The Nonlinear Quasi-Geostrophic Equation: Existence and Uniqueness of Solutions on a Bounded Domain

Abstract

The quasi-geostrophic theory leads to a single nonlinear partial differential equation for a streamfunction giving geostrophic velocity fields presumed to resemble the synoptic scales of atmospheric motion. This article is concerned with demonstrating that the quasi-geostrophic problem is well-posed mathematically, in the sense that solutions exist, and that they are continuously dependent on the initial data. The model studied is comprised of the quasi-geostrophic equation subject to the severe boundary condition that an isentrope coincides with the earth's surface. The main technique is the use of the eigenfunctions of an elliptic operator appearing within the quasi-geostrophic equation. These eigenfunctions provide the basis for a spectral model, which can be truncated to include a finite number of scales. The convergence properties of the solutions to the truncated model allow the existence of solutions to the entire model to be inferred with the methods of functional analysis. Thus, the conclusions reached are relative to generalized solutions and to the usual norm in the Hilbert space of quadratically integrable functions. The results have applications to the study of atmospheric predictability.