Linear theory of compressible convection in rapidly rotating spherical shells, using the anelastic approximation

The onset of compressible convection in rapidly rotating spherical shells is studied in the anelastic approximation. An asymptotic theory valid at low Ekman number is developed and compared with numerical solutions of the full equations. Compressibility is measured by the number of density scale heights in the shell. In the Boussinesq problem, the location of the onset of convection is close to the tangent cylinder when there is no internal heating only a heat flux emerging from below. Compressibility strongly affects this result. With only a few scale heights or more of density present, there is onset of convection near the outer shell. Compressibility also strongly affects the frequencies and preferred azimuthal wavenumbers at onset. Compressible convection, like Boussinesq convection, shows strong spiralling in the equatorial plane at low Prandtl number. We also explore how higher-order linear modes penetrate inside the tangent cylinder at higher Rayleigh numbers and compare modes both symmetric and antisymmetric about the equator.