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REVIEW 3 major objections 4 minor 83 references

Viscous Heating and Instabilities in the Partially Ionized Solar Atmosphere

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Neutral viscosity can dominate magnetic diffusion in the solar chromosphere and, combined with Hall diffusion or a slight viscosity anisotropy, turn shear flows into new wave instabilities.

desk verdict The heating analysis and the viscous-Hall/gyroviscous criteria are genuinely new and useful, but the threshold-free anisotropic-viscosity instability is undone by a sign error in Eq. (48); the paper deserves refereeing after a major revision. read the letter →

arxiv 2411.08242 v1 pith:LAM77COB submitted 2024-11-12 astro-ph.SR physics.plasm-phphysics.space-ph

B.P. Pandey , Mark Wardle This is my paper · ORCID
classification astro-ph.SRphysics.plasm-phphysics.space-ph
keywords partiallyionizedplasmasolarchromosphereBraginskiiviscosityambipolardiffusionHallMHDinstabilitiescoronalheatingshearflows
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

This paper argues that viscosity, not just magnetic diffusion, can be a dominant non-ideal transport process in the partially ionized solar atmosphere. In weak fields (up to about 50 G), neutral parallel and perpendicular viscosities exceed the Ohm, Hall, and ambipolar diffusivities from the middle chromosphere upward; in stronger fields they dominate mainly in the upper chromosphere and transition region. The same comparison puts viscous damping on par with ambipolar diffusion as a plasma-heating mechanism and yields wave-energy fluxes large enough to matter for coronal heating. The paper also derives new shear-driven instabilities: a viscous-Hall instability requiring parallel viscosity plus Hall diffusion, and an instability driven by the small difference between the two perpendicular viscosities, which can grow for arbitrarily small shear. If these results hold, observed chromospheric vortices are not just passive tracers but active generators of waves and turbulence.

What carries the argument

The load-bearing objects are the five Braginskii viscosity coefficients—parallel $\nu_0$, perpendicular $\nu_1$ and $\nu_2$, and gyroviscosities $\nu_3$ and $\nu_4$—compared with the Ohm, Hall, and ambipolar magnetic diffusivities through Prandtl numbers such as $\mathrm{Pr} = \max(\nu_0, \nu_3)/\max(\eta_O, \eta_H, \eta_A)$. The instability analysis runs on a linearized, Fourier-analyzed, nearly incompressible (Boussinesq, $\omega \ll k c_s$) single-fluid MHD system with background shear $v = s x \hat{y}$ and a uniform field $\mathbf{B} = (0, B_y, B_z)$, yielding a quartic dispersion relation whose coefficients separate into purely viscous parts $C_j$ and diffusion-plus-mixed parts $E_j$. The geometry is encoded in the obliqueness $\mu = \hat{k} \cdot \hat{b}$ and the topological switch $g = -\hat{k}_x \hat{k}_z b_y b_z$, which controls whether shear energy can couple to waves. The named new mechanism is the viscous-Hall instability, whose necessary condition (Eq. 54) requires both parallel viscosity and Hall diffusion and whose growth rate scales with the ratio $R_H = \eta_H/\nu_0$.

What would settle it

A local numerical solution of the linearized Boussinesq system with uniform $\mathbf{B} = (0, B_y, B_z)$, shear $v = s x \hat{y}$, and $\nu_1 \ne \nu_2$ but zero magnetic diffusion should show Alfvénic growth for any $s > 0$; failure to find growth there would falsify the anisotropic-viscosity instability. Separately, high-cadence chromospheric observations resolving swirls with vorticity near $0.1\!-\!0.2\,\mathrm{s^{-1}}$ should detect growing transverse fluctuations with growth times of about a minute if the extrapolated instability operates.

