EXPLOSIVE RESONANT INTERACTION OF BAROCLINIC ROSSBY WAVES AND STABILITY OF MULTILAYER QUASI-GEOSTROPHIC FLOW

The amplitude equations governing the nonlinear interaction among norm al modes are derived for a multilayer quasi-geostrophic channel. The s et of normal modes can represent any wavy disturbance to a parallel sh ear flow, which may be stable or unstable. Orthogonality in the sense of pseudomomentum or pseudoenergy is used to obtain the amplitude equa tions in a direct fashion, and pseudoenergy and pseudomomentum conserv ation laws permit the properties of the interaction coefficients to be deduced. Particular attention is paid to triads exhibiting explosive resonant interaction, as they lead to nonlinear instability of the bas ic flow. The relationship between this mechanism and the most recently discovered nonlinear stability conditions is discussed. Situations in which the basic velocity is constant in each layer are treated in det ail. A particular formulation of the stability condition is given that emphasizes the close connection between linear and nonlinear stabilit y. It is established that this stability condition is also a necessary condition: when it is not satisfied, and when the flow is linearly st able, explosive resonant interaction of baroclinic Rossby waves acts a s a destabilizing mechanism. Two- and three-layer models are specifica lly considered; their stability features are presented in the form of stability diagrams, and interaction coefficients are calculated in par ticular cases.