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Characteristics of acoustic-wave heating in simulations of the quiet Sun chromosphere

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Pith's one-line read In two 3D radiative MHD simulations, the energy lost by upward-propagating acoustic waves exceeds the viscous dissipation above 0.75 Mm, placing a major part of chromospheric heating at the beta=1 layer where waves steepen into shocks.

desk verdict A transparent analysis of Bifrost runs that credibly localizes acoustic heating near the beta=1 layer, with an unsigned flux proxy that a referee should ask them to tighten. read the letter →

arxiv 2505.21047 v1 pith:ZNJEMNEG submitted 2025-05-27 astro-ph.SR

Elias R. Udn{ae}s , Tiago M. D. Pereira This is my paper
classification astro-ph.SR
keywords quietSunchromosphereacousticwaveschromosphericheatingshockdissipationcut-offfrequencyradiativeMHDsimulationswavefluxcoronalhole
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

The paper's aim is to show that acoustic waves do not merely pass through the quiet Sun chromosphere but deposit a significant fraction of their energy there. Using two long 3D radiative magnetohydrodynamic simulations, one of the quiet Sun and one of a coronal hole, the authors find that upward-propagating acoustic waves near the acoustic cut-off frequency steepen into shocks around the height where plasma $\beta$ equals one, and that the height gradient of the acoustic flux exceeds the simulated viscous dissipation above 0.75 Mm. They identify frequencies of about 3 to 6 mHz and horizontal scales larger than about 1 Mm as the main carriers, and one tracked shock deposits a peak heating rate of about 4 kW $m^{-2}$. If correct, acoustic-wave heating is an important mechanism in the chromospheric energy balance, complementing magnetic heating rather than being a negligible leftover.

What carries the argument

The central object is the isothermal acoustic-gravity dispersion relation, $k_z^2 = c_s^{-2}(\omega^2 - \omega_a^2) - (\omega^2 - \omega_g^2) k_h^2/\omega^2$, with the acoustic and gravity cut-off frequencies $\omega_a = \gamma g/(2 c_s)$ and $\omega_g = \sqrt{\gamma-1}\, g/c_s$. It separates the Fourier power of measured vertical velocity into propagating acoustic waves, evanescent p-mode ridges, and internal gravity waves, and thereby defines which part of the measured flux counts as acoustic. The vertical mechanical flux is then $F_m = \rho_0 u_z^2 c_s$ from the azimuthally averaged power spectrum, and the height derivative of the acoustic flux is compared with the simulated viscous dissipation. Coherence between the flux and the dissipation term, computed with the paper's Eq. (6), identifies the frequencies and heights where wave energy actually becomes heat. The simulations themselves provide the velocity, density, sound speed, and dissipation fields, and one tracked shock front shows the same heating in physical space.

What would settle it

Compute the acoustic flux gradient using a fully 3D, non-isothermal dispersion relation that includes local temperature fluctuations and magnetic-field effects; if the corrected gradient no longer exceeds the viscous dissipation above 0.75 Mm, the central claim fails. A simpler check is to map where shocks actually form in the simulations and compare those heights with the beta=1 layer; if most shocks dissipate well above or below that layer, the localization claim is wrong.

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Extended reading notes

Core claim

In two long-duration 3D radiative magnetohydrodynamic simulations of the quiet Sun and a coronal hole, vertically propagating acoustic waves near the acoustic cut-off frequency carry a mechanical flux of about 6 kW $m^{-2}$ at 0.5 Mm. As this flux rises through the chromosphere it declines steeply, and its height gradient closely matches the time-averaged viscous dissipation between about 0.75 and 1.7 Mm. The dissipation is concentrated around 1 Mm, the height where the plasma $\beta$ equals one and the sound speed roughly equals the Alfvén speed; there the authors find coherence between the mechanical flux and viscous heating for frequencies of 3 to 6 mHz and horizontal wavenumbers below about 1 $Mm^{-1}$. Following one individual shock shows a peak integrated heating rate of about 4 kW $m^{-2}$, with viscous dissipation 6.3 times the Ohmic dissipation. The authors conclude that the energy lost by acoustic waves is more than enough to account for the heat generated by viscous dissipation above 0.75 Mm, making acoustic-wave heating an important mechanism in the chromospheric energy balance of the quiet Sun and coronal holes.

