Lateral heterogeneity and the geoid: the importance of the surface kinematic constraints.

We apply the Green function technique to describe the internal load induced deformation of 3-D, self-gravitating, spherical shells of incompressible Newtonian fluid consisting either of a single constant-viscosity shell or of 2 adjacent shells of different viscosity. Using this method we derive closed form, analytic expressions for the kernel functions connecting the internal lateral heterogeneity of density to the horizontal divergence of the surface flow and to the non-hydrostatic geoid. Although the geoid constrains only the ratio of the upper and lower mantle viscosities, the horizontal divergence field constrains their absolute values. We find that both surface divergence and geoid fields are best fit with only a factor of 8 viscosity increase at 1200km depth. We point out, however, that the coupling of poloidal and toroidal flow in the Earth's mantle, which is required to understand surface velocity spectra, will most probably allow us to reduce the viscosity increase required by the geoid data.-Authors