ALBERT

Notes: For incompressible viscous creeping flows that occur in a number of geophysical situations, the velocity field may be expressed as the curl of a vector potential function, the use of which allows the momentum equation to be written as a biharmonic equation. the 3-D Cartesian formulation for a constant viscosity fluid is summarized here with special reference to two important types of boundary condition: the stress-free boundary (with zero normal velocity and zero tangential stress) and the rigid boundary (with all components of velocity zero). Fast algorithms for inversion of the biharmonic operator with all boundaries stress-free are well established. There also exists a fast method for the solution of the biharmonic equation with a parallel pair of rigid boundaries with the other boundaries stress-free. This method has not previously been applied, but it is a relatively straightforward extension of the Fourier transform based algorithm for the stress-free problem, using an analytical solution to enforce the required boundary conditions for each horizontal harmonic component. the method is easily vectorized and allows solutions to be obtained that compare very favourably in accuracy and solution time with those for the stress-free problem. the errors are of comparable magnitude given the differing harmonic content required by the boundary conditions, and the solution requires between 20 per cent (for a 3-D problem) and 30 per cent (for a 2-D problem) more processing time than does the solution of a comparable stress-free problem.

Notes: A mantle plume is probably a complex 3-D thermal structure that possesses approximate axisymmetry as it approaches the base of the lithosphere from below, but followed down towards the base of the layer, probably consists of a triple-junction or quadruple-junction of connected hot sheets. A relatively weak hot sheet rising only part way through the layer probably connects two neighbouring mantle plumes. These conclusions are suggested by numerical experiments on a 3-D constant-viscosity, plane layer with stress-free boundaries, which detail the gradational change in the planform of a convecting layer from the top of the layer to its base. the planform of a convecting layer is a map in the horizontal plane of the principal thermal anomalies in the layer. These anomalies are the main sources of positive (for hot fluid) or negative (for cold fluid) buoyancy, and therefore they drive the convective flow. They may appear in cross-section as structures with either axial symmetry (columns), planar symmetry (sheets) or some complex asymmetric form. When convection is driven at least partially by basal heating, the planform near the top of the layer may be described as a network of cold sinking sheets and isolated hot columns, while near the base of the layer it appears as a network of hot rising sheets and isolated cold columns. the hot columns near the upper surface arise from the vertices or nodes of the network of hot sheets on the lower surface, and similarly the cold columns at the base of the layer form below the vertices of the network of cold sheets near the upper surface. Near the upper surface, the apparent planform of this experiment is analogous to that of mantle convection, the cold sheets compared to subduction zones and the hot columns compared to mantle plumes. the hot plumes impinging on the upper surface produce approximately axisymmetric temperature anomalies, surface uplift and extensional stress fields. However, the relatively minor deviations from axisymmetry of surface observables reflect the deep structure of the mantle plume, formed by the junction of three or four hot sheets on the base of the layer. It seems likely that the commonly occurring triple-junction form of continental rifts may reflect an underlying structure that is implicit in the convective circulation of the mantle beneath.