Real world ocean rogue waves explained without the modulational instability

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 second-order bound nonlinearities and not the modulational instability. This implies that rogue waves are likely to be rare occurrences of weakly nonlinear random seas.

Real world ocean rogue waves explained without the modulational instability

R A

P

Francesco Fedele,, Joseph Brennan, Sonia Ponce de Len, John Dudley & Frdric Dias

laboratory experiments and in mathematical studies, but there is no consensus on what actually takes

According to the most commonly used denition, rogue waves are unusually large-amplitude 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 decade16. Several physical mechanisms have been proposed to explain the occurrence of such waves7, including the two competing hypotheses of nonlinear focusing due to third-order quasi-resonant wave-wave interactions8, and purely dispersive focusing of second-order non-resonant or bound harmonic waves, which do not satisfy the linear dispersion relation9,10.

In particular, recent studies propose third-order quasi-resonant interactions and associated modulational instabilities11,12 inherent to the Nonlinear Schrdinger (NLS) equation as mechanisms for rogue wave formation3,8,1315. Such nonlinear eects cause the statistics of weakly nonlinear gravity waves to signicantly dier from the Gaussian structure of linear seas, especially in long-crested or unidirectional (1D) seas8,10,1619. The late-stage

evolution of modulation instability leads to breathers that can cause large waves1315, especially in 1D waves. Indeed, in this case energy is trapped as in a long wave-guide. For small wave steepness and negligible dissipation, quasi-resonant interactions are eective in reshaping the wave spectrum, inducing large breathers via nonlinear focusing before wave breaking occurs16,17,20,21. Consequently, breathers can be observed experimentally in 1D wave elds only at sufficiently small values of wave steepness2022. However, wave breaking is inevitable when the steepness becomes larger: breathers do not breathe23 and their amplication is smaller than that predicted by the NLS equation, in accord with theoretical studies24 of the compact Zakharov equation25,26 and numerical studies of the Euler equations27,28.

Typical oceanic wind seas are short-crested, or multidirectional wave elds. Hence, we expect that nonlinear focusing due to modulational eects is diminished since energy can spread directionally16,18,29. Thus, modulation instabilities may play an insignicant role in the wave growth especially in nite water depth where they are further attenuated30.

Tayfun31 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 second-order non-resonant or bound nonlinearities. Further, the surface statistics follow the Tayfun32 distribution32 in agreement with observations9,10,31,33. This is conrmed by a recent data quality control and statistical analysis of single-point measurements from xed sensors mounted on oshore platforms, the majority of which were recorded in the North Sea34. The analysis of

School Institut FEMTO-ST CNRS-

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Andrea Draupner Killard

Signicant wave height Hs [m] 10.0 11.2 11.4 Dominant wave period Tp [s] 14.3 15.0 17.2 Mean zero-crossing wave period T0 [s] 11.6 12.1 14.0 Mean wavelength L0 [m] 209 219 268 Depth d [m], k0d 74, 2.23 70, 2.01 58, 1.36 Spectral bandwidth 0.35 0.36 0.37 Angular spreading 0.37 0.39 0.34 Parameter

=

2 240 0.56 0.59 0.42 Benjamin Feir Index BFI in deep water8 0.24 0.23 0.18 Depth factor S40 0.31 0.36 0.04

Tayfun NB skewness 3,NB41 0.159 0.165 0.145 Mean skewness 3 from HOS simulations 0.141 0.146 0.142 Maximum NB dynamic excess kurtosis d40, max

R /2

29 2.3103 2.1103 2.7104 Janssen NB bound excess kurtosis NB

b

40,

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 threshold2 1.25Hs occurs on average once every Nr~104 waves with Nr referred to as the return period of a rogue wave and Hs is the signicant wave height. Some even more recent measurements o the west coast of Ireland35 revealed similar statistics with 13 rogue events out of an ensemble of 750873 individual waves and Nr~6104.

