Character and stability of axisymmetric thermal convection in spheres and spherical shells

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Abstract
Nonlinear axisymmetric convective motions of self-gravitating, infinite Prandtl number fluids in spheres and thick spherical shells are determined for a variety of shell sizes and for different modes of heating. For one combination of heating from within and from below the onset of convection is governed by a self-adjoint system of equations and boundary conditions. For two other heating modes, heating only from within or only from below, the linearized equations and boundary conditions are not self-adjoint. The properties of the self-adjoint solutions together with a parameter which quantifies the departure from self-adjointness provide a theoretical framework for organizing, understanding, and generalizing the heat transfer characteristics of the non-self-adjoint cases. The variations in heating and shell size that are considered yield 6 different patterns of steady convection—flows with 1 and 3 meridional cells, and pairs of oppositely rotating flows with 2 and 4 meridional cells. The heat transport properties of the different convection patterns and the stability of the flows to general axisymmetric and nonaxisymmetric disturbances are ascertained.

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