Varying the spherical shell geometry in rotating thermal convection

The effect of spherical shell geometry on rapidly-rotating thermal convection is studied in a suite of high resolution three-dimensional numerical simulations. The geometry is characterized by the radius ratio, chi = r(i) /r(o) , where r(i) is the inner shell radius, and r(o) is the outer shell radius. In this study, chi is varied over the broad range 0.10 to 0.92 in calculations of Boussinesq rotating convection subject to isothermal, rigid boundary conditions. Simulations are performed at Prandtl number Pr = 1 and for Ekman numbers E = 10(-3) , 3 x 10(-4) and 10(-4) . Near the onset of convection, the flow takes the form of rolls aligned parallel to the rotation axis and situated adjacent to the inner shell equator. The dimensionless azimuthal wavelength, lambda(c) , of the rolls is found to be independent of the shell geometry, only varying with the Ekman number. The critical wave number, m(c) , of the columnar rolls increases in direct proportion to the inner boundary circumference. For our simulations the critical Rayleigh number Ra-c at which convection first occurs varies in proportion to E-1.16 ; a result that is consistent with previous work on rotating convection. Furthermore, we find that Ra-c is a complex function of chi. We obtain the relation , which adequately fits all our results. In supercritical convection calculations the flows form quasi-geostrophic sheet-like structures that are elongated in the radial direction, stretching from the inner boundary toward the outer boundary.