To date, it is still under debate if in typical oceanic seas second-order nonlinearities dominate the dynamics of extreme waves as indicated by ocean measurements31,33, or if third-order nonlinear eects play also a signicant, if not dominant, role in rogue-wave 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 third-order nonlinearities. We rely on WAVEWATCH III hindcasts and High Order Spectral (HOS) simulations of the Euler equations for water waves36. In our study, we consider the famous Draupner and Andrea rogue waves and the less well known Killard rogue wave35. 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 Ekosk platform in the North Sea in a water depth d=74m4,5. 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 laser-based wave sensor37. The Killard wave was measured by ESB International on 28 January 2014 by a Waverider buoy o the west coast of Ireland in a water depth d = 39 m. In Table1 we summarize the wave parameters of the three sea states in which the rogue wave occurred and we refer to the Methods section for denitions and details. As one can see, the actual crest-to-trough (wave) heights H and crest heights h meet the classical criteria2 H/Hs> 2

and h/Hs > 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 presented33 together with the essential elements of Tayfuns32 second-order theory for the wave skewness and Janssens8 formulation for the excess kurtosis of multidirectional seas29. Then, we present and compare second-order and third-order 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 rogue-wave predictions.

Non-resonant and resonant wave-wave interactions cause the statistics of weakly nonlinear gravity waves to signicantly dier from the Gaussian structure of linear seas8,10,1618,38. The relative importance of ocean nonlinearities and the increased occurrence of large waves can be measured by integral statistics such as the wave skewness 3 and the excess kurtosis 40 of the zero-mean surface elevation (t):

4 4

= = .

/ , / 3 (1)

40

45 0.065 0.074 0.076 Mean excess kurtosis 40 from HOS simulations 0.041 0.032 0.011

Janssen NB setdown STNB/Hs45, predicted HOS setdown 0.12, 0.08 0.1, 0.11 0.1, 0.07 Maximum crest height h/Hs: observed, numerical 1.63, 1.71 1.55, 1.54 1.44, 1.57 Maximum crest-to-trough (wave) height H/Hs: observed, numerical 2.30, 2.51 2.10, 2.23 2.00, 2.28 Maximum trough-to-crest (wave) height H/Hs: observed, numerical 2.49, 2.67 2.15, 2.09 2.32, 2.29

Table 1. Wave parameters and various statistics of the simulated sea states labelled Andrea, Draupner and Killard. The Killard rogue wave occurred on a water depth of 39m, however the hincast input spectrum could only be computed at an averaged water depth of 58m. Statistical parameters are from an ensemble of 50 HOS simulations of sea states. We refer to the Methods section for the denitions of the wave parameters. Note that two wave heights are given for each wave: the zero-downcrossing value (crest to trough) and the zero-upcrossing value (trough to crest).

3 3

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Here, overbars imply statistical averages and is the standard deviation of surface wave elevations. Here and in the following we refer to the Methods section for the denitions of the wave parameters and details.

The skewness coefficient represents the principal parameter with which we describe the eects of second-order bound nonlinearities on the geometry and statistics of the sea surface with higher sharper crests and shallower more rounded troughs9,32,33. The excess kurtosis comprises a dynamic component due to third-order quasi-resonant wave-wave interactions and a bound contribution induced by both second- and third-order bound nonlinearities,9,10,32,33,39,40. In order to compare the relative orders of nonlinearities, we consider the characteristic wave steepness m= km, where km is the wavenumber corresponding to the mean spectral frequency m32.

To describe the statistics of rogue waves, we consider the conditional return period Nh() of a wave whose crest height exceeds the threshold h=Hs, namely

N = > =

h H P

( ) 1

Pr[ ]

s

where P() is the probability of a wave crest height exceeding Hs. Equation(2) implies that the threshold Hs, with Hs=4, is exceeded on average once every Nh() waves.

