Reevaluating thermal instability in a uniform plasma: an extended analysis of instability domains

TL;DR

We reexamine thermal instability in an infinite, uniform, non-magnetized plasma by including heating, radiative cooling, and Spitzer-type thermal conduction. Building on Field's formalism and Waters & Proga's domain classification, we map stability and mode structure across short and long wavelengths and introduce finite Field-length effects via the parameter \lambda_F. A corrected and extended instability-domain table shows how conduction reshapes the classic seven regions, with clear criteria in terms of $R$ and $\lambda_F/\lambda_{\mathrm{th}}$; for small Field lengths the Waters & Proga scheme holds, while large Field lengths erase a simple, universal classification. We discuss implications for coronal rain and catastrophic cooling, and outline how nonlinear saturation and magnetic fields could modify the homogeneous, linear results in more realistic astrophysical settings.

Abstract

Thermal instability plays a crucial role in the dynamics of astrophysical plasmas. Building upon the foundational work of Field (1965) and the subsequent analysis by Waters and Proga (2019), this study revisits the characteristics of thermal instability in a uniform, non-magnetic medium. Our primary aim is to reevaluate and expand the understanding of instability domains, focusing on the classification and characteristics of thermal and acoustic modes in the presence of heating, radiative cooling, and thermal conduction. Except for Spitzers expression for parallel thermal conductivity, the heating and cooling processes are unspecified. Additionally, we investigate the existence of isobaric and isochoric thermal modes across the extreme limits of very short and very long wavelengths, as well as at intermediate wavelengths. We perform an in-depth analysis of the dispersion relation for an infinite, uniform hydrodynamic medium, as initially derived by Field (1965). This approach enables the generation of growth-rate and dispersion diagrams, providing insight into the behaviour of thermal instability across different wavelength ranges. With the inclusion of thermal conduction, our study refines the classification of the seven instability regions previously outlined by Waters and Proga (2019). Our findings confirm that their classification holds when the Field length is smaller than or comparable to the thermal wavelength. For larger Field lengths, a simplified classification becomes impractical. Furthermore, we discuss the potential implications of the catastrophic cooling instability (Waters and Stricklan 2025) in the context of cool coronal rain formation. Our work not only validates certain classifications, but also proposes new insights into complex instability behaviours, thereby enhancing the theoretical framework established by Field and others.

Figures (11)

• Figure 1: Left column, top and bottom panels: real and imaginary parts of the growth rate versus wavelength around the critical wavelengths $\lambda_\pm$ for $\lambda_\mathrm{F} = 0$. The solid blue, solid red and dotted green lines are the three solutions of the dispersion relation. The vertical gray dotted lines are the values given by Eq. (14) of waters2019non. Right column: same as left column for $\lambda_\mathrm{F} = 0.2 \lambda_\mathrm{th}$; this is approximately the value used by waters2019non in their Fig. 1. The other parameter values are $R = 0.05$, $N_{\rho 0} = -10$ (hence $N_{\mathrm{p}0} = -2.5/3$).
• Figure 2: Same as Fig. \ref{['fig:lambda_pm']} for $\lambda_\mathrm{F}=0.8$, $N_{\rho 0}=-10$ and $R=0.12$. This value of $R$ is slightly larger than 1/9 and so, according to Table \ref{['table:R_and_lambda']}, there should be no critical wavelengths instead of two. This disparity is caused by $\lambda_\mathrm{F}$ not being zero and will be explored in Sect. \ref{['sect:arbitrary_lambdaF']}.
• Figure 3: Number of critical wavenumbers as a function of $N_{\mathrm{p}0}$ and $N_{\rho 0}$ for a fixed $\lambda_\mathrm{F} / \lambda_\mathrm{th}$. This panel shows the cases $\lambda_\mathrm{F} / \lambda_\mathrm{th} = 0.2$, 1, 5, 20, with other values of this ratio shown in the accompanying movie. The $R = 1/9$ line does not appear straight because of the symmetric logarithmic scale used in both axes.(Movie Online)
• Figure 4: Case $R < 0$, $N_{\rho 0} > 0$, $N_{\mathrm{p}0} < 0$. Top and bottom panels: real and imaginary parts of the growth rate versus wavelength. The solid blue, solid red and dotted green lines are the three solutions of the dispersion relation. The legends refer to the asymptotic curves in the limits $\lambda \rightarrow 0$ (dominant term of Eq. (\ref{['eq:lambda0_condens_n']}) and Eq. (\ref{['eq:lambda0_acoustic_n']})) and $\lambda \rightarrow \infty$ (dominant terms of Eqs. (\ref{['eq:k0_condens_n']}) and (\ref{['eq:k0_condens_n_Rlt0']})). The vertical gray dotted lines mark the position of the relevant wavelengths, the horizontal grey dotted line in the top panel corresponds to $n_\mathrm{R} = 0$. The parameter values are $N_{\rho 0} = 10$, $R = -2$ and $\lambda_\mathrm{F} = 0.2 \lambda_\mathrm{th}$.
• Figure 5: As in Fig. \ref{['fig:row1']} but for case $R < 0$, $N_{\rho 0} < 0$, $N_{\mathrm{p}0} > 0$. The legends refer to the asymptotic curves in the limits $\lambda \rightarrow 0$ (dominant term of Eq. (\ref{['eq:lambda0_condens_n']}) and Eq. (\ref{['eq:lambda0_acoustic_n']})) and $\lambda \rightarrow \infty$ (dominant terms of Eqs. (\ref{['eq:k0_condens_n']}) and (\ref{['eq:k0_condens_n_Rlt0']})). The parameter values are $N_{\rho 0} = -10$, $R = -2$ and $\lambda_\mathrm{F} = 0.2 \lambda_\mathrm{th}$.