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The physics of anomalous (‘rogue’) ocean waves

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Reports on Progress in Physics Rep. Prog. Phys. 77 (2014) 105901 (17pp)

doi:10.1088/0034-4885/77/10/105901

Review Article

The physics of anomalous (‘rogue’) ocean waves Thomas A A Adcock and Paul H Taylor Department of Engineering Science, University of Oxford, UK E-mail: [email protected] Received 11 May 2014, revised 13 August 2014 Accepted for publication 20 August 2014 Published 14 October 2014 Invited by G Gillies Abstract

There is much speculation that the largest and steepest waves may need to be modelled with different physics to the majority of the waves on the open ocean. This review examines the various physical mechanisms which may play an important role in the dynamics of extreme waves. We examine the evidence for these mechanisms in numerical and physical wavetanks, and look at the evidence that such mechanisms might also exist in the real ocean. Keywords: rogue wave, freak wave, ocean wave, non-linear (Some figures may appear in colour only in the online journal)

This review deals only with wind generated ocean waves, whose periods are generally less than ∼20 s. Waves with longer periods than this exist in the ocean (tides, storm surges, tsunamis) but these are outside the scope of this study other than where we consider the interaction between wind waves and currents. The subject of rogue waves is of great practical significance to engineers and naval architects. In many locations worldwide, wave loading is the dominant environmental load and for a fixed structure it is important to raise the deck of a platform, or the blades of a wind turbine, above the level at which they will encounter ‘green water’ (i.e. not just spray but the ‘solid’ wave itself). For ships and floating platforms the period and shape of the largest waves may be as significant for stability as any extra amplitude from the mechanisms discussed in this review, and thus it is important to determine not just whether large waves are anomalous in amplitude but also in their shape. Some excellent reviews of rogue waves have been written [3–6] as well as one complete book [7]; inevitably this review overlaps with these but in general we have endeavoured to approach the topic in a different way. This article confines itself to discussing rogue waves in the ocean. Some of the developments made in this field have been applied to rogue waves in other media: notably in optics

1. Introduction

Ocean waves are studied by mathematicians, physicists, and engineers who all, in their different ways, seek to model and understand them. Theoretical understanding of water waves started with the work of Airy [1] and Stokes [2]. However, serious research into understanding what actually happens in the open ocean, and applying theory to understand this, only really started during the Second World War driven by the need for amphibious landings over open beaches in locations exposed to sizable waves. Theories and models for explaining and forecasting waves have developed which, byand-large, show good agreement with measurements in the real ocean. However, there remain doubts, partly due to the lack of sufficient field data, as to whether the standard models developed account for all that goes on in the real ocean: are there waves where to obtain a satisfactory understanding we have to include physical aspects which can be ignored for most waves—and might such physics produce waves which are bigger, maybe even far bigger, than our standard models would predict? This article reviews some of the physical mechanisms which it has been suggested might lead to anomalous waves with the focus on whether such physics is actually likely to be important given our knowledge of how large waves form in the open ocean. 0034-4885/14/105901+17$88.00

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Figure 1. Examples of directional wave spectra taken from PaciOOS. Left—example of a ‘following’ sea state; right—example of a crossing sea state.

• The rms directional spreading parameter σ which we present in degrees rather than radians.

[8, 9]. An recent review paper discusses the different media in which rogue waves can occur [10]. There are analogies between the rogue waves in different media but caution must be applied. In many multi-dimensional media the governing equations which can produce instabilities are ‘focusing’ or ‘defocusing’ in two or more dimensions, whereas in the ocean wave problem we have focusing in the mean wave direction and de-focusing laterally. However, some other media do have similar properties [11].

It is normally assumed that the wave spectrum is stationary for a finite period over which we can collect statistics of individual waves. This is a major assumption and should perhaps be questioned more than it is in the literature (and more than we will in this review!). The sea state will change due to three dominant processes: energy input from wind; non-linear redistribution of energy within the spectrum; and dissipation due to wave breaking. In the most extreme seas, which are the ones we are perhaps most interested in when thinking about rogue waves, all of these terms may be large. Note that saying each of these terms is large does not in-ofitself imply the sea state is changing—these processes could be in perfect balance—however, if they are not then the spectrum might change rapidly enough to make our assumption of a stationary spectrum highly questionable. It is also possible that whilst one parameter such as Hs may appear stationary, this could mask quite substantial changes in the spectral shape or directionality. Spectra may be obtained either from a direct field measurement, or be inferred from satellite measurements, or from a numerical model (e.g. [15, 16]). In practice, these may be combined so that satellite data may be assimilated into the wave model, or the wave model (particularly for nearshore locations) is tuned by comparing to direct measurements. Models may be run using historical wind data to predict waves in the past (hindcasting), or take wind model predictions to predict the waves at present (nowcasting), or in the future (forecasting). Numerical models of wave spectra generally show excellent accuracy although they may struggle in unusual sea states such as cyclones. A practical issue of using such spectra is their directional resolution as these tend to be rather coarse and several of the mechanisms discussed in this review are very sensitive to small changes in the directional bandwidth. Numerical models may also overestimate the spectral bandwidth of the frequency spectrum [17] again with implications for rogue wave prediction and analysis.

2. The background sea state

The study of rogue waves must start with understanding the sea state—i.e. how much energy there is in waves on the ocean surface and how is this distributed in frequency and direction. This information is usually described by the spectrum. Figure 1 shows two example spectra: a ‘following’ sea when most of the wave energy is concentrated in one direction; and a ‘crossing’ sea where waves are propagating in different directions. Numerous attempts have been made to parametrize spectra (see discussion in [12]). In this article the following parameters are used: √ • The significant wave-height Hs defined as 4 m0 , where m0 is the
zeroth-moment of the power spectrum given ∞ by m0 = 0 S(ω) dω and equal to the variance of the vertical surface time history. The significant wave-height is usually approximately the average height of the largest third of the waves. • A representative peak frequency for the spectrum and peak wavenumber1 . • The JONSWAP spectrum [14] is used as a model for the spectral shape. The important parameter of this is the peak enhancement factor, γ above a background Pierson– Moskowitz spectrum. The relationship between this factor and the spectral bandwidth is discussed in section 6. 1 In this review we assume peak frequency and peak wavenumber are related by the linear dispersion relation although this is not formally always true [13].

