Analysis of a class of simplified models for nonlinear gravity wave interactions

Abstract

Kolmogorov-type cascade processes have long been postulated to explain power-law falloff of the spectrum of gravity wave autocorrelations. In an effort to understand these processes more deeply, simplified models of the nonlinear Hasselmann resonant four-wave interaction have been introduced. These models respect the fundamental structural aspects of that interaction, but are otherwise chosen for computational simplicity. In this paper the concept of quasilocality within the context of such models is extended to its logical limit in one spatial dimension. This results in differential rather than integral equations for the wave fields. One scaling solution to the new equations has a spectral exponent independent of the details of the model interaction, and is stable to small perturbations. In addition, under weak restrictions on the first and second derivatives of the model interactions, a further pair of two-parameter families of scaling solutions exist. The spectral exponents in this case depend on the details of the model, but unstable as well as stable perturbations to these solutions are supported.