Scaling laws of convection for cooling planets in a stagnant lid regime

Scaling laws used in 1-D parametrized thermal models to describe the cooling of planets have mainly been derived from steady-state convection models. Here, fully dynamic 2-D and 3-D models are employed to compute the interior thermal evolution using strong temperature-dependent viscosity, mixed heating with radioelement decay, spherical geometry. The results of these models are compared to those of parametrized models, where we distinguish between the upper thermal boundary layer (TBL) and the stagnant lid, in a Monte Carlo framework to derive scaling laws valid for the entire planet evolution. The upper TBL thickness depends on the Rayleigh number to the power beta(u), while the lid base is determined by the mantle temperature and the rate of viscosity change with temperature multiplied by a prefactor a(rh). By varying beta(u) and a(rh) we find that, although the heating conditions change as a function of time, the thermal evolution of a cooling planet in a stagnant lid regime can be represented by a wide range of combinations of a(rh) and beta(u). Suitable fits are found for different values of a(rh) and beta(u) that vary depending on the model set-up. The observed relationship between suitable values of a(rh) and beta(u) is explained by a trade-off between the lid and the TBL thicknesses. Only when considering a specific definition of the stagnant lid thickness in 3-D simulations, suitable combinations for all models converge to specific values of a(rh) and beta(u), Using a stagnant lid defined by the intersection between the depth axis and the tangent to the velocity profile at the depth corresponding to the maximum velocity gradient, we find best-fit values of 2.54 and 0.335 for a(rh) and beta(u), respectively.