## Abstract

The ‘New Year Wave’ was recorded at the Draupner platform in the North Sea and is a rare high-quality measurement of a ‘freak’ or ‘rogue’ wave. The wave has been the subject of much interest and numerous studies. Despite this, the event has still not been satisfactorily explained. One piece of information that was not directly measured at the platform, but which is vital to understanding the nonlinear dynamics is the wave’s directional spreading. This paper investigates the directionality of the Draupner wave and concludes it might have resulted from two wave-groups crossing, whose mean wave directions were separated by about 90^{°} or more. This result has been deduced from a set-up of the low-frequency second-order difference waves under the giant wave, which can be explained only if two wave systems are propagating at such an angle. To check whether second-order theory is satisfactory for such a highly nonlinear event, we have run numerical simulations using a fully nonlinear potential flow solver, which confirm the conclusion deduced from the second-order theory. This is backed up by a hindcast from European Centre for Medium-Range Weather Forecasts that shows swell waves propagating at approximately 80^{°} to the wind sea. Other evidence that supports our conclusion are the measured forces on the structure, the magnitude of the second-order sum waves and some other instances of freak waves occurring in crossing sea states.

## 1. Introduction

On 1 January 1995, a large wave was recorded at the Draupner platform in the North Sea (figure 1). This has been variously referred to as the ‘Draupner wave’ or the ‘New Year Wave’. The free surface was recorded with a downward pointing laser sampling at 2.1 Hz. The platform, which stands in 70 m water-depth, has a sparse structure that would have had a negligible effect on the motion of the wave, and the record is, therefore, assumed to be an undisturbed field measurement.

The giant wave had a crest of mean 18.6 m above mean see level and was 25.6 m in height. Other properties of the individual wave are given in Guedes Soares *et al.* (2004). Despite being very steep, there is no obvious sign in the data that the wave was breaking. The storm properties were approximately stationary over a 5 h period, with *H*_{s}=12 m and *T*_{z}=12.5 s. The data are not continuous, and were recorded for only 20 min in every hour. For more information on the structure, the instrumentation, the meteorological conditions and the wave record, see Haver (2004). Haver discusses at length the presumed nature of the wind and wave fields and observes that minor structural damage to robust equipment below main deck level on the platform was found after the storm. This damage was at an elevation consistent with the size of the large crest, which is the subject of this paper. So we may be reasonably confident that green water reached this level and the wave record is reliable. Numerous others have studied the properties of this wave: see, for instance, Trulsen (2001), Walker *et al.* (2005), Jensen (2005) and Clauss & Klein (2009).

The most common explanation given for the Draupner wave is that resonant wave–wave interactions caused a nonlinear focusing of wave energy. This focusing has been shown to occur for narrow-banded uni-directional waves in deep water (Janssen 2003; Onorato *et al.* 2009). However, these interactions reduce considerably if finite depth and directional spreading are taken into account. The non-dimensional water depth for the Draupner wave was *kd*=1.6, which is close to the finite depth limit of the narrow-banded approximation to the Zakharov equation (*kd*=1.36), where the Benjamin-Feir Index goes to zero (Onorato *et al.* 2006*a*; Janssen & Onorato 2007). This implies that nonlinear wave–wave focusing, as we currently understand it, will be small in this sea state. Directional spreading also reduces these nonlinear interactions (Waseda *et al.* 2009). The sea state at Draupner is unlikely to have had a mean directional spreading less than 20^{°} r.m.s. (see §2). The numerical simulations of Gramstad & Trulsen (2007) and the result reported in Toffoli *et al.* (2010) suggest that with this spreading there will be no extra elevation, even in deep water, over that expected by second-order theory.

A number of other explanations have been proposed to explain freak waves (Kharif & Pelinovsky 2003), which we will briefly consider here. The water depth at Draupner is too deep for the bathymetric steering to be significant and thus ray-focusing may be dismissed as a possible cause. Owing to the strong wind there will be a wind-driven current. Wind-driven currents are difficult to quantify accurately, but we can be fairly certain that it would have been going in broadly the same direction as both the wind and the waves and will therefore not increase the chances of a freak wave (Hjelmervik & Trulsen 2009). The tidal current at the Draupner platform at the time of the giant wave is predicted to be less than 0.2 m s^{−1} by the Admiralty’s Total Tide software, which is too small to induce significant wave–current interactions. The platform is too far west to be affected by the Norwegian Current. A final possibility, that localized wind–wave interactions might cause a localized increase in wave amplitude, is, at the time of writing, poorly understood.

