| Summary: | Plane-layer geometry convection models remain useful for modelling planetary mantle dynamics however they yield significantly warmer mean temperatures than spherical shell models. For example, in a uniform property spherical shell with the same radius ratio, f, as the Earth's mantle; a bottom heating Rayleigh number, Ra, of 10^7 and a nondimensional internal heating rate, H, of 23 (arguably Earth-like values) are insufficient to heat the mean temperature, θ, above the mean of the non-dimensional boundary value temperatures (0.5), the temperature in a plane-layer model with no internal heating. This study investigates the impact of this geometrical effect in convection models featuring uniform and stratified viscosity.
To address the effect of geometry, heat sinks are implemented to lower the mean temperature in 3D plane-layer isoviscous convection models. Over 100 models are analyzed, and their mean temperatures are used to derive a single equation for predicting θ, as a function of Ra, H and f in spherical and plane-layer systems featuring free-slip surfaces.
The inclusion of first-order terrestrial characteristics is introduced to quantitatively assess the influence of system geometry on planetary scale simulations. Again, over 100 models are analyzed featuring a uniform upper mantle viscosity and a lower mantle viscosity that increases by a factor of 30 or 100. An effective Rayleigh number, Raη, is defined based on the average viscosity of the mantle. Equations for the relationship between θ, Raη, and H are derived for convection in a spherical shell with f = 0.547 and plane-layer geometries.
These equations can be used to determine the appropriate heating rate for a plane-layer convection model to emulate spherical shell convection mean temperatures for effective Rayleigh numbers comparable to the Earth’s value and greater. Comparing cases with the same H and Raη, the increased lower mantle viscosity amplifies the mismatch in mean temperatures between spherical shell and plane-layer models. These findings emphasize the importance of adjusting heating rates in plane-layer geometry models and have important implications for studying convection with temperature-dependent parameters in plane-layer systems. The findings are particularly relevant to the study of convection in super-Earths where full spherical shell calculations remain intractable.
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