Load-bearing premise

The load-bearing premise is that a simple isothermal wave theory, using horizontally averaged sound speed and density, correctly identifies which measured oscillations are genuinely propagating acoustic waves rather than non-propagating p-mode ridges; if that identification is wrong, the inferred flux, dissipation heights, and the beta=1 heating claim weaken.

Editorial extensions

If this is right

  • The acoustic flux at 0.5 Mm is about 6 kW m^-2, large enough to compete with the canonical 4.6 kW m^-2 radiative loss of the chromosphere, so wave heating cannot be dismissed on energy grounds.
  • Above roughly 0.75 Mm, the loss of acoustic flux exceeds the simulated viscous heating, meaning acoustic shocks can supply the mid and upper chromospheric heating in these simulations without invoking magnetic dissipation.
  • The dissipative region lies near the beta=1 layer at frequencies of 3 to 6 mHz, giving a concrete prediction for where shock heating should be concentrated.
  • Internal gravity waves carry little flux in the chromosphere, so acoustic waves dominate mechanical energy transport above the temperature minimum.
  • In a coronal hole, the magnetic field alters the wave mode and reduces the acoustic-heating coherence, so the local magnetic geometry controls how much p-mode energy is deposited versus converted and lost.

Reading between the lines

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

  • A direct follow-up would be to count and integrate every shock front in the two simulations; the authors note they did not do this systematic accounting, and it would bridge the gap between per-event heating of 4 to 10 kW m^-2 and the mean viscous dissipation profile.
  • The same flux-gradient method could be applied to high-cadence chromospheric observations, with the expectation that unresolved spatial scales would make observed acoustic fluxes lower than simulated ones; the paper's numbers provide a target for resolution-corrected estimates.
  • If the beta=1 localization is real, chromospheric heating models should treat the beta=1 surface as a preferred dissipation site, a prediction testable by comparing shock emission locations with magnetic field inversions.
  • The coronal hole result hints that some p-mode energy is converted to other MHD modes and escapes the chromosphere; tracking wave energy across the beta=1 layer in the simulations would quantify that loss.
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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

3 major / 6 minor

Summary. This paper analyzes two 3D radiative MHD Bifrost simulations, one of the quiet Sun and one of a coronal hole, to study acoustic-wave heating in the chromosphere. The authors use an isothermal acoustic-gravity dispersion relation to separate vertical-velocity power into propagating acoustic flux, internal gravity-wave flux, and evanescent power, and then integrate the acoustic contribution to obtain a height-dependent acoustic flux (Eq. 8, Fig. 4). They compare the height gradient of this flux with the average viscous dissipation in the simulations (Fig. 5), compute the coherence between the flux proxy and viscous heating (Fig. 6), and examine one individual shock event in detail (Figs. 7-8). The central claim is that acoustic waves with frequencies near the acoustic cut-off dissipate much of their energy by shock formation near the plasma-β = 1 layer, making acoustic-wave heating an important component of the chromospheric energy balance in the quiet Sun and coronal holes.

