Onset of convection in a rapidly rotating compressible fluid spherical shell

We consider a rapidly rotating compressible fluid spherical shell, and study linear perturbations of a polytropic equilibrium state. Instead of considering the fully compressible problem, we make the anelastic approximation. We start from Boussinesq solutions and study the effect of introducing compressibility for different choices of the Prandtl number, Pi, comparing our results with those for a similar model of Glatzmaier and Gilman (1981). For Pr = 1 and 10, the results are similar. As compressibility is increased, convection becomes localised near to the inner boundary, an effect which is magnified by increasing the rotation rate. When we consider Pr = 0.1 we End different results. As compressibility is introduced, the critical Rayleigh number, R(c) decreases sharply and becomes negative. This behaviour was not found by Glatzmaier and Gilman.