For weakly nonlinear random seas, the probability P can be described by the (third-order) TF, (second-order Tayfun) T or (linear Rayleigh) R distributions. In particular33,

= > = +

P h H

( ) Pr[ ] exp( 8 )[1 (4 1)], (3)

TF s 0

0

232. Here, the wave steepness =3/3 is of O(m) and it is a measure of second-order bound nonlinearities as it relates to the skewness of surface elevations9. The relationship 3=3 is originally due to Tayfun31, who derived it for narrowband nonlinear waves that display a vertically asymmetric prole with sharper and higher crests and shallower and more rounded troughs. As such this sort of asymmetry is also reected in a quantitative sense in the skewness coefficient 3 of surface elevations from the mean sea level. Although the relationship was thought to be appropriate to only narrowband waves, Fedele & Tayfun9 have more recently veried that it is also valid for broadband waves. In simple terms, =3/3 serves as a convenient relative measure of the characteristic crest-trough asymmetry of ocean waves. For narrowband (NB) waves in intermediate water depth, Tayfun41 derived a compact expression that reduces to the simple form 3,NB = 3m in deep water32 (see Methods section for details). The parameter in Eq.(3) is a measure of third-order nonlinearities as a function of the fourth order cumulants of the wave surface33. Our studies show that it is approximated by appr=840/3 (see Methods section). For second-order seas, hereaer referred to as Tayfun sea states42, = 0 only and PTF in Eq.(3) yields the Tayfun (T) distribution32

where 0 follows from the quadratic equation

2

= =

T 0

For Gaussian seas, =0 and =0 and PTF reduces to the Rayleigh (R) distribution

= .

P ( ) exp( 8 ) (5)

R

2

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 dened as =3/39. In the following, we will not dwell on wave heights43,44 as our main focus will be the statistics of crest heights in oceanic rogue sea states.

For third-order nonlinear random seas the excess kurtosis

= + (6)

d b 40 40 40

comprises a dynamic component d40 due to nonlinear quasi-resonant wave-wave interactions8,40 and a Stokes bound harmonic contribution b4045. Janssen45 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) in45 and Methods section). In deep water it reduces to the simple form

2

= =

18 2

m NB

2 2, which is a dimensionless measure of the multidirectionality of dominant waves, with the spectral bandwidth and the angular spreading40,47. As waves become 1D waves R tends to zero. Note that the R values for the three rogue sea states in Table1 range from 0.4 to 0.6.

For deep-water narrowband waves characterized by a Gaussian type directional spectrum, the six-fold integral can be reduced to a one-fold integral, so that the dynamic excess kurtosis is computed as29

R /2

d t 40

2

h

1 ( ) , (2)

2

2

2

0

= +

2

0 0

2

+ +

P ( ) exp( 8 ) exp ( 1 1 8 )

2 (4)

2

.

2 40,45,46 where

3,NB=3m9,32,33. As for the dynamic component, Fedele29 recently revisited Janssens8 weakly nonlinear formulation for d40. In deep water, this is given in terms of a six-fold integral that depends on the Benjamin-Feir index

BFI and the parameter

=

NB

b

40,

3,

6 Im 11 2 3 1 2 3

BFI

2

m

=

,

0 2 2 2

+ + +

i iR R

d

(7)

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i 1 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 eld grows, attaining a maximum at the intrinsic time scale

where m is the mean spectral frequency, the spectral bandwidth, =

2 . 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 simulations48. The dynamic excess kurtosis maximum is well approximated by29

t R

1/ 3

3 (which corrects a misprint in29) and b= 2.48. In contrast, in the defocusing regime (R>1) the dynamic excess kurtosis is always negative. It reaches a minimum at tc 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 elds (R>0).

In unidirectional (R= 0) seas, energy is trapped as in a long wave-guide. An initially homogeneous Gaussian wave eld evolves as the dynamic excess kurtosis monotonically increases toward an asymptotic non-zero value given by

= BFI

3 /(3 3)

d NLS

40,

2 from Eq. (8)49. Clearly, wave energy cannot spread directionally, and quasi-resonant interactions induce nonlinear focusing and large breather-type waves initiated by modulation instability16,17,2023,50. However, realistic oceanic wind seas are typically multidirectional (short-crested) and energy can spread directionally. As a result, nonlinear focusing due to modulational instability eects diminishes16,18,29,51 or becomes essentially insignicant under realistic oceanic conditions29. Indeed, the large excess kurtosis transient observed during the initial stage of evolution is a result of the unrealistic assumption that the initial wave eld is homogeneous Gaussian whereas oceanic wave elds are usually statistically inhomogeneous both in space and time. Further, for time scales

t tc, starting with initial homogeneous and Gaussian conditions becomes irrelevant as the wave eld tends to a non-Gaussian state dominated by bound nonlinearities as the total kurtosis of surface elevations asymptotically approaches the value represented by the bound component52,53.