Significance. If the quantitative comparison is valid, the paper would provide a strong simulation-based case that acoustic waves from the convection zone can supply a significant fraction of the energy required to heat the quiet-Sun chromosphere, complementing ongoing observational debates. The study is transparent about many of its limitations, uses no parameters fitted to the target result, and combines independent diagnostics (flux gradients, viscous dissipation, coherence, and a single-shock case study). The qualitative picture and the shock event analysis are plausible and valuable. However, as detailed below, the central quantitative claim rests on interpreting the height derivative of an unsigned velocity-power proxy as the divergence of net acoustic energy flux, and that interpretation is not currently justified.

major comments (3)
  1. [§4–§5, Eq. (8), Fig. 5] The proxy F_m = ρ0 û_z² c_s is an unsigned power spectrum of vertical velocity, not a signed vertical energy flux. Upward and downward propagating waves both contribute with the same positive sign, and the paper itself shows shocks reflecting off the transition region (§6) and discusses mode conversion at β≈1 (§7). Therefore ∂F_ac/∂z is not the energy lost by the upward-propagating acoustic wave field. In a standing wave produced by reflection, |u_z|² can increase with height even with zero dissipation, so the inequality '∂F_ac/∂z > acoustic wave heating' stated in §5 is not guaranteed. The comparison in Fig. 5, and the Section 8 statement that 'the energy lost by acoustic waves is more than enough to account for the heat generated by viscous dissipation', rest on a flux measure that has not been validated as a divergence of net energy flux. I recommend computing a signed flux (e.g., the horizontal average of p' v_z, or a decomposition into upward and downward vertical wavenumber components) and comparing its divergence with q_visc. This is a necessary check for the central claim.
  2. [§5, Fig. 5] Even setting aside the sign issue, F_m = ρ0 c_s û_z² varies with height through the background quantities ρ0(z) and c_s(z). The derivative plotted in Fig. 5 therefore contains the term (d/dz)(ρ0 c_s) û_z², which is not a wave-energy loss and can be significant in a stratified atmosphere. The authors acknowledge this effect for the transition-region spike at 2 Mm, but the same background term may contaminate the 0.7–1.7 Mm range where the comparison is made. I ask that the authors either compute the divergence of the actual energy flux, or estimate and subtract this stratification term, or show explicitly that it is negligible in the region of interest.
  3. [§3, Eq. (1), Figs. 3–6] The isothermal, adiabatic dispersion relation evaluated with time- and horizontally-averaged c_s is used to define the propagating acoustic region. The authors state in §7 that 'an accurate dispersion relation for wave propagation is not feasible' and that the fluxes should be taken as proxies. However, the quantitative conclusion in Section 8 depends on the integrated acoustic flux and its gradient; if the filter misclassifies part of the evanescent p-mode ridge (visible below the cut-off in Fig. 3) as propagating, the values of F_ac and ∂F_ac/∂z will be biased. Please add a sensitivity test (e.g., varying the cut-off frequency within the range spanned by local temperature fluctuations, or using the local c_s field) to show that the height range and magnitude of the dissipation peak are robust. Without such a test, the energy-budget comparison is not established.
minor comments (6)
  1. [§3, Eq. (6)] Equation (6) and the definitions below it appear to have lost complex conjugates and averaging brackets. As written, K² uses F(k,ω)G(k,ω) without conjugation and S_{f,f}=⟨F(k,ω)⟩², which is not the standard coherence and can exceed one. Please check against Vigeesh et al. (2017) and correct the expression.
  2. [§3, DFT paragraph] The phrase 'the amplitude of the wave with frequency k' should read 'frequency f' (or 'angular frequency ω'), since k is used for wavenumber.
  3. [§2] The sentence 'Both simulation boxes have 7683 grid points' should read '768³ grid points' if that is the intended meaning.
  4. [Fig. 2] The axis label 2Hk_x uses H, which is not defined in the text or caption; please define the pressure scale height.
  5. [§4, Eq. (8)] Please clarify the normalization of û². If û is the Fourier amplitude of a harmonic component, a factor of 1/2 may be needed for the time-averaged acoustic energy flux (ρ0⟨u²⟩c_s). The current text does not make this factor explicit.
  6. [§7] The statement 'our flux estimates are a lower bound' should be qualified. Because Eq. (8) is unsigned, it is an upper bound on the net upward flux for the measured vertical component; the 'lower bound' wording presumably refers to the neglect of inclined propagation or other wave modes. Please clarify which meaning is intended.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the flux, dissipation, and coherence diagnostics are independently measured from the simulations and no fitted parameter is renamed as a prediction.

