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Rogue Waves and Large Deviations in Deep Rogue waves and large deviations in deep sea Giovanni Dematteisa,b, Tobias Grafkea,c, and Eric Vanden-Eijndena,1 aCourant Institute of Mathematical Sciences, New York University, New York, NY 10012; bDipartimento di Scienze Matematiche, Politecnico di Torino, I-10129 Torino, Italy; and cMathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved December 11, 2017 (received for review June 13, 2017) The appearance of rogue waves in deep sea is investigated by random initial data drawn from a Gaussian distribution (30). The using the modified nonlinear Schrodinger¨ (MNLS) equation in one spectrum of this field is chosen to have a width comparable to spatial dimension with random initial conditions that are assumed that of the Joint North Sea Wave Project (JONSWAP) spectrum to be normally distributed, with a spectrum approximating real- (31, 32) obtained from observations in the North Sea. We cal- istic conditions of a unidirectional sea state. It is shown that one culate the probability of occurrence of a large amplitude solu- can use the incomplete information contained in this spectrum as tion of MNLS out of these random initial data and thereby also prior and supplement this information with the MNLS dynamics estimate the tail of the surface height distribution. These calcu- to reliably estimate the probability distribution of the sea sur- lations are performed within the framework of large deviations face elevation far in the tail at later times. Our results indicate theory (LDT), which predicts the most likely way by which large that rogue waves occur when the system hits unlikely pockets of disturbances arise and therefore also explains the mechanism of wave configurations that trigger large disturbances of the surface rogue wave creation. Our results are validated by comparison height. The rogue wave precursors in these pockets are wave pat- with brute-force Monte Carlo simulations, which indicate that terns of regular height, but with a very specific shape that is iden- rogue waves in MNLS are indeed within the realm of LDT. Our tified explicitly, thereby allowing for early detection. The method approach therefore gives an efficient way to assess the probability proposed here combines Monte Carlo sampling with tools from of large waves and their mechanism of creation. large deviations theory that reduce the calculation of the most likely rogue wave precursors to an optimization problem that can 1. Problem Setup be solved efficiently. This approach is transferable to other prob- Our starting point will be the MNLS equation for the evolution APPLIED lems in which the system’s governing equations contain random of the complex envelope u(t; x) of the sea surface in deep water MATHEMATICS initial conditions and/or parameters. (18), in terms of which the surface elevation reads η(t; x) = i(k0x−!0t) < u(t; x)e (here k0 denotes the carrier wave number, Laplace method j JONSWAP spectrum j peregrine soliton j intermittency j p !0 = gk0, and g is the gravitational acceleration). Measuring Monte Carlo −1 −1 u and x in units of k0 and t in !0 , we can write MNLS in nondimensional form as ogue waves, long considered a figment of sailors’ imagina- tions, are now recognized to be a real, and serious, threat for 1 i 2 1 3 i 2 R @t u + @x u + @ u − @ u + juj u x x boats and naval structures (1, 2). Oceanographers define them as 2 8 16 2 [1] deep-water waves whose crest-to-trough height H exceeds twice 3 2 1 2 i 2 + juj @x u + u @x u¯ − j@x j juj = 0; x 2 [0; L]; the significant wave height Hs , which itself is four times the SD of 2 4 2 the ocean surface elevation. Rogue waves appear suddenly and where the bar denotes complex conjugation. We will consider unpredictably and can lead to water walls with vertical size on Eq. 1 with random initial condition u0(x) ≡ u(0; x), constructed the order of 20–30 m (3, 4), with enormous destructive power. via their Fourier representation, Although rare, they tend to occur more frequently than predicted by linear Gaussian theory (5, 6). While the mechanisms under- lying their appearance remain under debate (7–9), one plausible Significance scenario has emerged over the years: It involves the phenomenon of modulational instability (10, 11), a nonlinear amplification Quantifying the departure from Gaussianity of the wave- mechanism by which many weakly interacting waves of regular height distribution in the seas and thereby estimating the size can create a much larger one. Such an instability arises in the likelihood of appearance of rogue waves is a long-standing