Strongly variable viscosity flows in mantle convection

Convection in the Earth's mantle is a complicated phenomenon that causes various tectonic activities and affects mantle evolution on geologic time scales (billions of years). It is a subject as yet not fully understood. The early success of the high Rayleigh number constant viscosity theory was later tempered by the absence of plate motion when the viscosity is more realistically strongly temperature dependent. A similar problem arises if the equally strong pressure dependence of viscosity is considered, since the classical isothermal core convection theory would then imply a strongly variable mantle viscosity, which is inconsistent with results from postglacial rebound studies. We consider a mathematical model for Rayleigh-Bénard convection in a basally heated layer of a fluid whose viscosity depends strongly on both temperature and pressure, defined in an Arrhenius form. The model is solved numerically for extremely large viscosity variations across a unit aspect ratio cell, and steady solutions are obtained. To improve the efficiency of numerical computation, we introduce a modified viscosity law with a low temperature cut-off. We demonstrate that this simplification results in markedly improved numerical convergence without compromising accuracy. Continued numerical experiments suggest that narrow cells are preferred at extreme viscosity contrasts. We are then able to determine the asymptotic structure of the solution, and it agrees well with the numerical results. Beneath a stagnant lid, there is a vigorous convection in the upper part of the cell, and a more sluggish, higher viscosity flow in the lower part of the cell. We then offer some comments on the meaning and interpretation of these results for planetary mantle convection.