Global linear analysis of the magneto-thermal instability in a stratified spherical model of the intracluster medium

TL;DR

This work presents a global linear analysis of the magneto-thermal instability (MTI) in a stratified intracluster medium (ICM) using a compressible Braginskii-MHD framework with anisotropic heat flux. By formulating a 2D equatorial, spherical model under an NFW gravitational potential and deriving a single governing ODE for MTI eigenmodes, the authors connect global MTI modes to local elevator modes and to MRI channel modes, aided by fast-diffusion Sturm–Liouville theory and WKBJ scalings. The study is validated with a Chebyshev pseudo-spectral solver and direct numerical simulations using IDEFIX, achieving growth-rate agreement to better than 1% for several modes and establishing a robust numerical benchmark for Braginskii heat-flux codes. The results reveal turning-point localization of global MTI eigenfunctions in the inner ICM, clarify the local-global connection, and provide practical insights into MTI-driven turbulence and energetics, with implications for interpreting ICM observations and guiding nonlinear saturation studies.

Abstract

The buoyancy stability properties of the ICM are modified because of the anisotropic transport of heat along the magnetic field lines. This feature gives rise to the MTI when the temperature gradient is aligned with the gravity, which occurs in the outskirts of galaxy clusters. Most previous linear analyses of the MTI adopted a local, Boussinesq approach. However, the conduction length, which sets the characteristic length scale of the MTI, might be a non-negligible fraction of the scale height in the ICM. We want to assess the impact of locality assumptions on the linear physics of the MTI. Another goal is to unveil the deeper connections between these global MTI modes and their MRI counterparts in accretion discs. Our third objective is to provide a new benchmark against which any numerical code implementing the Braginskii heat flux in spherical geometry can be tested. We perform a global linear analysis of the MTI in a spherical stratified model of the ICM. We use a combination of analytical results, corroborated by numerical results obtained with both a pseudo-spectral solver and IDEFIX, to better explain the physics of the global MTI modes. We obtain scaling laws and approximate expressions for the growth rates of the global modes. We show that the associated functions are confined within an inner region, limited by a turning point, where the mode is allowed to grow. The most unstable local MTI modes correspond to the portion of the global mode localised near the turning point. This phenomenology is similar to that of the global MRI modes. Finally, direct simulations successfully reproduce the global MTI modes and their growth rates, with errors smaller than 1%. Overall, this study provides us with new insights on the linear theory of the global MTI in the ICM, and a useful numerical test bench for any astrophysical fluid dynamics code embedding anisotropic heat flux.

Figures (6)

• Figure 1: Local MTI frequency profile, $M_0$, as a function of the radius, $r$.
• Figure 2: Illustrative numerical global MTI eigenmode with $m=50$ and $n=20$. Top: varying radial wavenumber, $k_r = \sqrt{E - V_\mathrm{ftd}}$, as a function of the radius, $r$. Bottom: a high-order global MTI mode. The turning point, $r_\mathrm{c}$, of the mode is indicated by black dashed vertical lines. It can be found as the single solution to the equation $k_r(r)=0$, indicated by the grey dashed horizontal line on the top panel.
• Figure 3: Numerical solutions to the eigenproblem english Eq Éq s. (\ref{['eq:drho']}) , (\ref{['eq:dvphi']}) , (\ref{['eq:dvr']}) , (\ref{['eq:dT']}) , (\ref{['eq:dA']}) , (\ref{['eq:']}), english and et (\ref{['eq:']}) . From top to bottom: global MTI eigenfunctions for azimuthal order $m=30,50,100$, respectively. From left to right: global MTI eigenfunctions for mode number $n=3,12$, respectively. From top to bottom in each subfigure: perturbations of density, azimuthal velocity, radial velocity, and temperature as a function of the radius. The potential vector perturbation is not shown because it simply relates to that of radial velocity through english Eq Éq s. (\ref{['eq:dA']}) , (\ref{['eq:']}) , (\ref{['eq:']}) , (\ref{['eq:']}) , (\ref{['eq:']}) , (\ref{['eq:']}), english and et (\ref{['eq:']}) . Real parts of the modes are in plain lines, while imaginary parts are in dashed lines. The part whose corresponding line type is absent is zero. The modes oscillate in space up to their respective turning points, $r_\mathrm{c}$, and evanescent beyond.
• Figure 4: Dispersion relations of the global MTI eigenmodes. Top: growth rates of the first six MTI modes as a function of the azimuthal order, $m$. The curve $\sigma=M_1$ is indicated in dotted dark line. It seems to be a common asymptote to all curves, $\sigma_n(m)$. Inset: same growth rates, $\sigma_n$, again (one over two for the sake of visibility though), together with their WKBJ approximations, english Eq Éq s. (\ref{['eq:wkbjcrude']}) , (\ref{['eq:']}) , (\ref{['eq:']}) , (\ref{['eq:']}) , (\ref{['eq:']}) , (\ref{['eq:']}), english and et (\ref{['eq:']}) , in dashed lines. Bottom: growth rates of the MTI eigenmodes with $m=100$ and $m=50$ as a function of the mode number, $n$, in plain lines. Their WKBJ approximations, english Eq Éq s. (\ref{['eq:wkbjcrude2']}) , (\ref{['eq:']}) , (\ref{['eq:']}) , (\ref{['eq:']}) , (\ref{['eq:']}) , (\ref{['eq:']}), english and et (\ref{['eq:']}) , are shown too, in dashed lines. The curve $\sigma=M_2$ is indicated in dotted grey line.
• Figure 5: Comparison between the analytical dispersion relation (in plain lines) and the numerically measured growth rates (markers). Black diamonds are the numerical growth rates of the most unstable modes with $n=0$, found in the first simulation seeded with low amplitude random white noise on the velocity field. They match the theoretical dispersion relation to within $\zeta \lesssim 1\%$. The blue star depicts the MTI eigenmode with $m=92$ and $n=1$, which could be successfully isolated in the same simulation. The purple circle stands for the numerical growth rate of the MTI eigenmode with $m=101$ and $n=0$. It agrees with the theoretical dispersion relation to within $\zeta\approx 2.0\%$ only. The green triangle represents the numerical growth rate of the exact same mode, but in a simulation seeded with a single mode. The error is very small, $\zeta\approx0.027\%$.