Variable viscosity thermal convection at infinite Prandtl number in a thick spherical shell.

摘要

Variable viscosity thermal convection at infinite Prandtl number in a thick spherical shell is modeled using the finite element method. The discretized tensor Navier-Stokes equations are solved with the multigrid method applied to the spherical elements. Numerically convergent solutions can be obtained when specially tailored matrix dependent transfer is introduced in the multigrid method and line Jacobi relaxation is applied in the radial direction of the shell. The conjugate gradient method is used to correct both pressure and velocity fields simultaneously to satisfy the incompressibility condition. Numerical calculations are performed on a Cray T3D to investigate internally heated time dependent thermal convection for Rayleigh numbers between 10{dollar}sp5{dollar} and 10{dollar}sp6{dollar} with both Newtonian and non-Newtonian viscosity which can be dependent on temperature and pressure (depth). Linear downwellings and diffuse plume-like upwellings are dominant features in the numerical calculations. When Newtonian rheology is applied throughout the shell, depth- dependence of viscosity is the dominant rheological influence controlling the flow characteristics--lateral variation of viscosity plays a secondary role. Both a high viscosity lid and a viscosity increase below 660 km depth provide a general shift to lower harmonic degrees in both thermal anomalies and kinetic energy. With non-Newtonian rheology, narrow linear downwellings appear at the surface, and the upper boundary layer becomes very mobile. Thermal heterogeneities are greatly reduced throughout the shell. When non-Newtonian rheology is limited to the top 300 km as suggested by Karato and Wu (1993), the lateral flow increases because of the change in relative viscosity around downwelling regions at the transition depth. The downwelling network becomes more irregular and exhibits irregular plate-like characteristics. The planform of upwellings is also greatly affected by the large lateral flow and, in some case, shows narrow extended shapes. A significant increase in toroidal energy (toroidal-poloidal rms velocity ratio is 19-35% compared to less than 10% for Newtonian viscosity cases) is obtained with this hybrid rheology. The knowledge acquired from the simulations is applied to various depth regions in the global seismic tomography model proposed by Su et al. (1994) with a geodynamical interpretation.