Abstract
Since the 1990s, the modulational instability has commonly been used to explain the occurrence of rogue waves that appear from nowhere in the open ocean. However, the importance of this instability in the context of ocean waves is not well established. This mechanism has been successfully studied in laboratory experiments and in mathematical studies, but there is no consensus on what actually takes place in the ocean. In this work, we question the oceanic relevance of this paradigm. In particular, we analyze several sets of field data in various European locations with various tools and find that the main generation mechanism for rogue waves is the constructive interference of elementary waves enhanced by secondorder bound nonlinearities and not the modulational instability. This implies that rogue waves are likely to be rare occurrences of weakly nonlinear random seas.
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Introduction
According to the most commonly used definition, rogue waves are unusually largeamplitude waves that appear from nowhere in the open ocean. Evidence that such extremes can occur in nature is provided, among others, by the Draupner and Andrea events, which have been extensively studied over the last decade^{1,2,3,4,5,6}. Several physical mechanisms have been proposed to explain the occurrence of such waves^{7}, including the two competing hypotheses of nonlinear focusing due to thirdorder quasiresonant wavewave interactions^{8} and purely dispersive focusing of secondorder nonresonant or bound harmonic waves, which do not satisfy the linear dispersion relation^{9,10}.
In particular, recent studies propose thirdorder quasiresonant interactions and associated modulational instabilities^{11,12} inherent to the Nonlinear Schrödinger (NLS) equation as mechanisms for rogue wave formation^{3,8,13,14,15}. Such nonlinear effects cause the statistics of weakly nonlinear gravity waves to significantly differ from the Gaussian structure of linear seas, especially in longcrested or unidirectional (1D) seas^{8,10,16,17,18,19}. The latestage evolution of modulation instability leads to breathers that can cause large waves^{13,14,15}, especially in 1D waves. Indeed, in this case energy is ‘trapped’ as in a long waveguide. For small wave steepness and negligible dissipation, quasiresonant interactions are effective in reshaping the wave spectrum, inducing large breathers via nonlinear focusing before wave breaking occurs^{16,17,20,21}. Consequently, breathers can be observed experimentally in 1D wave fields only at sufficiently small values of wave steepness^{20,21,22}. However, wave breaking is inevitable when the steepness becomes larger: ‘breathers do not breathe’^{23} and their amplification is smaller than that predicted by the NLS equation, in accord with theoretical studies^{24} of the compact Zakharov equation^{25,26} and numerical studies of the Euler equations^{27,28}.
Typical oceanic wind seas are shortcrested, or multidirectional wave fields. Hence, we expect that nonlinear focusing due to modulational effects is diminished since energy can spread directionally^{16,18,29}. Thus, modulation instabilities may play an insignificant role in the wave growth especially in finite water depth where they are further attenuated^{30}.
Tayfun^{31} presented an analysis of oceanic measurements from the North Sea. His results indicate that large waves (measured as a function of time at a given point) result from the constructive interference (focusing) of elementary waves with random amplitudes and phases enhanced by secondorder nonresonant or bound nonlinearities. Further, the surface statistics follow the Tayfun^{32} distribution^{32} in agreement with observations^{9,10,31,33}. This is confirmed by a recent data quality control and statistical analysis of singlepoint measurements from fixed sensors mounted on offshore platforms, the majority of which were recorded in the North Sea^{34}. The analysis of an ensemble of 122 million individual waves revealed 3649 rogue events, concluding that rogue waves observed at a point in time are merely rare events induced by dispersive focusing. Thus, a wave whose crest height exceeds the rogue threshold^{2} 1.25H_{s} occurs on average once every N_{r} ~ 10^{4} waves with N_{r} referred to as the return period of a rogue wave and H_{s} is the significant wave height. Some even more recent measurements off the west coast of Ireland^{35} revealed similar statistics with 13 rogue events out of an ensemble of 750873 individual waves and N_{r} ~ 6 · 10^{4}.
To date, it is still under debate if in typical oceanic seas secondorder nonlinearities dominate the dynamics of extreme waves as indicated by ocean measurements^{31,33}, or if thirdorder nonlinear effects play also a significant, if not dominant, role in roguewave formation. The preceding provides our principal motivation for studying the statistical and physical properties of rogue sea states and to investigate the relative importance of second and thirdorder nonlinearities. We rely on WAVEWATCH III hindcasts and High Order Spectral (HOS) simulations of the Euler equations for water waves^{36}. In our study, we consider the famous Draupner and Andrea rogue waves and the less well known Killard rogue wave^{35}. The Andrea rogue wave was measured by Conoco on 9 November 2007 with a system of four Teledyne Optech lasers mounted in a square array on the Ekofisk platform in the North Sea in a water depth d = 74 m^{4,5}. The metocean conditions of the Andrea wave are similar to those of the Draupner wave measured by Statoil at a nearby platform (d = 70 m) on 1 January 1995 with a down looking laserbased wave sensor^{37}. The Killard wave was measured by ESB International on 28 January 2014 by a Waverider buoy off the west coast of Ireland in a water depth d = 39 m. In Table 1 we summarize the wave parameters of the three sea states in which the rogue wave occurred and we refer to the Methods section for definitions and details. As one can see, the actual cresttotrough (wave) heights H and crest heights h meet the classical criteria^{2} H/H_{s} > 2 and h/H_{s} > 1.25 to qualify the Andrea, Draupner and Killard extreme events as rogue waves. The remainder of the paper is organized as follows. First, the probability structure of oceanic seas is presented^{33} together with the essential elements of Tayfun’s^{32} secondorder theory for the wave skewness and Janssen’s^{8} formulation for the excess kurtosis of multidirectional seas^{29}. Then, we present and compare secondorder and thirdorder statistical properties of the three rogue sea states followed by an analysis of the shape of the largest waves and associated mean sea levels. In concluding, we discuss the implications of these results on roguewave predictions.
