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We use the machinery usually employed for studying the onset of Rayleigh--B��nard 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.
10 pages, 1 figure
Fluid Dynamics (physics.flu-dyn), FOS: Physical sciences, Galaxy cluster, Physics - Fluid Dynamics, Convection, Magnetothermal instability, Astrophysics - Astrophysics of Galaxies, Heat-flux-driven buoyancy instability, Intracluster medium, Linear stability analysis, Astrophysics of Galaxies (astro-ph.GA)
Fluid Dynamics (physics.flu-dyn), FOS: Physical sciences, Galaxy cluster, Physics - Fluid Dynamics, Convection, Magnetothermal instability, Astrophysics - Astrophysics of Galaxies, Heat-flux-driven buoyancy instability, Intracluster medium, Linear stability analysis, Astrophysics of Galaxies (astro-ph.GA)
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