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

A numerical model of two-dimensional Rayleigh-Benard convection is used to study the relationship between the surface heat how (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 proportional t o Ra(BL)(1/3)b(eff)(-4/3), where b(eff) is the effective dependence of visc osity with temperature at the base of the upper thermal boundary layer and Ra-BL is the Rayleigh number calculated with the viscosity upsilon(BL) (or the effective viscosity) at the base of the upper thermal boundary layer. T he 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 lay ers. These thermals are advected along the large-scale how. The above relat ionship is not valid at low heat flow where a stationary regime prevails an d for simulations forced into steady state. In these cases the Nusselt numb er follows a trend Nu proportional to Ra(BL)(1/5)b(eff)(-1), for a Newtonia n rheology, as predicted by the boundary layer theory. We argue that the eq uilibrium lithospheric thickness beneath old oceans or continents is contro lled by the development of thermals detaching fi om the thermal boundary la yers. Assuming this, we can estimate the viscosity at the base of the stabl e oceanic lithosphere. If the contribution of secondary convection to the s urface heat flux amounts to 40 to 50 mW m(-2), the asthenospheric viscosity is predicted to be between 10(18) and 2x10(19) Pa s.