On the baroclinic dynamics, hamiltonian formulation and general stability characteristics of density-driven surface currents and fronts over a sloping continental shelf

Abstract

A new theory is presented to describe the baroclinic dynamics of density-driven currents and fronts over a sloping continental shelf. The frontal dynamics is geostrophic to leading-order but not quasi-geostrophic since the dynamic frontal height is not small in comparison with the scale frontal thickness. The evolution of the underlying slope water is modelled quasigeostrophically and includes the influence of a background vorticity gradient due to the sloping bottom . The two layers are coupled together via baroclinic vortex-tube stretching associated with the perturbeddensity-driven current. The current dynamics includes the advection of mean flow vorticity . The model equations are obtained in a formal asymptotic expansion of there levant two-layer shallow-water equation sand boundary conditions. It is shown that the governing equations for the model can be put in to non-canonical hamiltonian form. A comprehensive analysis of the general linear and nonlinear stability characteristics of the governing equations is given. The normal mode problem associated with steady along-shore currents is studied and sufficient stability and necessary instability conditions are presented. It is shown that a zero in the frontal vorticity gradient is not needed for instability. Jump conditions for the perturbation frontal thickness are systematically derived associated with the continuity of pressure and normal mass flux for steady frontal configurations that possess discontinuities in the velocity or vorticity, and rigorous regularity conditions are obtained for the perturbation thickness on outcroppings. The formal stability of arbitrary steady currents is studied. It is shown how to obtain general steady current solutions as a variational solution to a suitably constrained hamiltonian. General criteria are obtained for establishing the linear stability of these steady density-driven currents in the sense of Liapunov. In the limit of steady parallel along-shore flow, the formal stability results reduce to the sufficient conditions found for the normal modes. Finally, the nonlinear stability of steady density-driven currents and fronts is studied. Based on the formal stability analysis, appropriate convexity hypothesis are found that rigorously establish nonlinear stability of steady currents in the sense of Liapunov, and establish nonlinear saturation bounds on the perturbation flow with respect to a potential enstrophy/energy norm .