The Rotating Shallow-Water Equations (Particle Methods)
This is joint work with Jason Frank from the CWI in Amsterdam on numerical methods for geophysical fluids. We are in particular interested in balanced motion and the advection of potential vorticity.
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