Simulation of wave interactions and turbulence in one-dimensional water waves

The weak- or wave-turbulence problem consists of finding statistical states of a large number of interacting waves. These states are obtained by forcing and dissipating a conservative dispersive wave equation at disparate scales to model physical forcing and dissipation, and by predicting the spectrum, often as a Kolmogorov-like power law, at intermediate scales. The mechanism for energy transfer in such systems is usually triads or quartets of waves. Here, we first derive a small-amplitude nonlinear dispersive equation (a finite-depth Benney-Luke-type equation), which we validate, analytically and numerically, by showing that it correctly captures the main deterministic aspects of gravity wave interactions: resonant quartets. Benjamin-Feir-type wave-packet stability, and wave-mean flow interactions. Numerically, this equation is easier to integrate than either the full problem or the Zakharov integral equation. Some additional features of wave interaction are discussed such as harmonic generation in shallow water. We then perform long time computations on the forced-dissipated model equation and compute statistical quantities of interest, which we compare to existing predictions. The forward cascade yields a spectrum close to the prediction of Zakharov, and the inverse cascade does not.