context of the focusing nonlinear Schrodinger¨ (NLS) equation problem with important practical implications for boats and (11–17) or its higher-order variants (18–22), which are known to naval structures. Here, a procedure is introduced to identify be good models for the evolution of a unidirectional, narrow- ocean states that are precursors to rogue waves, which could banded surface wave field in a deep sea. Support for the descrip- permit their early detection. Our findings indicate that rogue tion of rogue waves through such envelope equations recently waves obey a large deviation principle—i.e., they are dom- came from experiments in water tanks (23–26), where Dysthe’s inated by single realizations—which our method calculates modified NLS (MNLS) equation in one spatial dimension (18, by solving an optimization problem. The method generalizes 19) was shown to accurately describe the mechanism creating to estimate the probability of extreme events in other deter- coherent structures which soak up energy from its surroundings. ministic dynamical systems with random initial data and/or While these experiments and other theoretical works (27, 28) parameters, by using prior information about the nature of give grounds for the use of MNLS to describe rogue waves, they their statistics. have not addressed the question of their likelihood of appear- ance. Some progress in this direction has been recently made in Author contributions: G.D., T.G., and E.V.-E. designed research, performed research, ana- ref. 29, where a reduced model based on MNLS was used to esti- lyzed data, and wrote the paper. mate the probability of a given amplitude within a certain time, The authors declare no conflict of interest. and thereby compute the tail of the surface height distribution. This article is a PNAS Direct Submission. These calculations were done by using an ansatz for the solu- Published under the PNAS license. tions of MNLS, effectively making the problem 2D. The purpose 1To whom correspondence should be addressed. Email: [email protected]. of this work is to remove this approximation and study the prob- This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. lem in its full generality. Specifically, we consider the MNLS with 1073/pnas.1710670115/-/DCSupplemental. www.pnas.org/cgi/doi/10.1073/pnas.1710670115 PNAS j January 30, 2018 j vol. 115 j no. 5 j 855–860 Downloaded by guest on September 27, 2021 1=2 2 2 X ikn x ^ ^ −kn =(2∆ ) 1 ? 2 ? u0(x) = e (2Cn ) θn ; Cn = Ae ; [2] I (z(λ)) = ku (λ)k ; z(λ) = F (u(T ; u (λ))): [9] T 2 0 C 0 n2Z ? where u0 (λ) denotes the minimizer obtained in Eq. 8. It is easy where kn = 2πn=L, θn are complex Gaussian variables with ¯ ¯ ¯ to see from Eqs. 6 and 8 that ST (λ) is the Legendre transform of mean zero and covariance Eθn θm = δm;n , Eθn θm = Eθn θm = 0. IT (z) since: This guarantees that u0(x) is a Gaussian field with mean zero and 0 0 P ikn (x−x ) (u0(x)¯u0(x )) = 2 e C^n . To make contact with 1 2 E n2Z ST (λ) = sup(λz − IT (z)) = sup λz − inf ku0kC ; u 2Ω(z) the observational data, the amplitude A and the width ∆ in Eq. z2R z2R 2 0 ^ 2 are picked so that Cn has the same height and area as the [10] JONSWAP spectrum (31, 32); see Supporting Information for details. Because the initial data for Eq. 1 are random, so is the solution 3. Results at time t > 0, and our aim is to compute We considered two sets of parameters. In set 1, we took −5 −2 A = 5:4 · 10 k0 and ∆ = 0:19k0. Converting back into dimen- PT (z) ≡ P F (u(T )) ≥ z ; [3] −1 sional units by using k0 = 36 m consistent with the JONSWAP where P denotes probability over the initial data and F is a spectrum (31, 32), this implies a significant wave height u T > 0 p scalar functional depending on at time . Even though Hs = 4 C (0) = 3:3 m classified as a “rough sea” (33). It also our method is applicable to more general observables, here we p yields a Benjamin–Feir index (BFI) = 2 2C (0)=∆ = 0:34, (32, will focus on 34), meaning that the modulational instability of a typical ini- F (u(T )) = max ju(T ; x)j: [4] tial condition is of medium intensity. In set 2, we took A = 3:4 · x2[0;L] −4 −2 10 k0 and ∆ = 0:19k0, for which Hs = 8:2 m is that of a “high sea” and the BFI is 0.85, meaning that the modulational instabil- 2. LDT Approach ity of a typical initial condition is stronger. A brute-force approach to calculate Eq. 3 is Monte Carlo sam- Fig. 1, Upper shows the time evolution of ju(t; x)j start- pling: Generate random initial conditions u0(x) by picking ran- ing from an initial condition from set 1 optimized so that dom θn ’s in Eq.
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