Technological University Dublin ARROW@TU Dublin Articles School of Mathematics 2021 Five-Wave Resonances in Deep Water Gravity Waves: Integrability, Numerical Simulations and Experiments Dan Lucas Keele University Marc Perlin Texas A&M University Dian-Yong Liu Dalian Maritime University See next page for additional authors Follow this and additional works at: https://arrow.tudublin.ie/scschmatart Part of the Mathematics Commons Recommended Citation Lucas, D. et al. (2021) Five-WaveResonances in Deep Water Gravity Waves: Integrability, Numerical Simulations and Experiments. Fluids2021,6, 205. DOI:10.3390/fluids6060205 This Article is brought to you for free and open access by the School of Mathematics at ARROW@TU Dublin. It has been accepted for inclusion in Articles by an authorized administrator of ARROW@TU Dublin. For more information, please contact [email protected], [email protected]. This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 4.0 License Authors Dan Lucas, Marc Perlin, Dian-Yong Liu, Shane Walsh, Rossen Ivanov, and Miguel D. Bustamante This article is available at ARROW@TU Dublin: https://arrow.tudublin.ie/scschmatart/308 fluids Article Five-Wave Resonances in Deep Water Gravity Waves: Integrability, Numerical Simulations and Experiments Dan Lucas 1 , Marc Perlin 2,†, Dian-Yong Liu 3,‡, Shane Walsh 4, Rossen Ivanov 5 and Miguel D. Bustamante 4,* 1 School of Computing and Mathematics, Keele University, Staffordshire ST5 5BG, UK; [email protected] 2 Department of Ocean Engineering, Texas A&M University, 727 Ross Street, College Station, TX 77843, USA; [email protected] 3 Naval Architecture and Ocean Engineering College, Dalian Maritime University, Dalian 116026, China; [email protected] 4 School of Mathematics and Statistics, University College Dublin, Belfield, D04 V1W8 Dublin, Ireland; [email protected] 5 School of Mathematical Sciences, Technological University Dublin—City Campus, Grangegorman Lower, D07 ADY7 Dublin, Ireland; [email protected] * Correspondence: [email protected] † Former Affiliation: Department of Naval Architecture and Marine Engineering, University of Michigan, Ann Arbor, MI 48109, USA. ‡ Former Affiliation: State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian 116023, China; experiments performed at the University of Michigan. Abstract: In this work we consider the problem of finding the simplest arrangement of resonant deep- water gravity waves in one-dimensional propagation, from three perspectives: Theoretical, numerical and experimental. Theoretically this requires using a normal-form Hamiltonian that focuses on Citation: Lucas, D.; Perlin, M.; Liu, 5-wave resonances. The simplest arrangement is based on a triad of wavevectors K1 + K2 = K3 D.-Y.; Walsh, S.; Ivanov, R.; (satisfying specific ratios) along with their negatives, corresponding to a scenario of encountering Bustamante, M.D. Five-Wave wavepackets, amenable to experiments and numerical simulations. The normal-form equations for Resonances in Deep Water Gravity these encountering waves in resonance are shown to be non-integrable, but they admit an integrable Waves: Integrability, Numerical Simulations and Experiments. Fluids reduction in a symmetric configuration. Numerical simulations of the governing equations in natural 2021, 6, 205. https://doi.org/ variables using pseudospectral methods require the inclusion of up to 6-wave interactions, which 10.3390/fluids6060205 imposes a strong dealiasing cut-off in order to properly resolve the evolving waves. We study the resonance numerically by looking at a target mode in the base triad and showing that the energy Academic Editor: Alexander I. transfer to this mode is more efficient when the system is close to satisfying the resonant conditions. Dyachenko We first look at encountering plane waves with base frequencies in the range 1.32–2.35 Hz and steepnesses below 0.1, and show that the time evolution of the target mode’s energy is dramatically Received: 6 April 2021 changed at the resonance. We then look at a scenario that is closer to experiments: Encountering Accepted: 27 May 2021 wavepackets in a 400-m long numerical tank, where the interaction time is reduced with respect Published: 1 June 2021 to the plane-wave case but the resonance is still observed; by mimicking a probe measurement of surface elevation we obtain efficiencies of up to 10% in frequency space after including near-resonant Publisher’s Note: MDPI stays neutral contributions. Finally, we perform preliminary experiments of encountering wavepackets in a 35-m with regard to jurisdictional claims in long tank, which seem to show that the resonance exists physically. The measured efficiencies via published maps and institutional affil- iations. probe measurements of surface elevation are relatively small, indicating that a finer search is needed along with longer wave flumes with much larger amplitudes and lower frequency waves. A further analysis of phases generated from probe data via the analytic signal approach (using the Hilbert transform) shows a strong triad phase synchronisation at the resonance, thus providing independent experimental evidence of the resonance. