Understanding the effects of mantle compressibility on geoid kernels

We develop a quasi-analytical solution to compute geoid kernels for a compressible mantle with Newtonian rheology. By separating the stresses induced by self-gravitation from the stresses resulting from viscous flow, we simplify the equations and gain some insight. For realistic variations in the background density field ρ_o(r), the solution, obtained using propagator matrices, converges rapidly. Compressibility enters into the flow problem directly, through the continuity equation, and indirectly, by influencing parameters such as gravitational acceleration g(r) and density contrasts across compositional boundaries. In order to understand all these effects, we introduce them sequentially, starting with an incompressible earth model and ending up with a realistic compressible model that includes a compressible inner and outer core, phase changes in the transition zone, and an ocean. The largest effects on geoid kernels are from different assumptions for g(r); possible effects of transformational superplasticity and differences in assumptions for density contrasts at the surface and at the core-mantle boundary are next in importance. The effects of compressibility on the flow itself are somewhat smaller, followed by the effect of compressibility of the outer core. A gravitationally consistent treatment of the ocean layer yields geoid kernels that are very similar to those for a 'dry' planet. The compressibility of the inner core has a negligible impact on the geoid kernels. The largest effects from compressibility are comparable to the effects of a moderate (40 per cent) change in the viscosity contrast between the upper and lower mantle.