On the theory of nonlinear wave-wave interactions among geophysical waves

Abstract

The problem of nonlinear wave-wave interactions is reformulated, in a Eulerian framework, for two classical geophysical systems: barotropic Rossby waves and internal gravity waves on a vertical plane. The departure of the dynamical fields from the equilibrium state is expanded in the linear-problem eigenfunctions, using their properties of orthogonality and completeness. The system is then completely described by the expansion amplitudes, whose evolution is controlled by a system of equations (with quadratic nonlinearity) which is an exact representation of the original model equations. There is no a priori need for the usual multiple-time-scale analysis, or any other perturbation expansion, to develop the theory. These or other approximations (like truncation of the expansion basis or the Boltzmann equation for a stochastic description) can, if desired, be performed afterwards.The evolution of the system is constrained mainly by the conservation of energy E and pseudo-momentum P, properties related to time and space homogeneity of the model equations. Conservation of E and P has, in turn, some interesting consequences: (a) a generalization of Fjortoft's theorem, (b) a class of exact nonlinear solutions (which includes the system of one single wave), and (c) conservation of E and P in an arbitrarily truncated system (which is useful in the development of approximations of the problem).The properties of all possible resonant triads are shown and used to estimate the order of magnitude of off-resonant coupling coefficients.The results are used in two different problems: the stability of a single wave (maximum growth rates are evaluated in both the strong- and weak-interactions limits) and the three-wave system. The general solution (for any initial condition and for both the resonant and off-resonant cases) of the latter is presented.