Asymptotic solutions of convection in rapidly rotating non-slip spheres

Abstract

Asymptotic solutions describing the onset of convection in rotating, self-gravitating Boussinesq fluid spheres with no-slip boundary conditions, valid for asymptotically small Ekman numbers and for all values of the Prandtl number, are derived. Central to the asymptotic analysis is the assumption that the leading-order convection can be represented, dependent on the size of the Prandtl number, by either a single quasi-geostrophic-inertial-wave mode or by a combination of several quasi-geostrophic-inertial-wave modes, and is controlled or influenced by the effect of the oscillatory Ekman boundary layer. Comparisons between the asymptotic solutions and the corresponding fully numerical simulations show a satisfactory quantitative agreement.