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Figure 2. Extreme wave crest and trough from the Hurricane Camille data set.

This review mainly discusses abnormally high wavecrests. A highly related problem is that of abnormally large wave-troughs (‘holes in the sea’). Many of the physical mechanisms discussed in this review apply to extreme wave troughs as well as crests. For floating objects (e.g. ships) holes in the sea can be at least as dangerous as large crests. Figure 2 shows examples of extreme wave crests and extreme wave troughs taken from the Hurricane Camille data set and measured using a Baylor wave staff on magnetic tape and subsequently digitized [19, 20] (see also Santo et al [21]). Up to this point, and indeed for most of the rest of this review, we base our thinking of a rogue wave on the timeseries of the free surface at a single point (an Eulerian point measurement). Of course, in reality the wave is a 2D time varying surface. Once we start to think about the problem in two dimensions plus time, defining a rogue wave becomes still more difficult: even what constitutes an individual wave is poorly defined. In this review we argue that to understand rogue waves one needs to understand how the wave forms spatially and temporally. A starting point for defining a rogue wave when additional information is available is however the same as above—if the wave deviates significantly from the evolution expected under a model based on linear wave dynamics then we may consider it a rogue wave.

3. Definition of a rogue wave

The terms ‘rogue’ wave and ‘freak’ wave are used interchangeably both in the vernacular and in the technical literature on this topic2 . The term freak wave was introduced scientifically by Draper [18] although his paper argues that many waves which are called freaks in the vernacular are in fact predictable. The key point that Draper makes is that very large waves will occur in sea states in which the significant wave-height is very large—a ‘freak’ wave should be one which is much larger than the waves surrounding it. It has become conventional to define a freak or rogue wave as one which exceeds some criterion (or criteria) based on the background sea state defined by the significant waveheight, Hs . The most commonly used criteria are H /Hs > 2

and/or

ηc /Hs > 1.25,

(1)

where H is the crest to trough height of an individual wave and ηc is the height of wave crest above still water level. To some extent these definitions are unsatisfactory. In a standard linear model of wave dynamics, where the extra elevation due to bound second-order waves is accounted for, these criteria will be exceeded approximately once in every 3000 waves. As such, a rogue wave is not an unusual occurrence. If one observed the water level at one spot for 24 h during a storm with zero-upcrossing period, Tz of 10 s then the probability of not observing a rogue wave is 0.055— you are almost certain to observe some freak waves using the definition above. What interests scientists and engineers is when there are more waves which exceed the freak wave criteria than are expected by a simple linear, or even second-order, model of ocean wave dynamics. Haver and Andersen [3] have proposed this as the definition of what constitutes a rogue wave.

4. Evidence for rogue waves in the open ocean 4.1. Anecdotal evidence

The Authors are unable to recall a conversation with a mariner in which the mariner has not recounted to them a story of their interaction with a rogue wave. This should be unsurprising given that most mariners will have encountered many more than 3000 storm waves and are therefore very likely to have encountered what would be classed as a rogue wave.

2 Recently some publications have adopted the term freak or rogue wave to refer to any steep wave but generally this is not used in the mainstream literature.

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There is limited scientific use to such observations beyond a general reassurance to scientists and engineers that what we measure in the ocean and in our laboratories has some basis in reality. There are some common themes in anecdotal accounts. Many speak of ‘walls of water’ being formed—but it is difficult to know whether this corresponds to the width of the extreme wave crest actually being wider than expected by linear theory. Perhaps more convincing are accounts of abnormal waves propagating in directions different to those in the rest of the sea state—but allowance must be made for the vulnerability of any ship to a wave hitting it broadsides, meaning any incident like this would receive special notice by mariners. We note that many losses at sea are attributed to rogue waves. Doubtless some ships are lost to such encounters, but it is obviously sometimes convenient to blame a rogue wave rather than inadequate maintenance or poor seamanship.

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we can try to use physical models to examine how the largest waves in a sea state will differ from the majority of waves.

4.2. One off measurements

It is obvious that one cannot conclude that a statistical abnormality has occurred based only on a single extreme wave event. It is however possible to identify events which do not fit with our standard physical models. For instance, if a wave had a steepness that implies the wave theoretically should have broken, and yet the wave has not broken, this would be considered abnormal. A much studied example of a wave falling into the category of an observation which does not fit our standard physical models is the ‘New Year Wave’ or ‘Draunper’ wave. The time history of the Draupner wave is shown in figure 3. This wave had a crest of amplitude 18.6 m in a sea state with significant wave-height 12 m. Full background details of this event are given by Haver [22]. Careful analysis of the record identified two unusual characteristics to this wave which at first sight do not appear to fit within our normal understanding of the physics of extreme waves. The first of these is the wave steepness which was sufficiently steep that it could not be reproduced in physical experiments by [23] or using the potential flow numerical solver of Yan and Ma [24]. The second unusual aspect was the ‘set-up’ under the wave— a low-frequency increase in the water level under the giant wave observed by Walker et al [25]. Under a giant wave a ‘set-down’, or low frequency trough, would normally be expected. Both these unusual phenomena can be explained if the wave arose as the interaction of two waves crossing at 90◦ or more [26]—although this explanation raises as many questions as it answers.