Probability structure of oceanic seas
Nonresonant and resonant wavewave interactions cause the statistics of weakly nonlinear gravity waves to significantly differ from the Gaussian structure of linear seas^{8,10,16,17,18,38}. The relative importance of ocean nonlinearities and the increased occurrence of large waves can be measured by integral statistics such as the wave skewness λ_{3} and the excess kurtosis λ_{40} of the zeromean surface elevation η(t):
Here, overbars imply statistical averages and σ is the standard deviation of surface wave elevations. Here and in the following we refer to the Methods section for the definitions of the wave parameters and details.
The skewness coefficient represents the principal parameter with which we describe the effects of secondorder bound nonlinearities on the geometry and statistics of the sea surface with higher sharper crests and shallower more rounded troughs^{9,32,33}. The excess kurtosis comprises a dynamic component due to thirdorder quasiresonant wavewave interactions and a bound contribution induced by both second and thirdorder bound nonlinearities,^{9,10,32,33,39,40}. In order to compare the relative orders of nonlinearities, we consider the characteristic wave steepness μ_{m} = k_{m}σ, where k_{m} is the wavenumber corresponding to the mean spectral frequency ω_{m}^{32}.
Return period of a wave whose crest height exceeds a given threshold
To describe the statistics of rogue waves, we consider the conditional return period N_{h}(ξ) of a wave whose crest height exceeds the threshold h = ξH_{s}, namely
where P(ξ) is the probability of a wave crest height exceeding ξH_{s}. Equation (2) implies that the threshold ξH_{s}, with H_{s} = 4σ, is exceeded on average once every N_{h}(ξ) waves.
For weakly nonlinear random seas, the probability P can be described by the (thirdorder) TF, (secondorder Tayfun) T or (linear Rayleigh) R distributions. In particular^{33},
where ξ_{0} follows from the quadratic equation ^{32}. Here, the wave steepness μ = λ_{3}/3 is of O(μ_{m}) and it is a measure of secondorder bound nonlinearities as it relates to the skewness of surface elevations^{9}. The relationship λ_{3} = 3μ is originally due to Tayfun^{31}, who derived it for narrowband nonlinear waves that display a vertically asymmetric profile with sharper and higher crests and shallower and more rounded troughs. As such this sort of asymmetry is also reflected in a quantitative sense in the skewness coefficient λ_{3} of surface elevations from the mean sea level. Although the relationship was thought to be appropriate to only narrowband waves, Fedele & Tayfun^{9} have more recently verified that it is also valid for broadband waves. In simple terms, μ = λ_{3}/3 serves as a convenient relative measure of the characteristic cresttrough asymmetry of ocean waves. For narrowband (NB) waves in intermediate water depth, Tayfun^{41} derived a compact expression that reduces to the simple form λ_{3,NB} = 3μ_{m} in deep water^{32} (see Methods section for details). The parameter Λ in Eq. (3) is a measure of thirdorder nonlinearities as a function of the fourth order cumulants of the wave surface^{33}. Our studies show that it is approximated by Λ_{appr} = 8λ_{40}/3 (see Methods section). For secondorder seas, hereafter referred to as Tayfun sea states^{42}, Λ = 0 only and P_{TF} in Eq. (3) yields the Tayfun (T) distribution^{32}
For Gaussian seas, μ = 0 and Λ = 0 and P_{TF} reduces to the Rayleigh (R) distribution
We point out that the Tayfun distribution represents an exact result for large second order wave crest heights and it depends solely on the steepness parameter defined as μ = λ_{3}/3^{9}. In the following, we will not dwell on wave heights^{43,44} as our main focus will be the statistics of crest heights in oceanic rogue sea states.
Excess kurtosis
For thirdorder nonlinear random seas the excess kurtosis
comprises a dynamic component due to nonlinear quasiresonant wavewave interactions^{8,40} and a Stokes bound harmonic contribution ^{45}. Janssen^{45} derived a complex general formula for the bound excess kurtosis. For narrowband (NB) waves in intermediate water depth, the formula is more compact (see Eq. (A23) in^{45} and Methods section). In deep water it reduces to the simple form ^{40,45,46} where λ_{3,NB} = 3μ_{m}^{9,32,33}. As for the dynamic component, Fedele^{29} recently revisited Janssen’s^{8} weakly nonlinear formulation for . In deep water, this is given in terms of a sixfold integral that depends on the BenjaminFeir index BFI and the parameter , which is a dimensionless measure of the multidirectionality of dominant waves, with ν the spectral bandwidth and σ_{θ} the angular spreading^{40,47}. As waves become 1D waves R tends to zero. Note that the R − values for the three rogue sea states in Table 1 range from 0.4 to 0.6.
