Weakly nonlinear mode interactions in spherical Rayleigh-Bénard convection

In an annular spherical domain with separation d, the onset of convective motion occurs at a critical Rayleigh number Ra = Rac. Solving the axisymmetric linear stability problem, shows that degenerate points (d = dc, Rac) exist where two modes simultaneously become unstable. Considering the weakly nonlinear evolution of these two modes, it is found that spatial resonances play a crucial role in determining the preferred convection pattern for neighbouring modes (`, `±1) and non-neighbouring even modes (`, `±2). Deriving coupled amplitude equations relevant to all degeneracies, we outline the possible solutions and the influence of changes in d, Ra and Prandtl number Pr . Using direct numerical simulation (DNS) to verify all results, time periodic solutions are also outlined for small Pr . The 2 : 1 periodic signature observed to be general for oscillations in a spherical annulus, is explained using the structure of the equations. The relevance of all solutions presented is determined by computing their stability with respect to non-axisymmetric perturbations at large Pr