Watch

Extended reading notes

Core claim

The central claim is that viscous momentum transport should be treated as a first-order non-ideal effect in the partially ionized chromosphere, not a small correction. Working from a single-fluid MHD description with the full Braginskii viscous stress tensor and a realistic density-temperature atmosphere, the paper computes Prandtl numbers comparing parallel and gyroviscosities with Ohm, Hall, and ambipolar diffusivities. For footpoint fields $B_0 = 20\,\mathrm{G}$ the Prandtl number exceeds unity above about $1.4\,\mathrm{Mm}$, so viscosity dominates magnetic diffusion throughout the middle and upper chromosphere and transition region; for $B_0 = 50\!-\!100\,\mathrm{G}$ this happens only in the upper chromosphere and transition region. From a quartic dispersion relation for waves in a homogeneous shear flow with a uniform oblique magnetic field, the paper identifies two new instability channels: the viscous-Hall instability (necessary condition Eq. 54), in which parallel viscosity and Hall diffusion together channel shear energy into wave growth for positive shear gradients, and an anisotropic-viscosity instability in which the small $\nu_1 - \nu_2$ difference destabilizes Alfvén waves even for $s > 0$, with magnetic diffusion setting a wavelength cutoff. It further shows gyroviscosity destabilizes waves in the upper chromosphere and transition region, with stability controlled by $\alpha = \nu_3/\nu_4$ and the shear magnitude.

Load-bearing premise

The instabilities are derived from a local plane-wave analysis of a homogeneous background with a linear shear flow, and the growth-rate estimates are then extrapolated to the strongly stratified solar atmosphere using shear values that may not satisfy the same conditions.

Editorial extensions

If this is right

  • In quiet-Sun regions with $B_0 \lesssim 100\,\mathrm{G}$, viscous damping—not ambipolar diffusion—is the dominant wave-heating channel in the upper chromosphere and transition region, so heating models that omit viscosity understate the heating rate.
  • The estimated MHD wave-energy flux is on the order of $10^8\,\mathrm{erg\,cm^{-2}\,s^{-1}}$ for $B_0 \sim 100\,\mathrm{G}$ and far larger for kG fields, sufficient to balance quiet- and active-region coronal radiative losses.
  • Isotropic viscosity suppresses the Hall and ambipolar shear instabilities at short wavelengths, confining growth to long wavelengths; the small $\nu_1 - \nu_2$ anisotropy re-opens instability across wavelengths, with Ohmic, Hall, or ambipolar diffusion providing a cutoff.
  • The viscous-Hall instability operates across the chromosphere where $0 < R_H \lesssim 1$, with peak growth for nearly vertical fields and field-aligned wavevectors; the sign of the shear gradient selects between Hall instability and viscous-Hall instability.
  • In the upper chromosphere and transition region, gyroviscosity makes stability depend on $\alpha = \nu_3/\nu_4$ and shear $s$: for $1/2 \le \alpha \le 1$ an unstable band exists for $1/\alpha < s < 1/(1-\alpha)$, with maximum growth requiring $s > 2$.

Reading between the lines

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

  • Editorial inference: if the Prandtl-number ordering is correct, quiet-Sun chromospheric heating models that include only ambipolar diffusion are missing a comparable or dominant term; including viscous heating should raise predicted temperatures in the upper chromosphere and transition region.
  • Editorial inference: because the viscous-Hall instability needs positive shear while the pure Hall instability needs negative shear, vortex pairs with opposite rotation senses in the same magnetic topology should show asymmetric wave growth—an observationally distinguishable signature.
  • Editorial inference: a testable extension is a local 3D simulation with the full Braginskii tensor plus Hall and ambipolar diffusion, which should reproduce the predicted purely growing and overstable branches and their wavelength cutoffs in the chromosphere.
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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 / 4 minor

Summary. The manuscript investigates viscous momentum transport in the partially ionized solar atmosphere, using a single-fluid MHD description with Braginskii viscosities and Ohm/Hall/ambipolar diffusivities. It compares viscous and magnetic diffusion scales in a Fontenla et al. (1993) model atmosphere with a magnetic field that follows B ∝ n_n^0.3, finds that viscosities can dominate magnetic diffusivities in weak-field chromospheres and in the transition region, estimates viscous heating rates and energy fluxes, and derives a general dispersion relation for waves in a homogeneous plasma with a linear shear flow. It then analyzes several limiting cases: isotropic viscosity with Hall or ambipolar diffusion, anisotropic perpendicular viscosities, parallel viscosity combined with Hall or ambipolar diffusion, and gyroviscosity, claiming new instabilities including a viscous-Hall instability and an anisotropic-viscosity instability of Alfvén waves.