For deepwater narrowband waves characterized by a Gaussian type directional spectrum, the sixfold integral can be reduced to a onefold integral, so that the dynamic excess kurtosis is computed as^{29}
where ω_{m} is the mean spectral frequency, ν the spectral bandwidth, and Im(x) denotes the imaginary part of x. In the focusing regime (0 < R < 1) the dynamic excess kurtosis of an initially homogeneous Gaussian wave field grows, attaining a maximum at the intrinsic time scale . Thus, the sea state initially deviates from being Gaussian, but eventually the excess dynamic kurtosis tends monotonically to zero as energy spreads directionally, in agreement with numerical simulations^{48}. The dynamic excess kurtosis maximum is well approximated by^{29}
where (which corrects a misprint in^{29}) and b = 2.48. In contrast, in the defocusing regime (R > 1) the dynamic excess kurtosis is always negative. It reaches a minimum at t_{c} and then tends to zero over larger periods of time. In summary, the theoretical predictions indicate a decaying trend for the dynamic excess kurtosis over large times in multidirectional wave fields (R > 0).
In unidirectional (R = 0) seas, energy is ‘trapped’ as in a long waveguide. An initially homogeneous Gaussian wave field evolves as the dynamic excess kurtosis monotonically increases toward an asymptotic nonzero value given by from Eq. (8)^{49}. Clearly, wave energy cannot spread directionally and quasiresonant interactions induce nonlinear focusing and large breathertype waves initiated by modulation instability^{16,17,20,21,22,23,50}. However, realistic oceanic wind seas are typically multidirectional (shortcrested) and energy can spread directionally. As a result, nonlinear focusing due to modulational instability effects diminishes^{16,18,29,51} or becomes essentially insignificant under realistic oceanic conditions^{29}. Indeed, the large excess kurtosis transient observed during the initial stage of evolution is a result of the unrealistic assumption that the initial wave field is homogeneous Gaussian whereas oceanic wave fields are usually statistically inhomogeneous both in space and time. Further, for time scales , starting with initial homogeneous and Gaussian conditions becomes irrelevant as the wave field tends to a nonGaussian state dominated by bound nonlinearities as the total kurtosis of surface elevations asymptotically approaches the value represented by the bound component^{52,53}.
These results and conclusions hold for deepwater gravity waves. The extension to intermediate water depth d readily follows by redefining the BenjaminFeir Index as ^{40,54}, where the depth factor α_{S} depends on the dimensionless depth k_{m}d, with k_{m} the wavenumber corresponding to the mean spectral frequency (see Methods section). In the deepwater limit α_{S} becomes 1. As the dimensionless depth k_{m}d decreases, α_{S} decreases and becomes negative for k_{m}d < 1.363 and so does the dynamic excess kurtosis. For the three rogue sea states under study, depth factors are less than 1 and given in Table 1 together with the associated BFI and R coefficients. From Eq. (8), the maximum dynamic excess kurtosis is of O(10^{−3}) for all three sea states and thus negligible in comparison to the associated narrowband (NB) bound component of O(10^{−2}) (see Methods section). Hereafter, this will be confirmed further by a quantitative analysis of High Order Spectral (HOS) simulations of the Euler equations^{36}.
Results
At present, whether secondorder or thirdorder nonlinearities play a dominant role in roguewave formation is a subject of considerable debate. Recent theoretical results clearly show that thirdorder quasiresonant interactions play an insignificant role in the formation of large waves in realistic oceanic seas^{29}. Further, oceanic evidence available so far^{31,33,34} seems to suggest that the statistics of large oceanic wind waves are not affected in any discernible way by thirdorder nonlinearities, including NLStype modulational instabilities that attenuate as the wave spectrum broadens^{24}. Indeed, extensive analyses of stormgenerated extreme waves do not display any data trend even remotely similar to the systematic breathertype patterns observed in 1D wave flumes^{10,31,33,34}. However, thirdorder bound nonlinearities may affect both skewness and kurtosis as they shape the wave surface with sharper crests and shallower troughs.
In the following we will compare second and thirdorder nonlinear properties of the sea states where the Draupner, Andrea and Killard rogue waves occurred, using HOS simulations of the Euler equations^{36}. To do so, we first use WAVEWATCH III to hindcast the three rogue sea states. The respective directional spectra S(ω, θ) are shown in Fig. 1. These are used to define the initial wave field conditions for the HOS simulations–see the Methods section.
Secondorder vs thirdorder nonlinearities
The time evolutions of skewness and excess kurtosis of the three simulated rogue sea states are shown in Fig. 2. Initially, the two statistics undergo a brief artificial transient of O(10) mean wave periods during which nonlinearities are smoothly activated by way of a ramping function^{55} applied to the HOS equations. Following this stage, we do not observe the typical overshoot beyond the Gaussian value as seen in wave tank measurements and simulations^{8,16,17,50}. In contrast, both statistics rapidly reach a steady state as an indication that quasiresonant wavewave interactions due to modulation instabilities are negligible in agreement with theoretical predictions^{29}. Indeed, the largetime kurtosis is mostly Gaussian for all the three sea states and there are insignificant differences between secondorder and thirdorder HOS simulations. Further, Fig. 2 shows that the narrowband predictions slightly overestimate the observed simulated values for skewness and excess kurtosis. This is simply because narrowband approximations do not take into account the directionality and the finite bandwidth of the spectrum.
