G. 1992, Nature, 360, 241, doi: 10.1038/360241a0

    work page doi:10.1038/360241a0 1992

  22. [22]

    1983, Journal of Computational Physics, 49, 357, doi: 10.1016/0021-9991(83)90136-5

    Harten, A. 1983, Journal of Computational Physics, 49, 357, doi: 10.1016/0021-9991(83)90136-5

    work page doi:10.1016/0021-9991(83)90136-5 1983

  23. [23]

    2016, A&A, 591, A112 , doi: 10.1051/0004-6361/201628215

    Hillier, A., Takasao, S., & Nakamura, N. 2016, A&A, 591, A112 , doi: 10.1051/0004-6361/201628215

    work page doi:10.1051/0004-6361/201628215 2016

  24. [24]

    Hollweg, J. V. 1971, J. Geophys. Res., 76, 5155, doi: 10.1029/JA076i022p05155

    work page doi:10.1029/ja076i022p05155 1971

  25. [25]

    Huba, J. D. 2003, in Space Plasma Simulation, ed. J. B¨ uchner , C. Dum, & M. Scholer, Vol. 615 (Springer, Berlin, Heidelberg ), 166–192, doi: 10.1007/3-540-36530-3_9

    work page doi:10.1007/3-540-36530-3_9 2003

  26. [26]

    2012, ApJ, 747, 87, doi: 10.1088/0004-637X/747/2/87

    Khomenko, E., & Collados, M. 2012, ApJ, 747, 87, doi: 10.1088/0004-637X/747/2/87

    work page doi:10.1088/0004-637x/747/2/87 2012

  27. [27]

    2014, Phys ics of Plasmas, 21, 092901, doi: 10.1063/1.4894106

    Khomenko, E., Collados, M., D ´ ıaz, A., & Vitas, N. 2014, Phys ics of Plasmas, 21, 092901, doi: 10.1063/1.4894106

    work page doi:10.1063/1.4894106 2014

  28. [28]

    Khomenko, E., Collados, M., & D ´ ıaz, A. J. 2016, ApJ, 823, 132 , doi: 10.3847/0004-637X/823/2/132

    work page doi:10.3847/0004-637x/823/2/132 2016

  29. [29]

    Khomenko, E., Collados, M., Vitas, N., & Gonz´ alez-Morales , P. A. 2021, Philosophical Transactions of the Royal Society of London Series A, 379, 20200176, doi: 10.1098/rsta.2020.0176 Kra´ skiewicz, J., & Murawski, K. 2025, A&A, 698, A74, doi: 10.1051/0004-6361/202554085

    work page doi:10.1098/rsta.2020.0176 2021

  30. [30]

    2023, Sol

    Kraskiewicz, J., Murawski, K., Zhang, F., & Poedts, S. 2023, Sol. Phys., 298, 11, doi: 10.1007/s11207-022-02095-1

    work page doi:10.1007/s11207-022-02095-1 2023

  31. [31]

    Kreiss, H., Oliger, J., & Committee., G. A. R. P. J. O. 1973, Methods for the approximate solution of time dependent problems, GARP publications series ; no.10 (Geneva: International Council of Scientific Unions, W orld Meteorological Organization)

    1973

  32. [32]

    1999, ApJ, 514, 493, doi: 10.1086/306930

    Kudoh, T., & Shibata, K. 1999, ApJ, 514, 493, doi: 10.1086/306930

    work page doi:10.1086/306930 1999

  33. [33]

    Kumar, M., Murawski, K., Kadowaki, L., Ku´ zma, B., & Kilpua, E. K. J. 2024, A&A, 681, A60, doi: 10.1051/0004-6361/202245638

    work page doi:10.1051/0004-6361/202245638 2024

  34. [34]

    2003, Sol

    Kumar, N., & Roberts, B. 2003, Sol. Phys., 214, 241, doi: 10.1023/A:1024299029918 Ku´ zma, B., W´ ojcik, D., Murawski, K., Yuan, D., & Poedts, S. 2020, A&A, 639, A45, doi: 10.1051/0004-6361/201937260

    work page doi:10.1023/a:1024299029918 2003

  35. [35]

    Laming, J. M. 2004, ApJ, 614, 1063, doi: 10.1086/423780 —. 2015, Living Reviews in Solar Physics, 12, 2, doi: 10.1007/lrsp-2015-2

    work page doi:10.1086/423780 2004

  36. [36]