When analysing data for evidence that waves exist which do not fit our standard models it is first necessary to define these standard models. The standard model assumes that the phase of all wave-components are uncorrelated, that freely propagating waves can be added linearly, and that the dynamics of each component is governed by the linear dispersion equation. Arising out of the work on electrical noise by Rice [27], Longuet–Higgins showed that the waveheight of narrow banded random waves will follow a Rayleigh distribution [28]. This remains the basis for all future analyses. Numerous corrections have been suggested to account for the broad-banded nature of real world spectra based either on theory or empirical fits to field or laboratory measurements. Distributions are also available which account for finite depth limiting the size of waves [29]. For moderately severe storms, wavecrests will not follow the Rayleigh distribution due to the second-order ‘bound’ waves—wave components which do not modify the underlying dynamics and arise from the non-linear free surface boundary conditions. Figure 4 shows an example of the contribution to the free-surface crest height of ‘bound’ waves, where the data is taken from the potential flow simulations of [30]. The even harmonics in the data have been found by analysing the sum of wave groups which are 180◦ out of phase and by filtering in the frequency domain [31]. Again, numerous distributions have been proposed to account for this non-linearity such as [32, 33]. Thus we define as the standard statistical model of wave-height and wave-crest distributions as those models which derive from the assumption of linear superposition of waves where the free surface is modified by second-order bound waves. There are a number of ways in which observed statistics can depart from the standard statistical model. To aid discussion later in the article we define four types of random wave statistics which show up in both field and laboratory measurements of wave-height and wave-crest statistics. We categorize these into four different ‘types’. These are described

4.3.1. Expected characteristics.

4.3. Measured statistics

Making robust and scientifically accurate observations of wave-heights in the offshore environment is difficult. Ideally, scientists and engineers would like many measurements of both the temporal and spatial structure of rogue waves on which to base their understanding, and would like the measurements to be taken over long time periods to allows robust statistics to be built up. Unfortunately, such measurements do not exist. What we do have is robust statistics of less extreme waves and 4

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Figure 5. Examples of four different types of wave-height distributions. Crosses show random wave simulations of wave-height; black line show Rayleigh distribution.

below, and with examples of each drawn from random wave simulations show in figure 5.

evidence of wave breaking limiting the magnitude of these waves. • Type 4—The wave-height is underestimated by the Rayleigh distribution (or broad-banded equivalent) for probabilities of exceedence less than 10−3 but the Rayleigh distribution over-estimates the size of the waves at probability of exceedence 10−5 as wave-heights are limited due to wave breaking.

• Type 1—Wave height statistics follow a Rayleigh distribution (or broad-banded equivalent). There is no evidence of wave breaking of waves for with a probability of exceedence greater than 10−5 . • Type 2—Wave height statistics of smaller waves follows a Rayleigh distribution (or broad-banded equivalent) but this distribution over-estimates the magnitude of the larger waves in a sea state, usually attributed to wave-breaking. • Type 3—The magnitude of waves with probabilities of exceedence greater than 10−5 is underestimated Rayleigh distribution (or broad-banded equivalent) and there is no

The thresholds presented in the definitions have been chosen due to the way short-term statistics are used in the Tromans and Vandeschuren [34] method to calculate the magnitude of a wave at a location for a given return period. When this method is used the result is typically insensitive 5

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to the short-term distribution of waves with exceedence probabilities greater than 10−3 or less than 10−5 . An important difficulty arises in analysing field measurements for a departure from the standard model. For practical engineering calculations and for understanding the physics of extreme waves we are interested in waves with probability of exceedance of less than 10−4 . To collect reliable statistics at the 10−4 level we need to observe significantly more than 10000 waves. This would mean you typically needed around a day of data; however, it is very unlikely that the seastate statistics will remain the same for this long. The usual approach is to combine a large amount of data from different storms into one analysis by normalizing by the significant waveheight—however, this suffers from the problem that individual storms may have very different characteristics. For instance, if all the data in figure 5 are combined into one analysis then they all appear to form a type 3 distribution, which is obviously misleading. A further difficulty when combining different sea states into a single graph is that the sea states being combined will have different steepness and non-dimensionalized water depths, which would give differing second-order crest heights with a standard model.

waves, and it is reasonable to speculate that these might be the most severe sea states with the steepest and largest waves (as indicated by the results of [42]), or perhaps the sea states which result in waves going in multiple directions which are of greatest concern to ships or other floating structures which use ‘weathervaning’ to reduce the load on, and motion of, the ship. We will return to the subject of in situ measurements and whether specific mechanisms can be linked to observations of rogue waves later in the review. Further issues often arise when trying to work back from a sea states with evidence of an unusual number of rogue waves to deduce the cause. For instance, the Benjamin–Feir instability (section 6) would be expected to trigger rogue waves in seas with a very narrow spectrum; however, seas with very narrow spectra will tend to occur when there is a high wind input, but the interaction with wind may itself might be the cause of rogue waves (section 10). Thus it would be difficult to work out which mechanism was causing the unusual number of rogue waves. Satellite synthetic aperture radar (SAR) are widely used for determining sea-state parameters. Whilst not generally as good as in situ measurements, satellite measurements are extremely useful in validating numerical models of sea states and in engineering design. Attempts have been made to detect individual extreme waves using this technique (e.g. Rosenthal and Lehner [43]), however, at present these are not robust enough for practical scientific use [44]. Unquestionably, satellite based remote sensing will improve over time and this will lead to a revolution in the way in which rogue waves are understood.

4.3.3. Satellite measurements.

4.3.2. In situ measurements.

Various methods are used to measure the changing water level due to the presence of waves. Traditionally buoys have been used—however, whilst these generally give accurate measurements of integrated quantities such as significant wave-height, they are thought to underestimate the magnitude of the largest waves as their lateral movement potentially allows them to skirt around the largest crests. Rather more reliable, though by no means perfect, are wave-staffs and downward pointing lasers. These obviously have to be mounted on a support structure and, when analysing data for freak waves, care needs to be taken in analysing measurements that the support structure has not caused a significant change to the properties of the waves and, of course, these cannot be deployed in very deep water. Details of different measuring systems are given in [12] with intercomparisons between different sensors given in [35] with a useful review also being given in [36]. The general principle should be remembered that most sensors are designed for measuring, and are calibrated on, the normal population and may be expected to be rather less accurate at measuring extreme events. Studies which analysed the height and crest statistics of a large number of waves in different regions include Japan [37], North Sea [38], and the Campos Basin, Brazil [39]. Due to the short duration and rapidly changing wave-field of hurricanes it is not generally possible to collect robust statistics in cyclones, although Forristall carried out an important analysis of cyclone data [40]. A very thorough study combining wave data from all around the world was carried out by Christou and Ewans [41]. All four types of distribution defined in this review have been observed in the open ocean. An overall conclusion is that the standard model for wave heights and crests generally over-predicts the number of waves exceeding the criteria given in equation (1). This does not preclude there being sea states where the standard model underestimates the number of freak

