A HIGH-ORDER FINITE-DIFFERENCE METHOD APPLIED TO LARGE RAYLEIGH NUMBER MANTLE CONVECTION

A variable-grid, high-order finite difference (FD) method is applied t o the modeling of mantle convection in both two- and three-dimensional geometries. The algorithm combines extreme simplicity in programming with a very high degree of accuracy. Memory requirements are low and g row almost linearly with the total number of grid points in three dime nsions, regardless of the increase in grid points in the vertical dire ction. Higher-order methods, such as eighth order, yield significantly better results than a second-order method for the same grid size, wit h only a modest increase in memory requirements. This is particularly important for high Rayleigh number convection, where the large number of grid points required to obtain an accurate enough solution with sec ond-order schemes would make the computation extremely costly. The sma ll-scale Features in the hard-turbulent regime under high-Rayleigh num ber situations can greatly stress low-order methods, and in these situ ations a high-order method is definitely needed. We have numerically s imulated three-dimensional time-dependent convection for constant prop erties up to Ra = 10(8), using an eighth-order FD scheme. Both purely base-heated and partially internally heated situations have been consi dered. The hot plumes are broader near the surface with internal heati ng. Detailed studies of the three-dimensional constant viscosity plume s indicate that no small-scale circulation takes place in the ascendin g plume heads regardless of the heating configuration in accordance wi th predictions from boundary layer theory.