Upwind residual distribution for shallow-water ocean modelling

This article describes residual distribution for the rotating shallow water equations arising in oceanographic and meteorological modelling. The method is similar to continuous/discontinuous finite elements in that it is well suited for unstructured, locally refined meshes – therefore promising to be a viable alternative to more traditional methods for shallow-water ocean modelling. It has, however, two main advantages over finite-element methods. First, it creates a framework in which nonlinear dynamics can be represented very naturally. Second, by combining the treatment of the flux and source terms, it makes the preservation of certain balance properties – especially hydrostatic balance – easier to guarantee. The methods considered in this article have been previously shown to preserve many of the important physical properties of the original equations, such as conservation, oscillation-free behaviour and the exact preservation of hydrostatic balance. This work is intended as the first step into investigating the method’s suitability for modelling geophysical fluids. This is done through a number of carefully-chosen test cases, which include both f0f0-plane and ββ-plane approximations as well as non-flat bottom topography.

► We apply the residual distribution to the rotating shallow-water equations. ► Both nonlinear steady-state and nonlinear time-dependent test examples are used. ► The schemes are in exact hydrostatic balance. ► The time-dependent scheme is unconditionally stable and locally positive. ► The schemes are shown not to suffer from spurious numerical artefacts.