5. Formation of extreme waves from random sea states 5.1. Difference between uni-directional and directionally spread seas

The non-linear process by which an extreme wave forms is fundamentally different in a directionally spread sea state when compared to a uni-directional one. All realistic seastates are directionally spread, with non-linear wave–wave interactions keeping them thus. Extreme caution must be applied when results from uni-directional models are used to explain observations in real sea states. The are several ways of examining extreme waves to show that the non-linear evolution is fundamentally different in directionally spread seas compared with uni-directional. If we consider the resonant, and near-resonant, wave–wave interactions which are discussed further in section 6, then these are fundamentally different if the wavenumber vectors are confined to a single direction (see for instance Gibson and Swan [45]). A second difference between 1D and 2D can be seen if we examine the linear formation of an extreme event in both a directionally spread and a uni-directional sea state. To allow us to quote an analytical result we will assume that the wave-spectrum, and hence the shape of the wave-group is 6

Maximum amplitude of wave−group

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narrowbanded and Gaussian in form (although the conclusion would be no different if a more realistic spectrum were chosen). The linear evolution of the complex envelope of such a wavegroup is given in Kinsman [46], with a mis-print corrected in [47]. In the frame of reference moving with the wave-group this is U (x, y, t)  −0.5S 2 y 2 −0.5S 2 x 2 −ı0.5S 2 B tx 2 exp 1+S 4 Bx2 t 2 + 1+S 4xB 2xt 2 + 1+S 4 By2 t 2 + x x x x y y =A   2 1 − ıSx Bx t 1 + ıSy2 By t

ı0.5Sy2 By ty 2 1+Sy4 By2 t 2

validating these results and bringing this approach into offshore engineering practice where they became known as ‘NewWave’ [50–52]. The theory appears to be robust even in shallow water up until the point at which there is significant wave breaking [53]. In the limit, as the expected wave gets large relative to those around it, the most probable shape of the wave is given by the auto-correlation function of the wave-spectrum as   N n=1 S(ωn ) cos knx · x + kny · y − ωn t . ηNewWave = A N n=1 S(ωn ) (3) Figure 7 presents NewWave type groups in the time domain for JONSWAP spectra with γ = 3.3 and γ = 1. These are compared with the results of random wave simulations (generated following [54]). We show both the median shape of waves with H > 2Hs as well as the envelope which 95% of the waves lie within. It can be seen that there is excellent agreement between the median shape of the waves exceeding the waveheight criteria given by equation (1). However, there is a substantial variation in the timeseries around the giant peak (and thus also the spatial shape of the wave-group). Thus, it is possible that a wave fulfilling the criteria for a ‘rogue’ wave could occur immediately after a comparatively small wave with virtually no trough (see the work on ‘unexpected waves’ by Gremmrich and Garrett [55]). It is also possible for a rogue wave to be preceded (or followed) by a very large wave. From our linear random simulations, for a JONSWAP spectrum with γ = 3.3, we have derived probability density plots for the trough and the crest immediately preceding a rogue wave. This is shown in figure 8. It may be seen that there is significant variability about what wave precedes a rogue wave. Note, of course, that this analysis assumes linear waves—real waves would be modified slightly from this by second-order bound waves and possibly by other non-linearities discussed in the rest of this article. Note also that, as these are derived using linear theory and are time-reversible, these plots must, on average, be identical in the sense that these could be the trough/crest following a rogue wave as well. We note that there is no analytical relationship between the spectrum and the zero-up-cross period of a rogue wave [56]. However, it is straightforward to show from equation (3) that the instantaneous frequency3 at the crest of the linear freak wave does tend to the mean frequency of the spectrum although there is a significant amount of variation from one rogue wave to another.

 , (2)

where A is the amplitude of the wave-group at focus, Bx and By are constants and the spectral bandwidth in x and y is given by Sx and Sy . In a uni-directional sea the lateral bandwidth, Sy , is equal to zero. Figure 6 shows how the maximum amplitude of the wave-group envelope changes as a uni-directional and directionally Gaussian wave-group approach focus under linear evolution. It can be seen that the amplitude of the uni-directional group remains very large for much longer, implying that the evolution is likely to be inherently more non-linear. It should be noted that the timescale over which the Benjamin–Feir instability develops is proportional to amplitude−2 for the same spectrum. This raises the question as to whether analysis of unidirectional waves is of any value in explaining extreme events in the real ocean? This is doubtless a matter of debate. The view of the authors is that analysis of uni-directional waves can give some insight into the physics of extreme wave events, but direct conclusions should not be drawn from these to explain phenomena in the real ocean. For this review we have adopted this approach and where possible confine our discussion to results where wave directionality is accounted for; however, in some areas only uni-directional results are available and so our discussion ends up resorting to these. 5.2. The most probable shape of an extreme waves

As shown in the previous subsection, the shape a wave-group forms as it focusses up to form a giant wave is fundamental to understanding its dynamics. Connected with this is the shape of the extreme wave and the expected shape of the waves around them. The mathematical theory for the most probable shape surrounding a maxima in a random Gaussian field was developed by Lindgren [48] and applied to the ocean waves by Boccotti [49]. A series of papers were written in the 1990s

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Instantaneous frequency is here defined as 1/(2π )d/dt (arg(ηa (t))) where ηa (t) is the analytic signal of the free surface time-history.

