Bifurcation and Stability of Convective Flows in a Rotating or not Rotating Spherical Shell

In this paper bifurcation theory and group theoretical methods are applied to the analysis of the stationary convection of a fluid filling a spherical shell which is rotating (or not rotating) about an axis with a constant angular speed. In the case with rotation, an analytical relation is found between the Rayleigh number and the Taylor number, for which a transcritical branch of stationary and axisymmetric (about the axis of rotation) solutions occur. At fixed Taylor number, these solutions are stable supercritically. When the shell doesn’t rotate, a two-parameter family of axisymmetric solutions is found to bifurcate supercritically, these solutions being deduced one from the other by a simple rotation. Under an assumption on the sign of a certain coefficient, these solutions are “orbitally stable.”