Significance. If the central results held, the paper would provide a useful quantitative baseline for including viscosity alongside ambipolar diffusion in chromospheric heating models and would identify new shear-driven instabilities in partially ionized plasmas. The algebraic derivation is detailed and the paper makes concrete numerical predictions, e.g., that parallel and perpendicular viscosities dominate over magnetic diffusivities for B0 ≲ 50 G in the middle/upper chromosphere, and that viscous heating can supply the coronal radiative loss flux. However, the anisotropic-viscosity instability claim contains a sign error that invalidates one of the paper's headline conclusions as stated; the remaining instability results are conditional on a local, homogeneous, Boussinesq analysis whose applicability to the strongly stratified solar atmosphere is not quantitatively established.

major comments (3)
  1. [§3, Case II (Eqs. 47–48)] The inequality in Eq. (48) is obtained from Eq. (47) by dividing by ω1Δ1, but Δ1 = ω1 − ω2 < 0 in the solar atmosphere, so the direction of the inequality must reverse. The correct consequence of C0 < 0 for μ = 1 is s² < ω_A^4/(ω1Δ1), which cannot be satisfied because the right-hand side is negative; equivalently, C0 = ω_A^4 + ω1(ν2 − ν1)k^4s² > 0. Thus the conclusion that "even a small difference between the perpendicular viscosities destabilizes the Alfvén wave" is not supported for parallel propagation. The numerical example in Fig. 9 uses bz = 1, kz = 0.4 (i.e., μ = 0.4), not μ = 1, so the analytic statement preceding it does not describe the plotted configuration. Please correct the sign, identify the oblique-wave regime in which the instability can actually occur, and revise the corresponding claims in §4 and Summary item 5.
  2. [§4 (vorticity paragraph) and Figs. 8–12, 16] The growth rates are computed for dimensionless shear values s = 2–10 (e.g., Figs. 9, 11, 12, 16), but the final discussion states that observed or simulated vorticities of 0.1–0.2 s⁻¹ are sufficient for instability "within approximately one minute" without translating these values into the normalized shear s ν0/v_A² used in the dispersion relation. Because the threshold conditions (e.g., Eqs. 38, 44, 48, 75) depend on the ratio of shear to Alfvénic and viscous frequencies, the paper should give the physical values of s ν0/v_A² for the heights considered and verify that the unstable modes satisfy the Boussinesq condition ω ≪ k c_s. As written, the extrapolation from dimensionless growth rates to solar conditions is not demonstrated.
  3. [§2.3 and Appendix A] All instability results are derived for a homogeneous background with uniform B and linear shear v = s x ŷ, neglecting stratification, gravity, and background gradients. The solar chromosphere is strongly stratified over the same heights where the Prandtl number exceeds unity. The authors should state the range of wavelengths and heights over which the local approximation is valid, and ideally check the growth-rate results against a stratified model or at least show that k L ≫ 1 and ω ≪ k c_s for the unstable modes in Figs. 8–12 and 16.
minor comments (4)
  1. [§3, after Eq. (34)] When defining the viscous frequencies, the text writes "ω_3 = k²ν3 and ω_4 = k²ν3"; the second expression should be ω_4 = k²ν4.
  2. [Fig. 7 caption] The caption states that the energy flux is plotted for "100G (solid curve) and 5 kG fields," while the text in §2.4 refers to 1 kG fields; please reconcile the figure caption with the text.
  3. [Fig. 9 caption] The caption says the ratio of Ohm (ηO) and Hall (ηA) diffusivities to total viscosity is plotted, but panel (c) is labeled PrH and the surrounding text discusses ηH; the caption should read ηO and ηH.
  4. [§2.4, Eq. (26)] The applicability of the Braginskii heating formula, Eq. (26), to a partially ionized plasma is asserted in one sentence; please provide a derivation or a reference that establishes this extension.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the instability and heating predictions are derived from a general dispersion relation, with self-citations appearing as limit checks rather than load-bearing inputs.

