Temperature and heat flux scaling laws for isoviscous, infinite Prandtl number mixed heating convection

Thermal evolution of terrestrial planets is controlled by heat transfer through their silicate mantles. A suitable framework for modelling this heat transport is a system including bottom heating (from the core) and internal heating, for example, generated by secular cooling or by the decay of radioactive isotopes. The mechanism of heat transfer depends on the physical properties of the system. In systems where convection is able to operate, two different regimes are possible depending on the relative amount of bottom and internal heating. For moderate internal heating rates, the system is composed of active hot upwellings and cold downwellings. For large internal heating rates, the bottom heat flux becomes negative and the system is only composed of active cold downwellings. Here, we build theoretical scaling laws for both convective regimes following the approach of Vilella & Kaminski (2017), which links the surface heat flux and the temperature jump across both the top and the bottom thermal boundary layer (TBL) to the Rayleigh number and the dimensionless internal heating rate. Theoretical predictions are then verified against numerical simulations performed in 2-D and 3-DCartesiangeometry, and covering a large range of the parameter space. Our theoretical scaling laws are more successful in predicting the thermal structure of systems with large internal heating rates than that of systems with no or moderate internal heating. The differences between moderate and large internal heating rates are interpreted as differences in the mechanisms generating thermal instabilities. We identified three mechanisms: conductive growth of the TBL, instability impacting, and TBL erosion, the last two being present only for moderate internal heating rates, in which hot plumes are generated at the bottom of the system and are able to reach the surface. Finally, we apply our scaling laws to the evolution of the early Earth, proposing a new model for the cooling of the primordial magma ocean that reconciles geochemical observations and magma ocean dynamics.