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Figure 7. Examples of NewWave type wavegroup for JONSWAP spectra with γ = 3.3 (left) and right γ = 1 (right) shown with continuous line. Median shape of waves exceeding the rogue wave amplitude criteria from linear random simulations shown with thick dotted line. Envelope between which 95% of random simulation of freak waves fall between shown in grey. 2

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Figure 8. Probability density of the trough preceding a rogue wave (grey line) and the crest preceding a rogue wave (black line) from linear simulations of a JONSWAP spectrum with γ = 3.3.

energy from those around it. In the laboratory the Benjamin– Feir instability is well attested. Much research has gone into understanding whether this mechanism can create abnormal waves in the open ocean. The strength of the Benjamin–Feir instability, in unidirectional waves is a function of steepness of the waves and the bandwidth—with the instability being more likely in sea states which are higher in amplitude and narrower in bandwidth. This is captured in the Benjamin–Feir index introduced by Janssen [60]. Using standard spectral parameters in deep water this is √ 2k0 Hs , (4) BFI = 4σω

6. Non-linear instabilities

The evolution of ocean waves, as modelled using potential flow theory, is non-linear. In deep water, the most important parameter describing the degree of non-linearity is the wave steepness of the sea state. As discussed above, at the second order there is no change to the underlying dynamics of the waves. At higher orders waves will interact non-linearly [57]. If these interactions cause the sea state to be unstable (or unstable to a small perturbation) then there can be a rapid transfer of energy between spectral components potentially causing large waves to grow to a greater extend than they would under linear evolution.

where k0 is the dominant wavenumber and σω is the normalized bandwidth of the spectrum. In uni-directional seas, the spectrum becomes unstable and undergoes continual evolution when the Benjamin–Feir index is greater than 1. In practice, for a realistic spectrum, the evaluation of this index is ambiguous due to the potential sensitivity of the σω term the high frequency end of the spectrum. Serio et al [61] suggest using a ‘quality factor’ (defined in [62]) in this calculation. Using this approach on a JONSWAP spectrum, and dividing the BFI by

6.1. Physical mechanism

The Benjamin–Feir instability has been suggested as a mechanism for rogue wave generation. Study of this dates back to experiments done at the National Physical Laboratory, Teddington where it was found that regular wave-trains are unstable Benjamin and Feir [58] (see also Yuen and Lake [59]). Visually this phenomena causes one wave to grow, drawing 8

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BFI/(Hs k p)

3.5 3 2.5 2 1.5 1

1

2

3

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Figure 9. Benjamin–Feir index divided by significant steepness for different shapes of JONSWAP spectra.

the significant steepness gives the non-dimensional parameter shown as a function of JONSWAP peak enhancement factor, γ in figure 9. Other than sea states with small waves, γ is generally less than 3.3. Significant wave height multiplied by wavenumber is usually a maximum of 0.3, so in practice the criteria of BFI > 1 will not be exceeded (even before the points described below are included). In the ocean, sea states with large Hs will tend to be more broad-banded [63] (smaller γ ) thus somewhat inhibiting this mechanism. More importantly, in all directionally spread sea states the theoretical magnitude of this instability is greatly reduced [64, 65] leading to small, if any, extra wave elevation. It is important to note that Benjamin–Feir type instabilities are primarily a deep water phenomena. The magnitude of Benjamin–Feir index is depth dependent with the phenomena switching from ‘focussing’ to ‘de-focussing’ for water depths less than kd = 1.36 [66, 67]. However, for particular combinations of wave-numbers waves can theoretically become unstable in shallower water-depths [68]. Zakharov [69] found that the BFI was applicable in finite depths provided (ak))2  (kd)5 . In practice, waves meeting this criteria are improbable as they will have broken [70]. To evaluate whether the Benjamin–Feir instability causes an excess of rogue waves it is necessary to investigate ‘following’ seas (where the energy is confined to a narrow direction). Any experiment with a directional spread much less than 15◦ is unlikely to be representative of a real sea state (with the exception of low-steepness swells). If the Benjamin– Feir instability does cause rogue waves then one would expect that the number of rogues waves would decrease with increased directional spread, with increased bandwidth, and in shallower water.

demanding, and secondly the models break down when waves become steep enough to break. Thus, numerical studies of extreme waves using potential flow solvers have concentrated on isolated wave-groups. Examples include [30, 45]. These have found very dramatic changes to the local spectrum around an extreme wave, but only predict small extra elevation above that predicted by second-order theory, for waves which are just below the steepness that would cause them to break. Simulations of the evolution of the envelope of random waves have been carried out with a wide variety of numerical models with various orders of non-linearity. These are generally versions of the Zakharov equations [72], the simplest being the non-linear Schr¨odinger equation (which tends to overestimate the non-linear dynamics [73]). When realistic directional spreading parameters are employed it is generally found that no extra elevation is present and the statistical distribution of waves is similar to the type 1 distribution described above [74, 75]. Evidence from physical wave-tanks. Numerous physical experiments have looked at random, directionally spread, waves in wave tanks. There are some standard experimental difficulties encountered in these experiments. Firstly, the experiments must be done at scale with Froude number similarity. This does not appear to be too serve a limitation—although processes which do not scale (viscous flow, surface tension, and air entrapment) are important in how waves break, the onset of wave breaking is very weakly dependent on these parameters [76]. A further problem with physical tests is that it is rarely possible to have a wavetank deep enough to simulate truly deep water, which is of significance given the depth-dependence of the Benjamin–Feir instability. It is also very difficult to remove all reflections off walls in a confined tank, particularly of low frequency waves and there will almost certainly be greater energy in freely propagating waves at low frequencies in a wave-tank than in the open sea (e.g. [77]). Despite these difficulties, high quality wave-tanks are a vital tool for investigating what happens in the real ocean. Further discussion of how well physical wave-tanks model the real ocean is given by Buchner and Forristall [78]. The results of two independent sets of experiments, undertaken in Europe and Japan, were summarized by Onorato et al [79]. For directionally spread-sea states, both of these experiments found no significant deviation from standard

6.3.1.

