The energy cascade of surface wave turbulence: toward identifying the active wave coupling

TL;DR

This study experimentally investigates surface wave turbulence, revealing weak nonlinear interactions and the absence of an inverse cascade, challenging some theoretical predictions about wave energy transfer.

Contribution

The paper provides experimental evidence on the nature of wave interactions, specifically showing weak 4-wave coupling and the lack of inverse cascade in surface wave turbulence.

Findings

Weak 4-wave coupling observed via tricoherence analysis

No evidence of inverse cascade in the experimental data

Water deformation comprises linear waves and bound waves from non-resonant interactions

Abstract

We investigate experimentally turbulence of surface gravity waves in the Coriolis facility in Grenoble by using both high sensitivity local probes and a time and space resolved stereoscopic reconstruction of the water surface. We show that the water deformation is made of the superposition of weakly nonlinear waves following the linear dispersion relation and of bound waves resulting from non resonant triadic interaction. Although the theory predicts a 4-wave resonant coupling supporting the presence of an inverse cascade of wave action, we do not observe such inverse cascade. We investigate 4-wave coupling by computing the tricoherence i.e. 4-wave correlations. We observed very weak values of the tricoherence at the frequencies excited on the linear dispersion relation that are consistent with the hypothesis of weak coupling underlying the weak turbulence theory.

Tables

Table 1. Table 1: Parameters of the three datasets. f p subscript 𝑓 𝑝 f_{p} is the frequency of the peak of the spectrum. k p subscript 𝑘 𝑝 k_{p} is the wavenumber corresponding to f p subscript 𝑓 𝑝 f_{p} following the linear dispersion relation. σ η subscript 𝜎 𝜂 \sigma_{\eta} is the elevation variance. ϵ p subscript italic-ϵ 𝑝 \epsilon_{p} is the wave steepness computed as ϵ p = 2 ​ k p ​ σ η subscript italic-ϵ 𝑝 2 subscript 𝑘 𝑝 subscript 𝜎 𝜂 \epsilon_{p}=2k_{p}\sigma_{\eta} (see Aub2 ).

Dataset	$f_p$ [Hz]	$k_p$ [m-1]	${\sigma }_{\eta }$ [m]	${ϵ}_p$
\svhline weak	0.65	1.83	0.0294	0.11
strong	0.76	2.4	0.0339	0.16
short	1.5	9.05	0.0131	0.24

Equations

$$E^{\eta }(k)$$

$$\text{or }{E}^{\eta }(\omega )$$

$${\omega }_{LDR}=\sqrt{gk+\frac{\gamma k^3}{\rho }},$$

$${\omega }^{(\pm n)}={\omega }_{LDR}(k\mp n{k}_p)\pm n{\omega }_p,$$

$${\omega }_{LDR1}+{\omega }_{LDR2}={\omega }_{LDR3}\text{,}{\mathbf{k}}_1+{\mathbf{k}}_2={\mathbf{k}}_3$$

$${\omega }_{LDR1}+{\omega }_{LDR2}={\omega }_{LDR3}+{\omega }_{LDR4}\text{,}{\mathbf{k}}_1+{\mathbf{k}}_2={\mathbf{k}}_3+{\mathbf{k}}_4$$

$$C({\omega }_1,{\omega }_2,{\omega }_3,{\omega }_4)$$

Full text

1 Introduction

Wave Turbulence is a general framework that aims at describing the statistical properties of a large ensemble of waves. Although no general theory exists, the Weak Turbulence Theory (WTT) focusses on the case of vanishing non linearity in very large systems Zakh ; Naz ; New . It predicts an energy cascade in scale space between the large scale of forcing down to small scales at which dissipation dominates. Due to weak nonlinearity energy transfer occurs among resonant waves. Oceanic waves is the natural field of application of the theory following the work of Hasselman Has that assumes transfer among 4 resonant waves. A major result of the WTT is that analytic solutions of the wave Fourier spectrum can be exhibited in many cases, the so-called Kolmogorov-Zakharov spectra. For gravity surface waves the prediction of the wave elevation spectrum is Naz :

[TABLE]

where $g$ is the gravity acceleration, $P$ is the energy flux, $k$ is the wave number and $\omega$ the angular frequency. Although some field measurements of the spectra appear compatible with this prediction, laboratory experiments fail to reproduce this prediction. The observed spectral exponents of the frequency spectrum are significantly steeper than the $-4$ theoretical value NaLu ; Deik ; Aub2 . Our goal is to investigate further the statistical properties of the wave field recorded experimentally to obtain some insight on the reasons for the discrepancy between theory, observations and laboratory data. For surface gravity wave, due to the 4-wave coupling, the theory predicts also an inverse cascade of wave action Naz that maybe responsible for the long wave generation by the wind.

2 Experimental setup

Waves are generated by two wedge wavemakers in a circular tank of 13 m diameter and 0.9 m depth (the Coriolis facility located in Grenoble, France). Wave elevation is recorded by a set of 10 capacitive wave gauges that provide a local measurement and a stereoscopic system that provides a space and time resolved measurement of the wave elevation over a surface $2\times 2.6$ m2 (Fig. 1).