In an annular spherical domain with separation d, the onset of convective motion occurs
at a critical Rayleigh number Ra = Rac. Solving the linear stability problem, it is shown
that degenerate points (d = dc; Rac) exist where two modes simultaneously become
unstable. Considering the weakly nonlinear evolution of these 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 at all degeneracies we outline the in
uence of
changes in d; Ra and Prandtl number Pr. A particular conclusion is that only even
modes have pure mode solutions, and that odd modes exist only as a component of mixed
mode solutions. The mode-dependent in
uence of Pr on the saturation of mixed mode
solutions is shown to be markedly di erent in the limits Pr ! 0 and Pr ! 1. 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 of oscillations
in a spherical annulus, is explained using the structure of the equations derived.