ON THE NONLINEAR-INTERACTIONS OF GEOPHYSICAL WAVES IN SHEAR FLOWS

The nonlinear interactions between waves propagating in sheared basic flows are studied in an Eulerian framework using an expansion of the nonlinear motion equations in the normal modes of the linearized system. The orthogonality of the normal modes in the sense of pseudomomentum or pseudoenergy provides the necessary relations to deduce the interaction coefficients, and naturally relates the amplitude equations found with the system's conservation laws. Conservation of pseudomomentum and pseudoenergy leads to relations between the interaction coefficients inside a triad. These relations generally differ from those previously found for basic slates at rest. However, they are the same for resonant triads and therefore Hasselmann's criterion for wave instability through resonant interaction can be extended to shear flows. Three geophysical systems are considered within an unique formalism: barotropic Rossby waves on a beta-plane, Rossby-Haurwitz waves on a sphere, and internal gravity waves in a vertical plane. In each case, numerical evaluation of the interaction coefficients for triads of regular waves shows that the interaction properties depend strongly on the basic shear. The conditions for explosive resonant interaction are also examined and they are expressed in terms of the pseudomomentum of the triad members.