Free surface calculations in mantle convection

Abstract:

Geodynamic simulations increasingly rely on simulations with a true free surface to investigate questions of dynamic topography, tectonic deformation, gravity perturbations, and global mantle convection. However, implementations of free surface boundary conditions have proven challenging from a standpoint of accuracy, robustness, and stability. In particular, free surfaces tend to suffer from sloshing instabilities, also known as the "drunken sailor" instability, which severely limit time step sizes. Several schemes have been proposed in the literature to deal with these instabilities.

Here we analyze the problem of creeping viscous flow with a free surface and discuss the origin of these instabilities. We demonstrate their cause and how existing stabilization schemes work to damp them out. Our analysis is based on formulating a generalized eigenvalue problem for the relaxation spectra of the linearized free surface problem.

We also propose a new scheme for removing instabilites from free surface calculations. It does not require modifications to the system matrix, nor additional variables, but is instead an explicit scheme based on nonstandard finite differences. It relies on a single stabilization parameter which may be identified with the smallest relaxation timescale of the free surface.

We analyze the stability and accuracy of the nonstandard finite difference scheme, and describe its implementation in the open source mantle convection software Aspect. We also provide comparisons between the nonstandard finite difference scheme and the quasi-implicit scheme proposed by Kaus, Muhlhaus, and May (2010).