These results and conclusions hold for deep-water gravity waves. The extension to intermediate water depth d readily follows by redefining the Benjamin-Feir Index as

=

BFI BFI

S S

b40, of O(102) (see Methods section). Hereaer, this will be conrmed further by a quantitative analysis of High Order Spectral (HOS) simulations of the Euler equations36.

Results

At present, whether second-order or third-order nonlinearities play a dominant role in rogue-wave formation is a subject of considerable debate. Recent theoretical results clearly show that third-order quasi-resonant interactions play an insignicant role in the formation of large waves in realistic oceanic seas29. Further, oceanic evidence available so far31,33,34 seems to suggest that the statistics of large oceanic wind waves are not aected in any discernible way by third-order nonlinearities, including NLS-type modulational instabilities that attenuate as the wave spectrum broadens24. Indeed, extensive analyses of storm-generated extreme waves do not display any data trend even remotely similar to the systematic breather-type patterns observed in 1D wave umes10,31,33,34.

However, third-order bound nonlinearities may aect both skewness and kurtosis as they shape the wave surface with sharper crests and shallower troughs.

In the following we will compare second and third-order nonlinear properties of the sea states where the Draupner, Andrea and Killard rogue waves occurred, using HOS simulations of the Euler equations36. To do so, we rst use WAVEWATCH III to hindcast the three rogue sea states. The respective directional spectra S(, ) are shown in Fig.1. These are used to dene the initial wave eld conditions for the HOS simulationssee the Methods section.

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 articial transient of O(10) mean wave periods during which nonlinearities are smoothly activated by way of a ramping function55

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 simulations8,16,17,50. In contrast, both statistics rapidly reach a steady state as an indication that quasi-resonant wave-wave interactions due to modulation instabilities are negligible in agreement with theoretical predictions29. Indeed, the large-time kurtosis is mostly Gaussian for all the three sea states and there are insignicant dierences between second-order and third-order 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 nite bandwidth of the spectrum.

Our main conclusion is that second-order bound nonlinearities mainly aect the large-time skewness 3 whereas excess kurtosis is smaller since it is of

O( )

3

2 39,40 (see also Methods section). Clearly, second-order eects are the dominant factors in shaping the probability structure of the random sea state with a minor contribution of

c m c

1 , 0 1,

2 2 +

0

= =

d40, max

3 (2 )

BFI b R

R bR R

(8)

where

=

R 3 3/4

0

2 2 40,54, where the depth factor S depends on the dimensionless depth kmd, with km the wavenumber corresponding to the mean spectral frequency (see Methods section). In the deep-water limit S becomes 1. As the dimensionless depth kmd decreases, S decreases and becomes negative for kmd< 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 Table1 together with the associated BFI and R coefficients. From Eq.(8), the maximum dynamic excess kurtosis is of O(103) for all three sea states and thus negligible in comparison to the associated narrowband (NB) bound component NB

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Figure 1. WAVEWATCH III hindcast directional wave spectra S(, ) used as input for the HOS simulations. Here, is the angular frequency and the direction in degrees. (Le) Andrea, (center) Draupner, (right) Killard. The spectra have been normalized with respect to spectral peak values.

Figure 2. Time evolution of skewness 3 and excess kurtosis 40 for (le) Andrea, (center) Draupnerand (right) Killard sea states; HOS second-order (black solid), HOS third-order (red solid) averages and theoretical predictions of the narrowband Tayfun skewness and Janssen excess bound kurtosis (blue solid, see Eq. (9) in Methods Section). 95% condence bands (dashed) are also shown. Time is normalized by the mean wave period Tm. The statistical parameters are estimated from an ensemble of 50 HOS simulations.

