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REVIEW 2 major objections 6 minor 70 references

Two-fluid simulations of waves in the solar chromosphere II. Propagation and damping of fast magneto-acoustic waves and shocks

T0 review · 2 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The damping of fast magneto-acoustic waves in the solar chromosphere is set by linear ion–neutral decoupling, and shock steepening dramatically amplifies it.

desk verdict A careful two-fluid wave-damping study whose linear part is well supported by a WKB comparison, and whose nonlinear-heating conclusion needs a grid-convergence check before it is fully convincing. read the letter →

arxiv 1908.05262 v1 pith:EJRBKX4G submitted 2019-08-14 astro-ph.SR

B. Popescu Braileanu , V. S. Lukin , E. Khomenko , A. de Vicente This is my paper · ORCID
classification astro-ph.SR
keywords Sun:chromospherewavesmagneticfieldnumericalsimulationstwo-fluidplasmaion-neutralcouplingmagneto-acousticwavedamping
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Using simulations that evolve charged and neutral fluids separately, this paper tracks fast magneto-acoustic waves and shocks through a model solar chromosphere with a horizontal magnetic field. The authors show that waves and neutrals, coupled at the photosphere, progressively decouple with height, and that this decoupling controls wave damping: an analytic WKB solution of the linearized two-fluid equations reproduces the simulated damping in the linear regime rather precisely. On that basis the paper concludes that the damping observed in such simulations is determined by linear effects of ion–neutral decoupling, and that nonlinear steepening of wave fronts into shocks dramatically increases the collisional damping. If the claim holds, ion–neutral friction is a primary control on where high-frequency wave energy is deposited in the chromosphere, connecting wave heating to the poorly constrained collision frequency.

What carries the argument

The central object is a two-fluid system for a hydrogen plasma with ionized charges and neutrals coupled by elastic collisions. The momentum exchange is $\mathbf{R}_n = \alpha \rho_n \rho_c(\mathbf{u}_c-\mathbf{u}_n)$ and the energy exchange contains a frictional heating term $\tfrac12\alpha\rho_n\rho_c(\mathbf{u}_c-\mathbf{u}_n)^2$ plus thermal exchange, with the collisional parameter $\alpha$ built from ion-neutral and electron-neutral collision frequencies. The argument is carried by a WKB (slowly varying amplitude) solution of the linearized two-fluid equations, which reduces the problem to a fourth-order dispersion relation in the vertical wavenumber $k$; the imaginary part of $k$ as a function of height gives the damping length. The decisive comparison is between the simulated wave amplitude and this analytic WKB amplitude, which separates linear decoupling effects from nonlinear shock effects.

What would settle it

The paper itself shows that raising the collisional coefficient by a factor of $10^4$ eliminates decoupling; the claim would be falsified if a two-fluid simulation with realistically strong coupling still showed comparable damping, or if the analytic WKB damping failed to track the linear numerical solution in the same stratified hydrogen atmosphere. A concrete test is to measure the height where a 1 s fast wave is damped in the strong-field profile and compare it with the height at which the imaginary part of the WKB wavenumber becomes negative; a mismatch larger than the WKB approximation error would indicate another damping mechanism.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the amplitude loss of fast magneto-acoustic waves in a stratified, partially ionized chromosphere is governed by the linear decoupling of the charged and neutral fluids. The evidence is the close match between numerical solutions of the linearized two-fluid equations and an analytic WKB solution of the same system; the WKB solution's imaginary wavenumber gives the damping length. Because this agreement holds before shocks form, the paper attributes the damping to linear effects; in the nonlinear regime, the same collisional term acts on the shortened spatial scales of shock fronts, so shock steepening strongly increases the damping. The decoupling height responds sensitively to wave period, amplitude, and field strength, and the frictional heating term raises the background temperature at a steady rate proportional to the square of the velocity difference.

