Free Surface Formulation of Mantle Convection, Part 1: basic theory and implication to plumes

Shijie Zhong, Michael Gurnis, and Louis Moresi

Seismological Laboratory, 252-21, California Institute of Technology, Pasadena, CA 91125

In mantle convection models, the top surface is traditionally approximated as a free slip boundary and dynamic topography is obtained by assuming that the normal stress on the free slip boundary is compensated through surface deformation. Previously, it has been shown that this approximation is valid for long-wavelength topography. Based on both viscous and visco-elastic models with a free surface, we have found that the characteristic time for topographic growth is comparable to the time scales of mantle convection (10^6 year) for short and intermediate wavelengths (10^3 km or less) and/or a high effective lithospheric viscosity (>10^24 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. Compared with topography from free slip calculations, dynamic models of mantle plumes with free surface boundaries show that surface relaxation retards topography at intermediate and short wavelengths and produces a smoother topography. 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 rising plumes than models with free slip boundaries. In modeling a free surface, we use an Eulerian finite element formulation in which the mesh is not deformed; 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 less than 1%. This numerical technique can be applied to a variety of geophysical problems with free surfaces.