Modal growth and coupling in three-dimensional spherical convection

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Abstract
The growth of spherical harmonic modes with increasing Rayleigh number Ra (a nondimensional measure of convective vigor) in numerical simulations of three-dimensional, Boussinesq, infinite Prandtl number, basally heated, spherical-shell convection is analyzed. Two regular polyhedral convective patterns with tetrahedral and cubic symmetry are examined. Apart from the dominant spherical harmonic modes which define the polyhedral patterns, the most important modes (in terms of modifying the convection as Ra increases) are ones whose wavenumbers (i.e., spherical harmonic degree and order) are exactly triple those of the dominant modes. Modes with wavenumbers that are five times those of the dominant modes also maintain large growth rates with increasing Ra. These results indicate the possibility that the spherical harmonic modes which are primarily responsible for modifying convection (e.g., narrowing the boundary layers) with increasing Rayleigh number, occur at wavenumbers that are odd integer multiples of those of the dominant modes. This suggests that an extended mean field method—wherein solutions in which only these modes are kept—may reasonably represent steady convection with regular polyhedral patterns up to relatively high Ra; such a method would entail a significant simplification in the analysis of nonlinear convection.