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The Universal Route to Rogue Waves

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The Universal Route to Rogue Waves PHYSICAL REVIEW X 9, 041057 (2019) Featured in Physics Experimental Evidence of Hydrodynamic Instantons: The Universal Route to Rogue Waves Giovanni Dematteis ,1,2 Tobias Grafke,3 Miguel Onorato ,2,4 and Eric Vanden-Eijnden5 1Dipartimento di Scienze Matematiche, Politecnico di Torino, Corso Duca degli Abruzzi 24, I-10129 Torino, Italy 2Dipartimento di Fisica, Universit`a degli Studi di Torino, Via Pietro Giuria 1, 10125 Torino, Italy 3Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom 4INFN, Sezione di Torino, Via Pietro Giuria 1, 10125 Torino, Italy 5Courant Institute, New York University, 251 Mercer Street, New York, New York 10012, USA (Received 3 July 2019; revised manuscript received 2 October 2019; published 18 December 2019) A statistical theory of rogue waves is proposed and tested against experimental data collected in a long water tank where random waves with different degrees of nonlinearity are mechanically generated and free to propagate along the flume. Strong evidence is given that the rogue waves observed in the tank are hydrodynamic instantons, that is, saddle point configurations of the action associated with the stochastic model of the wave system. As shown here, these hydrodynamic instantons are complex spatiotemporal wave field configurations which can be defined using the mathematical framework of large deviation theory and calculated via tailored numerical methods. These results indicate that the instantons describe equally well rogue waves created by simple linear superposition (in weakly nonlinear conditions) or by nonlinear focusing (in strongly nonlinear conditions), paving the way for the development of a unified explanation to rogue wave formation. DOI: 10.1103/PhysRevX.9.041057 Subject Areas: Fluid Dynamics, Nonlinear Dynamics, Statistical Physics I. INTRODUCTION nonlinear focusing effects take over linear dispersion and are known to be responsible for increasing the likelihood A fascinating phenomenon observed in a wide class of of the rogue waves. This leads to non-Gaussian fat-tailed nonlinear dispersive systems is the occurrence of rogue waves with abnormally large amplitude; they are found in statistics for their amplitude [2,10], as opposed to the sea surface gravity waves [1,2], nonlinear fiber optics [3], Gaussian statistics observed in the dispersive regime. plasmas [4], and Bose-Einstein condensates. Rogue waves In the present article, we propose a statistical theory of have received a lot of attention in the past 20 years, and rogue waves and test it against experiments performed in different mechanisms for their formation have been put the one-dimensional setting of the wave flume. We show forward, but a definite explanation has yet to be agreed that, in the full range of experimental conditions tested, the upon [2,5–9]. To settle this question, studies in wave flumes rogue waves we observe closely resemble hydrodynamic – or basins are interesting because they permit us to create instantons [11 16]: these are specific spatio-temporal and measure wave states by means of mechanical wave configurations of the wave field which we define within generators under controlled conditions meant to mimic the framework of large deviation theory (LDT) as the (after rescaling) those in the sea. The water surface in the minimizers of an action associated with the random wave tank can be monitored accurately with high space-time model used to describe the system. Here we focus on the resolution, and abundant statistics can be collected. In one- nonlinear Schrödinger equation (NLSE) with random dimensional experiments that mimic an idealized long- initial data, but the approach is generalizable to more crested rescaled sea, if the surface is sufficiently energetic, complicated models. The finding that instantons explain experimental rogue waves for a wide range of surface conditions in the tank is striking because it offers a unified description of these waves. In particular, our approach Published by the American Physical Society under the terms of encompasses two of the main existing theories for the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to rogue wave creation: (i) the theory of quasideterminism the author(s) and the published article’s title, journal citation, [17,18], which predicts that the rogue wave is created by and DOI. linear superposition effects and its shape is given by the 2160-3308=19=9(4)=041057(12) 041057-1 Published by the American Physical Society GIOVANNI DEMATTEIS et al. PHYS. REV. X 9, 041057 (2019) autocorrelation function of the wave field, and (ii) the deviation theory is presented in Sec. IV, where we also semiclassical theory [19,20], which asserts instead that describe how we compute the instanton for the rogue localized perturbations in the wave field can lead to the waves. Theory