Lagrangian tracking methods applied to free surface boundaries in numerical geodynamic models

A desirable characteristic of mantle convection models is the ability to determine surface topography evolution over time on a global scale. This capability is increasingly important due to growing interest in coupling planetary geodynamics with climate, landscape, habitats, and biological evolution systems. A common way of achieving this in numerical geodynamic models with a fixed Eulerian grid is through the implementation of a free-surface boundary condition using the sticky air method in conjunction with an appropriate method of tracking the free surface. Although existing methods for tracking the interface between the air and rock layers are available, they often struggle to provide high-resolution results on a global scale without incurring significant computational costs. We propose a method for representing surfaces directly using Lagrangian markers that can track the location of a free surface in 2D and 3D models and test it using the finite-volume mantle convection code StagYY. This approach offers a direct, high-resolution representation of the surface without the need for a large number of high-cost volumetric markers throughout the model domain. Benchmarks demonstrate the effectiveness of this method compared to the commonly-used marker-in-cell method. The direct representation of the surface enables additional features such as the direct tracking of sea levels over time, potential for coupling with surface process models on the global scale, and enables the implementation of alternative discretisations of the Stokes equations to improve Stokes solver accuracy near the free surface boundary.