The initial articial transients are excluded from the ensemble averages as they are the result of a ramping function55 applied to the HOS equations to smoothly activate nonlinearities. See Methods section for details and denitions of wave parameters.

excess kurtosis eects. Such dominance is seen in Fig.3, where the HOS numerical predictions of the conditional return period Nh() of a crest exceeding the threshold Hs are compared against the theoretical predictions based on the linear Rayleigh (R), second-order Tayfun (T) and third-order (TF) models from Eq.(3). In particular, Nh()

follows from Eq.(2) as the inverse 1/P() of the empirical probabilities of a crest height exceeding the threshold Hs. An excellent agreement is observed between simulations and the third-order TF model, which is nearly the same as the second-order T model. This indicates that second-order eects are dominant, whereas the linear Rayleigh model underestimates the empirical return periods.

For both second- and third-order nonlinearities, the return period Nr of a wave whose crest height exceeds the rogue threshold 1.25Hs is nearly 2104 for the Andrea, Draupner and Killard sea states. Oceanic rogue wave measurements34 indicate that the rogue threshold for crest heights is exceeded on average once every Nr ~ 104 waves. Similarly, recent measurements o the west coast of Ireland35 yield Nr~6104. In contrast, in a Gaussian sea the same threshold is exceeded more rarely and on average once every 3105 waves.

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Figure 3. Crest height scaled by the signicant wave height () versus conditional return period (Nh) for the (le) Andrea, (center) Draupner and (right) Killard rogue sea states: HOS numerical predictions ()

in comparison with theoretical models (T=second-order Tayfun (light solid lines), TF=third-order(red solid lines) and R=Rayleigh distributions (dark dashes)). Condence bands are also shown (light dashes). Nh() is the inverse of the exceedance probability P()=Pr[h>Hs]. Horizontal lines denote the rogue threshold 1.25Hs2.

Note that all three rogue waves have crest heights that exceed the threshold 1.5Hs. This is exceeded on average once every 5105 waves in second- and third-order seas and extremely rarely in Gaussian seas, i.e. on average once every 6107 waves. This implies that the three rogue wave crest events are likely to be rare occurrences of weakly second-order random seas, or Tayfun sea states42. Our results clearly conrm that rogue wave generation is the result of the constructive interference (focusing) of elementary waves enhanced by second-order nonlinearities in agreement with the theory of stochastic wave groups proposed by Fedele and Tayfun9, which relies on Boccottis43 theory of quasi-determinism43. Our conclusions are also in agreement with observations9,10,31,33.

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. Figure4 compares the actual wave proles (thin solid line) with the largest second-order (thin dotted-dashed line) and third-order (thick solid line) simulated waves. While there are small dierences between the two orders, second-order nonlinearities are sufficient in predicting the observed proles with sufficient accuracy.

In the same gure, the simulated mean sea level (MSL) below the crests is also shown. The estimation of the MSL follows by low-pass ltering the measured time series of the wave surface with frequency cuto fc ~ fp/2,

where fp is the frequency of the spectral peak56. Note that the time series must be long enough and contain at least ~200 waves for a statistically robust estimation of wave-wave interactions. In shorter time series, a set-up 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 third-order nonlinearities predicting a setdown below the large crests as expected from theory57. However, the observed Draupner set-up (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 STNB (see Table1 and Methods section for denitions). The actual MSL for Andrea is not available, and buoy observations give neither a set-up nor a set-down for Killard.

Taylor et al.58 reported that for the Draupner wave the hindcast from the European Centre for Medium-Range 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 second-order theory. This would explain the set-up observed under the large wave56 instead of the second-order set-down normally expected57. Note that the angle between the two dominant sea directions lies outside the range ~2060 degrees where modulation instability is enhanced59.