While a large wave might be caused by the random superposition of linear waves, a wave crest of this magnitude is unlikely in this sea state. Assuming wave crests follow a Forristall distribution (Forristall 2000), one would expect a crest of this magnitude only once in more than 3×10^{6} waves (third-order bound harmonics may reduce this figure slightly (Zheng & Moan 2010) but the wave remains improbable). No concrete physical explanation for the giant Draupner wave has been found. One aspect of the wave which suggests very strongly that it was in some way abnormal was the low-frequency set-up under the giant wave as first observed by Walker *et al.* (2005), whereas typically one observes a set-down under a large wave. This aspect has not been thoroughly explored.

In this paper, we try to reconstruct the directional spreading of the Draupner wave, which was not directly measured, but which is vital to understanding the nonlinear dynamics of large waves. In doing this, we explain the low-frequency ‘set-up’ under the giant wave. We show, both analytically and with numerical simulations, that this only occurs when two wave systems cross at an angle greater than around 90^{°} (see also Toffoli *et al.* (2007) and Christou *et al.* (2009)). We find that this is consistent with other evidence including the hindcast of the sea state from ECMWF and the measured forces on the structure. Finally, we consider other freak waves that have occurred in ‘crossing’ sea states.

## 2. Spreading of the Draupner sea state

No directional information was recorded at the Draupner platform. The nearest location known to the authors where directional data for this storm was recorded is the Auk platform 180 km away (K.C. Ewans 2004, personal communication). The meteorological conditions suggest that this was part of the same storm, and the other sea-state parameters are similar. At Auk, the directional spreading had a standard deviation about the mean direction of 20^{°} when a weighted average is taken over all frequencies. This measurement is in agreement with the analysis of Adcock & Taylor (2009*a*) who used the second-order bound waves to estimate the directionality of the sea state over the duration of the storm. It should, however, be noted that this analysis broke down in the vicinity of the giant wave, this is discussed further in §4*b*. Hindcast data from ECMWF suggest that a slightly higher spreading of 25^{°} although this was for several hours before the storm reached its maximum intensity.

A more detailed analysis of the ECMWF results shows that 3 h before the giant wave, in addition to the wind sea, there was also a small swell present at the Draupner location. This was in a markedly different direction to the wind sea, as shown in figure 2. Although this sea state is timed at 12.00 h, it should be recalled that the output from a hindcast is of significant waveheight (*H*_{s}) averaged over both space and time. So short duration peaks in *H*_{s} may be missed. Also Haver (2004) describes the sea state as arising from a large winter depression and a short duration ‘Arctic bomb’, a much smaller but more violent wind field with a significantly different mean direction.

Although the ERA hindcast from ECMWF is produced using more recent version of the WAM model than that described in the book by Komen *et al.* (1994), there must still be some uncertainty in the accuracy of sea states arising from fast varying local wind fields. In this book Holthuijsen (§II.10) discusses this problem, which may be intrinsic in area and time-averaged models. An interesting more recent discussion of at least some of the problems associated with hindcasting sea states in regions with fast varying winds is given in Jensen *et al.* (2006).

The idea of crossing sea components is consistent with the analysis of SAR images by Rosenthal & Lehner (2008).

## 3. The spectrum of the Draupner sea state

The spectrum of the Draupner wave record has been studied in detail by Ewans & Buchner (2008). Here, we consider the Fourier energy spectrum around the Draupner wave and show it is significantly different to that of the rest of the storm. Figure 3 shows estimated mean spectra for all six of the hourly 20 min records available during the storm, the 20 min record containing the Draupner wave, and a 5 min and 2 min record centred on the Draupner wave, albeit these last two being rather coarse representations.

It can be seen that for 5 min of data around the giant wave, the spectrum is similar to that for longer time periods, which in turn are similar in form to a JONSWAP spectrum with a peak at around 0.067 Hz. In the 2 min spectrum, there is a clear second peak at 0.083 Hz. We call the peak around 0.067 Hz—peak 1, and the peak near 0.0833 Hz—peak 2.

A broadening of the spectrum owing to nonlinear wave–wave interactions is known to occur around large waves in deep water. This result is found in experimental studies (Johannessen & Swan 2001), numerical studies (Gibbs & Taylor 2005), analytically (Adcock 2009; Adcock & Taylor 2009*b*) and in field data by Krogstad *et al.* (2006). However, this effect is greatly reduced in finite depth (Adcock & Yan 2010; Katsardi & Swan 2011). Even if the spectral broadening is due to resonant wave–wave interactions, we would not normally expect these to form a double peak. However, the window used is short and so this may be an artefact of the coarse resolution of the Fourier transform. We will return to this double peak feature in §4*b*.

## 4. Bound waves

Freely propagating, or linear, waves will interact to produce ‘bound’ waves, which do not move independently. To second order, these waves will occur at the sum and difference of the frequencies of the linear waves. The freely propagating waves may be written as
4.1where *a*_{n} is the Fourier coefficient, *N* the number of Fourier components used, and
4.2where *ξ* gives the relative phase of the component.

