Linear Analyses of Thermal Instability in Stratified Medium

TL;DR

The study addresses how thermal instability operates in a gravitationally stratified circum-galactic medium, extending the classical uniform-medium analysis to include buoyancy via the Brunt-Väisälä frequency. By deriving both exact and approximate dispersion relations in stratified settings, the authors show that convective instability can enhance TI, enabling growth at wavelengths shorter than the Field length, while convectively stable cases can exhibit over-stable, oscillatory modes and two unstable TI-like families. They provide practical formulas for the most unstable wavelength and growth rate, and apply a simplified CGM model to estimate HVC properties and gas accretion rates, suggesting TI-driven accretion can sustain star formation in the Galactic disk. The results offer a robust theoretical framework for TI in stratified, gravitationally bound media and have direct implications for interpreting CGM observations and guiding high-resolution simulations.

Abstract

Thermal instability in the circum-galactic medium (CGM) can be responsible for the existence of cold clouds (e.g., high-velocity clouds) embedded in a hot diffuse medium (e.g., X-ray emitting gas). While many previous studies have analyzed thermal instability in uniform medium, the instability mechanism in gravitationally stratified medium like CGM has not been fully analyzed. This study investigates how gravity affects the behavior of thermal instability through linear perturbation analyses.We find that in stratified medium, thermal instability can drive over-stable modes, a behavior distinctly different from the monotonic growth of thermal instability in a uniformmedium. Furthermore, we find that the combination of buoyancy and thermal instability drives other two unstable modes. Applying our results to a simplified model of the CGM, we estimate the gas accretion rate from the CGM to the Galactic disk and the typical size of high-velocity cloud driven by thermal instability to be a few solar masses per year. This gas accretion rate is comparable to the observed star formation rate, and hence, the mass in the Galactic disk can be maintained. Our results provide a theoretical framework for understanding the formation of multi-phase gas, particularly in the CGM.

Figures (10)

• Figure 1: The growth rates as a function of the wave number for $\alpha = 0.9,\ 0.5$, and 0.1. The results are shown for a fixed $\gamma = 5/3$ with varying the effect of thermal conduction, $\beta = 10^{-3},\ 10^{-2},\ 10^{-1},\ 10^0$. The horizontal and vertical axes are the normalized wave number and growth rate of the unstable mode. The difference of lines denotes the difference of the effects of the thermal conduction, $\beta$.
• Figure 2: The dispersion relation under the short wavelength approximation in Equation \ref{['eq:1D:Linear_DR:short']}. The orange lines represent the approximate solutions, while the black solid lines are the exact solutions as given by Equation \ref{['eq:1D:Linear_DR']}.
• Figure 3: The dispersion relation under the long wavelength approximation, i.e., $\mathcal{L}_n \delta n + \mathcal{L}_T \delta T + \kappa k^2 \delta T = 0$ in Equation \ref{['eq:1D:Linear_DR:long']}. The sky blue lines represent the approximate solutions, while the black solid lines are the exact solutions.
• Figure 4: Approximation for the maximum growth rate and the most unstable wavelength $(k_{\rm max, TI},\ \sigma_{\rm max, TI})$ for each parameters. The grey curves show the exact dispersion relations, and the cross markers indicate the approximate estimates obtained from Equations \ref{['eq:1D:Linear_DR:most_derivative1']} and \ref{['eq:1D:Linear_DR:most_wn']}.
• Figure 5: Dispersion relation in the convectively unstable case, for parameters $\alpha = 0.5$, $\beta = 0.01$, $\gamma = 5/3$, and $k_z = 0$. The grey solid line shows the result for a uniform medium, while the colored lines represent the behavior in a gravitationally stratified medium. The red curves correspond to thermally unstable modes in the uniform case, while the blue curves represent modes that originate from vertical oscillations governed by Equation \ref{['eq:2D:Linear_EoMz']}. Different line styles reflect different values of the normalized $Brunt-V\ddot{a}is\ddot{a}l\ddot{a}$ frequency $\tilde{\omega}_{\mathrm{BV}}^2$. Note that larger wavenumber modes are stabilized by viscosity as shown in Section \ref{['subsubsec:discuss:viscosity']}.