    E., Lukin, V

    Leake, J. E., Lukin, V. S., Linton, M. G., & Meier, E. T. 2012, ApJ, 760, 109, doi: 10.1088/0004-637X/760/2/109

    work page doi:10.1088/0004-637x/760/2/109 2012

  37. [37]

    Lighthill, M. J. 1960, Philosophical Transactions of the Ro yal Society of London Series A, 252, 397, doi: 10.1098/rsta.1960.0010

    work page doi:10.1098/rsta.1960.0010 1960

  38. [38]

    Multi-fluid multi-species models for inverse FIP-effect

    Maneva, Y. G., Alvarez Laguna, A., Lani, A., & Poedts, S. 2017 , ApJ, 836, 197, doi: 10.3847/1538-4357/aa5b83 Mart ´ ınez-G´ omez, D. 2025a, ApJ, 982, 4, doi: 10.3847/1538-4357/adb713 —. 2025b, Physics of Plasmas, 32, 112101, doi: 10.1063/5.0293493 Mart ´ ınez-G´ omez, D., Soler, R., & Terradas, J. 2016, ApJ, 832, 101, doi: 10.3847/0004-637X/832/2/101 —. ...

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.3847/1538-4357/aa5b83 2017

  39. [39]

    A., de Moortel, I., & Hood, A

    McLaughlin, J. A., de Moortel, I., & Hood, A. W. 2011, A&A, 527, A149, doi: 10.1051/0004-6361/201015552

    work page doi:10.1051/0004-6361/201015552 2011

  40. [40]

    T., & Shumlak, U

    Meier, E. T., & Shumlak, U. 2012, Physics of Plasmas, 19, 072508, doi: 10.1063/1.4736975

    work page doi:10.1063/1.4736975 2012

  41. [41]

    1956, MNRAS, 116, 503, doi: 10.1093/mnras/116.5.503

    Mestel, L., & Spitzer, L., J. 1956, MNRAS, 116, 503, doi: 10.1093/mnras/116.5.503

    work page doi:10.1093/mnras/116.5.503 1956

  42. [42]

    D., Giannios, D., & Mimica, P

    Mouschovias, T. C., Ciolek, G. E., & Morton, S. A. 2011, MNRAS, 415, 1751, doi: 10.1111/j.1365-2966.2011.18817.x

    work page doi:10.1111/j.1365-2966.2011.18817.x 2011

  43. [43]

    M., et al

    Murabito, M., Stangalini, M., Laming, J. M., et al. 2024, Phys. Rev. Lett., 132, 215201, doi: 10.1103/PhysRevLett.132.215201

    work page doi:10.1103/physrevlett.132.215201 2024

  44. [44]

    2022, Ap&SS, 367, 111, doi: 10.1007/s10509-022-04152-4

    Kadowaki, L. 2022, Ap&SS, 367, 111, doi: 10.1007/s10509-022-04152-4

    work page doi:10.1007/s10509-022-04152-4 2022

  45. [45]

    2024, A&A, 691, A25 4, doi: 10.1051/0004-6361/202449941

    Niedziela, R., Murawski, K., & Poedts, S. 2024, A&A, 691, A25 4, doi: 10.1051/0004-6361/202449941

    work page doi:10.1051/0004-6361/202449941 2024

  46. [46]

    Osterbrock, D. E. 1961, ApJ, 134, 347, doi: 10.1086/147165

    work page doi:10.1086/147165 1961

  47. [47]

    Pandey, B. P. 2018, MNRAS, 476, 344, doi: 10.1093/mnras/sty201

    work page doi:10.1093/mnras/sty201 2018

  48. [48]

    P., & Dwivedi, C

    Pandey, B. P., & Dwivedi, C. B. 2015, MNRAS, 447, 3604, doi: 10.1093/mnras/stu2503

    work page doi:10.1093/mnras/stu2503 2015

  49. [49]

    Cunha, J

    Pandey, B. P., & W ardle, M. 2008, MNRAS, 385, 2269, doi: 10.1111/j.1365-2966.2008.12998.x —. 2022, MNRAS, 513, 1842, doi: 10.1093/mnras/stac1028

    work page doi:10.1111/j.1365-2966.2008.12998.x 2008

  50. [50]

    2021, A&A, 652, A114, doi: 10.1051/0004-6361/202141262 —

    Pelekhata, M., Murawski, K., & Poedts, S. 2021, A&A, 652, A114, doi: 10.1051/0004-6361/202141262 —. 2023, A&A, 669, A47, doi: 10.1051/0004-6361/202244671

    work page doi:10.1051/0004-6361/202141262 2021

  51. [51]