6.2. Ocean measurements

Perhaps due to the difficulty in accurately evaluating the Benjamin–Feir index (see also [71]) there is limited evidence of sea-state conditions with a high Benjamin–Feir index actually contain an increased number of rogue waves [41] found no correlation between spectral shape and the chances of encountering a rogue wave. 6.3. Evidence from numerical wave-tanks

Simulating random wave fields using potential flow solvers leads to two difficulties: firstly these are very computationally 9

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Figure 10. Shape of extreme wave-groups under linear evolution (left) and non-linear evolution (right). Reproduced with permission

from [83].

theory, and thus no evidence that non-linear instabilities were causing extra elevation. The statistics measured in these tests were of type 1 or type 2 distributions, where these distributions are described above. Recently Latheef and Swan [80] carried out high quality experiments on directionally spread random waves. They did record some cases where there appeared to be extra elevation beyond that predicted by second-order theory and where the wave-crest distribution was similar to type 3, though only for cases when rms directional spreading was 15◦ and not when it was 30◦ suggesting some form of instability which was stronger when energy was travelling in a confined direction4 . Some of the cases simulated were more non-linear than would normally be expected in the open ocean and in these wave breaking tended to limit the largest wave crest leading to a type 4 pattern on the amplitude of extreme waves. It should however be noted that these experiments were carried out at a water-depths (kd between 2 and 3) which would significantly inhibit any Benjamin–Feir type instability.

0 −2000

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Figure 11. Variation in wave height relative to wave height at northern boundary for English Channel off Arromanches in France at low tide. White indicates waves have broken. Black indicates land.

Whilst there is only limited evidence that non-linear wave–wave interactions cause an increase in the number of rogue waves observed there is some evidence large waves are affected by non-linearity in other ways. For instance, there appears to be a local reduction in the directional spreading and a widening of the crest width [30, 74, 81]. Figure 10 examines the non-linear evolution of two isolated NewWave type wave-groups in deep water. Both wave-groups have the same initial conditions. The wave-group on the left shows the shape of the wave-group if evolution was linear, with that on the right showing the shape of the wave-group under non-linear evolution (using the potential flow solver of [82]). The difference in the shape of the wavegroup under non-linear evolution is dramatic. There is also some evidence that non-linear interactions may lead to a slowing of the time-scale over which a large

6.3.2. Other non-linear changes to extreme waves.

wave-group evolves potentially leading to an increase in the number of large waves that would be expected (derived for uni-directional waves in [47] but shown numerically to occur in directionally spread seas as well). 7. Shallow water wave focusing

In shallow water (depth less than half the wavelength is a common criterion) waves will start to ‘feel’ the sea-bed— their propagation and characteristics become modified by the bathymetry. Where the wavevector is not normal to the contours of the beach this will lead to the waves to change direction, which in turn can lead to energy being focused on certain areas (‘hotspots’). A classic example of this is Mavericks in Half Moon Bay, California. Figure 11 shows an example of this focusing of energy, where the physics has been modelled using [84], for the coast offshore of Arromanches in Normandy (where the British Mulberry Harbour was located in the Second World War) (see [85] for further details).

4 It should be noted that a quick calculation shows that the difference in wave crest elevation between 15◦ and 30◦ spreading cannot be attributed to second-order bound waves.

10

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The concentration of wave energy due to refraction is not, in itself, a cause for an excess of rogue waves since the focussing would be expected to increase the significant wave-height rather than just individual large waves. In practice, individual waves with slightly different directions and frequency will interact with the water depth slightly differently and this might lead to a departure from the standard wave statistics at a given spatial location. There appear to be no field studies investigating this phenomena although with the interest in putting wave energy devices in such locations such studies would appear to be needed. A numerical study was made by Janssen and Herbers [86] which found that refraction could lead to a distribution wave-heights of the type 3 described above.

Figure 12. Schematic showing spatial region where additional wave elevation may occur. Wave direction shown with large arrow.

cause abnormal waves in this region these would have been documented scientifically by the time of writing.

8. Wave/current interactions

8.3. Experimental evidence

There is very strong evidence, as detailed below, that wave interactions with currents can lead to abnormal waves. An obvious point should be made that if we wish to predict these abnormal waves in the open ocean we need to not only understand the interactions between currents and waves, but also be able to understand and predict the currents.

Many studies examining wave interactions with currents have used uni-directional wave simulations (e.g. [95]) or directional spreads far smaller than are observed in the real ocean (e.g. [96]). These studies should be treated with caution as ‘caustics’ are inhibited by a directionally spread wave-field. However, these clearly show that waves in the presence of currents will have abnormal properties—the nature of these being dependent on the temporal and spatial variation in the current. These results support the more realistic random wave simulations carried out by Hjelmervik and Trulsen [97].

8.1. Physical mechanisms

Currents can be generated in the open ocean from a variety of mechanisms. These can vary from being spatially and/or temporally uniform to being highly variable. Ocean currents typically have some form of sheared profile although this will vary depending on the local bathymetry properties. Ocean waves will interact with currents changing the properties of the waves and thus the expected distribution of wave-heights. As waves propagate across spatially changing currents the properties of the waves will change. If waves travel from water with no net current into an area with a current opposing the waves there will be a build-up of energy which will modify waves of different frequencies differently. If the current approaches the group velocity of the waves then the waves can be made to ‘stop’, in which areas very abnormal waves can be expected. Even if they do not completely stop wave-current interactions can induce instabilities [87, 88]. An additional mechanism is that waves may be steered by the current to create areas of increased energy concentration similar to that described in section 7 [89–91].

9. Waves at the top of slopes

This is a relatively new suggestion and to date has only been investigated for uni-directional waves and so should be treated with some caution. It was observed experimentally that waves travelling from deeper water to shallower water seem to have an excess of rogue waves at the top of the slope, shown schematically in figure 12 [98]. Numerical experiments have shown qualitative agreement [99]. No fully satisfactory explanation has been made of these results. As the results have been found in Boussinesq type numerical simulations the physical mechanism must obviously be something which can be modelled by the Boussinesq simplification to the full water wave equations. We are not aware of any published field measurements which are directly relevant to this proposed mechanism. 10. Wind/wave interactions

8.2. Ocean observations

Ocean waves, of the type considered in this review, draw their energy from the wind. It seems intuitively obvious that larger waves will have a greater interaction with the wind than smaller waves. It is therefore plausible that this could lead to amplification of the largest waves, possibly causing more or larger freak waves.

