Scaling of time-dependent stagnant lid convection: Application to small-scale convection on Earth and other terrestrial planets

Small-scale convection associated with instabilities at the bottom of the lithospheric plates on the Earth and other terrestrial planets occurs in the stagnant lid regime of temperature-dependent viscosity convection. Systematic numerical simulations of time-dependent, internally heated stagnant lid convection suggest simple scaling relationships for a variety of convective parameters and in a broad range of power law viscosities. Application of these scaling relationships to the Earth's oceanic lithosphere shows that for either diffusion or dislocation viscosity of olivine, convective instabilities occur in the lower part of the lithosphere between 85 and 100 km depth (the rheological sublayer). "Wet" olivine satisfies constraints on the heat flux and mantle temperature better than "dry" olivine, supporting the view that the upper mantle of the Earth is wet. This is also consistent with the fact that the rheological sublayer is located below the Gutenberg discontinuity which was proposed to represent a sharp change in water content. The viscosity of asthenosphere is (3-6)x10(18) Pa s, consistent with previous estimates. The velocities of cold plumes are relatively high reaching several meters per year in the dislocation creep regime. A low value of the heat flux in old continental cratons suggests that continental lithosphere might be convectively stable unless it is perturbed by processes associated with plate tectonics and hot plumes. The absence of plate tectonics on other terrestrial planets and the low heat transport efficiency of stagnant lid convection can lead to widespread melting during the thermal evolution of the terrestrial planets. If the terrestrial planets are dry, small-scale convection cannot occur at subsolidus temperatures.