A new spherical shallow-water model: building in wave–vortex decomposition

A new approach to simulating spherical shallow-water flows is described and illustrated. The approach exploits the distinction between slow vortical motions and fast gravity waves at leading order. Contour advection is used for the vortical part, specifically for the potential vorticity. This allows an accurate representation of the vortical dynamics at modest grid resolution. New variables are used to represent the gravity-wave part, at leading order in Froude number. These variables represent the departure from geostrophic balance. Their use requires the solution of several spatially elliptic equations for the original variables (fluid depth and velocity). These equations come from the definitions of the new variables and the potential vorticity. They simplify to balance conditions for the vortical flow when the “wave” variables are neglected. Simulations of rotating turbulence at moderate to high Froude numbers demonstrate that these new variables help to preserve the underlying vortical flow in these flows. Moreover, they reduce the erroneous numerical generation of gravity waves arising from a misrepresentation of the underlying vortical flow.