full rationale

The paper's central results are not equivalent to their inputs by construction. The general dispersion relation (Eq. 25, Appendix B) is derived from linearized momentum and induction equations with a standard Braginskii viscous stress tensor and non-ideal MHD diffusivities; no target instability is inserted as an assumption. The viscous-Hall condition (Eq. 54) follows algebraically from the quartic, and the limit omega_0=0 reduces to PW13 Eq. (37), which is a consistency check rather than a circular import. Similarly, the ambipolar/viscous criteria reduce to PW13 limits, and the gyroviscous dispersion relation reduces to PW22 Eq. (57) for alpha=1/2, mu=1. The transport coefficients (viscosities and diffusivities) are taken from PW22, Braginskii (1965), and the F93 atmospheric model; these are external inputs, and the paper does not fit those coefficients to the predicted instability thresholds. No uniqueness theorem or ansatz from the authors' prior work is invoked to force a choice. There is a separate algebra concern in Case II: Eq. (47) with mu=1, G2=0, and G0=-omega_1 Delta_1 s^2 gives omega_A^4 < omega_1 Delta_1 s^2, which with Delta_1<0 is impossible, whereas Eq. (48) prints the opposite inequality and is trivially satisfied. This is a correctness issue, not a circularity, and would need referee verification; it does not raise the circularity score.

Assumptions & free parameters 4 free parameters · 7 assumptions · 0 invented entities

The paper's conclusions rest on an empirical atmospheric model (F93), an empirical B-n_n scaling, a single-fluid strongly-coupled MHD closure, and a local Boussinesq linear analysis. Four quantities are chosen by hand: B0, the shear s, the B-n_n exponent (from the cited literature), and δB/B for the heating estimate. These are not fitted to the target instabilities, so the circularity burden is low, but they do control where and whether the predicted instabilities operate.

free parameters (4)
  • Exponent in B-n_n relation = 0.3
    Empirical exponent in Eq. (2), B = B0 (n_n/n_0)^0.3, taken from Martinez et al. (1997); the height at which the Prandtl number exceeds unity (Figs. 2-3) is sensitive to this exponent.
  • Footpoint magnetic field B0 = 20, 50, 100, 1000 G
    Chosen to represent quiet Sun and active regions; the stratification of the viscosity and diffusivity balance and all instability thresholds depend on B0.
  • Shear gradient s = s = 2, 3, 10 (dimensionless) in instability figures; s ~ 0.2 s^-1 in the chromospheric estimate
    The growth rates and necessary conditions for the viscous instabilities are functions of the assumed linear shear; the paper scans s over a wide range, and the 'likely' conclusion in Section 4 assumes observed or simulated vorticity values.
  • Wave magnetic amplitude δB/B = 0.1
    Assumed in Eq. (34) and Fig. 7 for the wave heating flux estimate; the conclusion that the flux greatly exceeds coronal radiative losses scales quadratically with this value.
assumptions (7)
  • domain assumption Boussinesq approximation: ω << k c_s, fluid nearly incompressible
    Invoked in Section 2.3; the dispersion relation (Eq. 25) and all limiting-case instabilities are derived in this low-frequency limit, excluding compressible and high-frequency modes.
  • domain assumption Homogeneous background with linear shear flow v = s x y and uniform field B = (0, B_y, B_z)
    Appendix A sets the equilibrium as homogeneous and unstratified; stratification and gravity are neglected, so the derived stability criteria are local and may not capture global or surface modes.
  • domain assumption Single-fluid strongly-coupled MHD description of the partially ionized plasma (Pandey & Wardle 2022)
    Section 2.2 adopts the PW22 single-fluid equations, valid when neutral-ion collisions are frequent; the Hall frequency and Larmor radius are rescaled by fractional ionization, which sets the gyroviscosity regime.
  • standard math Barotropic closure P = c_s^2 ρ
    Section 2.2 closes the momentum equation with a barotropic relation; an isothermal sound speed is assumed.
  • domain assumption Model atmosphere: density and temperature from Fontenla et al. (1993) and B = B0 (n_n/n_0)^0.3
    The computed Prandtl numbers and heating rates (Figs. 2, 5-7) depend on these empirical profiles; other atmosphere models would shift the layer boundaries.
  • domain assumption Equipartition temperatures T_i = T_e = T_n = T for the viscosity coefficients
    Section 2.2 states that the viscosity expressions from PW22 assume equal temperatures; the ratio ν3/ν4 in Eq. (6) and its height dependence (Fig. 4) rely on this.
  • standard math Plane-wave perturbations exp(i k·x + σ t) and local dispersion analysis
    The stability analysis uses Fourier modes in a homogeneous medium, standard for local linear stability but neglecting background spatial variation.

