The Cauchy Problem and the Continuous Limit for the Multilayer Model in Geophysical Fluid Dynamics

Abstract

We study a multilayer model in geophysical fluid dynamics that approximately governs the large-scale motions of the atmosphere or the ocean. The model consists of n two-dimensional Euler equations which represent the evolution of n layers of liquid. These equations are written using the potential vorticity. The potential vorticity in each layer is obtained from the velocity potential of the adjacent layers. We show that the Cauchy problem for this model is globally well-posed in time for smooth initial data. In the second part of the paper, we let the number of layers tend to infinity while their thickness tends to zero. We write the system as a suitable finite-element approximation of a continuous model and show the convergence of this approximation to the classical quasi-geostrophic model.