Remarks on the planetary geostrophic model of gyre scale ocean circulation

Abstract

We study in this article the mathematical formulation of the planetary geostrophic (PG) equations of large-scale ocean circulation, in the case where small-scale processes are parameterized by the traditional Laplacian eddy diffusion and eddy viscosity. We prove the existence and uniqueness of global in time strong solutions of these equations with either $L^\infty$ or $H^2$ initial data. Due essentially to the high nonlinearity (comparable to a squared gradient) of the equations, two problems remain open. First, the existence of more regular solutions with $L^\infty \cap H^1$ initial data is still unknown, although more regular solutions are obtained with $H^2$ initial data. Second, the existence of global attractor and its dimension estimates are open, and related to that are the time uniform boundedness of the norm in $H^2$ and higher order Sobolev spaces of the solutions.