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Pith. "Pith review of Viscous Heating and Instabilities in the Partially Ionized Solar Atmosphere." pith.science (2026). https://pith.science/paper/LAM77COB

@misc{pith2026241108242,
  author       = {Pith},
  title        = {Pith review of: Viscous Heating and Instabilities in the Partially Ionized Solar Atmosphere},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LAM77COB}},
  note         = {Machine review of arXiv:2411.08242}
}
abstract

In weak magnetic fields ($\lesssim 50 \,\mbox{G}$), parallel and perpendicular viscosities, mainly from neutrals, may exceed magnetic diffusivities (Ohm, Hall, ambipolar) in the middle and upper chromosphere. Ion-driven gyroviscosity may dominate in the upper chromosphere and transition region. In strong fields ($\gtrsim 100\, \mbox{G}$), viscosities primarily exceed diffusivities in the upper chromosphere and transition region. Parallel and perpendicular viscosities, being similar in magnitude, dampen waves and potentially compete with ambipolar diffusion in plasma heating, potentially inhibiting Hall and ambipolar instabilities when equal. The perpendicular viscosity tensor has two components, $\nu_1$ and $\nu_2$, which differ slightly and show weak dependence on ion magnetization. Their differences, combined with shear, may destabilize waves, though magnetic diffusion introduces a cutoff for this instability. In configurations with a magnetic field $\bf{B}$ having vertical ($b_z=B_z/|\bf{B}|$) and azimuthal ($b_y=B_y/|\bf{B}|$) components, and a wavevector $\bf{k}$ with radial ($\kx=k_x/|\bf{k}|$) and vertical ($\kz=k_z/|\bf{k}|$) components, parallel viscosity and Hall diffusion can generate the viscous-Hall instability. Gyroviscosity further destabilizes waves in the upper regions. These findings indicate that the solar atmosphere may experience various viscous instabilities, revealing complex interactions between viscosity, magnetic fields, and plasma dynamics across different atmospheric regions.

Figures

Figures reproduced from arXiv: 2411.08242 by the authors.

