Effects of strongly variable viscosity on three‐dimensional compressible convection in planetary mantles

A systematic investigation into the effects of temperature dependent viscosity on three-dimensional compressible mantle convection has been performed by means of numerical simulations in Cartesian geometry using a finite volume multigrid code, with a factor of 1000–2500 viscosity variation, Rayleigh numbers ranging from 105–107, and stress-free upper and lower boundaries. Considerable differences in model behavior are found depending on the details of rheology, heating mode, compressibility, and boundary conditions. Parameter choices were guided by realistic Earth models. In Boussinesq, basally heated cases with viscosity solely dependent on temperature and stress-free, isothermal boundaries, very long wavelength flows (∼25,000 km, assuming the depth corresponds to mantle thickness) with cold plumes and hot upwelling sheets result, in contrast to the upwelling plumes and downwelling sheets found in small domains, illustrating the importance of simulating wide domains. The addition of depth dependence results in small cells and reverses the planform, causing hot plumes and cold sheets. The planform of temperature-dependent viscosity convection is due predominantly to vertical variations in viscosity resulting from the temperature dependence. Compressibility, with associated depth-dependent properties, results in a tendency for broad upwelling plumes and narrow downwelling sheets, with large aspect ratio cells. Perhaps the greatest modulation effect occurs in internally heated compressible cases, in which the short-wavelength pattern of time-dependent cold plumes commonly observed in constant-viscosity calculations completely changes into a very long wavelength pattern of downwelling sheets (spaced up to 24,000 km apart) with time-dependent plumelike instabilities. These results are particularly interesting, since the basal heat flow in the Earth's mantle is usually thought to be very low, e.g., 5–20% of total. The effects of viscous dissipation and adiabatic heating play only a minor role in the overall heat budget for constant-viscosity cases, an observation which is not much affected by the Rayleigh number. However, viscous dissipation becomes important in the stiff upper boundary layer when viscosity is temperature dependent. This effect is caused by the very high stresses occurring in this stiff lid, typically 2 orders of magnitude higher than the stresses in the interior of the domain for the viscosity contrast modeled here. The temperature in the interior of convective cells is highly sensitive to the material properties, with temperature-dependent viscosity and depth-dependent thermal conductivity strongly increasing the internal temperature, and depth-dependent viscosity strongly decreasing it. The sensitivity of the observed flow pattern to these various complexities clearly illustrates the importance of performing compressible, variable-viscosity mantle convection calculations with rheological and thermodynamic properties matching as closely as possible those of the Earth.