Further studies and a high resolution hindcast of the Draupner sea state are needed to clarify if it was a crossing-seas situation as our WAVEWATCH III hindcast spectrum does not indicate so. Concerning the disagreement for the Draupner wave on the set-up, 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 set-up. As a matter of fact, whether one obtains a set-up or a set-down depends on the angle between the crossing seas. As the angle increases, the set-down turns into a set-up see Fig.5. However, we still nd that second-order eects are dominant and third-order 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

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Figure 4. Third-order HOS simulated extreme wave proles (red thin solid), second-order HOS proles (blue thin solid) and mean sea levels (MSL) (thin dashed) versus the dimensionless time t/Tp for (le)

Andrea, (center) Draupner and (right) Killard waves. For comparisons, measurements (thick solid) and actual MSLs (thin solid) are also shown. Note that the Killard MSL is insignicant and the Andrea MSL is not available. Tp is the dominant wave period (see Methods section for denitions).

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 second-order Tayfun sea state32.

Discussion

Since the 1990s, modulational instability11,12 of a class of solutions to the NLS equation has been proposed as a mechanism for rogue wave formation3,8,1315. The availability of exact analytical solutions of 1D NLS breathers13 via the Inverse Scattering Transform60 enormously stimulated new research on rogue waves. In particular, it has been found that in 1D wave elds, the late-stage evolution of modulation instability leads to large waves in the form of breathers1315. Indeed, in such situations energy is trapped as in a long wave-guide, and quasi-resonant interactions are eective in inducing large breathers via nonlinear modulation before wave breaking occurs16,17,20,21.

However, rogue waves in the form of breathers can be observed experimentally in 1D waves only at sufficiently small values of wave steepness (~0.010.09)2022. Indeed, wave breaking is inevitable for wave steepness larger than 0.1: breathers do not breathe23, and their amplication is smaller than that predicted by the NLS equation, as conrmed by numerical simulations27,28.

Clearly, typical oceanic wind seas are short-crested, or multidirectional wave elds and their dynamics is more free than the 1D long-wave-guide counterpart. Indeed, energy can spread directionally and as a result nonlinear focusing due to modulational instability is diminished16,18,29. Our results suggest that in typical oceanic elds third-order nonlinearities do not play a signicant role in the wave growth.

Furthermore, we found that skewness eects on crest heights are dominant in comparison to bound kurtosis contributions and statistical predictions can be based on second-order models32,33,61. 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 eects in agreement with observations9,10,31,33 and with the

theory of stochastic wave groups9. This theory about the mechanics of wave groups shows that they can be thought of as genes of a non-Gaussian sea dominated by second-order nonlinearities, when interested in the dynamics of large surface displacements. The space-time 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 sets34.

Methods

The signicant wave height Hs is dened as the mean value H1/3 of the highest one-third of wave heights. It can be estimated either from a zero-crossing analysis or more easily from the wave omnidirectional spectrum

S S

( ) ( , )d

0

0 2 0 . The associated wavelength L0=2/k0 follows from the linear dispersion relation = gk k d

tanh( )

0 0 0 , with d the water depth. The mean spectral frequency is dened as m=m1/m032 and the associated mean period Tm is equal to 2/m. A characteristic wave steepness is dened as m=km, where km is the wavenumber corresponding to the mean spectral frequency m32. The following quantitites are also introduced: qm=kmd, Qm=tanhqm, the phase velocity cm=m/km, the group velocity cg=cm[1+2qm/ sinh(2qm)]/2. The spectral bandwidth =

2 1/2 gives a measure of the frequency broadening. The

= 2 as Hs 4, where = m0 is the standard deviation of surface elevations, mj=S()jd are spectral moments and S(, ) is the directional wave spectrum.

The dominant wave period Tp=2/p refers to the frequency p of the spectral peak. The mean zero-crossing wave period T0 is equal to 2/0, with = m m

/

m m m

( / 1)

0 2 1

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Figure 5. Upper row: crossing directional wave spectra S(, ) computed using two identical JONSWAP spectra with Draupner spectral properties. Lower row: extreme wave proles simulated with third order HOS (red lines) and second order HOS (black lines). In addition, the corresponding mean sea levels are shown (dashed lines). The mean sea levels are scaled by three for emphasis. Crossing angles from le to right: /2, /4, and /8. Note that for the nal case, the relatively small crossing angle results in the spectrum appearing to contain only one dominant peak.

angular spreading is estimated as = +

a b m

(2(1 / ))

2 2

0

1/2, where

= 2 0 and

a d d

S

cos( ) ( , )

0

=

62. Note that

= +

1

m

2

b d d

S

sin( ) ( , )

0

0

0

2.

