A “vortex in cell” model for quasi-geostrophic, shallow water dynamics on the sphere

The vortex in cell (VIC) method has been found to be an attractive computational choice for solving a number of fluid dynamical problems. The Lagrangian advection of particles leads to stability of the method even for long time steps. Moreover, conservation of particle properties, such as potential vorticity, can be enshrined at the heart of the numerical procedure. In this paper we describe a numerical implementation of the VIC method for a reduced gravity, quasi-geostrophic model and make use of its novel aspects to explore the interaction of waves and turbulence on the sphere. The Lagrangian advection of particles is performed using a fourth order Runge–Kutta method and the stream-function is obtained by the inversion of potential vorticity using an underlying Eulerian grid. The scheme is tested in the simulation of Rossby and Rossby–Haurwitz waves. Encouraging results are obtained for various radii of deformations corresponding to both the atmosphere and ocean.