Heat transport in stagnant lid convection with temperature‐ and pressure‐dependent Newtonian or non‐Newtonian rheology

A numerical model of two‐dimensional Rayleigh‐Bénard convection is used to study the relationship between the surface heat flow (or Nusselt number) and the viscosity at the base of the lithosphere. Newtonian or non‐Newtonian, temperature‐ and pressure‐dependent rheologies are considered. In the high Rayleigh number time‐dependent regime, calculations yield Nu ∝ Ra BL 1/3 b eff −4/3 where b eff is the effective dependence of viscosity with temperature at the base of the upper thermal boundary layer and Ra BL is the Rayleigh number calculated with the viscosity ν BL (or the effective viscosity) at the base of the upper thermal boundary layer. The heat flow is the same for Newtonian and non‐Newtonian rheologies if the activation energy in the non‐Newtonian case is twice the activation energy in the Newtonian case. In this chaotic regime the heat transfer appears to be controlled by secondary instabilities developing in thermal boundary layers. These thermals are advected along the large‐scale flow. The above relationship is not valid at low heat flow where a stationary regime prevails and for simulations forced into steady state. In these cases the Nusselt number follows a trend Nu ∝ Ra BL 1/5 b eff −1 for a Newtonian rheology, as predicted by the boundary layer theory. We argue that the equilibrium lithospheric thickness beneath old oceans or continents is controlled by the development of thermals detaching from the thermal boundary layers. Assuming this, we can estimate the viscosity at the base of the stable oceanic lithosphere. If the contribution of secondary convection to the surface heat flux amounts to 40 to 50 mW m −2 , the asthenospheric viscosity is predicted to be between 10 18 and 2×l0 19 Pa s.