Nonlinear stability of intermediate baroclinic flow on a sloping bottom

Abstract

The governing equations describing the dynamics of mesoscale gravity currents or coupled density fronts and steadily-travelling coherent cold eddies on a sloping bottom are shown to possess a non-canonical hamiltonian structure. We exploit the hamiltonian formalism to obtain a variational principle that describes arbitrary steady solutions in terms of a suitably constrained hamiltonian. Two Arnol’d-like stability theorems are obtained which can establish the linear stability in the sense of Liapunov of these steady solutions. Based on this analyses two a priori estimates are derived which bound the disturbance energy and the Liapunov norm with respect to the initial disturbance potential enstrophy and energy. In the limit of parallel shear flow solutions corresponding to a current flowing along isobaths, the first formal stability theorem reduces to a previously established normal-mode stability result. Based on the formal stability analysis, convexity conditions are given for the constrained hamiltonian that can rigorously establish nonlinear stability in the sense of Liapunov for the steady current solutions. A variational principle is also presented which can describe steadily-travelling isolated cold eddy solutions of the model. The principle is based on constraining the hamiltonian with appropriately chosen Casimir and momentum invariants. It is shown that a suitably extended form of Andrews’ theorem holds for our model equations. Therefore, the stability of the steadily travelling isolated eddy solutions cannot be established using the energy-Casimir analysis developed here.