Useful time-stepping methods for the Coriolis term in a shallow water model

The attempt to simultaneously optimize stability, accuracy, and efficiency in an ocean model leads to a wide range of methods that are potentially useful. For some models, a major issue is the efficient integration of the Coriolis term when the underlying numerical model uses a semi-implicit time integration. Existing numerical models treat this integration with a variety of methods including explicit Adams–Bashforth schemes and implicit schemes. The semi-implicit approach is useful in that it provides a method to remove restrictive stability constraints and mode splitting errors. Published literature would suggest that many if not most standard explicit methods for the Coriolis term are unstable. On the other hand, implicit integration of the Coriolis term is very inefficient for staggered grid models with normal velocity degrees of freedom as it leads to inversion of a large, sparse velocity matrix. Our purpose is to explore and compare a variety of explicit or essentially explicit treatments of the Coriolis term. An analysis of the discretized shallow water equations indicates that some of the explicit methods are stable under somewhat relaxed conditions. Of these, the third-order Adams–Bashforth and the FBT scheme show good behavior. Numerical examples of coastal and deep ocean simulations illustrate the findings.