Derivation of the Critical Rayleigh Number for Convection in Crystalline Solids

The classical theory of thermal convection applies to viscous materials whose linear flow behavior can be described by a Newtonian coefficient of viscosity. When applied to the earth's mantle and polar ice sheets, this theory should be modified so that it holds for viscoplastic materials whose nonlinear flow behavior is described by a non-Newtonian coefficient of effective viscosity. When this is done, the critical Rayleigh number for nonlinear convection flow depends on the thermal stress driving convection and on the parameters of the flow law which describes creep deformation of the viscoplastic material.