Load-bearing premise

The central conclusions stand on the assumed collisional coupling between charges and neutrals, set by fixed cross sections in a hydrogen-only atmosphere; if the real chromospheric collision frequencies differ, the decoupling heights, damping lengths, and heating rates all move.

Editorial extensions

If this is right

  • High-frequency waves (periods 1–5 s) are damped within the modeled chromosphere; in the stronger-field profile a 1 s wave essentially disappears by about 0.8 Mm.
  • The height where decoupling begins falls as the wave period decreases and as the magnetic field strengthens, yet the damping length is shorter for the weaker field.
  • Frictional heating accumulates linearly in time, raising the temperature of both fluids and reducing the background magnetic field, at a rate proportional to the squared velocity difference.
  • In the linear regime the analytic WKB solution matches the simulations closely, indicating that reflection and nonlinearities are not needed to explain the damping.
  • Nonlinear shock steepening markedly increases collisional damping, and the effect is stronger for shorter-period waves because their shocks form at lower heights.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the collision cross sections are revised, the decoupling heights, damping lengths, and heating rates shift systematically; observed damping of high-frequency chromospheric waves could therefore serve as a constraint on the ion-neutral collision frequency.
  • Because the control run with artificially increased collisions eliminates decoupling, the same setup offers a clean numerical testbed for any future code that adds ionization, radiation, or conduction: those additions should preserve the linear WKB damping baseline if the present claim is right.
  • The 1D horizontal-field geometry likely understates the role of shock-front geometry; in multi-dimensional simulations with inclined fields, the effective front scale varies along the front, which would spread the damping over a range of heights rather than a single sharp layer.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 6 minor

Summary. The paper presents two-fluid simulations of fast magneto-acoustic waves in a stratified, purely hydrogen model of the solar chromosphere, using the MANCHA3D code. Waves are driven at the base of the chromosphere for periods 1–20 s and amplitude factors A = 0.5–100, in two horizontal magnetic-field profiles. The authors quantify the height-dependent decoupling between ion and neutral velocities, the associated collisional damping, and frictional heating. They derive a WKB solution of the linearized two-fluid equations and compare it with linear and nonlinear numerical runs. The principal claims are that (i) in the linear regime the damping is well described by the WKB solution and arises from ion-neutral decoupling, and (ii) nonlinear steepening dramatically increases collisional damping. Time-averaged temperature increases are attributed to frictional heating.

Significance. If fully established, the paper would provide a useful quantitative framework for two-fluid damping and heating of chromospheric fast waves, including a WKB tool for estimating linear damping and a separation of linear and nonlinear contributions. The model is transparent, the governing equations are given explicitly, and the linear comparison (Fig. 14) and the α×10^4 control (Fig. 8) are strong internal consistency checks. However, the nonlinear part of the central claim currently rests on runs without shock capturing and without a grid-convergence study, so the enhanced damping in Fig. 16 could be partly numerical. The quantitative outputs also depend on fixed collision cross sections, and no sensitivity study is presented. These issues are addressable with additional numerical experiments.