full rationale

The claimed derivation chain is a set of simulation diagnostics, not a parametric fit. Eq. (8) defines the mechanical flux proxy Fm = rho0 uhat^2 cs from the vertical-velocity Fourier power, and Eq. (1) supplies an isothermal acoustic-gravity dispersion relation used only as a band filter. The acoustic flux Fac is the integral of Fm over the 'propagating' region; its height gradient is compared with the viscous dissipation qvisc, which is computed directly from the Bifrost heating terms (Section 5). No parameter is adjusted to make dFac/dz exceed qvisc, and the inequality is an empirical result. The paper explicitly labels these quantities as proxies ('these fluxes are not totally accurate and should be taken as proxies', Section 4) and lists reflection, mode conversion, and radiation as alternative loss channels that could reduce the flux without heating. The Bifrost code and the coronal hole simulation are community/in-house products, but they are datasets, not circular inputs to the wave-heating conclusion. The unsigned nature of Fm means dFac/dz is an upper-bound proxy for net upward energy loss rather than a signed flux divergence; that is a validity caveat about whether the comparison supports the central claim, not a definitional equivalence. No step in the paper reduces the prediction to its input by construction, and no load-bearing argument rests on a self-citation chain.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters were fitted to produce the acoustic flux or dissipation profiles; gamma=5/3 and standard physical constants are fixed by the simulation setup, and the Tukey window and 80% coherence threshold are analysis choices rather than fitted parameters. The main assumptions are domain assumptions about adiabatic propagation, the applicability of an isothermal dispersion relation to a dynamic 3D atmosphere, vertical-only propagation, and the physical interpretation of numerical viscous dissipation. No new particles, forces, or conserved quantities are introduced.

assumptions (5)
  • domain assumption The acoustic-gravity dispersion relation Eq. (1) for an isothermal atmosphere, with acoustic and gravity cut-offs from Eqs. (2) and (3), adequately represents wave propagation in the simulated chromosphere.
    Used in Sections 3 and 4 to separate acoustic from internal gravity and evanescent wave power; the authors note that non-isothermal and radiative effects modify the propagation boundaries, so this is a simplifying model assumption.
  • domain assumption Adiabatic wave propagation is a fair approximation for the flux estimates.
    Section 7 argues that radiation counteracts the vertical temperature gradient, pushing propagation closer to the adiabatic case; without this assumption, the cut-off frequency and the wave classification would change.
  • domain assumption The vertical mechanical flux can be approximated by rho0 times u_z squared times c_s, following Lighthill (1978), using horizontal averages of density and sound speed.
    Equation (8) defines the flux; it ignores horizontal velocity components, uses horizontal averages, and treats only vertically propagating waves, which the authors acknowledge makes the estimates lower bounds.
  • domain assumption Bifrost's viscous dissipation term is a credible proxy for heating by resolved and numerically broadened shock fronts.
    In Section 6 and the Discussion, shocks are smeared over five to six grid points by numerical diffusion; the paper assumes this smoothing yields a lower bound on physical heating rather than a spurious numerical artifact.
  • standard math Fourier power spectra and coherence computed with Eq. (6) are valid estimators for the sampled, apodized signals.
    Standard signal processing assumptions, with the paper noting Gibbs phenomenon artifacts in the Fourier transform of the spatially sharp viscous dissipation field.

