Scaling laws for convection with temperature-dependent viscosity and grain-damage

Numerical experiments of convection with grain-damage are used to develop scaling laws for convective heat flow, mantle velocity and plate velocity across the stagnant lid and plate-tectonic regimes. Three main cases are presented in order of increasing complexity: a simple case wherein viscosity is only dependent on grain size, a case where viscosity depends on temperature and grain size, and finally a case where viscosity is temperature and grain size sensitive, and the grain-growth (or healing) is also temperature sensitive. In all cases, convection with grain-damage scales differently than Newtonian convection; whereas the Nusselt number (Nu), typically scales with the reference Rayleigh number, Ra-0, to the 1/3 power, for grain-damage this exponent is larger because increasing Ra-0 also enhances damage. In addition, Nu, mantle velocity, and plate velocity are also functions of the damage to healing ratio, (D/H); increasing D/H increases Nu because more damage leads to more vigorous convection. For the fully realistic case, numerical results show stagnant lid convection, fully mobilized convection that resembles the temperature-independent viscosity case, and partially mobile or transitional convection, depending on D/H, Ra-0, and the activation energies for viscosity and healing. Applying our scaling laws for the fully realistic case to Earth and Venus we demonstrate that increasing surface temperature dramatically decreases plate speed and heat flow, essentially shutting down plate tectonics, due to increased healing in lithospheric shear zones, as proposed previously. Contrary to many previous studies, the transitional regime between the stagnant lid and fully mobilized regimes is large, and the transition from stagnant lid to mobile convection is gradual and continuous. Thus planets could exhibit a full range of surface mobility, as opposed to the bimodal distribution of fully mobile lid planets and stagnant lid planets that is typically assumed.