    Piddington, J. H. 1956, MNRAS, 116, 314, doi: 10.1093/mnras/116.3.314 Nonlinear Alfv ´ enic waves in weakly ionized plasmas 17

    work page doi:10.1093/mnras/116.3.314 1956

  52. [52]

    2012, A&A, 544, A66, doi: 10.1051/0004-6361/201219019 Popescu Braileanu, B., & Keppens, R

    Pinto, C., Verdini, A., Galli, D., & Velli, M. 2012, A&A, 544, A66, doi: 10.1051/0004-6361/201219019 Popescu Braileanu, B., & Keppens, R. 2021, A&A, 653, A131, doi: 10.1051/0004-6361/202140872 Popescu Braileanu, B., Lukin, V. S., Khomenko, E., & de Vicen te, ´A. 2019a, A&A, 627, A25, doi: 10.1051/0004-6361/201834154 —. 2019b, A&A, 630, A79, doi: 10.1051/0...

    work page doi:10.1051/0004-6361/201219019 2012

  53. [53]

    Priest, E. R. 1984, Solar magneto-hydrodynamics (Dordrech t, Springer)

    1984

  54. [54]

    T., & Samson, J

    Rankin, R., Frycz, P., Tikhonchuk, V. T., & Samson, J. C. 1994 , J. Geophys. Res., 99, 21291, doi: 10.1029/94JA01629

    work page doi:10.1029/94ja01629 1994

  55. [55]

    1963, USSR Computational Mathematics and Mathematical Physics, 3, 1537, doi: https://doi.org/10.1016/0041-5553(63)90256-8

    Sarmin, E., & Chudov, L. 1963, USSR Computational Mathematics and Mathematical Physics, 3, 1537, doi: https://doi.org/10.1016/0041-5553(63)90256-8

    work page doi:10.1016/0041-5553(63)90256-8 1963

  56. [56]

    1991, The Numerical Method of Lines: Integrat ion of Partial Differential Equations (Elsevier Science)

    Schiesser, W. 1991, The Numerical Method of Lines: Integrat ion of Partial Differential Equations (Elsevier Science). https://books.google.es/books?id=1vLFQgAACAAJ

    1991

  57. [57]

    Schunk, R. W. 1977, Reviews of Geophysics and Space Physics, 15, 429, doi: 10.1029/RG015i004p00429

    work page doi:10.1029/rg015i004p00429 1977

  58. [58]

    2021, A&A, 645, A81, doi: 10.1051/0004-6361/202039667

    Snow, B., & Hillier, A. 2021, A&A, 645, A81, doi: 10.1051/0004-6361/202039667

    work page doi:10.1051/0004-6361/202039667 2021

  59. [59]

    2026, A&A, 708, A68, doi: 10.1051/0004-6361/202659417

    Soler, R. 2026, A&A, 708, A68, doi: 10.1051/0004-6361/202659417

    work page doi:10.1051/0004-6361/202659417 2026

  60. [60]

    Soler, R., Carbonell, M., & Ballester, J. L. 2013a, ApJS, 209 , 16, doi: 10.1088/0067-0049/209/1/16

    work page doi:10.1088/0067-0049/209/1/16

  61. [61]

    L., & Terradas, J

    Soler, R., Carbonell, M., Ballester, J. L., & Terradas, J. 20 13b, ApJ, 767, 171, doi: 10.1088/0004-637X/767/2/171

    work page doi:10.1088/0004-637x/767/2/171

  62. [62]

    Soler, R., Terradas, J., Oliver, R., & Ballester, J. L. 2016, A&A, 592, A28, doi: 10.1051/0004-6361/201628722

    work page doi:10.1051/0004-6361/201628722 2016

  63. [63]

    Song, P., & Vasyli¯ unas, V. M. 2011, Journal of Geophysical Research (Space Physics), 116, A09104, doi: 10.1029/2011JA016679

    work page doi:10.1029/2011ja016679 2011

  64. [64]

    K., Singh, A., Singh, B., et al

    Srivastava, A. K., Singh, A., Singh, B., et al. 2024, Philoso phical Transactions of the Royal Society of London Series A, 382, 20230220, doi: 10.1098/rsta.2023.0220

    work page doi:10.1098/rsta.2023.0220 2024

  65. [65]

    Stix, T. H. 1992, W aves in plasmas (New York: American Institute of Physics)

    1992

  66. [66]

