Theoretical and numerical study of a thermal convection problem with temperature-dependent viscosity in an infinite layer

A convection problem with temperature-dependent viscosity in an infinite layer is presented. This problem has important applications in mantle convection. The existence of a stationary bifurcation is proven together with a condition to obtain the critical parameters at which the bifurcation takes place. A numerical strategy has been developed to calculate the critical bifurcation curves and the most unstable modes for a general dependence of viscosity on temperature. An exponential dependence of viscosity on temperature has been considered in the numerical calculations. Comparisons with the classic Rayleigh-Bénard problem with constant viscosity indicate that the critical temperature difference threshold decreases as the exponential rate parameter increases. The vertical velocity of the marginal mode exhibits motion concentrated in the region where viscosity is smaller.