Stability and accuracy of Runge-Kutta-based split-explicit time-stepping algorithms for free-surface ocean models

Most ocean model time integration algorithms rely on mode splitting techniques to deal with the large difference in timescales between the fast surface waves and the slower dynamics of the inner ocean. With these techniques, the surface wave dynamics are considered as independent of the vertical, are modelled as 2D, and are treated with a dedicated time-stepping scheme, often different from that used for the interior dynamics, which are fully 3D. However, such a separation is inaccurate because the surface wave dynamics are not strictly depth independent, due to stratification effects. As a result of the splitting errors, instabilities arise which are usually kept under control by some form of dissipation applied to the 2D dynamics.
In this talk we present an analysis of the stability and accuracy of time integration algorithms based on Runge-Kutta schemes for the 3D part and explicit integration for the 2D part (i.e. the so-called split-explicit approach). We investigate how the viability of the algorithms, measured by the amount of dissipation required to stabilise them, varies with different ways of articulating the 2D/3D coupling within Runge-Kutta schemes. The analysis is based on the theoretical framework developed by Demange et al. (2019), which is based on an eigenvector decomposition using the true depth-dependent barotropic mode. The analysis allows us to identify the most promising algorithms, whose performance is verified with idealised linear and non-linear numerical experiments.