and experiment are then compared in Sec. V, formation of a Peregrine soliton via nonlinear focusing with special focus on the quasilinear and highly nonlinear instability. Our approach reconciles these two, apparently limiting cases. We conclude in Sec. VI by discussing the incompatible, theories and smoothly interpolates between implications of our results in the context of a unified theory them as the experimental control parameters are varied: of rogue waves. when the nonlinear effects are weak, the shape of the instantons converges to the autocorrelation function pre- II. EXPERIMENTAL SETUP dicted by the theory of quasi-determinism, and when the nonlinear effects are strong, their shape converges to that The experimental data were recorded in the 270-m-long of the Peregrine soliton. Because the instanton calculus wave flume at Marintek (Norway) [41,42], schematically proposed in this paper uses as limiting parameter the represented in Fig. 1. At one end of the tank a plane-wave maximal wave amplitude itself, without condition on model generator perturbs the water surface with a predefined parameters or regimes in the NLSE, it allows us to assess random signal. These perturbations create long-crested the validity of the quasideterministic and semiclassical wave trains that propagate along the tank toward the theories by comparing them to the results of our approach opposite end, where they eventually break on a smooth in appropriate regimes. Our approach could also be useful beach that suppresses most of the reflections. The water in the context of other nonlinear theories for rogue waves surface ηðx; tÞ is measured by probes placed at different based on NLSE, like statistical approaches based on the distances from the wave maker (x coordinate). The signal at Alber and the Wigner equations [21–26]. We also stress the wave maker ηðx ¼ 0;tÞ ≡ η0ðtÞ is prepared according that the method proposed here can be generalized to the full to the stationary random-phase statistics with deterministic ω two-dimensional setting, as well as other relevant physical spectral amplitudes Cð jÞ: systems where an understanding of extreme events is XN qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi important [27,28] but made challenging by the complexity e η0ðtÞ¼ 2CðωjÞδω cosðωjt þ ϕjÞ: ð1Þ of the models involved combined with the stochasticity of j¼1 their evolution and the uncertainty of their parameters [27,29–32]. In this sense our approach adds to other rare Here the phases ϕj are mutually independent random events methods [33–40]. variables uniformly distributed on ½0; 2π, δω ¼ð2π=τÞ, The remainder of this paper is organized as follows. ωj ¼ jδω, and τ is the time-series length. This guarantees e We introduce the experimental setup in Sec. II. In Sec. III, that, for N and τ sufficiently large, η0ðtÞ is approximately a we explain how we extract extreme event data from the stationary Gaussian random field with energy spectrum experimental measurements. Our approach based on large CðωÞ > 0, i.e., FIG. 1. Wave flume experiment. The wave maker generates a random wave field with stationary Gaussian statistics with the JONSWAP energy spectrum observed in the oceans. The planar wave fronts propagate along the water tank, where the surface elevation η is measured by vertical probes. 041057-2 EXPERIMENTAL EVIDENCE OF HYDRODYNAMIC … PHYS. REV. X 9, 041057 (2019) XN e e 0 0 hη0ðtÞη0ðt Þi ¼ CðωjÞδω cos½ωjðt − t Þ 1 Zj¼ ∞ ∼ CðωÞ cos½ωðt − t0Þdω; ð2Þ 0 where the bracket denotes expectation with respect to the random phases ϕj. In the experiment, CðωÞ is taken to be the Joint North Sea Wave Project (JONSWAP) spectrum [43] of deep water waves observed in the ocean, 2 4 αg 5 ω0 2 2 2 exp ½−ðω−ω0Þ =2σ ω C ω exp − γ J 0 : 3 ð Þ¼ ω5 4 ω ð Þ −2 FIG. 2. JONSWAP spectra from Eq. (3) for the three exper- Here g ¼ 9.81 ms is the gravity acceleration, ω0 ¼ imental regimes of Table I (lines), compared to experimental −1 4.19 s is the carrier frequency (spectral peak), and σJ ¼ measurements at the x ¼ 10 m probe (dots). These spectra 0.07 if ω ≤ ω0 and σJ ¼ 0.09 if ω > ω0. These parameters remain roughly constant through the tank, except for small are fixed for all sea states, and we can use the dispersion changes that are the signature of non-Gaussian effects that relation of surface gravity waves in deep water to obtain the develop [42]. 2 −1 carrier wave number k0 ¼ ω0=g ¼ 1.79 m . The remain- α γ ing parameters and in Eq. (3) are dimensionless and (γ ¼ 3.3, Hs ¼ 0.13 m), and highly nonlinear (γ ¼ 6, vary according to weather conditions. In the experiments, Hs ¼ 0.15 m); see Table I. Note that these three regimes α 0 012 γ ¼ . throughout, while the enhancement factor have comparable significant wave heights Hs, but the ranges from 1 to 6, which is a realistic range of values for difference in their enhancement factors γ has significant the ocean measurements from calmer to rougher sea states.
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