On the Theory of Absorption of Sound Waves via the Bulk Viscosity in the Partially Ionized Solar Chromosphere

Bulk viscosity and thermodynamic variables of a hydrogen-helium cocktail: internal energy, enthalpy, pressure, their derivatives, heat capacities per constant density and pressure are obtained using temperature and density height profiles of the solar atmosphere [Avrett & Loeser, ApJS Vol. 175, 229 (2008)]. The qualitative evaluation for the necessary sound wave energy flux to heat the solar chromosphere is determined to be 320 kW/m$^2$. It is concluded that the bulk viscosity creates the dominating mechanism of acoustic waves damping and it is not necessary to introduce artificial viscosity or to conclude that shear viscosity is not sufficient for chromosphere heating; the volume viscosity induced wave absorption is sufficient.

💡 Research Summary

The paper presents a comprehensive theoretical investigation of bulk (volume) viscosity as the dominant mechanism for acoustic wave damping and heating in the partially ionized solar chromosphere. The authors begin by noting that most previous chromospheric heating studies have focused on shear viscosity, radiative losses, or have introduced artificial viscosity terms, while bulk viscosity has been largely ignored. They argue that, because the chromosphere is cool and only partially ionized, ionization‑recombination processes generate a bulk viscosity that can far exceed the shear viscosity.

The analysis proceeds in several steps. First, the authors derive the basic energetics of a longitudinal acoustic wave in a weakly stratified medium, defining the energy density, flux, and the relationship between the damping rate k″ and the dissipative coefficients (shear viscosity η, bulk viscosity ζ′(ω), thermal conductivity κ, and heat capacities Cv, Cp). Using the WKB approximation they connect the spatial damping to a force density and a heating term Qζ.

Next, they model the chromospheric plasma as a cold hydrogen‑helium cocktail, employing the temperature and density height profiles from the Avrett & Loeser (2008) Model C7. The ionization fraction α of hydrogen is obtained from the Saha equation, while helium is assumed essentially neutral (aHe≈0.1). From these quantities they compute the total particle number density, pressure, internal energy, and enthalpy. Detailed expressions for the derivatives of pressure and enthalpy with respect to temperature and density are derived, leading to explicit formulas for the Jacobian J, the specific heat capacities at constant volume (Cv) and pressure (Cp), and their difference ΔC. A key parameter D = (1−α)α²−α, which encapsulates the effect of ionization‑recombination on thermodynamics, appears throughout the formulas and modifies the usual γ = Cp/Cv ratio.

The bulk viscosity is then introduced via a frequency‑dependent form ζ′(ω)=ζ0/(1+ω²τ²). The zero‑frequency value ζ0 = p τ B A depends on pressure, a characteristic time τ, and dimensionless factors A and B that contain the ionization fraction and the ratio ι = I/T (ionization energy over temperature). The relaxation time τ is identified with the mean lifetime of a neutral hydrogen atom against electron‑impact ionization, τ = 1/(β ne), where β is the Maxwell‑averaged ionization cross‑section. The authors adopt a semi‑empirical expression for β based on Wannier’s theory and experimental constants (C_W≈2.7). In the low‑frequency limit relevant for chromospheric acoustic oscillations (≈5 mHz), ωτ≪1, so ζ≈ζ0 and the bulk viscosity dominates.

Numerical evaluation using the Model C7 profiles shows that the bulk viscosity exceeds the shear viscosity by two to four orders of magnitude throughout the lower chromosphere (heights <2.1 Mm). The bulk‑viscosity Prandtl number Pζ/η is correspondingly large. The damping rate simplifies to k″≈ω²ζ/(2ρc³) in this regime, and the heating rate Qζ = 2k″q is directly proportional to the acoustic energy flux q = cE. Integrating over height, the authors find that an acoustic flux of roughly 320 kW m⁻² is sufficient to balance radiative losses and maintain the observed temperature structure of the quiet chromosphere. This value is substantially larger than typical observed acoustic fluxes (a few hundred W m⁻²), but the authors argue that previous estimates underestimated the contribution of bulk viscosity; when properly accounted for, acoustic waves can indeed supply the required energy without invoking artificial viscosity or additional heating mechanisms.

The paper concludes that bulk viscosity, arising from ionization‑recombination dynamics, is the primary dissipative agent for acoustic waves in the partially ionized chromosphere. It calls for future work to incorporate magnetic field anisotropy, a broader frequency spectrum, and direct comparison with high‑resolution observations to refine the heating budget. The study also suggests that similar bulk‑viscosity‑driven heating may be relevant in other astrophysical partially ionized plasmas.