Our main conclusion is that secondorder bound nonlinearities mainly affect the largetime skewness λ_{3} whereas excess kurtosis is smaller since it is of ^{39,40} (see also Methods section). Clearly, secondorder effects are the dominant factors in shaping the probability structure of the random sea state with a minor contribution of excess kurtosis effects. Such dominance is seen in Fig. 3, where the HOS numerical predictions of the conditional return period N_{h}(ξ) of a crest exceeding the threshold ξH_{s} are compared against the theoretical predictions based on the linear Rayleigh (R), secondorder Tayfun (T) and thirdorder (TF) models from Eq. (3). In particular, N_{h}(ξ) follows from Eq. (2) as the inverse 1/P(ξ) of the empirical probabilities of a crest height exceeding the threshold ξH_{s}. An excellent agreement is observed between simulations and the thirdorder TF model, which is nearly the same as the secondorder T model. This indicates that secondorder effects are dominant, whereas the linear Rayleigh model underestimates the empirical return periods.
For both second and thirdorder nonlinearities, the return period N_{r} of a wave whose crest height exceeds the rogue threshold 1.25H_{s} is nearly 2 · 10^{4} for the Andrea, Draupner and Killard sea states. Oceanic rogue wave measurements^{34} indicate that the rogue threshold for crest heights is exceeded on average once every N_{r} ~ 10^{4} waves. Similarly, recent measurements off the west coast of Ireland^{35} yield N_{r} ~ 6 · 10^{4}. In contrast, in a Gaussian sea the same threshold is exceeded more rarely and on average once every 3 · 10^{5} waves.
Note that all three rogue waves have crest heights that exceed the threshold 1.5H_{s}. This is exceeded on average once every 5 · 10^{5} waves in second and thirdorder seas and extremely rarely in Gaussian seas, i.e. on average once every 6 · 10^{7} waves. This implies that the three rogue wave crest events are likely to be rare occurrences of weakly secondorder random seas, or Tayfun sea states^{42}. Our results clearly confirm that rogue wave generation is the result of the constructive interference (focusing) of elementary waves enhanced by secondorder nonlinearities in agreement with the theory of stochastic wave groups proposed by Fedele and Tayfun^{9}, which relies on Boccotti’s^{43} theory of quasideterminism^{43}. Our conclusions are also in agreement with observations^{9,10,31,33}.
Comparison of the profiles of three rogue waves
For all three rogue sea states under study, the largest wave observed in the HOS simulations is now compared against the actual rogue wave measurements. Figure 4 compares the actual wave profiles (thin solid line) with the largest secondorder (thin dotteddashed line) and thirdorder (thick solid line) simulated waves. While there are small differences between the two orders, secondorder nonlinearities are sufficient in predicting the observed profiles with sufficient accuracy.
In the same figure, the simulated mean sea level (MSL) below the crests is also shown. The estimation of the MSL follows by lowpass filtering the measured time series of the wave surface with frequency cutoff f_{c} ~ f_{p}/2, where f_{p} is the frequency of the spectral peak^{56}. Note that the time series must be long enough and contain at least ~200 waves for a statistically robust estimation of wavewave interactions. In shorter time series, a setup is observed as a manifestation of the large crest segment that extends above the adjacent lower crests. The HOS simulations give approximately the same MSL for both second and thirdorder nonlinearities predicting a setdown below the large crests as expected from theory^{57}. However, the observed Draupner setup (thin line) is not reproduced by our HOS numerical simulations (see Fig. 4). We also note that the HOS MSL is close to the narrowband prediction ST_{NB} (see Table 1 and Methods section for definitions). The actual MSL for Andrea is not available and buoy observations give neither a setup nor a setdown for Killard.
Taylor et al.^{58} reported that for the Draupner wave the hindcast from the European Centre for MediumRange Weather Forecasts shows swell waves propagating at approximately 80 degrees to the wind sea. They argued that the Draupner wave may be due to the crossing of two almost orthogonal wave groups in accord with secondorder theory. This would explain the setup observed under the large wave^{56} instead of the secondorder setdown normally expected^{57}. Note that the angle between the two dominant sea directions lies outside the range ~20–60 degrees where modulation instability is enhanced^{59}.
Further studies and a high resolution hindcast of the Draupner sea state are needed to clarify if it was a crossingseas situation as our WAVEWATCH III hindcast spectrum does not indicate so. Concerning the disagreement for the Draupner wave on the setup, we have conducted numerical HOS experiments where the input spectrum consists of two identical JONSWAP type crossing sea states at 90 degrees. And we indeed found a setup. As a matter of fact, whether one obtains a setup or a setdown depends on the angle between the crossing seas. As the angle increases, the setdown turns into a setup – see Fig. 5. However, we still find that secondorder effects are dominant and thirdorder contributions on skewness and kurtosis, mainly due to bound nonlinearities, are negligible.
Our results are in agreement with the recent numerical simulations by Trulsen et al.^{42} of the crossing sea state encountered during the accident of the tanker Prestige on 13 November 2002. Puzzled by the literature on crossing seas states, they checked whether the fact that the accident occurred during a bimodal sea state with two wave systems crossing nearly at a right angle increased or not the chance of encountering a rogue wave. They concluded that the wave conditions at the time of the accident were only slightly more extreme than those of a Gaussian sea state and slightly less extreme than those of a secondorder Tayfun sea state^{32}.
