Wave activity and baroclinic stability theorems for a two-dimensional stratified semi-geostrophic model

Abstract

Arnold's method is used to derive two finite-amplitude wave-activity invariants – pseudo-momentum and pseudo-energy – based on the Hamiltonian structure for the two-dimensional baroclinic semi-geostrophic (SG) model in physical coordinates. The corresponding Liapunov stability theorems are established. For small-amplitude disturbances the pseudo-momentum and pseudo-energy can be greatly simplified by introducing the quasi-potential-vorticity perturbation. The corresponding linear stability conditions, expressed explicitly in terms of the basic state, are obtained. The lateral boundary conditions are found to have a big impact on the stability of SG dynamics in both linear and nonlinear cases. The results show that nonlinear stability criteria are more complicated than linear stability criteria, in that they contain more conditions in the interior resulting from the inclusion of potential temperature and the nonlinearity of SG potential vorticity in the wave activity. It is also found that the nonlinear stability criteria for the SG model are very different from those for the quasi-geostrophic (QG) model. The failure to extend the Liapunov stability theorems to the three-dimensional SG model is discussed. It is found that this failure results from the third-order nonlinear terms in the three-dimensional SG potential vorticity. Since the variation of these terms in the wave activity is not sign-definite, nonlinear stability conditions cannot be obtained. Copyright © 2011 Royal Meteorological Society