Zonal shear flows with a free surface: Hamiltonian formulation and linear and nonlinear stability

Abstract

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.

Keywords:

• Zonal shear flows
• Stability
• Hamiltonian formulation

Acknowledgments

Our thanks are due to Dr Leon Kamp for helpful discussions and the referees for useful criticism. The authors are thankful to a referee for pointing out Nycander (1988) and Ripa (1992 and 1993). This work has been supported by the J.M. Burgers Centre.