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Numerical solutions for thermal convection of two superposed viscous fluid layers, with the same thicknesses as the Earth's upper and lower mantles and separated by a fixed horizontal interface, are obtained at the onset of convection and in the nonlinear regime. Several heating configurations and thermal boundary conditions are considered, including an isothermal bottom boundary, a prescribed heat flux bottom boundary, and internal heating with an insulated lower boundary. At the onset of convection and with the same material properties in each layer, the lower layer is more unstable than the upper layer and long horizontal wavelengths (proportional to the lower layer thickness) predominate. For layers with sufficiently different material properties, the upper layer can be more unstable and short horizontal wavelengths (proportional to the upper layer thickness) can predominate at the onset of convection. When the material properties are chosen to make the two layers equally unstable, there are two stability curve (Rayleigh number Ra versus wavelength) minima with the lower minimum at long wavelength. Linear stability results typify weakly nonlinear convection except when both layers are equally unstable. When the layers have unequal stability, the stable layer has a conductive temperature field and its flow is viscously driven by the convective flow in the unstable layer. When the two layers are equally unstable, both layers are separately convecting even at Rayleigh numbers of only twice the critical value for convection onset. At large enough supercritical Ra the lower layer convects with a long wavelength cell even when linear stability theory predicts short wavelength cells. The upper layer always begins convecting with short wavelength cells. However, the upper layer cell over the lower layer downwelling plume increases in size with increasing Ra. Thus, horizontal wavelength (plate dimension, for example) cannot be used to discriminate between whole mantle and layered mantle convection. Mechanical coupling of the layers (downwelling plume over upwelling plume) occurs for all cases with the same material properties in each layer and for the case with the same Ra in each layer. When the material properties of the layers are sufficiently different to produce short wavelength cells at the onset of convection, the upper layer has two thermally coupled cells (downwelling plume over downwelling plume and a shear zone at the interface) over both lower layer plumes. A large (50 per cent of total) temperature difference across the interface occurs in all cases examined, implying that two-layer mantle convection should also have a large temperature change across the 670 km discontinuity. However, temperature dependent viscosity may reduce the magnitude of this temperature difference. Interpretations of geoid anomalies over subduction zones and seismic traveltime anomalies of slabs should account for possible thermal coupling of separate upper and lower mantle convective systems.

We determine the variation of effective viscosity q across the lower mantle from models of the Gibb's free energy of activation G* and the adiabatic temperature profile. The variation of G* with depth is calculated using both an elastic strain energy model, in which G* is related to the seismic velocities, and a model which assumes G* is proportional to the melting temperature. The melting temperature is assumed t o follow Lindemann's equation. The adiabatic temperature profile is calculated from a model for the density dependence of the Gnineisen parameter. Estimates of q depend on whether the lower mantle is a Newtonian or power law fluid. In the latter case separate estimates of q are obtained for flow with constant stress, constant strain rate, and constant strain energy dissipation rate. For G* based on the melting temperature, increases in q with depth range from a factor of about 100 for Newtonian deformation or power-law flow with constant stress to about 5 for non-Newtonian deformation with constant strain rate. For G* based on elastic defect energy, increases in q with depth range from a factor of about 1500 for Newtonian deformation or power-law flow with constant stress to about 10 for non-Newtonian deformation with constant strain rate. Among these models, only a non-Newtonian lower mantle convecting with constant strain rate or constant strain energy dissipation rate is consistent with recent estimates of mantle viscosity from post-glacial rebound and true polar wander data.