Free surface methods applied to global scale numerical geodynamic models

The study of coupled Earth systems, and in particular the coupled interactions between the lithosphere, atmosphere, and biosphere, have received greater attention in recent years (Gerya et al. 2020). Interactions between these systems occur primarily at the surface, and are driven on the large scale by topographic and bathymetric evolution controlled by deep mantle processes. However, due to the large difference in length scales between the mantle and the surface, it is difficult to capture topographic evolution to a high degree of accuracy in existing global mantle convection models including a free surface boundary condition.

Global mantle convection models incorporating a free surface often employ a marker-in-cell technique with a layer of "sticky air" (i.e. material with the density of the air and sufficiently low viscosity, which is still much higher than that of real air) to characterise the surface. However, accurate topographic evolution using this method requires a high density of markers near the surface. This need for additional computational resources motivates alternative methods of tracking the interface between the air and rock layers, as is done frequently in existing multiphase fluid flow codes. Such interface tracking methods include the use of a Lagrangian surface marker chain with an appropriate remeshing procedure (van Keken, 1997), the use of a level-set function defined by the distance to the surface (Hieber & Koumoutsakos, 2005), or the introduction of a volume of fluid method with piecewise-linear interface reconstruction and a geometric advection routine (Katopodes, 2019).

We demonstrate toy models of these methods implemented in the Julia programming language, as well as benchmarks of a preliminary implementation within the existing advanced mantle convection code StagYY (Tackley, 2008). Models of global scale topography and evolution produced using StagYY may later be used as a tool for further studies on the coupling of mantle dynamics with modelling of the landscape, and the evolution of the atmosphere and biosphere.