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

The amplitude equations governing the nonlinear interaction among normal modes are derived for a multilayer quasi-geostrophic channel. The set of normal modes can represent any wavy disturbance to a parallel shear flow, which may be stable or unstable. Orthogonality in the sense of pseudomomentum or pseudoenergy is used to obtain the amplitude equations in a direct fashion, and pseudoenergy and pseudomomentum conservation 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 basic 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 detail. A particular formulation of the stability condition is given that emphasizes the close connection between linear and nonlinear stability. It is established that this stability condition is also a necessary condition: when it is not satisfied, and when the flow is linearly stable, explosive resonant interaction of baroclinic Rossby waves acts as a destabilizing mechanism. Two- and three-layer models are specifically considered; their stability features are presented in the form of stability diagrams, and interaction coefficients are calculated in particular cases.