Infinite Prandtl number thermal convection in a spherical shell

Abstract

A Galerkin method is used to calculate the finite amplitude, steady, axisymmetric convective motions of an infinite Prandtl number, Boussinesq fluid in a spherical shell. Convection is driven by a temperature difference imposed across the stress-free, isothermal boundaries of the shell. The radial gravitational field is spherically symmetric and the local acceleration of gravity is directly proportional to radial position in the shell. Only the case of a shell whose outer radius is twice its inner radius is considered. Two distinct classes of axisymmetric steady states are possible. The temperature and radial velocity fields of solutions we refer to as ‘even’ are symmetric about an equatorial plane, while the latitudinal velocity is antisymmetric about this plane; solutions we refer to as ‘general’ do not possess any symmetry properties about the equatorial plane. The characteristics of these solutions, i.e. the isotherms, streamlines, spherically averaged temperature profiles, Nusselt numbers, etc., are given for Rayleigh numbers Ra as high as about 10 times critical for the even solutions and 3 times critical for the general solutions. Linear stability analyses of the nonlinear steady states show that the general solutions are the preferred form of axisymmetric convection when Ra is less than about 4 times critical. Furthermore, while the preferred motion at the onset of convection is non-axisymmetric, axisymmetric convection is stable when Ra exceeds about 1·3 times the critical value.