Mathematical Theory of the Barotropic Model in Geophysical Fluid Dynamics

The two-dimensional barotropic model describes the motion of an ideal two- dimensional incompressible fluid on a uniformly rotating sphere. In this dissertation, we study the Hamiltonian structures and Lie-Poisson reduction for this model. Some classes of steady and traveling wave solutions, including modons, Liouville flows and discontinuous zonal vortices, are presented and how they are connected to the zonal flows are discussed. Sufficient conditions for the stability for these flows are established by a method (developed by Arnold, Smale, Marsden,..., etc) called the energy-momentum method. More specifically, the following results, which we believe to be new in the literature, are found: (1) Euler equation for fluids on a nonuniformly rotating sphere. (2) An exact nonlinear stability analysis for traveling wave solutions. (3) The existence of Liouville flows. (4) Stability analysis for a class of steady discontinuous zonal flows.