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 l ithospheric plates on the Earth and other terrestrial planets occurs in the stagnant lid regime of temperature-dependent viscosity convection. Systema tic numerical simulations of time-dependent, internally heated stagnant lid convection suggest simple scaling relationships for a variety of convectiv e parameters and in a broad range of power law viscosities. Application of these scaling relationships to the Earth's oceanic lithosphere shows that f or either diffusion or dislocation viscosity of olivine, convective instabi lities occur in the lower part of the lithosphere between 85 and 100 km dep th (the rheological sublayer). "Wet" olivine satisfies constraints on the h eat flux and mantle temperature better than "dry" olivine, supporting the v iew 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 disc ontinuity which was proposed to represent a sharp change in water content. The viscosity of asthenosphere is (3-6)x10(18) Pa s, consistent with previo us estimates. The velocities of cold plumes are relatively high reaching se veral meters per year in the dislocation creep regime. A low value of the h eat flux in old continental cratons suggests that continental lithosphere m ight be convectively stable unless it is perturbed by processes associated with plate tectonics and hot plumes. The absence of plate tectonics on othe r 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 co nvection cannot occur at subsolidus temperatures.