On the stability of the stagnant-lid regime in internally heated convection with temperature-dependent viscosity: Scaling approach and comparison of viscosity laws

Internally-heated convection for a material whose viscosity depends on temperature only is studied, using a large set of 2-D Cartesian numerical simulations. A Newtonian temperature-dependent viscosity is used, obeying either an Arrhenius law, with realistic parameters for an Earth-like mantle, or the so-called Frank-Kamenetskii (FK) approximation. When the viscosity depends very strongly on temperature, a rigid stagnant lid develops at the surface of the model. In the absence of plastic yielding, some surface deformation is still possible when the dependence of the viscosity on the temperature is weak. The transition between these two behaviors has been extensively studied with the FK approximation, but not with the Arrhenius law. In the present study, when the latter is used, the stagnant-lid regime is very stable. Non-negligible surface motion is limited to very low values of the activation energy , well below estimates for terrestrial mantles, or to unrealistically high values of the surface temperature. A new scaling approach is proposed to predict the internal temperature below the stagnant lid, as well as the thicknesses of the rigid lid and of the active thermal boundary layer (TBL) below the lid. The predicted values, with the two viscosity formulations used here, fit the observed results of numerical simulations. It is shown that surface deformation is non-negligible when the active TBL is thicker than the stagnant lid. While this criterion can be met when the FK approximation is used, even when considering otherwise realistic parameters for a terrestrial mantle, it is not observed in numerical simulations, nor predicted, when an Arrhenius viscosity with an activation energy larger than 40 kJ/mol is used.