On the Theory of Bulk Viscosity of Cold Plasmas

TL;DR

This work addresses the problem of quantifying bulk viscosity in cold plasmas in the regime $T \ll I_a$, where ionization-recombination kinetics governs dissipation. It develops a kinetic-theory framework that linearizes the ionization-state dynamics around LTE and couples them to harmonic pressure perturbations, yielding a complex, frequency-dependent bulk-viscosity through the complex polytropic index $\hat{\gamma}(\omega)$. The authors derive explicit expressions for the low- and high-frequency limits, validate the Mandelstam-Leontovich single-time-constant approximation against the exact solution, and provide a Drude-like representation $\hat{\zeta}(\omega)=\zeta_0/(1-i\omega\tau)$ with analytically computable $\zeta_0$ and $\tau$. They also present an analytical hydrogen–helium cocktail solution, illustrating that ML-like behavior remains exact for this common solar-plasma mixture and enabling practical estimates of bulk-viscosity–driven damping. The results have implications for acoustic heating in the solar chromosphere and offer a tractable route to laboratory validation using alkali–noble gas cocktails.

Abstract

Solving the kinetic equation for ionization-recombination processes in cold plasmas for temperatures much lower than the first ionization potentials, we derive an explicit expression for the bulk viscosity. We obtain that bulk viscosity can be many order of magnitude bigger than the shear viscosity. Our result for the relaxation time reveals that the Mandelstam-Leontovich approximation for the frequency dependence of the bulk viscosity is in practice an exact result for the cold plasmas. The illustrative numerical examples correspond to the plasma cocktail of the solar chromosphere at the height of the minimal polytropic index. The possible application for the acoustic heating of the inner solar atmosphere up to the transition region is shortly discussed together with the evaluation to confirm the theory by laboratory plasmas.

Figures (8)

• Figure 1: Semicircular Cole-Cole ColeCole plot, or Mandelstam-Leontovich (ML) fit of the relative polytropic index $\hat{\gamma}_\mathrm{rel}=\hat{\gamma}/\gamma_a$ for homogeneous plasma according to the AL08 profile Avrett:08 and the 6 component cocktail consisting of H, He, C, Mg, Si, Fe. On the abscissa the real part $\Re(\hat{\gamma}_\mathrm{rel})$ is given while in the ordinate it is the imaginary part $-\Im(\hat{\gamma}_\mathrm{rel})$. The line shows the exact calculation according to Eq. (\ref{['gamma_rel_1']}), while the open circles $\bigcirc$ correspond to the ML fit according to Eq. (\ref{['M-L_1']}) with fitting parameter parameter $\Gamma_{0,\infty}=0.654$. This arch is analogous to the Cole-Cole plot in the physics of liquids and dielectrics ColeCole.
• Figure 2: Reciprocal Q-factor of the ML approximation as a function of $x \equiv \omega\tau$ according to Eq. (\ref{['Q_fit_1']}) and Eq. (\ref{['M-L_1']}) $1/Q_{a,\mathrm{max}} \equiv Q_{a,\mathrm{min}} = 0.21$ at $x_\mathrm{max}=0.8235$. Independently using Eq. (\ref{['Q_factor_kinetics_1']}), we obtain $\omega_\mathrm{max}=0.9127$ µ rad/s and the corresponding time-constant is determined as $\tau_\mathrm{max}=x_\mathrm{max}/\omega_\mathrm{max} \approx 10.5$ days. In the general case $1/\mathcal{Q}^{\mathrm{(ML)}}$ has a maximum at $x_\mathrm{max}=\omega_\mathrm{max} \tau_\mathrm{max}=\sqrt{\Gamma_{0,\infty}}=c_0/c_\infty$, cf. LL6. Here we have plotted only the $1/\mathcal{Q}^\mathrm{(ML)}_a(\omega)$ from Eq. (\ref{['Q_fit_1']}) since the same calculation from Eq. (\ref{['Q_factor_kinetics_1']}) coincides with it as the Cole-Cole plot in Fig. \ref{['fig:gama']}
• Figure 3: Sound velocity ration $c_\mathrm{s}/c_\mathrm{_N}=\sqrt{\gamma^\prime}$ as function of frequency $f=\omega/2\pi$ according to the calculations by the general formula Eq. (\ref{['complex_gamma_1']}). For high frequencies $\sqrt{\gamma^\prime(\infty)}=\sqrt{5/3}=\sqrt{\gamma_a}$ (upper horizontal dashed line) this ratio reaches mono-atomic value. At low frequency $\sqrt{\gamma^\prime(0)}=\sqrt{\gamma(T,\rho)}\simeq1$ sound velocity is determined by the explicit thermodynamic expression Eq. (\ref{['gamma_thermodynamic_1']}). The minimal value $\gamma_0\approx 1$ (the lower horizontal dashed line) can be reached at $T\ll I_1$ for $\overline\alpha_1\simeq 1/2$.
• Figure 4: Frequency dependence of the real (absorptive) part of the bulk viscosity according to Eq. (\ref{['zeta_prime_1']}). On the ordinate is drawn $\zeta^\prime/\overline p$ with dimension of time, while on the abscissa is the angular frequency $\omega$.
• Figure 5: Frequency dependent dimensionless bulk viscosity $\mathcal{Z}=\zeta^\prime/\zeta_0$ as function of dimensionless frequency $\mathcal{W} \equiv \omega\tau$ according according to Eq. (\ref{['Bash_Equation_1']}). The curve is undistinguished by its Drude fit.