A Three-Dimensional Version of the Free Surface Capturing Discretization for Staggered Grid Finite Difference Schemes: Implementation into StagYY

<p>In order to treat a free surface in models of lithosphere and mantle dynamics that use a fixed Eulerian grid it is typical to use "sticky air", a layer of low-viscosity, low-density material above the solid surface (e.g. Crameri et al., 2012). This can, however, cause numerical problems, including poor solver convergence due to the huge viscosity jump and small time-steps due to high velocities in the air. Additionally, it is not completely realistic because the assumed viscosity of the air layer is typically similar to that of rock in the asthenosphere so the surface is not stress free.  </p><p>In order to overcome these problems, Duretz et al. (2016) introduced and tested a method for treating the free surface that instead detects and applies special conditions at the free surface. This avoids the huge viscosity jump and having to solve for velocities in the air. They applied it to a two-dimensional staggered grid finite difference / finite volume scheme, a discretization that is in common use for modelling mantle and lithosphere dynamics. Here I document the application of this approach to a three-dimensional spherical staggered grid solver in the mantle simulation code StagYY. Some adjustments had to be made to the two-dimensional scheme documented in Duretz et al. (2016) in order to avoid problems due to undefined velocities for certain boundary topographies. The approach was applied not only to the Stokes solver but also to the temperature solver, including the implementation of a mixed radiative/conductive boundary condition applicable to surface magma oceans/lakes.</p><p><strong>References</strong></p><p>Crameri, F., H. Schmeling, G. J. Golabek, T. Duretz, R. Orendt, S. J. H. Buiter, D. A. May, B. J. P. Kaus, T. V. Gerya, and P. J. Tackley (2012), A comparison of numerical surface topography calculations in geodynamic modelling: an evaluation of the ‘sticky air’ method, Geophysical Journal International,189(1), 38-54, doi:10.1111/j.1365-246X.2012.05388.x.</p><p>Duretz, T., D. A. May, and P. Yamato (2016), A free surface capturing discretization for the staggered grid finite difference scheme, Geophysical Journal International, 204(3), 1518-1530, doi:10.1093/gji/ggv526.</p>

Comments on ‘An improved free surface capturing method based on Cartesian cut cell mesh for water-entry and -exit problems’

In the paper entitled ‘An improved free surface capturing method based on Cartesian cut cell mesh for water-entry and -exit problems’ ( Wang & Wang 2009 Proc. R. Soc. A 465 , 1843–1868 ( doi:10.1098/rspa.2008.0410 )), the present authors' earlier work ( Qian et al . 2006 , Proc. R. Soc. A 462 , 21–42 ( doi:10.1098/rspa.2005.1528 )) has been specifically applied to the study of water-entry and -exit of solid objects. An extended boundary condition, retaining the term owing to acceleration of moving boundaries in the momentum equation, has been implemented for calculating the pressure gradient at solid surfaces and, based on their numerical experiments, it was concluded by Wang and Wang that without this term the calculation will substantially under-predict the impact forces and may even break down. Therefore, a more complex procedure based on the exact solution of a Riemann problem for moving boundaries was implemented. In this short comment, by applying the authors' free surface capturing code to the same flow problem of water-entry of a wedge, it can, however, be demonstrated that the results from implementing the new pressure boundary condition are nearly identical to that of employing the original boundary condition without the acceleration term, indicating that its effects on the simulation results are minimal. A further examination of the implementation details on the pressure boundary condition also supports this conclusion.