A grid-free vortex method for the quasi-geostrophic shallow-water dynamics on the sphere

Abstract

Navier-Stokes equations are able to explain the dynamics of fluids at all scales. However, small-scale phenomena control the time and space resolution of the problem, making thus impossible to capture the slowly varying large-scale phenomena. Therefore, scale separation becomes a need for the study of large-scale oceanic and atmospheric dynamics. The quasi-geostrophic (QG) equations, derived from Navier-Stokes equations by means of systematic scaling, offer a way to study fluid dynamics at the planetary scale by filtering out fast waves that manifest themselves at the small scale. Their unique feature is that they reduce the dynamic equations to a single prognostic equation by means of the geostrophic relationship. In this study, we aim to describe a numerical implementation of the ‘grid-free’ vortex method to solve the quasi-geostrophic shallow-water equations, and test the method's ability to simulate fundamental geophysical phenomena. The method appears to be an attractive way for solving the problem. The stability of the method is afforded by the Lagrangian advection of particles. Moreover, conservation of properties carried by particles can be easily expressed in the numerical procedure.