The Rotating Shallow-Water Equations

The shallow water equations in a frame of reference rotating with angular velocity f0/2 are:

Du/Dt = f0 v - g0 hx
Dv/Dt = -f0 u - g0 hy
Dh/Dt = -(H+h)(ux + vy),

where g0 is a gravitational parameter, (u,v) represents the velocity field, and h is the layer depth perturbation from a mean value of H. The shallow water equations provide a very simple dynamical model of the atmosphere.

A very important quantity in atmospheric dynamics is the potential vorticity, defined as

q = (vx - uy + f0)/(H + h),

which is conserved along fluid particle paths:

Dq/Dt = 0.

The focus of our research is the preservation of the above property in a computer simulation of the shallow water equations. We set f0 = 2pi, g0 = 4pi^2, H = 1$ and consider a double-periodic domain [-pi,+pi] x [-pi,+pi]. The equations are simulated over a period of 30 days using initial conditions that are in almost geostrophic balance. The method used is a particle-mesh method. The layer-depth is advected along Lagrangian particles using radial basis functions to solve the associated continuity equations. The overall method is Hamiltonian and respects a circulation theorem. To keep the solutions smooth the layer-depth is smoothed over the grid using an inverse Helmholtz operator. Below we report simulation results for the simulation of the SWE as described above and from a number of extensions (multi-layers, spherical geometry, non-hydrostatic vertical slice model):

barotropic instability simulation


two layer baroclinic instability simulation


simulation results for the SWE on the sphere


flow over a mountain in a non-hydrostatic vertical slice model

The simulations were performed using MATLAB and C (mex) programs for some of the subroutines. The code for the barotropic instability simulation can be downloaded from Jason's CWI web-page.