Energy spectrum of ensemble of weakly nonlinear gravity-capillary waves on a fluid surface

TL;DR

This paper derives an analytical energy spectrum for weakly nonlinear gravity-capillary waves, revealing how the spectrum depends on the ratio of surface tension to gravity, and connects this with existing models in nonlinear wave theory.

Contribution

It provides the first analytical expression for the energy spectrum of nonlinear gravity-capillary waves considering nonlinearity parameters where kinetic equations do not apply.

Findings

Derived explicit energy spectrum depending on surface tension and gravity ratio

Confirmed limits match pure capillary and gravity wave results

Linked D-cascade model with established nonlinear wave theories

Abstract

In this Letter we regard nonlinear gravity-capillary waves with parameter of nonlinearity being $\varepsilon \sim 0.1 \div 0.25$. For this nonlinearity time scale separation does not occur and kinetic wave equation does not hold. An energy cascade in this case is built at the dynamic time scale (D-cascade) and is computed by the increment chain equation method first introduced in \emph{Kartashova, \emph{EPL} \textbf{97} (2012), 30004.} We compute for the first time an analytical expression for the energy spectrum of nonlinear gravity-capillary waves as an explicit function depending on the ratio of surface tension to the gravity acceleration. It is shown that its two limits - pure capillary and pure gravity waves on a fluid surface - coincide with the previously obtained results. We also discuss relations of the model of D-cascade with a few known models used in the theory of nonlinear waves such as Zakharov's equation, resonance of the modes with nonlinear Stokes corrected frequencies and Benjamin-Feir index. These connections are crucial in the understanding and forecasting specifics of the energy transport in a variety of multi-component wave dynamics, from oceanography to optics, from plasma physics to acoustics.