Emulating the thermal structure of spherical shell convection in plane-layer geometry mantle convection models

In a uniform property spherical shell with the same inner to outer radius ratio, f, as the Earthʹs mantle, a bottom heating Rayleigh number, Ra, of 107 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 boundary value temperatures (nondimensional value 0.5). Thus, to attain spherical shell-type temperatures in a plane-layer geometry system, some degree of internal cooling is required. We present the findings from 68 calculations of three-dimensional plane-layer convection featuring a range of Rayleigh numbers and internal heating and cooling rates. Observed mean temperatures are fit with a power–law scaling and combined with the results from spherical shell geometry convection studies to obtain a single equation relating θ, Ra, H and f. For a given Rayleigh number, the derived expression can be used to calculate an appropriate heating or cooling rate for a plane-layer convection model in order to obtain the θ of a spherical system described by f. Encouragingly, we find that at a Rayleigh number consistent with estimates of the effective value of Ra for the Earthʹs mantle, geotherms are similar for an appropriately cooled plane-layer system and a spherical shell model featuring a value for H based on estimates of the present-day rate of mantle internal heating. In particular, our findings have important implications for plane-layer geometry numerical models of mantle convection featuring temperature-dependent parameters and laboratory tank models. For example, we conclude that disregarding internal heating is most appropriate for modelling terrestrial mantle convection in a plane-layer geometry. Moreover, because cooling a plane-layer model becomes increasingly relevant when emulating spherical shell convection at higher Rayleigh numbers, our study indicates important considerations for modelling super-Earth mantle dynamics in plane-layer convection studies.