major comments (2)
  1. [Section 4.1 (Fig. 16; Fig. 3 caption)] The conclusion that nonlinear steepening "dramatically increases" collisional damping is not yet numerically secured. The nonlinear runs are performed with a second-order scheme that, as the authors explicitly note for the A=100 cases (Fig. 3 caption), produces shock-overshoot artifacts because no shock-capturing algorithm is used. Since the A=10 runs in Fig. 16 also develop saw-tooth wave fronts, they are likely subject to the same numerical deficiency. No grid-convergence study is reported, and the paper does not demonstrate that the numerical dissipation at unresolved shock fronts is small compared with the physical collisional dissipation. The frictional-heating term Q ∝ (u_c−u_n)^2 in Eq. (25) amplifies spurious small-scale velocity differences at under-resolved fronts, so the enhanced amplitude decay in Fig. 16 and the temperature increase in Figs. 9–11 could be contaminated by numerical heating. Please provide a convergence study at two additional grid resolutions for a representative case (e.g., A=10, P=5 s) or a comparison with a shock-capturing/artificial-viscosity variant, and quantify the numerical contribution to the heating budget.
  2. [Section 2.1 (Eqs. 4–5) and Section 4.1] The quantitative results—decoupling heights, damping lengths, and heating rates—scale directly with the collision frequencies in Eq. (5), which adopt fixed cross sections σ_in = 5×10^−19 m^2 and σ_en = 10^−19 m^2 from Braginskii (1965). The only collisional sensitivity test is the α×10^4 control in Fig. 8, which is an extreme strong-coupling limit rather than a realistic uncertainty range. Because the paper makes specific quantitative statements (e.g., decoupling above 0.7–1 Mm, damping lengths for 1 s and 5 s waves), the authors should test the robustness of these heights and damping lengths to a factor-of-two variation in α (or in the cross sections). Without such a test, it is unclear how much of the quantitative output is an artifact of the assumed momentum-transfer cross sections.
minor comments (6)
  1. [Equation (35)] Equation (35) appears to contain a systematic typo: the terms proportional to −ik and −k^2 are written with V_{c1}(z) in place of the full amplitude V_c(z) (or V_{c0} at zeroth order). As printed, the zeroth-order terms would vanish and the dispersion relation (36) would not follow; I assume this is a typesetting error, but it should be corrected to make the derivation traceable.
  2. [Section 2.1, after Eq. (5)] The sentence "Additionally, both temperatures are assumed to be equal to Tc" is ambiguous: the model has separate T_n and T_c, and Eq. (3) includes a thermal exchange term (T_c−T_n). Presumably the authors mean that the initial equilibrium temperatures of charges and neutrals are equal; please clarify.
  3. [Section 4.1, Fig. 14] The WKB analytical solution is derived from the same linearized two-fluid equations and the same background atmosphere as the numerical runs, so the agreement in Fig. 14 is an internal consistency check rather than an independent validation. The paper should state this explicitly, so that readers do not overinterpret the comparison as a benchmark against an external model.
  4. [Section 2.2, Eq. (13)] The description of the integration constant C and the role of the parameter j (S-profile vs B-profile) is terse; the sentence "The integration constant dominates over the space-varying term" is difficult to follow. Please expand the explanation of how the two magnetic-field profiles are constructed and why the constant term makes the field almost flat.
  5. [Figure 12] The text refers to modes #1 and #4 as "unphysical" in the strongly coupled limit, but the criterion for this classification is not stated. Please define the criterion (e.g., extremely large k_I/k_R, or negative group velocity) so the reader can interpret the solution branches.
  6. [Section 3.2, Fig. 9] The statement that a temperature increase of about 0.5 K "is not negligible" relative to the 10 K oscillation amplitude would benefit from a dimensionless measure (e.g., a heating rate per wave period or a fraction of the local temperature), which is available in Fig. 10 and the text.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the WKB comparison is an internal consistency check, not a fitted prediction.

full rationale

The paper's central claims are extracted from numerical evolution of the two-fluid equations, and the analytical WKB solution is derived independently from the linearized versions of the same equations and the same background atmosphere. No free parameter is fitted to the simulation output in the comparison shown in Fig. 14: the analytical solution uses the same boundary velocity amplitude, wave frequency, and background collision frequencies as the runs, so it can fail to match the numerical solution in shape, height of damping onset, or damping length. The close agreement therefore tests whether the linear numerical solver implements the same physics as the WKB reduction, which is a legitimate consistency check rather than a circular prediction. The decoupling height and damping onset are emergent quantities; they are not used to define any input parameter. The control run with the collision parameter multiplied by 10^4 (Fig. 8) is a genuine parameter variation and the nonlinear-versus-linear comparison in Fig. 16 separates regimes without renaming fitted data. Self-citations to Popescu Braileanu et al. (2019) and the Braginskii (1965) cross sections support code implementation and physical inputs, respectively, and are not load-bearing for the target conclusion. The acknowledged absence of shock-capturing algorithms for the A=100 cases is a numerical accuracy risk for the nonlinear-damping conclusion, but it is a correctness concern, not circularity.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The central claims rest on the two-fluid collisional model and the chosen background atmosphere. The model parameters (magnetic field, wave amplitude, wave period) are input controls of the experiment rather than fitted constants. The physics inputs from prior literature are the collision cross sections and the VALC temperature profile; no new entities are introduced.

