The Rotating Shallow-Water Equations (Semi-Lagrangian Methods)

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 focus of our research is the development of an alternative implementation of the semi-implicit semi-Lagrangian method widely used in the meteorology community. The basic idea consists in a semi-Lagrangian implementation of the staggered time-stepping method originally proposed for the Hamiltonian particle-mesh (HPM) method. It was found, however, that the regularization procedure, used in the HPM method, needs to be modified in order to achieve dynamic equivalence between the newly proposed staggered regularized semi-Lagrangian method and the conventional semi-implicit semi-Lagrangian method on the level of linearized equations. Recent work has led us to the investigation of nonlinear balance relations to further improve the regularization procedure. Keep looking at my list of publications for the latest developments!

The following setting is used for all simulations. The equations are scaled such that one time unit corresponds to one day. The temporal time-step is equivalent to dt = 20 minutes. The spatial grid-size dx = dy = 2 pi/N, N the number of grid points, corresponds to 60 km. The spatial domain is restricted to a double-periodic domain [0,2pi] x [0,2pi]. The mean fluid depth is set equal to H = 1. The gravitational constant g0 is chosen such that

0.5 dt (g0 H)^(1/2) ~ (10)^(1/2) dx,

which corresponds to a typical setting used by the UK MetOffice in their global model.

Two different values are chosen for the Coriolis parameter f0. The baroclinic instability simulations use a midlatitude value of f0 = 4pi/(2)^(1/2), while a smaller value of f0 = pi is used for the vortex pair simulations. Note also that the baroclinic instability runs with N = 128 grid points, while a lower resolution of N = 64 is applied to the vortex pair (implying that a unit length scale corresponds to different horizontal extensions in dimensional variables!).

The time-stepping is implemented as a two-time level (one step) method and is symmetric except for the semi-Lagrangian departure point calculations. No explicit viscosity is added. The continuity equations is solved in its logarithmic form (no mass conservation is enforced). Spatial differencing was performed using FFT (spectral method). The simulations were performed using MATLAB.

Below we report results from the following two sets of simulations:

vortex pair


barotropic instability