Forced two layer beta-plane quasi-geostrophic flow

Abstract

We consider a model of quasigeostrophic turbulence that has proven useful in theoretical studies of large scale heat transport and coherent structure formation in planetary atmospheres and oceans. The model consists of a coupled pair of hyperbolic PDE’s with a forcing which represents domain-scale thermal energy source. Although the use to which the model is typically put involves gathering information from very long numerical integrations, little of a rigorous nature is known about long-time properties of solutions to the equations. In the first part of my dissertation we define a notion of weak solution, and show using Galerkin methods the long-time existence and uniqueness of such solutions. In the second part we prove that the unique weak solution found in the first part produces, via the inverse Fourier transform, a classical solution for the system. Moreover, we prove that this solution is analytic in space and positive time.