free parameters (4)
  • Base magnetic field strength Bx0(z0) = 1e-4 T
    Input parameter for the magneto-hydrostatic equilibrium; together with the integration constant C it sets the S and B profiles. Varied in the study, not fitted to output.
  • Magnetic field integration constant C = 0.988 Pa (S profile), 120.73 Pa (B profile)
    Chosen via Eq. (13) to set the height where neutral and magnetic pressures are equal; determines the vertical profile of Bx0.
  • Wave amplitude factor A = 0.5, 1, 2, 10, 100
    Driver amplitude at the base, set as a fraction of sound speed; scanned to study nonlinear effects.
  • Wave period P = 1, 2.5, 5, 7.5, 20 s
    Driver frequency; scanned to study frequency dependence of decoupling and damping.
assumptions (4)
  • domain assumption The solar plasma can be described as two distinct fluids, charges (ions plus electrons) and neutrals, with only elastic collisional coupling; no ionization/recombination, viscosity, thermal conduction, or radiation are included.
    This is the core modeling assumption of the two-fluid Mancha3D equations in Section 2.1.
  • domain assumption A pure hydrogen plasma is a valid model for the upper photosphere and chromosphere, with metal electrons neglected; the VALC temperature profile is used but densities are recomputed for hydrogen.
    Stated in Section 2.2; the model is restricted to heights z0 to zf where this is acceptable.
  • domain assumption The WKB approximation is valid: wavelengths are shorter than the stratification scale and amplitude/wavenumber gradients are small (Eq. 34).
    Used to derive the analytical damping solution in Section 4.1; its accuracy is verified a posteriori by comparison with linear numerical solutions.
  • domain assumption The driver at the lower boundary is accurately represented by the linearized single-fluid solution with ambipolar diffusion neglected at the bottom (Eqs. 17-22).
    The paper assumes strong collisional coupling at the base and uses this solution in ghost points to inject the wave.

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Cite this review

Pith. "Pith review of Two-fluid simulations of waves in the solar chromosphere II. Propagation and damping of fast magneto-acoustic waves and shocks." pith.science (2026). https://pith.science/paper/EJRBKX4G

@misc{pith2026190805262,
  author       = {Pith},
  title        = {Pith review of: Two-fluid simulations of waves in the solar chromosphere II. Propagation and damping of fast magneto-acoustic waves and shocks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EJRBKX4G}},
  note         = {Machine review of arXiv:1908.05262}
}
read the original abstract

Waves and shocks traveling through the solar chromospheric plasma are influenced by its partial ionization and weak collisional coupling, and may become susceptible to multi-fluid effects, similar to interstellar shock waves. In this study, we consider fast magneto-acoustic shock wave formation and propagation in a stratified medium, that is permeated by a horizontal magnetic field, with properties similar to that of the solar chromosphere. The evolution of plasma and neutrals is modeled using a two-fluid code that evolves a set of coupled equations for two separate fluids. We observed that waves in neutrals and plasma, initially coupled at the upper photosphere, become uncoupled at higher heights in the chromosphere. This decoupling can be a consequence of either the characteristic spatial scale at the shock front, that becomes similar to the collisional scale, or the change in the relation between the wave frequency, ion cyclotron frequency, and the collisional frequency with height. The decoupling height is a sensitive function of the wave frequency, wave amplitude, and the magnetic field strength. We observed that decoupling causes damping of waves and an increase in the background temperature due to the frictional heating. The comparison between analytical and numerical results allows us to separate the role of the nonlinear effects from the linear ones on the decoupling and damping of waves.