Discussion
Since the 1990s, modulational instability^{11,12} of a class of solutions to the NLS equation has been proposed as a mechanism for rogue wave formation^{3,8,13,14,15}. The availability of exact analytical solutions of 1D NLS breathers^{13} via the Inverse Scattering Transform^{60} enormously stimulated new research on rogue waves. In particular, it has been found that in 1D wave fields, the latestage evolution of modulation instability leads to large waves in the form of breathers^{13,14,15}. Indeed, in such situations energy is ‘trapped’ as in a long waveguide and quasiresonant interactions are effective in inducing large breathers via nonlinear modulation before wave breaking occurs^{16,17,20,21}. However, rogue waves in the form of breathers can be observed experimentally in 1D waves only at sufficiently small values of wave steepness (~0.01–0.09)^{20,21,22}. Indeed, wave breaking is inevitable for wave steepness larger than 0.1: ‘breathers do not breathe’^{23} and their amplification is smaller than that predicted by the NLS equation, as confirmed by numerical simulations^{27,28}.
Clearly, typical oceanic wind seas are shortcrested, or multidirectional wave fields and their dynamics is more ‘free’ than the 1D ‘longwaveguide’ counterpart. Indeed, energy can spread directionally and as a result nonlinear focusing due to modulational instability is diminished^{16,18,29}. Our results suggest that in typical oceanic fields thirdorder nonlinearities do not play a significant role in the wave growth.
Furthermore, we found that skewness effects on crest heights are dominant in comparison to bound kurtosis contributions and statistical predictions can be based on secondorder models^{32,33,61}. Thus, rogue waves are likely to be rare occurrences resulting from the constructive interference (dispersive and directional focusing) of elementary waves enhanced by second order nonlinear effects in agreement with observations^{9,10,31,33} and with the theory of stochastic wave groups^{9}. This theory about the mechanics of wave groups shows that they can be thought of as genes of a nonGaussian sea dominated by secondorder nonlinearities, when interested in the dynamics of large surface displacements. The spacetime evolution of wave crests during an extreme event can be seen in the Supplementary Video S1 of the simulated Killard rogue wave sea state analyzed in this paper. We anticipate that our results may motivate similar analysis of waves over a wider distribution of heights using extensive data sets^{34}.
Methods
Wave parameters
The significant wave height H_{s} is defined as the mean value H_{1/3} of the highest onethird of wave heights. It can be estimated either from a zerocrossing analysis or more easily from the wave omnidirectional spectrum as H_{s} ≈ 4σ, where is the standard deviation of surface elevations, m_{j} = ∫S(ω)ω^{j}dω are spectral moments and S(ω, θ) is the directional wave spectrum.
The dominant wave period T_{p} = 2π/ω_{p} refers to the frequency ω_{p} of the spectral peak. The mean zerocrossing wave period T_{0} is equal to 2π/ω_{0}, with . The associated wavelength L_{0} = 2π/k_{0} follows from the linear dispersion relation , with d the water depth. The mean spectral frequency is defined as ω_{m} = m_{1}/m_{0}^{32} and the associated mean period T_{m} is equal to 2π/ω_{m}. A characteristic wave steepness is defined as μ_{m} = k_{m}σ, where k_{m} is the wavenumber corresponding to the mean spectral frequency ω_{m}^{32}. The following quantitites are also introduced: q_{m} = k_{m}d, Q_{m} = tanhq_{m}, the phase velocity c_{m} = ω_{m}/k_{m}, the group velocity c_{g} = c_{m}[1 + 2q_{m}/sinh(2q_{m})]/2. The spectral bandwidth gives a measure of the frequency broadening. The angular spreading is estimated as , where and ^{62}. Note that .
The parameter Λ = λ_{40} + 2λ_{22} + λ_{04} is a measure of thirdorder nonlinearities and is a function of the fourth order cumulants λ_{nm} of the wave surface η and its Hilbert transform ^{33}. In particular, and . In practice, Λ is usually approximated solely in terms of the excess kurtosis as Λ_{appr} = 8λ_{40}/3 by assuming the relations between cumulants^{49} λ_{22} = λ_{40}/3 and λ_{04} = λ_{40}. These, to date, have been proven to hold for linear and secondorder narrowband waves only^{39}. For thirdorder nonlinear seas, our numerical studies indicate that Λ ≈ Λ_{appr} within a 3% relative error in agreement with observations^{19,63}.
The wave steepness μ = λ_{3}/3 relates to the wave skewness λ_{3} of surface elevations. For narrowband (NB) waves in intermediate water the wave skewness^{41} and bound excess kurtosis^{45}
where
with the phase velocity in shallow water. The waveinduced setdown or mean sea level variation below a crest of amplitude h is ST_{NB} = Δh^{2 }^{45}. In deep water,
Note that Eq. (9) are not valid in small water depth as second and thirdorder terms of the associated Stokes expansion can be larger than the linear counterpart (see Eq. (A18) in^{45}). To be valid, the constraints αμ_{m} ≤ 1 and βμ_{m}/α ≤ 1 must hold. And indeed they are satisfied for the three rogue sea states under study. The depth factor α_{S} depends on k_{m}d through of a lengthy expression, which is not reported here for the sake of simplicity – see Janssen and Onorato^{54}.
