Multiplicity of nonlinear thermal convection in a spherical shell

Abstract

Linear and weakly nonlinear thermal convection in a moderately thin spherical shell in the presence of a spherically symmetric gravity subject to a spherically symmetric boundary condition is systematically investigated through fully three-dimensional numerical simulations. The convection problem is self-adjoint and the linear convective stability is characterized by l, the degree of a spherical harmonics Yml (theta,phi). While the radial structure of the linear convection is determined by the stability analysis, there exists a (2l + 1)-fold degeneracy in the horizontal structure of the spherical convection. When l = O(10) , i.e., in a moderately thin spherical shell, the removal or partial removal of the degeneracy represents a mathematically difficult, physically not well-understood problem. By starting with carefully chosen initial conditions, we are able to obtain a variety of nonlinear convective flows at exactly the same parameters near the onset of convection, including steady axisymmetric convection, steady azimuthally periodic convection, steady azimuthally nonperiodic convection, equatorially asymmetric convection, and steady convection in the form of a single giant spiral roll covering the whole spherical shell which is stable and robust for a wide range of the Prandtl number.