Patterns of convection in spherical shells. Part 2

Abstract

The analysis by Busse (1975) of preferred patterns of convection in spherical shells is extended to include the case of odd degreeslof spherical harmonics. In the general part of the paper only the property of spherical symmetry of the basic state is used. The results are thus applicable to all bifurcation problems with spherical symmetry. Except in the case1= 1 a pattern degeneracy of the linear problem exists, which is partly removed by the solvability conditions that are generated when nonlinear terms are taken into account as perturbations. In each of the cases1considered so far, at least1physically different solutions have been found. The preferred solution among I existing ones is determined for1≥2 by a stability analysis. In the casel= 3 emphasized in this paper the axisymmetric solution is found to be always unstable, and the solution of tetrahedronal symmetry appears to be generally preferred. The latter result is rigorously established in the special case of a thin layer with nearly insulating boundaries treated in the second part of the paper.