Free-surface formulation of mantle convection-I. Basic theory and application to plumes

In mantle convection models, the top surface is traditionally approximated as a freeslip boundary, and the dynamic topography is obtained by assuming that the normal stress on the free-slip boundary is compensated instantaneously through surface deformation. It has already been shown that this approximation is valid for longwavelength topography. Based on both viscous and viscoelastic models with a free surface, we have found that the characteristic time for topographic growth is comparable to the timescales of mantle convection (lo6 year) for short and intermediate wavelengths ( lo3 km or less) and/or a high effective lithospheric viscosity (> Pa s). This suggests that the topography is history-dependent under these conditions and that a free-surface formulation is required to study the topography at these wavelengths. We have developed a new Eulerian finite-element technique to model a free surface. Since the technique is based upon an undeformable Eulerian grid, this enables us to study long-term, free-surface dynamics in the presence of evolving buoyancy. We have compared numerical with analytic solutions of viscous relaxation for fixed buoyancy problems. As long as the magnitude of topography is much smaller than the wavelength, we find that the finite-element method is very accurate, with relative errors of less than 1 per cent. This numerical technique can be applied to a variety of geophysical problems with free surfaces. In applying this technique to dynamic models of mantle plumes, we find that surface relaxation retards the topography at intermediate and short wavelengths and produces a smoother topography, compared with topography from free-slip calculations. This reduced topography has a significant influence on the geoid at the corresponding wavelengths. Moreover, free-surface models, by allowing vertical motion on the free surface, yield a hotter lithosphere over ascending plumes than models with free-slip boundaries.