Onset of rotating and non-rotating convection in compressible and anelastic ideal gases

A linear stability analysis for compressible convection in a plane layer geometry both with and without the influence of rotation is presented. For the rotating cases we employ the tilted f-plane geometry that allows for varying angles between the rotation and gravity vectors. The stability criteria for compressible and anelastic ideal gases is compared. As expected, the critical parameters for the compressible equations approach those of the anelastic equations as the background stratification approaches the adiabatic (anelastic) limit. For the rotating cases, we observe asymptotic scaling behavior in the critical parameters in both compressible and anelastic fluids as the Taylor number becomes large. In contrast to the incompressible limit, finite tilt angles between the gravity and rotation vectors result in propagating compressible Rossby waves as the most unstable eigenmode and the critical parameters are established for a range of stratification levels and Taylor numbers; all wave orientations are found to propagate in prograde and equatorward directions for non-isothermal background states. We also compare the linear stability of the thermodynamically rigorous anelastic equations with an anelastic model that replaces thermal diffusion with an entropy diffusion-like term in the energy equation; it is shown that the linear stability of the entropy diffusion model yields qualitatively similar results for the critical parameters in comparison to the full anelastic set. We show that a thermodynamically rigorous alternative to the entropy diffusion model is the isothermal adiabatic background state in which temperature and entropy become equivalent thermodynamic quantities and viscous heating becomes subdominant in the energy equation; the stability characteristics of this model are also presented.