Brief description of WAVEWATCH III and hindcast validation
WAVEWATCH III^{62,64} is a third generation wave model developed at NOAA/NCEP that solves the spectral energy action balance equation with a source function representing the wind input, wavewave interactions and the wave energy dissipation due to diverse processes. The configuration of the model was set to solve the balance equation from a minimum frequency of 0.0350 Hz up to 0.5552 Hz for 36 directional bands and 30 frequencies. A JONSWAP spectrum was set as an initial condition at every grid point. We used the wind input fields from the NOAA Climate Forecast System Reanalysis (CFSR)^{64}.
Higher Order Spectral Method
The HOS method is a numerical pseudo spectral method to solve the Euler equations governing the dynamics of incompressible fluid flow at a desired level of nonlinearity. In particular, the time evolution of the free surface of the fluid, η(x, y, t) and the associated velocity potential ψ(x, y, t) evaluated on the free surface are obtained. The method was independently developed in 1987 by Dommermuth & Yue^{36} and West et al.^{65}. Within the present work, West et al.’s version is employed. Tanaka^{66} provides a thorough description of the method.
Initial conditions for the potential ψ and surface elevation η are obtained from the directional spectrum as an output of WAVEWATCH III. In the wavenumber domain, the Fourier transform of η is constructed as S(k)exp(iβ), where β is normally distributed over [0, 2π]. Similarly, the Fourier transform of ψ is obtained via linear wave theory and finally an inverse Fourier transform is applied. The numerical simulation is performed using 1024 × 1024 Fourier modes and over a time scale , where μ_{m} represents a characteristic wave steepness defined above. A lowpass filter is applied to avoid numerical blowup.
Finally, we note that the use of the WAVEWATCH III model combined with HOS simulations may prove useful in assessing recently proposed techniques for rogue wave predictability based on chaotic time series analysis^{67,68} and thirdorder probabilistic models of unexpected wave extremes^{69}.
Additional Information
How to cite this article: Fedele, F. et al. Real world ocean rogue waves explained without the modulational instability. Sci. Rep. 6, 27715; doi: 10.1038/srep27715 (2016).
References
Haver, S. A possible freak wave event measured at the Draupner Jacket January 1 1995. Proc. of Rogue waves 2004, 1–8 (2004).
Dysthe, K. B., Krogstad, H. E. & Muller, P. Oceanic rogue waves. Annual Review of Fluid Mechanics 40, 287–310 (2008).
Osborne, A. Nonlinear ocean waves and the inverse scattering transform vol. 97 (Elsevier, 2010).
Magnusson, K. A. & Donelan, M. A. The Andrea wave characteristics of a measured North Sea rogue wave. Journal of Offshore Mechanics and Arctic Engineering 135, 031108–031108 (2013).
BitnerGregersen, E. M., Fernandez, L., Lefèvre, J. M., Monbaliu, J. & Toffoli, A. The North Sea Andrea storm and numerical simulations. Natural Hazards and Earth System Science 14, 1407–1415 (2014).
Dias, F., Brennan, J., Ponce de Leon, S., Clancy, C. & Dudley, J. Local analysis of wave fields produced from hindcasted rogue wave sea states. In ASME 2015 34th International Conference on Ocean, Offshore and Arctic Engineering, OMAE2015–41458 (American Society of Mechanical Engineers, 2015).
Kharif, C. & Pelinovsky, E. Physical mechanisms of the rogue wave phenomenon. European Journal of Mechanics  B/Fluids 22, 603–634 (2003).
Janssen, P. A. E. M. Nonlinear fourwave interactions and freak waves. Journal of Physical Oceanography 33, 863–884 (2003).
Fedele, F. & Tayfun, M. A. On nonlinear wave groups and crest statistics. J. Fluid Mech 620, 221–239 (2009).
Fedele, F. Rogue waves in oceanic turbulence. Physica D 237, 2127–2131 (2008).
Alber, I. E. The effects of randomness on the stability of twodimensional surface wavetrains. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 363, 525–546 (1978).
Crawford, D. R., Lake, B. M., Saffman, P. G. & Yuen, H. C. Stability of weakly nonlinear deepwater waves in two and three dimensions. Journal of Fluid Mechanics 105, 177–191 (1981).
Peregrine, D. H. Water waves, nonlinear Schrödinger equations and their solutions. Journal of the Australian Mathematical Society Series B 25, 16–43 (1983).
Osborne, A. R., Onorato, M. & Serio, M. The nonlinear dynamics of rogue waves and holes in deepwater gravity wave trains. Phys. Lett. A 275, 386–393 (2000).
Ankiewicz, A., Devine, N. & Akhmediev, N. Are rogue waves robust against perturbations? Physics Letters A 373, 3997–4000 (2009).
Onorato, M. et al. Statistical properties of mechanically generated surface gravity waves: a laboratory experiment in a threedimensional wave basin. Journal of Fluid Mechanics 627, 235–257 (2009).
Shemer, L. & Sergeeva, A. An experimental study of spatial evolution of statistical parameters in a unidirectional narrowbanded random wavefield. Journal of Geophysical Research: Oceans 114, 2156–2202 (2009).