Figures

Figures reproduced from arXiv: 1908.05262 by the authors.

Figure 1
Figure 1. Parameters of background model atmosphere as function of height. Panel (a): gas pressure of neutrals (blue), charges (orange), and magnetic pressures corresponding to the S magnetic field profile (green) and B profile (red). Panel (b): number density of neutrals (blue), and charges (orange). For comparison, corresponding number densities from VALC model atmosphere are plotted in green and red. Panel (c): temperature… view at source ↗
Figure 2
Figure 2. Height dependence of characteristic frequencies calcu￾lated for background atmospheric model. The blue and orange lines are ion-cyclotron frequencies corresponding to the S and B magnetic field profiles, red and green lines are ion-neutral and neutral-ion collision frequencies, and violet is the highest wave frequency used is our work, corresponding to the wave period of 1 s. where F should satisfy F(z) ≥ 0 and F(z0… view at source ↗
Figure 3
Figure 3. Height dependence of oscillations in velocity and temperature of charges and neutrals, and in magnetic field for fixed moment of time, obtained from numerical solution of two fluid equations for magnetic field profile S. Panel (a) and (b) are for the initial wave amplitude factor A = 1; panels (c) and (d) are for A = 100. The wave period is P = 20 s for panels (a) and (c), and P = 7.5 s for panels (b) and (d). Orang… view at source ↗
Figures from the paper (12 more)
Figure 4
Figure 4. Figure 4: Height dependence of oscillations in velocity and temperature of charges and neutrals, and in magnetic field for fixed moment in time, for initial wave amplitude factor A = 1. The format of the figure is the same as for [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: Decoupling between charges and neutral velocity as function of height for magnetic field profile S. Left panel: wave period 20 s, amplitude factors A = 1 (blue), A = 10 (orange), A = 100 (green). Right panel: wave period 1 s, amplitude factors A = 0.5 (blue), A = 1 (or…
Figure 6
Figure 6. Figure 6: Decoupling between charges and neutral velocity as func￾tion of height for magnetic field profile S and wave amplitude factor A = 1. Wave periods P = 20 s (violet), P = 7.5 s (red), P = 5 s (green), P = 2.5 s (orange), and P = 1 s (blue). 3.1. Decoupling The dependence…
Figure 7
Figure 7. Figure 7: Decoupling between charges and neutral velocity as func￾tion of height for amplitude factor A = 1 and wave period P = 1 s. The dependence on the magnetic field profiles is shown with different curves: S (blue) and B (orange). the simulations (when the waves reached the…
Figure 8
Figure 8. Figure 8: Decoupling between charges and neutral velocity as func￾tion of height for S magnetic field profile and amplitude factor A = 1. Blue line: nonlinear simulation; orange line: linear sim￾ulation; green line: linear simulation with artificially increased collisional param…
Figure 9
Figure 9. Figure 9: Time average of perturbed variables as function of height for simulation with S magnetic field profile, wave period P = 1 s and amplitude factor A = 2. From top to bottom: temperatures of neutrals (orange) and charges (blue), magnetic field, density of charges, density…
Figure 11
Figure 11. Figure 11 [PITH_FULL_IMAGE:figures/full_fig_p013_11.png]
Figure 12
Figure 12. Figure 12: shows the real and imaginary part of the four solutions, E, of the fourth order dispersion relation, Eq. (42), as a function of the non-dimensional variable F, which is the ratio between the collisional and the wave frequencies, scaled with ρ0. This dispersion relatio…
Figure 13
Figure 13. Figure 13: Solutions of dispersion relation (36) for two-fluid lin￾earized equations. The upper and lower panels show the real and imaginary part of k as a function of height, calculated for the magnetic field profile S and 5 s period. νni and νin. This is true for both magnetic…
Figure 14
Figure 14. Figure 14: Comparison between numerical solution in linear regime (black lines) and analytical solution of two-fluid equations (red dotted line). Individual panels show snapshots of the velocity of charges as a function of height at fixed time moments in the stationary regime of…
Figure 15
Figure 15. Figure 15: Left panel: imaginary part of k as function of wave period (horizontal axis) and height (vertical axis) obtained after solving dispersion relation for two-fluid equations (Eq. 36) for S magnetic field profile. Solution #3 from [PITH_FULL_IMAGE:figures/full_fig_p017_15.png]
Figure 16
Figure 16. Figure 16: Comparison between numerical solution in linear regime (black lines) and nonlinear regime (red dotted line) with A=10. Individual panels show snapshots of the velocity of charges as a function of height at fixed time moments in the stationary regime of the simulations…