The most commonly cited location where abnormal numbers of rogue waves are observed is in the Agulhas current off the coast of South Africa [92–94]. This area is a major international shipping lane and hence has received much attention. It is perhaps surprising that there is not greater evidence for abnormal waves in other areas. For instance, one area where abnormal waves might be expected is where waves interact with the Gulf Stream between Bermuda and Florida. However, given the perpetual interest in the ‘Bermuda Triangle’ one would have thought if the wave current interactions did

10.1. Physical mechanisms

Wave generation from wind is generally described by the ‘Miles–Phillips’ mechanism. The initial growth of waves 11

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is due to turbulent wind causing a pressure variation on the surface of still water [100]. Once the waves have developed, wind shear will continue to feed energy into the system as long as the velocity of the wind is greater than the phase velocity of the waves [101, 102]. This latter process is reasonably well understood on average in terms of the growth of significant wave-height and is parametrized in wave forecasting. Arguably, this is less well understood in detail on a wave-by-wave basis. It would, however, appear that the rate of energy transfer is too small to directly influence the wave dynamics over the timescales we are interested in for rogue wave generation (see also the timescale argument in Touboul et al [103]). A further possibility, which could potentially transfer energy at a rather faster rate involves the wind boundary layer separating as the wind passes over a crest (see the work of Jeffreys [104]). Whilst this mechanism is important for waves which are breaking (e.g. [105]) it is not clear whether this mechanism exists (i.e. whether separation will occur) for waves prior to breaking. There is the possibility that some other non-linear mechanism causes waves to become asymmetric and steep enough on one side for flow separation to take place. It can be argued intuitively that larger waves will stick up further into the atmospheric boundary layer than smaller waves, encountering higher wind speeds, and generally locally enhancing any wind/wave interaction. There is some evidence [106, 107] that energy input (such as from wind) can feed back into enhancing the Benjamin–Feir instability (discussed in section 6)5 . Thus, if a mechanism can be found which transfers energy to large waves rapidly enough this could potentially be supplemented by extra elevation from the Benjamin–Feir instability.

whereas in a laboratory there is generally much less, spray, froth and air entrapment at the surface (put another way, the ‘Hokusai’ wave [110] would look very different at laboratory scale). Various studies of the interaction of wind on extreme waves have been made by [111] (and other studies by the same set of authors). The general conclusion is that wind does interact more with larger waves than with smaller waves and can lead to a small increase in the amplitude of the largest waves, suggesting that a type 3 distribution could occur when a strong wind coincides with the dominant direction of the waves. 11. Crossing sea states

Recently, much attention has been paid to analysing rogue waves in crossing seas—that is sea states which are the superposition of two wave systems with differing directions. One key reason why crossing seas are promising for rogue wave hunters is that waves can reach much larger amplitudes without breaking. A further reason why understanding the dynamics of large waves in crossing seas is important is the interaction of such waves with floating objects, such as ships, which try to position themselves head on into the waves—this becomes impossible in seas where waves come from multiple directions. Low frequency bound waves, occurring at the difference between the frequencies of the freely propagating waves, may be different in a crossing sea to a more typical ‘following’ sea state where most of the wave energy is concentrated in a relatively narrow directional range. In following seas there will be a ‘set-down’ under an energetic wave group as shown in figure 4. However, in crossing seas this may be inverted, leading to a ‘set-up’ and hence to slightly larger wave-crests than would be expected on a wave-crest model that assumed a following sea [26, 112]. Excluding the extra height due to bound waves, there is however no established physical mechanism to explain why waves would be larger in crossing seas that in following ones. A number of studies have explored non-linear instabilities in crossing seas [113, 114] One possible reason why rogue waves are often observed in crossing sea is perhaps that these are actually rapidly developing sea states where stationarity cannot be assumed. It is also known that over a timescale of hours two systems of waves travelling at less than approximately 90◦ will interact to form a single system (‘shape stabilization’ of the spectrum see [115]) and as such many crossing sea states will tend to exist in areas where there is a rapidly changing spectrum and probably a rapidly changing weather system. When the mean wave directions are separated by more than 90◦ the non-linear interactions between the two systems becomes negligible [116]. We note that the definition of what constitutes a crossing sea is rather broad and imprecise. In this section, we are lumping together and sort of sea state which does not have all the wave energy travelling in one confined direction. There are clearly differences in the physics where two systems interact when travelling at 30◦ than 150◦ . The relative frequency of

10.2. Ocean observations

Various authors have looked for correlation between high winds and rogue waves. Generally, the results show no obvious correlation (e.g. [37, 41]) although data presented in [109] does find a correlation between strong winds and rogue waves in the North Sea. The obvious point should be made that when the wind is strongest the energy input into the sea state is high and thus the assumption of stationarity may be more questionable for some high wind conditions. Even if the sea state is statistically stationary, a high wind is likely to mean a rapidly growing sea state which is likely to have a narrower spectrum, making Benjamin–Feir type instabilities more plausible. Thus a correlation between wind and rogue waves does not necessarily mean that the rogue waves were directly caused by the wind. 10.3. Experimental evidence

There are some obvious problems of dimensional similarity when investigating wind interaction with gravity waves in the laboratory as it is impractical to keep both Froude number and Reynolds number similarity. A further difficulty is that the free surface of waves in the open ocean is typically ‘frothy’ 5 Conversely energy dissipation may suppress the Benjamin–Feir mechanism [108].

12

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Table 1. Table summarizing the different mechanisms for excess numbers of rogue waves.

