Abstract

The dynamic topography is the part of the surface topography that is caused by mantle flow. For instance low density anomalies rising towards the surface may cause topographic heights. In modelling the convection of the mantle, not only the velocity and the temperature are therefor unknowns of the system, but
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also the shape of the surface. Physically, the surface of the Earth is a free surface, where the velocity and surface deformation are constrained by a zero normal velocity, zero shear stress and a constant normal stress. The implementation in numerical models is not trivial and hence the top surface is modelled with a free-slip condition where the normal velocity and shear stress are set to zero. The normal stress varies at this surface and is interpreted as the presence of topography. For time dependent problems, this method assumes that the time scale of the formation of topography is much smaller than the time scale of mantle convection, such that the topography is approximately instantaneously compensated. Numerical models are able to reproduce results of Rayleigh-Benard convection within errors of one percent, whereby the rate of convergence is fast and the convergence range is large.
If time scale of mantle convection and the formation of topography are of the same order of magnitude, the history dependence of the topography should be taken into account. The vertical velocity is then not constraint, however a normal stress is applied on the top surface which reduces this velocity. The normal stress is updated each time step or iteration by the resulting normal velocity. Although this method is physically more realistic, for stationary problems, the accuracy of the solution is finite, since the velocity field is not influenced by small-scale variations in the applied normal stress. Furthermore, the convergence range of this boundary conditions is smaller. If this condition is applied on both the top and bottom of a domain it is extremely sensitive to pressure oscillations, easily caused by numerical errors.
It is preferred to model the top surface as a free-surface. In this case two of the three boundary conditions constrain the velocity field and the third (implicitly) the shape of the surface. Experiments are done where the normal velocity constrains the shape of the surface (kinematic conditions). These models are not able to reproduce the steady state solutions of Rayleigh-Benard convection, since there is no mechanism that prevents the topography to reach a maximum value. In the modelling of Rayleigh-Benard convection with a free-surface boundary conditions, it is necessary to use a normal stress balance (the surface is constraint by the normal stress boundary conditions). This implies both a natural as well as an essential boundary condition of the velocity field. In the used SEPRAN code, these implementation do not exist at the moment and hence a fully numerical description of the free-surface is not yet possible.
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