[Paper Review] Identifying four wave resonant interactions in a surface gravity wave turbulence experiment

Antoine Campagne, Roumaïssa Hassaini+5 more|arXiv (Cornell University)|Jun 14, 2019

Ocean Waves and Remote Sensing51 references12 citations

TL;DR

This study investigates four-wave resonant interactions in surface gravity wave turbulence using a 13-meter diameter tank with stereoscopic wave elevation measurements. By applying a space-time filter to separate free waves from bound waves generated via 3-wave coupling, the authors identify weak but significant 4-wave correlations, supporting the Weak Turbulence Theory as a valid framework for wave forecasting despite unresolved questions about bound wave contributions to dissipation and energy transfer.

ABSTRACT

The nonlinear dynamics of waves at the sea surface is believed to be ruled by the Weak Turbulence framework. In order to investigate the nonlinear coupling among gravity surface waves, we developed an experiment in the Coriolis facility which is a 13-m diameter circular tank. An isotropic and statistically stationary wave turbulence of average steepness of 10\% is maintained by two wedge wave makers. The space and time resolved wave elevation is measured using a stereoscopic technique. Wave-wave interactions are analyzed through third and fourth order correlations. We investigate specifically the role of bound waves generated by non resonant 3-wave coupling. Specifically, we implement a space-time filter to separate the dynamics of free waves (i.e. following the dispersion relation) from the bound waves. We observe that the free wave dynamics causes weak resonant 4-wave correlations. A weak level of correlation is actually the basis of the Weak Turbulence Theory. Thus our observations support the use of the Weak Turbulence to model gravity wave turbulence as is currently been done in the operational models of wave forecasting. Although in the theory bound waves are not supposed to contribute to the energy cascade, our observation raises the question of the impact of bound waves on dissipation and thus on energy transfers as well.

Motivation & Objective

• To investigate the role of four-wave resonant interactions in surface gravity wave turbulence.
• To isolate and analyze free wave dynamics from bound waves generated by non-resonant 3-wave coupling.
• To test the validity of Weak Turbulence Theory (WTT) in experimental wave turbulence with finite-size and dissipative effects.
• To assess whether bound waves, though not predicted to contribute to energy cascades in WTT, may still influence dissipation and energy transfer.

Proposed method

• Conducted experiments in the Coriolis facility, a 13-m diameter circular tank, with two wedge wave makers to generate isotropic, statistically stationary wave turbulence.
• Used stereoscopic imaging to measure space- and time-resolved wave elevation with high spatiotemporal resolution.
• Applied a space-time filter to separate free waves (adhering to linear dispersion relation) from bound waves (generated via 3-wave coupling).
• Analyzed third- and fourth-order wave correlations to detect resonant interactions.
• Used the linear dispersion relation ω² = gk for deep-water gravity waves and applied resonance conditions for four-wave interactions: ω₁ + ω₂ = ω₃ + ω₄ and k₁ + k₂ = k₃ + k₄.
• Compared observed correlations with theoretical predictions of Weak Turbulence Theory, particularly the Kolmogorov-Zakharov spectrum Eη(ω) ∝ gP¹ᐟ³ω⁻⁴.

Experimental results

Research questions

• RQ1 Do four-wave resonant interactions emerge in experimental surface gravity wave turbulence under weak nonlinearity?
• RQ2 To what extent do bound waves—generated by non-resonant 3-wave coupling—affect the observed wave correlations and energy transfer?
• RQ3 Is the observed level of 4-wave correlation consistent with the predictions of Weak Turbulence Theory?
• RQ4 How do finite-size effects and dissipation influence the detectability of resonant 4-wave interactions in laboratory experiments?
• RQ5 Can the separation of free and bound waves via a space-time filter improve the identification of true resonant wave dynamics?

Key findings

• The experiment observed weak but measurable four-wave correlations, consistent with the theoretical foundation of Weak Turbulence Theory.
• The space-time filter successfully isolated free wave dynamics, confirming that these waves drive the observed weak resonant 4-wave interactions.
• The level of 4-wave correlation observed is in line with the weak nonlinearity assumption central to WTT, supporting its use in operational wave forecasting models.
• Bound waves, though not expected to contribute to energy cascades in WTT, were found to be dynamically present and may influence dissipation processes.
• The observed wave spectrum exhibited steeper decay than predicted by the KZ spectrum, suggesting that dissipation and finite-size effects remain significant challenges in experimental validation.
• The results indicate that while WTT remains a valid framework, the role of bound waves in energy transfer and dissipation requires further investigation.