The onset of thermal convection in rotating spherical shells

Abstract

The correct asymptotic theory for the linear onset of instability of a Boussinesq fluid rotating rapidly in a self-gravitating sphere containing a uniform distribution of heat sources was given recently by Jones et al. (2000). Their analysis confirmed the established picture that instability at small Ekman number $E$ is characterized by quasi-geostrophic thermal Rossby waves, which vary slowly in the axial direction on the scale of the sphere radius $r_o$ and have short azimuthal length scale $O(E^{1/3}r_o)$. They also confirmed the localization of the convection about some cylinder radius $s\,{=}\,s_M$ roughly $r_o/2$. Their novel contribution concerned the implementation of global stability conditions to determine, for the first time, the correct Rayleigh number, frequency and azimuthal wavenumber. Their analysis also predicted the value of the finite tilt angle of the radially elongated convective rolls to the meridional planes. In this paper, we study small-Ekman-number convection in a spherical shell. When the inner sphere radius $r_i$ is small (certainly less than $s_M$), the Jones et al. (2000) asymptotic theory continues to apply, as we illustrate with the thick shell $r_i\,{=}\,0.35\,r_o$. For a large inner core, convection is localized adjacent to, but outside, its tangent cylinder, as proposed by Busse & Cuong (1977). We develop the asymptotic theory for the radial structure in that convective layer on its relatively long length scale $O(E^{2/9}r_o)$. The leading-order asymptotic results and first-order corrections for the case of stress-free boundaries are obtained for a relatively thin shell $r_i\,{=}\,0.65\,r_o$ and compared with numerical results for the solution of the complete PDEs that govern the full problem at Ekman numbers as small as $10^{-7}$. We undertook the corresponding asymptotic analysis and numerical simulation for the case in which there are no internal heat sources, but instead a temperature difference is maintained between the inner and outer boundaries. Since the temperature gradient increases sharply with decreasing radius, the onset of instability always occurs on the tangent cylinder irrespective of the size of the inner core radius. We investigate the case $r_i\,{=}\,0.35\,r_o$. In every case mentioned, we also apply rigid boundary conditions and determine the $O(E^{1/6})$ corrections due to Ekman suction at the outer boundary. All analytic predictions for both stress-free and rigid (no-slip) boundaries compare favourably with our full numerics (always with Prandtl number unity), despite the fact that very small Ekman numbers are needed to reach a true asymptotic regime.