The linear waves in equation (4.1) will interact to give a second-order sea state given by
4.3where
4.4where *κ*^{+} and *κ*^{−} are the interaction kernels given in equations (4.5). The kernels for finite depth were first given in Dean & Sharma (1981), see also Dalzell (1999) where corrections to the finite depth case are given.
4.5The magnitude of the wavenumber |**k**| and natural frequency *ω* is related by the linear dispersion relation. The angle between the interacting components is *θ*.

We now consider the form of these kernels for the water depth at Draupner of 70 m. The behaviour of the sum term is comparatively straightforward. For any two interacting waves, this will be a maximum when they are travelling in the same direction (unless the ratio of the frequencies is large, in which case the assumption under which these equations are derived is invalid). Figure 4*a* shows the variation in *κ*^{+} for the interactions with the first peak in the Draupner spectrum. It can be seen that the sum kernel goes to zero at around 80^{°} for all frequency interactions and is negative for angles greater than this. For the difference term, the behaviour is more complex. For waves travelling in the same direction, the kernel is always non-positive. However, if the waves are travelling at widely differing angles then the kernel is positive, the value at which the kernel is zero being strongly dependent on the frequencies of the waves interacting. This is shown in figure 4*b*.

### (a) Second-order sum waves

We can see evidence of second-order sum waves in figure 3. Three peaks can be seen at approximately twice the frequency of peak 1 (0.125 Hz), twice the frequency of the peak 2 (0.175 Hz) and the sum of the frequencies of the two peaks (0.15 Hz). Unfortunately, the poor frequency resolution limits our confidence in these assertions. It should also be noted that some linear waves will be in this part of the spectrum. The relative magnitude of these three second-order peaks is interesting. The peak at 0.125 Hz is much larger than the one at 0.175 Hz. This is to be expected as peak 1 is larger than peak 2. However, the peak at 0.15 Hz is substantially smaller again. This is unexpected if the wave energy from both peaks is travelling in the same direction. However, consideration of figure 4*a* suggests this would be expected if some of the energy in the peak 2 was travelling at an angle between perhaps 60^{°} and 150^{°} to the mean wave direction of peak 1.

Thus, the second-order sum term is consistent with a hypothesis that a substantial portion of the wave energy in peak 2 is travelling at a large angle to the waves in peak 1. We cannot go further than this when considering the sum term owing to the amount of linear information in the spectrum at these frequencies, the modification to the spectrum owing to third-order effects (Janssen 2009), and the poor resolution of the Fourier transform.

### (b) Second-order difference waves

Adcock & Taylor (2009*a*) developed an approach for estimating the spreading of a sea-state from the second-order difference record. As noted in §2, they found the whole storm to have a standard deviation of spreading of 20^{°}. However, their analysis did not work for a short section of time around the giant wave as described below.

The low-frequency waves that are present in the time history may be extracted by filtering in the frequency domain. These are shown in figure 5, where the original time series has been low pass filtered at 0.03 Hz. It can be seen that under the giant wave, the low pass-filtered waves are positive, i.e. there is a set-up. It is not expected that any significant free waves will be propagating at frequencies this low. Therefore, the waves propagating in this part of the spectrum may be predictable from the linear waves using the interaction kernel given by equation (4.5). This predicts that energetic wave groups in following seas will cause a set-down (Forristall 2000). It is, therefore, surprising that there is a significant set-up under the giant wave in the Draupner record as noted in Walker *et al.* (2005). A set-down is observed under all other sizable wave groups throughout the rest of the dataset. One possibility is that third-order (Madsen & Fuhrman 2006), or higher order interactions cause this set-up. To investigate this possibility, we have carried out fully nonlinear simulations of the Draupner wave in §5, and find that the low-frequency waves, even for a wave system up to breaking, have a set-down in the following sea, as predicted by the second-order theory. Therefore, the higher order nonlinearity is not the key factor leading to the set-up in the Draupner wave.

Walker *et al.* (2005) introduced an approach to linearizing a free surface time history so as to estimate the freely propagating, or linear, waves. These may then be used to estimate the low-frequency waves for given spreading as set out in Adcock & Taylor (2009*a*). Using this linearized data, we can use a model for spreading to predict the low-frequency second-order difference waves. If we use a frequency independent wrapped normal spreading function
4.6with *σ*, the standard deviation of spreading around the mean direction *θ*_{0}, of 20^{°}, then our estimate for the low-frequency wave is as shown in figure 5. This is clearly completely out of phase for around 2 min near the giant wave.