Figure 1
Figure 1. The ratios ν1/ν0 (solid curve 1) and ν2/ν0 (dashed curve 2) are plotted as functions of height for the photosphere￾chromosphere region (top panel) and the chromosphere-transition region (bottom panel). These ratios represent the relative mag￾nitudes of different viscous transport coefficients at various alti￾tudes. The altitude dependence of the magnetic field is derived from Eq. (2). The density and temperature pro… view at source ↗
Figure 4
Figure 4. The ratio of the gyro viscosities ν3/ν4 is plotted against height for B0 = 50 and 100 G field. Other parameters are the same as used in the previous figure. Their ratio is given by: ν3 ν4 = 1 2  1 + 3 1 + β 2 i  , (6) and is shown in Fig. (4) for the magnetic field profile Eq. (2). Since gyroviscosity manifests only when ions are mag￾netized, it becomes important in the upper chromosphere (& 2.19, Mm) and beyond, … view at source ↗
Figure 5
Figure 5. The total damping rate log10(Γ) [Eq. (31)] (solid curve) and Pedersen damping rate [Eq. (30)] ) dotted curve) are plotted against height for B = 20 G (top panel), and 100 G (bottom panel). are expressed in terms of ambipolar length scale LA. We see from Fig. (5) that in the case of a weak magnetic field (B = 20 G), which is characteristic of the quiet regions of the Sun, the total damping rate (log10(Γ)) [solid curv… view at source ↗
Figures from the paper (9 more)
Figure 6
Figure 6. Figure 6: The total damping rate log10(Γ) for 1 kG (dotted curve) and 100 G field (solid curve). 0 0.5 1.0 1.5 2.19 4 8 12 log10(Ex) 1 kG 100 G (a) 2.19 2.20 2.21 2.22 Height (Mm) 3 6 9 log10(Ex) 100 G 1 kG (b) [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]
Figure 7
Figure 7. Figure 7: The energy flux log10(Ex) is plotted against height for 100G (solid curve) and 5 kG fields (dotted curve). v ∼ vA, the energy flux of the wave Ex = 2 × 10−2 [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]
Figure 8
Figure 8. Figure 8: The growth rate σ(= σ ν0/v2 A) vs. k(= k vA) is plot￾ted for both the Hall-viscous (panel a) and the ambipolar-viscous (panel b) case. Each curve is labeled according to its respective viscosity value, ν0. The figure also displays values for s, kˆz, and bz. Here, kˆx =…
Figure 9
Figure 9. Figure 9: The ratio of Ohm (ηO) and Hall (ηA) diffusivities to total viscosity (ν0 + ν1 + ν2) is plotted versus height for the pho￾tosphere (a) and chromosphere (c). Panel (b) shows viscous insta￾bility versus k for ηO s/v2 A = 0 and 0.1, while panel (d) compares ηH s/v2 A = 0 a…
Figure 11
Figure 11. Figure 11: The growth rate σ vs. k is plotted for s = 10 and by = − p 1 − b 2 z . In panel (a), RH = ηH/ν0 (labeled on the curve) is varied for kˆz = bz = 0.9. In panel (b), with RH = 1 and kˆz = 0.9, the value of bz (also labeled on the curve) is varied. k 2 and the dispersion …
Figure 12
Figure 12. Figure 12: shows the dispersion relation (Eq. 25) with coeffi￾cients from Eq. (59), for s = 2 and bz = ˆkz = 0.5. Panel (a) demonstrates that while the purely growing ambipolar insta￾bility exists when ν0 = 0, introducing viscosity (ν0 = 0.1 , 1) constrains the instability to lo…
Figure 13
Figure 13. Figure 13: The coefficient C0 is plotted against s for fixed α in the above figure . Given the complexity of the dispersion relation, Eq. (69), we first analyze a simplified case to gain analytical insight. We consider: µ = 1 (purely vertical field) and kz = 1 (parallel propagat…
Figure 15
Figure 15. Figure 15: Normal modes in the region I-IV of Fig. (14) is shown in the above figure . when s 2 − 1 − 1 4 k 2 > 0 i.e. when s > 2 and k 2 < 2 (s − 2). 3. C0 > 0 (D > 0): Two negative roots, −ω 2 1 , −ω 2 1, or σ = ±i ω1 and ± i ω2 , (83) This is the only stable case. Now we anal…
Figure 17
Figure 17. Figure 17: Panel (a) shows magnetic diffusion coefficients, parallel viscosity, and FLR viscosities in the transition region for B0 = 100 G, while panel (b) shows these quantities for B0 = 1 kG. Balmaceda L., Vargas Domi´nguez S., Palacios J., Cabello I., & Domingo V. 2010, A&A,…

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