Time scales and structures of wave interaction exemplified with water waves

ABSTRACT Presently two models for computing energy spectra in weakly nonlinear dispersive media are known: kinetic wave turbulence theory, using a statistical description of an energy cascade over a continuous spectrum (K-cascade), and the D-model, describing resonant clusters and energy cascades (D-cascade) in a deterministic way as interaction of distinct modes. In this Letter we give an overview of these structures and their properties and a list of criteria, which model of an energy cascade should be used in the analysis of a given experiment, {\red using water waves as an example}. Applying time scale analysis to weakly nonlinear wave systems modeled by the focusing nonlinear Sch\"{o}dinger equation, we demonstrate that K-cascade and D-cascade are not competing processes but rather two processes taking place at different time scales, at different characteristic levels of nonlinearity and based on different physical mechanisms. Applying those criteria to data known from experiments with surface water waves we find, that the energy cascades observed occurs at short characteristic times compatible only with a D-cascade. The only pre-requisite for a D-cascade being a focusing nonlinear Sch\"{o}dinger equation, the same analysis may be applied to existing experiments with wave systems appearing in hydrodynamics, nonlinear optics, electrodynamics, plasma, convection theory, etc.