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Works this paper leans on

70 extracted references · 68 canonical work pages

  1. [1]

    Anderson , L. S. and Athay , R. G. 1989, , 336, 1089

    work page 1989

  2. [2]

    D., Brady , C

    Arber , T. D., Brady , C. S., and Shelyag , S. 2016, , 817, 94

    work page 2016

  3. [3]

    L., Alexeev , I., Collados , M., et al

    Ballester , J. L., Alexeev , I., Collados , M., et al. 2018 a , , 214, 58

    work page 2018

  4. [4]

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

    Ballester , J. L., Carbonell , M., Soler , R., and Terradas , J. 2018 b , , 609, A6

    work page 2018

  5. [5]

    2009, , 508, 941

    Bello Gonz \'a lez , N., Flores Soriano , M., Kneer , F., and Okunev , O. 2009, , 508, 941

    work page 2009

  6. [6]

    2010 a , , 522, A31

    Bello Gonz \'a lez , N., Flores Soriano , M., Kneer , F., Okunev , O., and Shchukina , N. 2010 a , , 522, A31

    work page 2010

  7. [7]

    2010 b , , 723, L134

    Bello Gonz \'a lez , N., Franz , M., Mart \' nez Pillet , V., et al. 2010 b , , 723, L134

    work page 2010

  8. [8]

    1994, Journal of Computational Physics, 114, 185

    Berenger , J.-P. 1994, Journal of Computational Physics, 114, 185

    work page 1994

Show all 70 references
  1. [9]

    J., Carlsson , M., Hansteen , V

    Bogdan , T. J., Carlsson , M., Hansteen , V. H., et al. 2003, , 599, 626

    2003

  2. [10]

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

    1965

  3. [11]

    Cally , P. S. 2006, Philosophical Transactions of the Royal Society of London Series A, 364, 333

    2006

  4. [12]

    Cally , P. S. and Goossens , M. 2008, , 251, 251

    2008

  5. [13]

    and Stein , R

    Carlsson , M. and Stein , R. F. 2002, in ESA Special Publication, Vol. 505, SOLMAG 2002. Proceedings of the Magnetic Coupling of the Solar Atmosphere Euroconference, ed. H. Sawaya-Lacoste , 293--300

    2002

  6. [14]

    Cheung , M. C. M. and Cameron , R. H. 2012, , 750, 6

    2012

  7. [15]

    Draine , B. T. 1980, , 241, 1021

    1980

  8. [16]

    Draine , B. T. and McKee , C. F. 1993, , 31, 373

    1993

  9. [17]

    T., Roberge , W

    Draine , B. T., Roberge , W. G., and Dalgarno , A. 1983, , 264, 485

    1983

  10. [18]

    2010, , 719, 357

    Felipe , T., Khomenko , E., and Collados , M. 2010, , 719, 357

    2010

  11. [19]

    and Carlsson , M

    Fossum , A. and Carlsson , M. 2006, , 646, 579

    2006

  12. [20]

    Gupta , G. R. 2014, AandA, 568, A96

    2014

  13. [21]

    2016, , 591, A112

    Hillier , A., Takasao , S., and Nakamura , N. 2016, , 591, A112

    2016

  14. [22]

    and Tripathi , D

    Isobe , H. and Tripathi , D. 2006, AandA, 449, L17

    2006

  15. [23]

