Zonal Shear Flows with a Free Surface: Hamiltonian Formulation and Linear and Nonlinear Stability

General theoretical results via a Hamiltonian formulation are developed for zonal shear flows with the inclusion of the vortex stretching effect of the deformed free surface. These results include a generalization of the Flierl-Stern-Whitehead zero angular momentum theorem} for localized nonlinear structures on or off the beta-plane, and sufficient conditions for linear and nonlinear stability in the Liapunov sense - the latter are derived via bounds on the equilibrium potential vorticity gradient.