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Pith. "Pith review of Characteristics of acoustic-wave heating in simulations of the quiet Sun chromosphere." pith.science (2026). https://pith.science/paper/ZNJEMNEG

@misc{pith2026250521047,
  author       = {Pith},
  title        = {Pith review of: Characteristics of acoustic-wave heating in simulations of the quiet Sun chromosphere},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZNJEMNEG}},
  note         = {Machine review of arXiv:2505.21047}
}
abstract

Understanding energy transfer through the chromosphere is paramount to solving the coronal heating problem. We investigated the energy dissipation of acoustic waves in the chromosphere of the quiet Sun using 3D radiative magnetohydrodynamic (rMHD) simulations. We analysed the characteristics of acoustic-wave heating and its dependence on height and magnetic field configuration. We find the typical heights where acoustic waves steepen into shocks and the frequencies and wavenumbers that most efficiently dissipate wave energy through this steepening. We combined a comprehensive large-scale analysis, spanning the entirety of the simulations for several solar hours, with a detailed view of an individual shock. We find that the flux of propagating acoustic waves correlates closely with viscous dissipation in the chromosphere above the temperature minimum. Acoustic waves with frequencies close to the acoustic cut-off frequency can efficiently heat the quiet Sun chromosphere at the plasma-$\beta$ = 1 interface and play an important role in the chromospheric energy balance.

Figures

Figures reproduced from arXiv: 2505.21047 by the authors.

Figure 1
Figure 1. Ratio of sound speed to Alfvén speeds in xz cuts taken from quiet Sun and coronal hole simulations. The simulation times are taken at 140 minutes and 40 seconds s and at y coor￾dinates 3.10 Mm. Overplotted are the magnetic field lines, with the thickness of the lines be￾ing scaled by the logarithm of the field strength, log10 ∥B∥. tions. On the other hand, Bello González et al. (2009) and Bello González et al. (2010… view at source ↗
Figure 2
Figure 2. Diagnostic diagram for acoustic-gravity waves in an isothermal atmosphere. Similar to Mihalas & Mihalas (1984, [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. kh −ν diagram of mechanical flux in quiet Sun and coronal hole simulations at five different heights in the atmosphere. The solid black lines are the dispersion relations calculated from time-averaged quantities at each height. The x-axes are divided by 2π and are therefore not in angular units. A common tool to analyse waves is the Fourier transform. As part of a discrete Fourier transform (DFT) method, we use the … view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Total integrated flux in quiet Sun and coronal hole simulations. Solid lines show the total mechanical flux; dashed lines show inte￾grated acoustic flux; dotted lines represent integrated internal gravity flux. Contributions are determined from the dispersion relation …
Figure 5
Figure 5. Figure 5: Gradient of vertical acoustic flux (solid lines) and time-averaged viscous dissipation (dashed lines) in simulations of the quiet Sun (blue) and coronal hole (red). means the acoustic flux is not damped around the temperature minimum. The coronal hole simulation has a …
Figure 6
Figure 6. Figure 6: kh − ν diagram of coherence Km,q between mechanical flux Fm and viscous dissipation Fq. Km,q, given by Eq. (6), is calculated at five different heights in the simulated chromospheres. The solid black lines are the dispersion relations calculated from time-averaged quan…
Figure 7
Figure 7. Figure 7: Vertical velocity, temperature, and dis￾sipation coefficients in a column of the coro￾nal hole simulation at x = 3.25 Mm and y = 3.06 Mm. netic field, which introduces magnetic field dependency to the acoustic waves lower down in the atmosphere. Therefore, the acoustic…
Figure 8
Figure 8. Figure 8: A column of the coronal hole simulation where a wave front develops into a shock. x-axes show simulation time in minutes, and y-axes show height above the surface in Mm. Panel (a) shows vertical velocity, (b) shows temperature, and (e) and (f) show the viscous and Ohmi…

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  1. On the Theory of Bulk Viscosity of Cold Plasmas and Thermodynamics of Alkali-Noble Gas Cocktails

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    For cold plasmas, the frequency-dependent bulk viscosity follows the Mandelstam-Leontovich/Drude form with high accuracy, and can exceed shear viscosity by orders of magnitude.

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