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

TL;DR

This work addresses solar chromospheric heating by acoustic waves, arguing that bulk viscosity in the partially ionized H–He plasma dominates damping. By deriving LTE-based thermodynamic and kinetic coefficients for a hydrogen–helium mixture and employing Avrett–Loeser atmospheric profiles, the authors obtain a large bulk-viscosity–driven damping rate and a Mandelstam–Leontovich-like time constant, linking wave dissipation to an energy flux on the order of $320\,\mathrm{kW\,m^{-2}}$. A coupled system for temperature and density profiles is solved with boundary conditions fixed from AL08, incorporating radiative losses $Q_r=\mathcal{P}(T)n_e n_p$ and an Arrhenius extrapolation at low $T$. The central result is that bulk viscosity is the dominant mechanism for chromospheric heating in the considered regime, quantified by a Prandtl number $P_{\zeta/\eta}\sim 10^{10}$, suggesting that existing models should include $\zeta$ to accurately capture acoustic heating. The study provides a pathway to more comprehensive 3D analyses and spectral investigations of chromospheric heating driven by acoustic damping.

Abstract

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.

Figures (6)

• Figure 1: The height dependence of dimensionless function $\lg\mathcal{D}(h)$ Eq. (\ref{['cal_D']}) which is via Avrett:08 for heights $h \equiv x <2.1$ Mm. This function is the main ingredient of all analytical results in which ionization-recombination processes are relevant. For comparison in the same logarithmic ordinate is given the profile of the dimensionless variable $\iota(h)\gg1$. The abrupt change of both variables is physically in the solar transition region and it is beyond the scope of the present study.
• Figure 2: The height $h$ profile of bulk viscosity Prandtl number Eq. (\ref{['visc_Prandtl_']}) $\mathrm{P}_{\zeta/\eta}\equiv\zeta/\eta$ via Avrett:08. One of the purposes of the present study is to emphasize that for heights $h \equiv x <2.1$ Mm from the solar photospere the bulk viscosity indispensable must be included in the considerations of acoustic wave heating of chromosphere.
• Figure 3: Height $h$ profiles of the frequencies $f_\mathrm{ML}\equiv 1/2\pi\tau \ll f \ll f_c\equiv f_\mathrm{ML}\sqrt{\mathrm{P}_{\zeta/\eta}}$ again via Avrett:08. The damping rate $k_\infty$ is frequency independent according to Eq. (\ref{['damping_inf']}) and determined by bulk viscosity $\zeta$ and volume heating. In this frequency interval the acoustic wave heating according to Eq. (\ref{['force_density_and-heating rate']}) is proportional to the total energy flux of sound waves.
• Figure 4: Radiative loss rate $\lg \mathcal{P}$ versus $1/T^\prime$; point are after Dere:09, photospheric abundance. The 4 smallest temperature points excellently approximate a straight line. For comparison are marked the lowest temperature in the chromosphere $T_\times^\prime= 4$ K and the photosphere temperature $T_\star^\prime= 6$ kK.
• Figure 5: Radiative loss rate $\lg \mathcal{P}$ versus $\lg T^\prime$ with CHIANTI 6 Dere:09 The dashed line is the Arrhenius extrapolation Eq. (\ref{['Arrhenius_extrapolation']}) which used for $T^\prime<T_\mathrm{min}^\prime=10$ kK, while the dash-dotted line is the CHIANTI 6 radiative loss rate.