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

Context. The buoyancy stability properties of dilute plasma, as found in the intracluster medium (ICM), are dramatically modified because of the anisotropic transport of heat along the magnetic field lines. This feature gives rise to the magneto-thermal instability (MTI) when the temperature gradient is aligned with the gravity, which systematically occurs in the outskirts of galaxy clusters. Aims. Most previous linear analyses of the MTI adopted a local approach and the Boussinesq formalism. 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 magneto-rotational instability (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. Methods. We perform a global linear analysis of the MTI in a spherical stratified model of the ICM, subject to a Navarro-Frenk-White gravitational potential of dark matter. We use a combination of analytical results from both the Sturm-Liouville theory and WKBJ approximations, corroborated by numerical results obtained with both a pseudo-spectral Chebyshev solver and the finite-volume code IDEFIX, to better explain the physics of the global MTI eigenmodes. Results. We obtain scaling laws and approximate expressions for the growth rates of the global modes. We show that the associated eigenfunctions 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 very similar to that of the global MRI modes in accretion discs. Finally, direct numerical simulations successfully reproduce the global MTI modes and their growth rates, with errors smaller than 1%. Conclusions. 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.