Geoid and topography for infinite Prandtl number convection in a spherical shell

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Abstract
Geoid anomalies and surface and lower‐boundary topographies are calculated for numerically generated thermal convection in an infinite Prandtl number, Boussinesq, axisymmetric spherical fluid shell with constant gravity and viscosity. Calculations were done for heating entirely from below (HFB) and entirely from within (HFW). Convective solutions were obtained for Rayleigh numbers Ra up to 20 times the critical Ra in the HFB case and 27 times critical in the HFW case. Bifurcations that occur with varying Ra are associated with different convective patterns. Boundary topography is positive at upwellings and negative at downwellings. Absolute topography is greater at the poles than at the equator due to poleward compression of the flow field; lower‐boundary topography is greater than at the surface due to radial compression of the flow field. The geoid is primarily dominated by the gravitational potential of surface masses due to surface boundary deformations; hence the geoid parallels surface undulations. In general, boundary deformation increases with increasing cell wavelength. This property may elucidate whether core‐mantle boundary (CMB) topography is due to mantle convection or mantle dregs (chemically dense material) which are only indirectly of dynamic origin, and hence are unlikely to have the same wavelength dependence in its associated topography. The ratios of boundary deformation and geoid to αΔTd (α, coefficient of thermal expansion; ΔT, characteristic temperature drop across the fluid shell; and d, shell thickness) maximize with increasing Ra at approximately 10 times Ra critical and then steadily decrease out to the maximum Ra calculated. Dimensionless geoid and topography in the HFB case are approximately 5 times greater than in the HFW case. At the highest Rayleigh numbers investigated, dimensionless peak‐to‐trough amplitudes of geoid and topography are weak functions of Ra, which lends some justification to estimating the dimensional values of these quantities with Earthlike parameters. For parameter values characteristic of the Earth's mantle, the geoid and upper and lower boundary topograpies are approximately 1 km, 10 km, and 15 km for the HFB case, and approximately 200 m, 2 km, and 3 km for the HFW case, respectively. The HFW values correlate with geophysical data more closely than the HFB values, suggesting a predominance of internal heating in the mantle. Although this result is qualfied by the low Rayleigh numbers and axisymmetry of the model calculations, our study emphasizes that dynamically induced topography and geoid are sensitive to the mode of heating in the Earth's mantle.