    B., Mathioudakis , M., Erd \'e lyi , R., et al

    Jess , D. B., Mathioudakis , M., Erd \'e lyi , R., et al. 2008, , 680, 1523

    2008

  16. [24]

    B., Morton , R

    Jess , D. B., Morton , R. J., Verth , G., et al. 2015, , 190, 103

    2015

  17. [25]

    2017, Plasma Physics and Controlled Fusion, 59, 014038

    Khomenko , E. 2017, Plasma Physics and Controlled Fusion, 59, 014038

    2017

  18. [26]

    and Calvo Santamaria , I

    Khomenko , E. and Calvo Santamaria , I. 2013, in Journal of Physics Conference Series, Vol. 440, Journal of Physics Conference Series, 012048

    2013

  19. [27]

    and Collados , M

    Khomenko , E. and Collados , M. 2006, , 653, 739

    2006

  20. [28]

    and Collados , M

    Khomenko , E. and Collados , M. 2009, , 506, L5

    2009

  21. [29]

    and Collados , M

    Khomenko , E. and Collados , M. 2012, , 747, 87

    2012

  22. [30]

    and Collados , M

    Khomenko , E. and Collados , M. 2015, Living Reviews in Solar Physics, 12, 6

    2015

  23. [31]

    2014 a , Physics of Plasmas, 21, 092901

    Khomenko , E., Collados , M., D \' az , A., and Vitas , N. 2014 a , Physics of Plasmas, 21, 092901

    2014

  24. [32]

    2014 b , , 565, A45

    Khomenko , E., D \' az , A., de Vicente , A., Collados , M., and Luna , M. 2014 b , , 565, A45

    2014

  25. [33]

    2017, , 604, A66

    Khomenko , E., Vitas , N., Collados , M., and de Vicente , A. 2017, , 604, A66

    2017

  26. [34]

    2018, ArXiv e-prints

    Khomenko , E., Vitas , N., Collados , M., and de Vicente , A. 2018, ArXiv e-prints

    2018

  27. [35]

    A., Tanner , S

    Klimchuk , J. A., Tanner , S. E. M., and De Moortel , I. 2004, , 616, 1232

    2004

  28. [36]

    E., DeVore , C

    Leake , J. E., DeVore , C. R., Thayer , J. P., et al. 2014, , 184, 107

    2014

  29. [37]

    S., and Linton , M

    Lee , E., Lukin , V. S., and Linton , M. G. 2014, AandA, 569, A94

    2014

  30. [38]

    and Zhang , J

    Li , T. and Zhang , J. 2012, , 760, L10

    2012

  31. [39]

    Luna , M., Terradas , J., Oliver , R., and Ballester , J. L. 2008, , 676, 717

    2008

  32. [40]

    G., Alvarez Laguna , A., Lani , A., and Poedts , S

    Maneva , Y. G., Alvarez Laguna , A., Lani , A., and Poedts , S. 2017, , 836, 197

    2017

  33. [41]

    2016, , 832, 101

    Mart \' nez-G \'o mez , D., Soler , R., and Terradas , J. 2016, , 832, 101

    2016

  34. [42]

    2017, , 837, 80

    Mart \' nez-G \'o mez , D., Soler , R., and Terradas , J. 2017, , 837, 80

    2017

  35. [43]

    2016, , 831, L1

    Mart \' nez-Sykora, J., De Pontieu, B., Carlsson, M., and Hansteen, V. 2016, , 831, L1

    2016

  36. [44]

    2012, , 753, 161

    Mart \' nez-Sykora , J., De Pontieu , B., and Hansteen , V. 2012, , 753, 161

    2012

  37. [45]

    A., De Moortel , I., Hood , A

    McLaughlin , J. A., De Moortel , I., Hood , A. W., and Brady, C. S. 2009, AandA, 493, 227

    2009

  38. [46]

    Mullan , D. J. 1971, , 153, 145

    1971

  39. [47]