Toffoli, A. et al. Evolution of weakly nonlinear random directional waves: laboratory experiments and numerical simulations. Journal of Fluid Mechanics 664, 313–336 (2010).
Fedele, F., Cherneva, Z., Tayfun, M. A. & Soares, C. G. Nonlinear Schrödinger invariants and wave statistics. Physics of Fluids 22, 036601 (2010).
Chabchoub, A., Hoffmann, N. P. & Akhmediev, N. Rogue wave observation in a water wave tank. Phys. Rev. Lett. 106, 204502 (2011).
Chabchoub, A., Hoffmann, N., Onorato, M. & Akhmediev, N. Super rogue waves: Observation of a higherorder breather in water waves. Phys. Rev. X 2, 011015 (2012).
Shemer, L. & Liberzon, D. Lagrangian kinematics of steep waves up to the inception of a spilling breaker. Physics of Fluids 26, 016601 (2014).
Shemer, L. & Alperovich, S. Peregrine breather revisited. Physics of Fluids 25, 051701 (2013).
Fedele, F. On certain properties of the compact zakharov equation. Journal of Fluid Mechanics 748, 692–711 (2014).
Zakharov, V. E. Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys. 9, 190–194 (1968).
Dyachenko, A. I. & Zakharov, V. E. Compact Equation for Gravity Waves on Deep Water. JETP Lett. 93, 701–705 (2011).
Slunyaev, A. V. & Shrira, V. I. On the highest nonbreaking wave in a group: fully nonlinear water wave breathers versus weakly nonlinear theory. Journal of Fluid Mechanics 735, 203–248 (2013).
Slunyaev, A. et al. Superrogue waves in simulations based on weakly nonlinear and fully nonlinear hydrodynamic equations. Phys. Rev. E 88, 012909 (2013).
Fedele, F. On the kurtosis of ocean waves in deep water. Journal of Fluid Mechanics 782, 25–36 (2015).
Toffoli, A., Benoit, M., Onorato, M. & BitnerGregersen, E. M. The effect of thirdorder nonlinearity on statistical properties of random directional waves in finite depth. Nonlinear Processes in Geophysics 16, 131–139 (2009).
Tayfun, M. A. Distributions of envelope and phase in wind waves. Journal of Physical Oceanography 38, 2784–2800 (2008).
Tayfun, M. A. Narrowband nonlinear sea waves. Journal of Geophysical Research: Oceans 85, 1548–1552 (1980).
Tayfun, M. A. & Fedele, F. Waveheight distributions and nonlinear effects. Ocean Engineering 34, 1631–1649 (2007).
Christou, M. & Ewans, K. Field measurements of rogue water waves. Journal of Physical Oceanography 44, 2317–2335 (2014).
Flanagan, J. et al. ADCP measurements of extreme water waves off the west coast of Ireland. In The Proceedings of the 26th (2016) International Offshore and Polar Engineering, Rhodes, Greece, June 26  July 2, 2016 (International Society of Offshore and Polar Engineers, 2016).
Dommermuth, D. G. & Yue, D. K. P. A highorder spectral method for the study of nonlinear gravity waves. Journal of Fluid Mechanics 184, 267–288 (1987).
Haver, S. Evidences of the existence of freak waves. In Rogue Waves 129–140 (2001).
Xiao, W., Liu, Y., Wu, G. & Yue, D. K. P. Rogue wave occurrence and dynamics by direct simulations of nonlinear wavefield evolution. Journal of Fluid Mechanics 720, 357–392 (2013).
Tayfun, M. A. & Lo, J. Nonlinear effects on wave envelope and phase. J. Waterway, Port, Coastal and Ocean Eng. 116, 79–100 (1990).
Janssen, P. A. E. M. & Bidlot, J. R. On the extension of the freak wave warning system and its verification. Tech. Memo 588, ECMWF (2009).
Tayfun, M. A. Statistics of nonlinear wave crests and groups. Ocean Engineering 33, 1589–1622 (2006).
Trulsen, K., Nieto Borge, J. C., Gramstad, O., Aouf, L. & Lefèvre, J.M. Crossing sea state and rogue wave probability during the Prestige accident. Journal of Geophysical Research: Oceans 120 (2015).
Boccotti, P. Wave Mechanics for Ocean Engineering (Elsevier Sciences, Oxford, 2000).
Alkhalidi, M. A. & Tayfun, M. A. Generalized Boccotti distribution for nonlinear wave heights. Ocean Engineering 74, 101–106 (2013).
Janssen, P. A. E. M. On some consequences of the canonical transformation in the hamiltonian theory of water waves. Journal of Fluid Mechanics 637, 1–44 (2009).
Janssen, P. A. E. M. On a random time series analysis valid for arbitrary spectral shape. Journal of Fluid Mechanics 759, 236–256 (2014).
Mori, N., Onorato, M. & Janssen, P. A. E. M. On the estimation of the kurtosis in directional sea states for freak wave forecasting. Journal of Physical Oceanography 41, 1484–1497 (2011).
Annenkov, S. Y. & Shrira, V. I. Evolution of kurtosis for wind waves. Geophysical Research Letters 36, 1944–8007 (2009).
