Three-dimensional thermal convection in a spherical shell

Abstract

Independent pseudo-spectral and Galerkin numerical codes are used to investigate three-dimensional infinite Prandtl number thermal convection of a Boussinesq fluid in a spherical shell with constant gravity and an inner to outer radius ratio equal to 0.55. The shell is heated entirely from below and has isothermal, stress-free boundaries. Nonlinear solutions are validated by comparing results from the two codes for an axisymmetric solution at Rayleigh number Ra = 14250 and three fully three-dimensional solutions at Ra = 2000, 3500 and 7000 (the onset of convection occurs at Ra = 712). In addition, the solutions are compared with the predictions of a slightly nonlinear analytic theory. The axisymmetric solution is equatorially symmetric and has two convection cells with upwelling at the poles. Two dominant planforms of convection exist for the three-dimensional solutions: a cubic pattern with six upwelling cylindrical plumes, and a tetrahedral pattern with four upwelling plumes. The cubic and tetrahedral patterns persist for Ra at least up to 70000. Time dependence does not occur for these solutions for Ra [les ] 70000, although for Ra > 35000 the solutions have a slow asymptotic approach to steady state. The horizontal and vertical structure of the velocity and temperature fields, and the global and three-dimensional heat flow characteristics of the various solutions are investigated for the two patterns up to Ra = 70000. For both patterns at all Ra, the maximum velocity and temperature anomalies are greater in the upwelling regions than in the downwelling ones and heat flow through the upwelling regions is almost an order of magnitude greater than the mean heat flow. The preferred mode of upwelling is cylindrical plumes which change their basic shape with depth. Downwelling occurs in the form of connected two-dimensional sheets that break up into a network of broad plumes in the lower part of the spherical shell. Finally, the stability of the two patterns to reversal of flow direction is tested and it is found that reversed solutions exist only for the tetrahedral pattern at low Ra.