    E., Rosner , R., Stein , R

    Musielak , Z. E., Rosner , R., Stein , R. F., and Ulmschneider , P. 1994, , 423, 474

    1994

  40. [48]

    Nye , A. H. and Thomas , J. H. 1976, , 204, 573

    1976

  41. [49]

    Sehgal, a

    Ofman, L., Nakariakov, V., and N. Sehgal, a. 2008, , 533, 1071

    2008

  42. [50]

    S., Khomenko , E., and de Vicente , \'A

    Popescu Braileanu , B., Lukin , V. S., Khomenko , E., and de Vicente , \'A . 2019, , 627, A25

    2019

  43. [51]

    M., and Lukin , V

    Provornikova , E., Laming , J. M., and Lukin , V. S. 2018, ArXiv e-prints

    2018

  44. [52]

    P., Przybylski , D., et al

    Rijs , C., Rajaguru , S. P., Przybylski , D., et al. 2016, , 817, 45

    2016

  45. [53]

    C., Khomenko , E., Collados , M., and de Vicente , A

    Santamaria , I. C., Khomenko , E., Collados , M., and de Vicente , A. 2017, , 602, A43

    2017

  46. [54]

    2016, , 819, L11

    Shelyag , S., Khomenko , E., de Vicente , A., and Przybylski , D. 2016, , 819, L11

    2016

  47. [55]

    and Hillier , A

    Snow , B. and Hillier , A. 2019, arXiv e-prints, arXiv:1904.12518

    2019 arXiv

  48. [56]

    L., and Terradas , J

    Soler , R., Carbonell , M., Ballester , J. L., and Terradas , J. 2013 a , , 767, 171

    2013

  49. [57]

    J., Ballester , J

    Soler , R., Díaz , A. J., Ballester , J. L., and Goossens , M. 2013 b , AandA, 551, A86

    2013

  50. [58]

    and Terradas , J

    Soler , R. and Terradas , J. 2015, , 803, 43

    2015

  51. [59]

    Soler , R., Terradas , J., Oliver , R., and Ballester , J. L. 2017, , 840, 20

    2017

  52. [60]

    and Vasyli \= u nas , V

    Song , P. and Vasyli \= u nas , V. M. 2011, Journal of Geophysical Research (Space Physics), 116, A09104

    2011

  53. [61]

    V., et al

    Tóth, G., van der Holst, B., Sokolov, I. V., et al. 2012, Journal of Computational Physics, 231, 870 , special Issue: Computational Plasma PhysicsSpecial Issue: Computational Plasma Physics

    2012

  54. [62]

    Vasquez , B. J. 2005, Journal of Geophysical Research (Space Physics), 110, A10S02

    2005

  55. [63]

    E., Avrett , E

    Vernazza , J. E., Avrett , E. H., and Loeser , R. 1981, , 45, 635

    1981

  56. [64]

    M., Ofman , L., and Deluca , E

    Verwichte , E., Nakariakov , V. M., Ofman , L., and Deluca , E. E. 2004, , 223, 77

    2004

  57. [65]

    Withbroe , G. L. and Noyes , R. W. 1977, Annual Review of Astronomy and Astrophysics, 15, 363

    1977

  58. [66]

    V., Khodachenko , M

    Zaqarashvili , T. V., Khodachenko , M. L., and Rucker, H. O. 2011, AandA, 529, A82

    2011

  59. [67]

    V., Khodachenko , M

    Zaqarashvili , T. V., Khodachenko , M. L., and Soler , R. 2013 a , , 549, A113

    2013

  60. [68]

    V., Khodachenko , M

    Zaqarashvili , T. V., Khodachenko , M. L., and Soler , R. 2013 b , , 549, A113

    2013

  61. [69]

    , " * write output.state after.block = add.period write newline

    ENTRY address author booktitle chapter edition editor howpublished institution journal key month note number organization pages publisher school series title type volume year label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence a...

  62. [70]

    write newline

    " write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 gl...

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.