Time scales and structures of wave interaction

TL;DR

This paper analyzes wave interaction structures and time scales in weakly nonlinear systems, demonstrating that kinetic and D-cascades operate at different timescales and are both relevant for understanding energy transfer.

Contribution

It provides a unified overview of wave interaction theory, clarifies the distinct roles of kinetic and D-cascades, and applies time scale analysis to various wave systems.

Findings

Kinetic and D-cascades occur at different time scales.

Energy cascades in water waves are faster than kinetic wave turbulence predictions.

D-model can be applied broadly to various wave systems.

Abstract

In this paper we give a general account of Wave Interaction Theory which by now consists of two parts: kinetic wave turbulence theory (WTT), using a statistical description of wave interactions, and the D-model recently introduced in \emph{Kartashova, PRE \textbf{86}: 041129 (2012)} describing interactions of distinct modes. Applying time scale analysis to weakly nonlinear wave systems modeled by the focusing nonlinear Sch\"{o}dinger equation, we give an overview of the structures appearing in Wave Interaction Theory, their time scales and characteristic times. We demonstrate that kinetic 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 due to different physical mechanisms. Taking surface water waves as an example we show that energy cascades in this system occur at much faster characteristic times than those required by the kinetic WTT but can be described as D-cascades. As D-model has no special pre-requisites, it may be rewarding to re-evaluate existing experiments in other wave systems appearing in hydrodynamics, nonlinear optics, electrodynamics, plasma, convection theory, etc. To appear in EPL