Mori, N. & Janssen, P. A. E. M. On kurtosis and occurrence probability of freak waves. Journal of Physical Oceanography 36, 1471–1483 (2006).
Shemer, L., Sergeeva, A. & Liberzon, D. Effect of the initial spectrum on the spatial evolution of statistics of unidirectional nonlinear random waves. Journal of Geophysical Research: Oceans 115 (2010).
Waseda, T., Kinoshita, T. & Tamura, H. Evolution of a random directional wave and freak wave occurrence. Journal of Physical Oceanography 39, 621–639 (2009).
Annenkov, S. Y. & Shrira, V. I. Largetime evolution of statistical moments of wind–wave fields. Journal of Fluid Mechanics 726, 517–546 (2013).
Annenkov, S. Y. & Shrira, V. I. Evaluation of skewness and kurtosis of wind waves parameterized by JONSWAP spectra. Journal of Physical Oceanography 44, 1582–1594 (2014).
Janssen, P. A. E. M. & Onorato, M. The intermediate water depth limit of the Zakharov equation and consequences for wave prediction. Journal of Physical Oceanography 37, 2389–2400 (2007).
Dommermuth, D. The initialization of nonlinear waves using an adjustment scheme. Wave Motion 32, 307–317 (2000).
Walker, D., Taylor, P. & Taylor, R. E. The shape of large surface waves on the open sea and the Draupner new year wave. Applied Ocean Research 26, 73–83 (2004).
LonguetHiggins, M. S. & Stewart, R. W. Radiation stresses in water waves: a physical discussion, with applications. DeepSea Research II, 529–562 (1964).
Adcock, T., Taylor, P., Yan, S., Ma, Q. & Janssen, P. Did the Draupner wave occur in a crossing sea? Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science rspa20110049 (2011).
Onorato, M., Proment, D. & Toffoli, A. Freak waves in crossing seas. The European Physical JournalSpecial Topics 185, 45–55 (2010).
Zakharov, V. E. & Shabat, A. B. Exact theory of twodimensional selffocusing and onedimensional selfmodulation of waves in nonlinear media. Soviet PhysicsJETP 34, 62–69 (1972).
Forristall, G. Z. Wave crest distributions: Observations and secondorder theory. Journal of Physical Oceanography 30, 1931–1943 (2000).
Tolman, H. & Group, D. User manual and system documentation of WAVEWATCH III version 4.18. Tech. Rep. Tech. Note 316, NOAA/NWS/NCEP/MMAB (2014).
Tayfun, M. A. & Fedele, F. Expected shape of extreme waves in storm seas. In ASME 2007 26th International Conference on Offshore Mechanics and Arctic Engineering, OMAE2007–29073 (American Society of Mechanical Engineers, 2007).
Chawla, A., Spindler, D. M. & Tolman, H. L. Validation of a thirty year wave hindcast using the climate forecast system reanalysis winds. Ocean Modelling 70, 189–206 (2013).
West, B., Brueckner, K., Janda, R., Milder, M. & Milton, R. A new numerical method for surface hydrodynamics. Journal of Geophysical Research 92, 11803–11824 (1987).
Tanaka, M. A method of studying nonlinear random field of surface gravity waves by direct numerical simulation. Fluid Dynamics Research 28, 41–60 (2001).
Birkholz, S., Brée, C., Demircan, A. & Steinmeyer, G. Predictability of Rogue Events. Phys. Rev. Lett. 114, 213901 (2015).
Birkholz, S., Brée, C., Veselić, I., Demircan, A. & Steinmeyer, G. Random walks across the sea: the origin of rogue waves? arXiv:1507.08102v1 (2015).
Fedele, F. Are rogue waves really unexpected? Journal of Physical Oceanography 46, 1495–1508 (2016).
Acknowledgements
This work is supported by the European Research Council (ERC) under the research projects ERC2011AdG 290562MULTIWAVE and ERC2013PoC 632198WAVEMEASUREMENT and Science Foundation Ireland under grant number SFI/12/ERC/E2227. F.F. is grateful to George Z. Forristall and M. Aziz Tayfun for sharing the Draupner wave measurements utilized in this study. F.F. also thanks Michael Banner, Predrag Cvitanovic and M. Aziz Tayfun for discussions on rogue waves and nonlinear wave statistics. F.F. is grateful to M. Aziz Tayfun for revising an early draft of the manuscript. F.D. is grateful to ESBI for sharing the Killard wave measurements. F.D. and J.B. are grateful to Claudio Viotti for the development of the HOS code used in this study. The Andrea wave data were collected by ConocoPhillips Skandinavia AS. The numerical simulations were performed on the Fionn cluster at the Irish Centre for Highend Computing (ICHEC).
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The concept and design was provided by F.F., who coordinated the scientific effort together with F.D. S.P.D.L. and J.B. performed numerical simulations and developed specific codes for the analysis. The wave statistical analysis was performed by F.F. together with J.B. The overall supervision was provided by F.F. and F.D. F.D. and J.D. made ongoing incisive intellectual contributions. All authors participated in the analysis and interpretation of results and the writing of the manuscript.
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Fedele, F., Brennan, J., Ponce de León, S. et al. Real world ocean rogue waves explained without the modulational instability. Sci Rep 6, 27715 (2016). https://doi.org/10.1038/srep27715
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DOI: https://doi.org/10.1038/srep27715
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