Convection in a variable viscosity fluid heated from within

A time-dependent theoretical model of finite amplitude thermal convection in a layer of fluid that has a variable viscosity and that is heated from within is developed. Numerical results are presented for the case where the viscosity increases exponentially with depth for a range of rates of increase with depth and a range of Rayleigh numbers that may be appropriate for the earth's mantle. It is found that convection is confined largely to the low-viscosity zone and that the cells are shortened in horizontal extent. A steady state was not reached in any case considered. Initial thermal structure is found to drastically affect the history and pattern of convection, and thus the theory cannot, provide much constraint to speculations about mantle convection.