    2004, ApJ, 610, 523, doi: 10.1086/421514

    Terradas, J., & Ofman, L. 2004, ApJ, 610, 523, doi: 10.1086/421514

    work page doi:10.1086/421514 2004

  67. [67]

    L., & Keppens, R

    Terradas, J., Oliver, R., Ballester, J. L., & Keppens, R. 200 8, ApJ, 675, 875, doi: 10.1086/526512

    work page doi:10.1086/526512

  68. [68]

    J., Oliver, R., & Balleste r, J

    Terradas, J., Soler, R., D ´ ıaz, A. J., Oliver, R., & Balleste r, J. L. 2013, ApJ, 778, 49, doi: 10.1088/0004-637X/778/1/49

    work page doi:10.1088/0004-637x/778/1/49 2013

  69. [69]

    F., & Araneda, J

    Terradas, J., Vi˜ nas, A. F., & Araneda, J. A. 2022, A&A, 660, A24, doi: 10.1051/0004-6361/202141914

    work page doi:10.1051/0004-6361/202141914 2022

  70. [70]

    2010, Space Sci

    Testa, P. 2010, Space Sci. Rev., 157, 37, doi: 10.1007/s11214-010-9714-3

    work page doi:10.1007/s11214-010-9714-3 2010

  71. [71]

    2023, ApJ, 944, 117, doi: 10.3847/1538-4357/acb343

    Testa, P., Mart ´ ınez-Sykora, J., & De Pontieu, B. 2023, ApJ, 944, 117, doi: 10.3847/1538-4357/acb343

    work page doi:10.3847/1538-4357/acb343 2023

  72. [72]

    H., & Drake, J

    Testa, P., Saar, S. H., & Drake, J. J. 2015, Philosophical Transactions of the Royal Society of London Series A, 373, 20140259, doi: 10.1098/rsta.2014.0259

    work page doi:10.1098/rsta.2014.0259 2015

  73. [73]

    Threlfall, J. W. 2012, PhD thesis, Saint Andrews University , UK

    2012

  74. [74]

    T., Rankin, R., Frycz, P., & Samson, J

    Tikhonchuk, V. T., Rankin, R., Frycz, P., & Samson, J. C. 1995 , Physics of Plasmas, 2, 501, doi: 10.1063/1.870975

    work page doi:10.1063/1.870975 1995

  75. [75]

    To, A. S. H., Laming, J. M., Reep, J., & Finley, A. J. 2026, arXi v e-prints, arXiv:2604.13174, doi: 10.48550/arXiv.2604.13174

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2604.13174 2026

  76. [76]

    R., Brizard, A

    Tracy, E. R., Brizard, A. J., Richardson, A. S., & Kaufman, A. N. 2014, Ray Tracing and Beyond (Cambridge University Press)

    2014

  77. [77]

    M., & Longbottom, A

    Verwichte, E., Nakariakov, V. M., & Longbottom, A. W. 1999, Journal of Plasma Physics, 62, 219–232, doi: 10.1017/S0022377899007771 W al´ en, C. 1944, Arkiv for Matematik, Astronomi och Fysik, 3 0A, 1 W alker, A. 2004, Series in Plasma Physics, 16, doi: 10.1201/9781420034004 W argnier, Q. M., Mart ´ ınez-Sykora, J., Hansteen, V. H., & De

    work page doi:10.1017/s0022377899007771 1999

  78. [78]

    2023, ApJ, 946, 115, doi: 10.3847/1538-4357/acbfb1 W atanabe, T

    Pontieu, B. 2023, ApJ, 946, 115, doi: 10.3847/1538-4357/acbfb1 W atanabe, T. 1961, Canadian Journal of Physics, 39, 1044, doi: 10.1139/p61-114

    work page doi:10.3847/1538-4357/acbfb1 2023

  79. [79]

    V., Khodachenko, M

    Zaqarashvili, T. V., Khodachenko, M. L., & Rucker, H. O. 2011 a, A&A, 529, A82, doi: 10.1051/0004-6361/201016326 —. 2011b, A&A, 534, A93, doi: 10.1051/0004-6361/201117380

    work page doi:10.1051/0004-6361/201016326 2011

  80. [80]

    V., Khodachenko, M

    Zaqarashvili, T. V., Khodachenko, M. L., & Soler, R. 2013, A& A, 549, A113, doi: 10.1051/0004-6361/201220272

    work page doi:10.1051/0004-6361/201220272 2013

Showing first 80 references.