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article Keywords: water gravity waves; 5-wave resonances; pseudospectral numerical simulations; water distributed under the terms and wave tank experiments conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). Fluids 2021, 6, 205. https://doi.org/10.3390/fluids6060205 https://www.mdpi.com/journal/fluids Fluids 2021, 6, 205 2 of 35 1. Introduction Historically, nonlinear resonant interactions in surface water waves have focused mainly on the so-called exact resonances, defined by the equations k1 ± ... ± kN = 0, w1 ± ... ± wN = 0, where N denotes the number of interacting waves, kj denote the wave-vectors, and wj denote the frequencies where wj = w(kj) is provided by a dispersion relation. Early theoretical developments for these interactions gave rise to resonance interaction theory, originated by Phillips (1960) and developed by several authors (see the reviews by [1,2], and references therein). The key working hypothesis of this theory is the smallness of the wave steepness, namely the product between wave amplitude and magnitude of the wave- vector. A formal perturbation theory based on multiple-scale methods is thus developed which allows one to obtain the exact resonances as necessary conditions to avoid secular behaviour. Solutions to these exact resonance equations can be found in many cases. Case N = 3, called triad resonances, applies to capillary-gravity waves (and also to a variety of wave systems such as Charney-Hasegawa-Mima, quasi-geostrophic equations, internal waves, inertial waves, etc.); case N = 4, known as quartet resonances, applies to gravity waves (including deep water). The case N = 5 is relevant in the important context of gravity waves in one-dimensional propagation, which is the topic to be discussed in this paper. In this context, it was shown in [3] that the interaction coefficients for the 4-wave resonances vanish identically. Later on, in [4] it was demonstrated that 5-wave resonances can exist and their interaction coefficients are nonzero. Then, in [5] (see also [6]) the explicit calculation of all 5-wave resonances along with their interaction coefficients was performed. Once solutions to the exact resonance equations are found, the dynamics of the wave amplitudes (at dominant orders) is dictated by coupled nonlinear PDE systems. In some simple scenarios the equations are integrable and can be solved analytically via the inverse scattering transform, leading to recurrent behaviour, showing periodic exchanges of energy between modes. In some scenarios, however, particularly in cases where multiple resonant interactions are coupled, the ODE and PDE systems obtained are known to display chaotic behaviour [7–10]. For practical reasons, the multiple resonant interaction dynamics is discussed us- ing stochastic approaches, quite successfully. Pioneered by Hasselmann [11–13] and Zakharov [14,15], the now established theory of wave turbulence describes the energy exchanges across scales due to resonant quartet interactions in the case of deep water surface gravity waves. A key assumption of this theory, in addition to the smallness of steepness and the limit of large box size, is an asymptotic closure of the hierarchy of cumu- lants [16], which leads to a set of evolution equations for the so-called spectrum variables, namely the individual quadratic energies of the spatial Fourier transforms of the original field variables. This approach assumes that the phases of the Fourier transforms do not play an important role in the dynamics of energy transfers. Interesting modern departures from these theories for water waves and other wave systems consider quasi-resonant sce- narios [17,18] and finite-amplitude regimes in discrete wave turbulence for other systems (such as the barotropic vorticity equation and more recently in wave-mean flow models of solar cycle modulations), where the phases interact with the spectrum variables producing interesting effects, such as precession resonance [19–21]. Up to and including the papers by [22–24], and the review by [1], only two experiments studied resonant interactions between wavetrains with comparable initial amplitudes and only