The destruction of geochemical heterogeneities by differential fluid motions during mantle convection

A two-dimensional numerical method is presented which permits the observation of deformation within convecting fluids at high Prandtl number. This method is applied to steady state convection, from which it is concluded that deformation is achieved by simple shear, when considered over long time-scales. In numerical examples over a wide range of Rayleigh numbers and heat sources this shear is shown usually to be prograde, with the periphery of the convection cell rotating slower than the core. Occasionally retrograde shear occurs, in which the reverse is true. When the method is applied to time-dependent convection, the deformation of discrete bodies occurs by lateral eddy diffusion of mass, and exponential increase of surface area. The rates of these two processes are coupled and vary as Ra0.5 over a wide range of Rayleigh numbers. It is shown that an eddy diffusive approximation is only appropriate for horizontal scales greater than about 3 times the depth of a convecting layer and for time-scales of at least 300 Myr in the upper mantle. The effects of variable viscosity and three-dimensionality are discussed, and the results of the experiments applied to the Earth's mantle. It is concluded that any convecting layer within the mantle must be well mixed on a lateral scale of at least 2000 km after a time of 0.5-1 Gyr, depending on the processes of magma extraction from a streaky parental material. The deformation rate depends only upon the layer depth and the fluid viscosity. Hence whole mantle convection is 5 times more effective in dispersing geochemical heterogeneities than is convection confined to the upper mantle. Various models for the spatial distribution of geochemical reservoirs are discussed and the only viable models are those in which geochemical reservoirs are identified with separate fluid layers. Whole mantle convection is not favoured by these conclusions. On other grounds, some layered models are discarded and we support a conventional model with separate convection in the upper and lower mantle with a boundary at about 700 km depth. Isotopic anomalies erupted at the surface must be relatively recent additions to the otherwise homogeneous upper mantle.