Stability analysis of split-explicit free surface ocean models : implication of the depth-independent barotropic mode approximation

The evolution of the oceanic free-surface is responsible for the propagation of fast surface gravity waves, which approximatively propagate at speed root gH (with g the gravity and H the local water depth). In the deep ocean, this phase speed is roughly two orders of magnitude faster than the fastest internal gravity waves. The very strong stability constraint imposed by those fast surface waves on the time-step of numerical models is handled using a mode splitting between slow (internal/baroclinic) and fast (external/barotropic) motions to allow the possibility to adopt specific numerical treatments in each component. The barotropic mode is traditionally approximated by the vertically integrated flow because it has only slight vertical variations. However the implications of this assumption on the stability of the splitting are not well documented. In this paper, we describe a stability analysis of the mode splitting technique based on an eigenvector decomposition using the true (depth-dependent) barotropic mode. This allows us to quantify the amount of dissipation required to stabilize the approximative splitting. We show that, to achieve stable integrations, the dissipation usually applied through averaging filters can be drastically reduced when incorporated at the level of the barotropic time stepping. The benefits are illustrated by numerical experiments. In addition, the formulation of a new mode splitting algorithm using the depth-dependent barotropic mode is introduced