A Statistical Dynamical Theory of Strongly Nonlinear Internal Gravity Waves

A statistical dynamical closure theory describing the interaction of strongly (and weakly) nonlinear two-dimensional internal waves in the presence of viscous dissipation and thermal conduction is derived. By applying renormalization methods originally formulated for quantum and classical statistical field theory, closures similar to the Direct Interaction and eddy-damped quasi-normal procedures of turbulence are derived. These methods are applied directly to the strongly nonlinear primitive field equations in Eulerian variables, thus avoiding the small amplitude assumptions inherent in the resonant interaction formalism. Propagator renormalization techniques provide formulas for the nonlinear internal wave frequency and spectral width in terms of the energy spectrum. The commonly used multiple time and space scale analysis is replaced by an analysis of the two-point correlation functions in terms of sum and difference variables. This permits the systematic development of a Landau equation. This generalization of the Boltzmann equation incorporates spatial variation of the group velocity and scattering due to spatial inhomogeneity. In the limit of weakly interacting waves and zero viscosity, the closures reduce to the resonant interaction approximation formalism. It is shown that the inviscid resonant interaction limit is singular in the sense that the quilibrium spectrum differs from that of the general inviscid nonlinear off-resonant case. This is due to the fact that in the resonant interaction limit there is an additional constant of motion, viz. z-momentum. The implications of these results are discussed.