The onset of thermal convection in an infinite Prandtl number, variable-viscosity, compressible Earth's mantle by the mean-field approximation

The values of the Rayleigh number critical for the onset of thermal instabi lity in an infinite Prandtl number, compressible Earth's mantle are determi ned against the aspect ratio (the ratio between the width and the depth of the cell) for constant and for pressure- and temperature-dependent viscosit y. The procedure adopted follows the experimental approach which determines the critical Rayleigh number as the value at which the heat flux across th e fluid exceeds the conductive one. We solve the single-mode mean-field equ ations for the conservation of mass,;momentum and energy and evaluate the N usselt number, i.e. the ratio between the actual heat transfer and the heat which would be transported only by conduction. The value of the Rayleigh n umber for which the Nusselt number exceeds unity can be identified as the c ritical value. Between incompressible and compressible fluids with constant viscosity the critical Rayleigh numbers differ at most by 10-20% for cells aspect ratios between 0.1 to about 30, while at larger aspect ratios compr essibility yields critical Rayleigh numbers one or more orders of magnitude larger. For variable viscosity, since a more viscous interior prevents mot ion, a much higher Rayleigh number is required for the onset of convection. It is also shown how viscous dissipation of heat increases the critical Ra yleigh number, while compressibility favours the onset of convection.