Comparison of spherical-shell and plane-layer mantle convection thermal structure in viscously stratified models with mixed-mode heating: implications for the incorporation of temperature-dependent parameters

Plane-layer geometry convection models remain a useful tool for investigating planetary mantle dynamics but yield significantly warmer geotherms than spherical-shell systems. Comparisons of uniform property plane-layer and spherical-shell models have provided insight into the role of geometry on temperature in convecting systems but the inclusion of first-order terrestrial characteristics is needed to quantitatively assess the influence of system geometry on more relevant mantle models. Here, we analyse the mean temperatures of over 160 spherical-shell and plane-layer convection models featuring a uniform upper-mantle viscosity and a lower mantle that increases in viscosity by a factor of 30 or 100. With the imposition of the stratified viscosity, an effective Rayleigh number, Ra-n(-), is defined based on the average viscosity of the mantle. We derive equations for the relationship between the mean temperature, theta, Ra-n(-) and the non-dimensional internal heating rate, H, for both convection in a spherical shell with Earth-like mantle geometry and plane-layer solution domains. These equations predict the mean temperatures in the corresponding systems to an accuracy of a few percent or better. Our equations can be combined to derive the appropriate heating rate for a plane-layer convection model to emulate the temperatures in a mixed heating mode spherical-shell convection model with effective Rayleigh number comparable to the Earth's value, or greater. When comparing cases with the same internal heating rate and effective Rayleigh number, we find that the increased lower mantle viscosity amplifies the mean temperature ratio of the plane-layer and spherical-shell systems relative to isoviscous convection. These findings imply that the disagreement between spherical-shell mantle convection and plane-layer geometry mantle convection thermal structure must be particularly accounted for in plane-layer geometry models featuring variable viscosities.