1 A Hamiltonian Particle - Mesh Method forthe Rotating Shallow Water

A new particle-mesh method is proposed for the rotating shallow-water equations. The spatially truncated equations are Hamiltonian and satisfy a Kelvin circulation theorem. The generation of non-smooth components in the layer-depth is avoided by applying a smoothing operator similar to what has recently been discussed in the context of-Euler models. The interplay, in atmospheric ows, between high speed, divergence-dominated gravity waves and slowly advected vortical structures presents a challenge to numerical modelling. The quantity of principle interest, potential vorticity, is advected materially along particle paths, making particle methods an attractive option. However, to prevent the generation of spurious gravity waves, one must ensure that diierentiated ow eld variables remain smooth. Particle methods have the additional advantage of being Hamiltonian, with the well-known consequences for long time dynamics that follow from this. The application of artiicial smoothing operators can destroy this property, though, if not done carefully. A 2D model of the atmosphere which still retains the important dynamic interactions mentioned above is the rotating shallow-water equations (SWEs): d dt u = ?f 0 u ? ? c 0 r x ;