Stability Analysis of Convection in the Intracluster Medium

We use the machinery usually employed for studying the onset of Rayleigh--B\'enard convection in hydro- and magnetohydro-dynamic settings to address the onset of convection induced by the magnetothermal instability and the heat-flux-buoyancy-driven-instability in the weakly-collisional magnetized plasma permeating the intracluster medium. Since most of the related numerical simulations consider the plasma being bounded between two `plates' on which boundary conditions are specified, our strategy provides a framework that could enable a more direct connection between analytical and numerical studies. We derive the conditions for the onset of these instabilities considering the effects of induced magnetic tension resulting from a finite plasma beta. We provide expressions for the Rayleigh number in terms of the wave vector associated with a given mode, which allow us to characterize the modes that are first to become unstable. For both the heat-flux-buoyancy-driven-instability and the magnetothermal instability, oscillatory marginal stable states are possible.