Parker/buoyancy instabilities with anisotropic thermal conduction, cosmic rays, and arbitrary magnetic field strength

We report the results of a local stability analysis for a magnetized, gravitationally stratified plasma containing cosmic rays. We account for cosmic-ray diffusion and thermal conduction parallel to the magnetic field and allow beta to take any value, where p is the plasma pressure and B is the magnetic field strength. We take the gravitational acceleration to be in the -z-direction and the equilibrium magnetic field to be in the y-direction, and we derive the dispersion relation for small-amplitude instabilities and waves in the large-|k_x| limit. We use the Routh-Hurwitz criterion to show analytically that the necessary and sufficient criterion for stability in this limit is n k_B dT/dz + dp_cr/dz + (1/8pi)dB^2/dz > 0, where T is the temperature, n is the number density of thermal particles, and p_cr is the cosmic-ray pressure. We present approximate analytical solutions for the normal modes in the low- and high-diffusivity limits, show that they are consistent with the derived stability criterion, and compare them to numerical results obtained from the full, unapproximated, dispersion relation. Our results extend earlier analyses of buoyancy instabilities in galaxy-cluster plasmas to the beta <= 1 regime. Our results also extend earlier analyses of the Parker instability to account for anisotropic thermal conduction, and show that the interstellar medium is more unstable to the Parker instability than was predicted by previous studies in which the thermal plasma was treated as adiabatic.