A co-volume scheme for the rotating shallow water equations on conforming non-orthogonal grids

Abstract

A co-volume scheme is introduced for the rotating shallow water equations, in which both velocity components are specified on cell edges, and the thickness variables evolve on both the primary and the dual cell centers. The scheme applies to generic, conforming and non-orthogonal staggered grids, including the widely used lat–lon quadrilateral grids and the Delaunay–Voronoi tessellations. It can be viewed either as coupled C-grid schemes on the primary and dual meshes, or as an generalization of the traditional E-grid scheme on a new non-overlapping grid. Linear dispersive wave analysis shows that, the dispersive relations resolved by either the primary or the dual mesh of a uniform quadrilateral staggered grid is the same as those of the Z-grid scheme. The total wavenumber space resolved by the staggered grid is twice as large, on which the co-volume behaves exactly like the E-grid scheme. On a uniform hexagon-triangular staggered grid, the co-volume has two steady modes and two inertial-gravity modes on the hexagonal mesh, and one steady mode, two inertial gravity modes, and two spurious modes on the triangular mesh. On the wavenumber space resolved by either the hexagonal or the triangular mesh, the inertial-gravity wave modes remain positive and largely monotone. For the nonlinear shallow water equations, the co-volume scheme is shown to preserve the potential vorticity dynamics and the total energy exactly. Numerical results are presented to corroborate and supplement the analyses.