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, X = ri/ro, where ri is the inner shell radius, and ro is the outer shell radius. In this study, X 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, λ c, of the rolls is found to be independent of the shell geometry, only varying with the Ekman number. The critical wave number, mc, of the columnar rolls increases in direct proportion to the inner boundary circumference. For our simulations the critical Rayleigh number Rac. 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 Rac is a complex function of X. We obtain the relation Rac E1.16 = 0.21/X2+ 22.4√(1 - x)(1 + x), 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. © 2004 Taylor and Francis Ltd.

The geometry of the liquid region of a planetary core can effect core convection and magnetic field generation processes. Varying the spherical shell radius ratio, χ = ri/ro, illustrates differences between planets with differing core radius ratios as well as how dynamo processes vary with time in an evolving planetary core. Here we study numerical models of thermally driven dynamo action in a rotating shell of outer radius ro with electrically conducting Boussinesq fluid that surrounds an equally conductive solid inner sphere of radius ri. Dynamo solutions are found for 0.15 ≤ χ ≤ 0.65 at Ekman number E = 3 × 10-4, Prandtl number Pr = 1, and magnetic Prandtl number Pm = 5, with mechanically rigid, isothermal boundary conditions. In cases with mildly supercritical Rayleigh numbers, Ra, convection occurs in the form of columnar plumes aligned with the rotation axis. In thick-shelled cases (χ = 0.15), onset of axial dipolar dynamo action occurs at relatively high Ra by the action of a single turbulent columnar plume. In thinner shells (χ ≥ 0.25), multi-plume columnar convection generates strongly dipolar dynamo fields near the onset of convection. Although some of our control parameters are far from planetary values, the outer boundary Elsasser number, Λ, (a non-dimensional measure of magnetic field strength) for thinner shell cases is comparable to that for the Earth's core-mantle boundary and estimates for the dynamos of Jupiter and Saturn. In contrast, thick shell, single plume dynamos produce relatively low Elsasser numbers at the outer shell boundary, possibly relevant to the magnetic field of Mercury. For a single evolving planet, these results imply that a transition during early inner core growth from a single plume or regional convective regime to a multi-plume or global convective regime could produce a significant increase in the exterior global magnetic field intensity. Our calculations are consistent with observations of axial dipole dominated planetary magnetic fields and support the concept of a geocentric axial dipolar field for much of Earth's history. © 2005 Elsevier B.V. All rights reserved.