The parameter = 40+ 222+ 04 is a measure of third-order nonlinearities and is a function of the fourth order cumulants nm of the wave surface and its Hilbert transform

33. In particular,

=

/ 1

22

2 2 4 and

=

/ 3

4 4 . In practice, is usually approximated solely in terms of the excess kurtosis as appr=840/3 by assuming the relations between cumulants49 22=40/3 and 04=40. These, to date, have been proven to hold for linear and second-order narrowband waves only39. For third-order nonlinear seas, our numerical studies indicate that appr within a 3% relative error in agreement with observations19,63.

The wave steepness =3/3 relates to the wave skewness 3 of surface elevations. For narrowband (NB) waves in intermediate water the wave skewness41 and bound excess kurtosis45

04

NB 6 ( ), 24 ( 2( ) ) 43 1 2( ) , (9)

m NB

= + = + + + =

+

b

2

3, 40,

2 2

3,

m NB

+ +

2

where

= = + = =

Q Q

Q Q

2

3

2 3

6

2 2

2 2

2

3 4 ,

m

m

24 3(1 )

64 , 2 ,

14 2

m

cc c

QQ q

S

S g

1 1 ,

(10)

+

m

m m

m

with =

c gd

S the phase velocity in shallow water. The wave-induced set-down or mean sea level variation below a crest of amplitude h is STNB=h245. In deep water,

= = = .

3 , 18 2 (11)

NB m NB

b

2

3, 40,

2

m NB

3,

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Note that Eq.(9) are not valid in small water depth as second and third-order terms of the associated Stokes expansion can be larger than the linear counterpart (see Eq. (A18) in45). To be valid, the constraints m1 and m/ 1 must hold. And indeed they are satised for the three rogue sea states under study. The depth factor S depends on kmd through of a lengthy expression, which is not reported here for the sake of simplicity see Janssen and Onorato54.

WAVEWATCH III62,64 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, wave-wave interactions and the wave energy dissipation due to diverse processes. The conguration of the model was set to solve the balance equation from a minimum frequency of 0.0350Hz up to 0.5552Hz 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 elds from the NOAA Climate Forecast System Reanalysis (CFSR)64.

The HOS method is a numerical pseudo spectral method to solve the Euler equations governing the dynamics of incompressible uid ow at a desired level of nonlinearity. In particular, the time evolution of the free surface of the uid, (x, y, t), and the associated velocity potential (x, y, t) evaluated on the free surface are obtained. The method was independently developed in 1987 by Dommermuth & Yue36 and West et al.65. Within the present work, West et al.s version is employed. Tanaka66 provides a thorough description of the method.

Initial conditions for the potential and surface elevation are obtained from the directional spectrum as an output of WAVEWATCH III. In the wavenumber domain, the Fourier transform

k

( ) of is constructed as S(k) exp(i), where is normally distributed over [0, 2]. Similarly, the Fourier transform

k

( ) of is obtained via linear wave theory, and nally an inverse Fourier transform is applied. The numerical simulation is performed using 1024 1024 Fourier modes and over a time scale

^

T T O

/ ( )

m m

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2 , where m represents a characteristic wave steepness dened above. A low-pass lter is applied to avoid numerical blow-up.

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 analysis67,68 and third-order probabilistic models of unexpected wave extremes69.

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This work is supported by the European Research Council (ERC) under the research projects ERC-2011-AdG 290562-MULTIWAVE and ERC-2013-PoC 632198-WAVEMEASUREMENT, 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 dra of the manuscript. F.D. is grateful to ESBI for sharing the Killard wave measurements. F.D.

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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 High-end Computing (ICHEC).

The concept and design was provided by F.F., who coordinated the scientic eort together with F.D. S.P.D.L. and J.B. performed numerical simulations and developed specic 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.

Supplementary information accompanies this paper at http://www.nature.com/srep

Competing nancial interests: The authors declare no competing nancial interests.

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).

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