On a class of ocean model instabilities that may occur when applying small time steps, implicit methods, and low viscosities

The horizontal grid size in ocean models may be gradually reduced with increasing computer power. This allows simulations with less viscosity. The hyperbolic, free wave nature of the problem will become more dominant. However, as viscosities are reduced, instabilities may more easily occur. To avoid such instabilities, it is well known from the literature on numerical methods for ordinary differential equations that implicit methods have attractive stability properties. Therefore, it may be tempting to add implicitness into the time stepping also in low viscosity ocean modelling to improve the robustness. However, when using methods with implicit features and low viscosity, it may happen that models are stable for longer time steps, but become unstable as the time step is reduced. In the present paper, an explanation for this phenomenon is given. It is shown that the explanation may be valid when solving the shallow water equations on a C-grid.