Benjamin–Feir instability Shallow water focussing Wave/current interaction Waves at top of slopes Wind/wave interactions Crossing sea state a

Physical mechanism understood

Experimental measurement evidence

Ocean evidence evidence

Satisfactory statistical model of waves

Yes Yes Yes No No No/Not clear

No Yes Yes Yes Not clear No

No Yes Yes No No Yes

OK a No No No No No

Not sufficient evidence to conclude standard model is wrong.

each group is also of obvious importance. It may also be important if significant energy is being fed into one of the systems from the wind. At present, we do not have sufficient data or theoretical understanding to sub-divide the different types of crossing seas into more refined categories. This is something we hope future work will address.

an excess in kurtosis6 in some crossing sea states which was maximized when the angle between the systems was between 40◦ and 60◦ . Petrova et al [124] found evidence of an increased number of rogue waves for sea states separated by 120◦ even after the removal of bound waves. 11.3. Numerical wave-tanks

11.1. Ocean observations

Reference [123] carried out a study using a relatively crude model and found results consistent with their findings in physical experiments. Gramstad and Trulsen [125] found that for wind waves interacting with ‘swell’ there was an interaction which could lead to an increased probability of a rogue wave, except for where the directions of the swell and the waves were orthogonal i.e. implying that in most crossing seas there would not be wave–wave interactions which would lead to crossing seas. Unpublished work by the Authors with Shiqiang Yan and Qingwei Ma used a potential flow model [126] to model the interaction of large NewWave type wave groups crossing at angles between 60◦ and 150◦ . No significant non-linear interactions or obvious elevation above that expected by a second-order model was observed.

There is other evidence that unusual and dangerous waves occur in crossing seas. Ferreira de Pinho et al [39] studied extreme waves in the Campos basin off Brazil. They only observed extreme waves (H /Hs > 2.4) in sea states which were highly directionally spread, and can be assumed to be crossing seas. Rosenthal and Lehner [43] also found evidence that freak waves occur in crossing sea states from satellite measurements, although as pointed out earlier these must be treated with caution [117] studied the sea states in which maritime accidents are known to have occurred. They found a very high proportion of these occurred in crossing seas as two systems of waves merged. This is unsurprising since crossing seas are notoriously difficult for ships to navigate through. However, what was surprising is the number of accidents that occurred when the swell was much smaller (less than 20% as energetic) than the wind sea. There are a number of anecdotal accounts of extreme waves having unusual directional properties. An account in a British Broadcasting Corporation Horizon programme of an unusual wave hitting the cruise ship Caledonian Star in 2001 states that this wave was travelling at 30◦ to the waves around it. Video recorded during filming for The Deadliest Catch apparently shows a fishing boat being hit broadside by a largewave travelling at around 90◦ to the main wave system. The giant wave which hit the Queen Mary in 1942 is described as hitting her broadside, so presumably this wave was travelling at roughly 90◦ to the majority of the waves. There is evidence that the wave which hit the Louis Majesty in the Mediterranean occurred in a crossing sea state [118, 119]. Similar accounts are given by the crews of the Gloucester dragger [120] and the RMS Etruria [121] and also in [122].

12. Discussion

There are still open questions about all the mechanisms discussed in this review. Given our limited ability to measure what really happens in the open ocean this is only to be expected, and whilst progress can certainly be made in the laboratory, data on real ocean waves is the only way we will fully understand rogue waves. To some extend there is a requirement for simply more data to improve our ability to draw meaningful statistical conclusions about the extreme events. However, we also hope than it we will soon be able to make quality measurements of the two-dimensional free surface and how this varies with time. Recently there has been encouraging progress in this area (e.g. [127]) and we look forward to the day when we will be able to examine the shape and structure of extreme waves. Table 1 presents a summary of the evidence discussed in this paper. The table is somewhat crude and simplistic and

11.2. Physical wave-tanks

6 Kurtosis is used by some as a proxy for sea states with enhanced rogue waves. Kurtosis can be useful in evaluating statistical distributions of waveheights.

A number of studies have recently been carried out looking at rogue waves in crossing sea states. Toffoli et al [123] found 13

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doubtless there are some details which others may disagree on or which new evidence will change within a few years. Nevertheless, we consider it a useful overview of the present state-of-play. The most urgent area for investigation in our view is for investigation of crossing seas. These are reasonably common in the open ocean and can certainly form in extreme events such as hurricanes. In all sea states the wave steepness (and thus the wave-height) will be limited by wave-breaking, but in crossing seas it is known that waves break at higher amplitudes than in following seas. Further reasons why engineers should worry particularly about crossing seas is the lack of set-down, and possible set-up, under the wave. This would increase the wave-crest elevation of the design wave significantly which is important for fixed structures. For floating structures, such as ‘weathervaning’ FPSOs, it is well known that these are vulnerable to waves coming from different angles. An obvious area to look at is the non-linear interactions which take place in rapidly evolving crossing seas. Gramstad and Stiassnie [128] considered this in a phase-averaged sense, but it is still impractical to examine the evolution of individual waves. There is a question as to what should be the goal of rogue wave research. The answer to this is perhaps different depending on whether you are a mathematician, a physicist, or an engineer. Speaking as engineers, the objective is to work towards a statistical model dependent on the sea-state, water depth, currents, etc. This has two uses: firstly, for short-term forecasting of rogue waves; secondly, for combining with longterm statistical distributions of sea-states (and currents, etc.) to derive design criteria. Note that an important part of this latter would be able to robustly predict type 2 or type 4 distributions where the amplitude of the largest waves is limited by wave breaking.

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13. Conclusions

At the time of writing scientists and engineers have insufficient measurements of ocean waves to determine whether, and under what conditions, more rogue waves may occur than are predicted by standard theory. In this review we have argued that the key to understanding rogue waves is to start with the analysis of linear waves (corrected for second-order effects) whose evolution can be satisfactorily modelled by the linear dispersion relationship. From this, we can examine how a large wave is likely to form by considering the random superposition of components described by a spectrum with realistic directional spreading and frequency bandwidth. The open question is then whether physics beyond linear theory is needed to describe the dynamics of the largest waves, or whether the non-linear processes which are known to occur in real waves have too long a timescale to significantly change the dynamics of an individual wave. Various possible mechanisms are discussed in this paper but, with the exception of the interaction of waves with currents, none has been clearly demonstrated to lead to an abnormal number of rogue waves. Further study, be it analytically, numerically or by physical modelling, of the mechanisms described in this article will improve our understanding, although we stress the importance 14

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