Thermal convection of an internally heated infinite prandtl number fluid in a spherical shell

Abstract

Finite amplitude, axisymmetric, steady convective motions of an infinite Prandtl number, Boussinesq fluid in a spherical shell are computed using a Galerkin technique. Two types of heating are considered. In one case, convection is driven both by internal heat sources in the fluid and by an externally imposed temperature drop across the boundaries of the shell. In the other case, only internal heat sources drive convection and the lower boundary of the shell is adiabatic. The boundaries of the shell are stress free. When heating is both from below and from within, the lower boundary is isothermal. The upper boundary is isothermal for both types of heating. The radial gravitational field is spherically symmetric and the local acceleration of gravity is directly proportional to radial position in the shell, as is appropriate to a sphere of constant density. 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 ranging from the critical value for the onset of convection to several times this value. Linear stability analyses of the nonlinear axisymmetric steady states show, when heating is both from below and from within, that near the onset of convection, a fully three-dimensional general motion is preferred, while at higher Rayleigh numbers, axisymmetric steady general motions are both preferred over axisymmetric steady even motions and stable against nonaxisymmetric perturbations. When heating is strictly from within and the lower boundary is adiabatic, the only possible steady motions are fully three-dimensional, i.e. all steady axisymmetric solutions are unstable to azimuthal perturbations.