Patterns of convection in spherical shells. Part 2
The analysis by Busse (1975) of preferred patterns of convection in spherical shells is extended to include the case of odd degrees l of 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 case 1 = 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 cases 1 considered so far, at least 1 physically different solutions have been found. The preferred solution among I existing ones is determined for 1[greater-than-or-equal]2 by a stability analysis. In the case l = 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.(Published Online April 20 2006)
(Received February 10 1981)
(Revised November 17 1981)
p1 Permanent address: Institute of Geophysics and Planetary Physics, University of California at Los Angeles.
p2 Present address: University of Illinois at Urbana–Champaign.