Zakharov and L’vov long ago invoked a Kolmogorov cascade hypothesis to describe the modal transfer of energy for surface gravity waves in deep water. Within this context the cascade hypothesis requires the existence of dynamical time scales in the evolution of the mode amplitudes which scale with wave number. The purpose of this Letter is to study the temporal evolution of individual modes, with the aim of identifying these dynamical time scales. The main result reported here is that mode-dependent time scales τ(k)∼${\mathit{k}}^{\mathrm{-}\mathrm{\beta }}$ can indeed be identified, with exponent β related to the exponent α which describes the spectrum of the waveheight autocorrelation function.

• Received 13 October 1992

DOI:https://doi.org/10.1103/PhysRevLett.71.1172