Abstract

In this paper, we numerically study the wave turbulence of surface gravity waves in the framework of Euler equations of the free surface. The purpose is to understand the variation of the scaling of the spectra with wavenumber$k$and energy flux$P$at different nonlinearity levels under different forcing/free-decay conditions. For all conditions (free decay and narrow-band and broad-band forcing) that we consider, we find that the spectral forms approach the wave turbulence theory (WTT) solution$S_\eta \sim k^{-5/2}$and$S_\eta \sim P^{1/3}$at high nonlinearity levels. With a decrease of nonlinearity level, the spectra for all cases become steeper, with the narrow-band forcing case exhibiting the most rapid deviation from WTT. We investigate bound waves and the finite-size effect as possible mechanisms causing the spectral variations. Through a tri-coherence analysis, we find that the finite-size effect is present in all cases, which is responsible for the overall steepening of the spectra and the reduced capacity of energy flux at lower nonlinearity levels. The fraction of bound waves in the domain generally decreases with the decrease of nonlinearity level, except for the narrow-band case, which exhibits a transition at a critical nonlinearity level below which a rapid increase is observed. This increase serves as the main reason for the fastest deviation from WTT with the decrease of nonlinearity in the narrow-band forcing case.

Highlights

  • The normal state of the ocean surface is characterized by a large number of waves at difference scales subject to nonlinear interactions in the presence of wind forcing and viscous dissipation

  • We find that the scalings of the wave spectra with wavenumber and energy flux both approach the weak turbulence theory (WTT) solution at sufficiently high nonlinearity levels

  • With a decrease of nonlinearity level, steeper spectra and reduced energy flux capacity can be observed indicating the deviation from WTT, with the largest deviation rate found in the narrow-band forcing case

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Summary

Introduction

The normal state of the ocean surface is characterized by a large number of waves at difference scales subject to nonlinear interactions in the presence of wind forcing and viscous dissipation. Existing work includes Dyachenko, Korotkevich & Zakharov (2004) and Lvov et al (2006) for forcing turbulence and Onorato et al (2002) and Yokoyama (2004) for free-decay turbulence of gravity waves in the context of Euler equations While all these works report a scaling Sη(k) ∼ k−5/2 consistent with (1.1), the simulation (in each of them) is conducted at a single nonlinearity level, and is not capable of resolving/understanding the sensitivity of the spectra to various conditions and their scaling with P. We conduct a numerical study of the spectral properties of gravity wave turbulence at different forcing (in terms of bandwidths and amplitudes) and free-decay (with relatively broad-band initial data) conditions.

Mathematical formulation
Results
Spectral slopes
Bound waves
Finite-size effect
Conclusions
Full Text
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