An implicit free surface algorithm for geodynamical simulations

Identifying the dominant controls on Earth's surface topography is of critical importance to understanding both the short- and long-term evolution of geological processes and past- and present-day dynamics of Earth's coupled mantle-lithosphere system. The ability to simulate a stress free - or a so-called 'free surface' - boundary condition is required to examine such processes via numerical models. However, at present, geodynamical models incorporating a free surface are limited, as most underlying free surface algorithms place severe restrictions on the computational timestep. Consequently, the simulations are often intractable. In this study, we introduce a new approach for incorporating a free surface within geodynamical models: an algorithm, in which free surface elevation is treated as an independent variable and is solved for in conjunction with the momentum and continuity equation, using implicit time integration. We demonstrate that the method is straightforward to implement in existing models and, using a series of analytical and benchmark comparisons, we show that it does not suffer from the timestep constraints of previous schemes. Furthermore, the scheme can be made second order accurate in time, at no additional cost. The method therefore dramatically improves the computational efficiency of geodynamical simulations including a free surface, whilst maintaining solution accuracy.