3-D Convection With Variable Viscosity

Abstract

Accepted 1990 September 6. Received 1990 July 16; in original form 1990 May 3 SUMMARY report numerical calculations for 3-D convection with variable viscosity. A hybrid spectral and finite difference method is used. The coupling of modes in the equation of motion, which is caused by lateral viscosity variations, is treated iteratively. Solutions for bimodal, hexagonal, square, triangular and spoke patterns are reported for bottom heated convection at infinite Prandtl number. The Rayleigh number, based on the viscosity at the mean of top and bottom temperature, is between critical and lo5, and temperature-induced viscosity contrasts up to 100 are considered (lo00 in one case). In agreement with results from laboratory experiments we find that at low Rayleigh number temperature-dependent viscosity favours flow patterns like squares or hexagons, where a columnar rising current is surrounded by sheet-like descending flow. The dichotomy in geometry between upwelling and sinking flow becomes more pronounced with increasing viscosity contrast. The temperature dependence of viscosity gives rise to a toroidal velocity component; however, it amounts only to a few per cent of the total velocity. In contrast, at the earth’s surface an approximate equipartitioning of poloidal and toroidal energy is found. We show that with non-Newtonian and depth-dependent rheology the toroidal component at the free surface can become significant, and a pattern reminiscent of plate motion can arise in a free convection model. Although these results are obtained in a parameter range which is not directly applicable to the earth, they support the conclusions that (i) upwelling flow in the mantle is unlikely to be sheet-like and will probably be in the form of columnar plumes, and that (ii) the toroidal motion found at the earth’s surface is due to the highly non-linear rheology which leads to the existence of mobile surface plates and is not caused by viscosity variations related to lateral temperature contrasts deeper in the mantle.