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 (equivalent barotropic model). These results include a generalization of the Flierl–Stern–Whitehead zero angular momentum theorem for localized nonlinear structures (whether or not on a β-plane), and sufficient conditions for linear and nonlinear stability in the Liapunov sense–the latter are given as estimates in terms of an L ²-type perturbation norm which are global in time and are derived via bounds on the equilibrium potential vorticity gradient.