Buoyancy instabilities in galaxy clusters: Convection due to adiabatic cosmic rays and anisotropic thermal conduction

Using a linear stability analysis and two- and three-dimensional nonlinear simulations, we study the physics of buoyancy instabilities in a combined thermal and relativistic (cosmic ray) plasma, motivated by the application to clusters of galaxies. We argue that the cosmic-ray diffusion time is likely to be long compared to the buoyancy time on large length scales, so that cosmic rays are effectively adiabatic. If the cosmic-ray pressure p _cr is ≳25% of the thermal pressure, and the cosmic-ray "entropy" p _cr/ρ^4/3 (where ρ is the thermal-plasma density) decreases outward, cosmic rays drive an adiabatic convective instability analogous to Schwarzschild convection in stars. Global simulations of galaxy cluster cores show that this instability saturates by reducing the cosmic-ray entropy gradient and driving efficient convection and turbulent mixing. At larger radii in cluster cores where cosmic-ray pressure is negligible, the thermal plasma is unstable to the heat-flux-driven buoyancy instability (HBI), a convective instability generated by anisotropic thermal conduction and a background conductive heat flux. The HBI saturates by rearranging the magnetic field lines to become largely perpendicular to the local gravitational field; the resulting turbulence also primarily mixes plasma in the perpendicular plane. Cosmic-ray-driven convection and the HBI may contribute to redistributing metals produced by Type Ia supernovae in clusters. Our calculations demonstrate that adiabatic simulations of galaxy clusters can artificially suppress the mixing of thermal plasma. When anisotropic thermal conduction is included, the buoyant response of the thermal plasma is not governed by the stable entropy gradient, and mixing (driven by mergers, cosmic ray buoyancy, etc.) is more effective. Such mixing may contribute to cosmic rays being distributed throughout the cluster volume.