This suggests that either there is unusual physics happening around the giant wave, or that our model of spreading is wrong. As pointed out by Toffoli *et al.* (2007) and shown in figure 4*b*, the interaction kernel can be positive if the two interacting components are propagating in different directions with a large separation angle (greater than 80^{°}). Thus, if there are two crossing wave systems in the New Year Wave record, a set-up could occur. The remainder of this section examines this possibility.

The spectrum of the wave record for the section where the long waves are unusual is shown in figure 6. This is 120 s before through to 75 s after the crest of the giant wave—note that this is a slightly different section of data to that used in figure 3. Shown also in figure 6 is the JONSWAP spectrum obtained by fitting the data excluding the period around the giant wave. We now assume that the linear waves are the sum of two systems of waves: one has a JONSWAP spectrum while the other has a spectrum given by the difference between the actual measured spectrum and JONSWAP spectrum (the area shaded in figure 6). We call these the ‘JONSWAP waves’ (J) and ‘transverse waves’ (t), respectively.

We have freedom to choose the phase of the individual Fourier components in the transverse wave. We can iterate these so as to try and recreate the difference wave observed in the original data. This iteration was carried out manually using the following assumptions:

— The sum of the main wave train and the transverse wave group equals the original linearized time series.

— The main wave train has a JONSWAP spectrum.

— For estimating the spreading, it was assumed that both sets of waves had a spread of 20

^{°}about their mean value. The angle between the mean direction of the two systems of waves was*ζ*.

Figure 7*a* shows the transverse wave that produces the difference waves shown in figure 8 when the angle between the mean direction of the wave trains is 120^{°}. The main wave train is shown in figure 7*b*. There is good agreement for around 60 s either side of the giant wave, regardless of exactly where the data are low pass filtered. The sensitivity of this may be examined by varying the parameters used. In figure 9*a*, we vary the angle (*ζ*) between the mean directions of the wave groups, in figure 9*b*, we vary the spreading of the transverse wave packet, in figure 9*c*, we vary the amplitude of the transverse waves and in figure 9*d*, we shift the phase of the transverse waves by ±20^{°}.

The phasing of the transverse wave is plainly very important to recreating a difference wave similar to that observed. There is less sensitivity to changes in the other parameters. There are other wave groups, with different phasing or frequency content, which we could use as the transverse wave group and which would give approximately correct second-order difference waves. However, we are certain that any such group must be propagating at an angle greater than 90^{°} to the JONSWAP waves for the predicted long wave components to be as observed under the giant wave.

Given the lack of confidence in the sea-state details at the time of the giant crest, we have not attempted to estimate the probability of a 8 m crest moving in one direction colliding with a 6 m crest in a second wave group moving at close to 90^{°}.

## 5. Numerical modelling of the Draupner wave

To further confirm the conclusion obtained in the above section, we have modelled the Draupner wave numerically using a fully nonlinear Quasi Arbitrary Lagrangian-Eulerian Finite Element Method (QALE-FEM) potential-flow solver. In the QALE-FEM method, the flow is governed by the fully nonlinear potential theory, where a boundary value problem for velocity potential is solved using the FEM. The main features of the QALE-FEM are that the unstructured mesh is moving at every time step using a specially developed spring analogy method for free surface problems and the velocity is calculated by a three-point technique suitable for any mesh structures (Ma 2008; Ma & Yan 2009; Yan & Ma 2010*a*). More importantly, the QALE-FEM can model highly nonlinear free surfaces and three-dimensional overturning waves. The QALE-FEM method has been compared with experimental data and other numerical for wave breaking cases and found to be accurate (Yan & Ma 2009; 2010*b*).

In all cases presented in this section, we use a water depth of 70 m. Average spatial discretization across the domain is 2.5 m, which reduced to approximately 0.8 m around the crest of the giant wave. This compares to a wavelength of approximately 230 m.

### (a) Modelling the Draupner wave as a wave in a following sea

We start by assuming that the Draupner wave occurred in a following sea state. To do this, we use an r.m.s. spreading of 20^{°} consistent with values discussed in §2. We take as our initial conditions a NewWave (Boccotti 1983; Tromans *et al.* 1991) which would have a linear amplitude (when focused) of 14.7 m (as estimated by Walker *et al.* (2005) for the Draupner wave) and use the ‘linearized’ spectrum (Walker *et al.* 2005; Adcock & Taylor 2009*a*) for a 10 min record around the giant wave. When the higher order harmonics are added, Walker *et al.* found this reproduced the measured free surface with the anomalous set-up filtered out. We then run the linear profile back in time for 100 s assuming linear evolution and use this as the initial conditions for the nonlinear simulation, after correcting the free surface and velocity potential using second-order theory.

The numerical simulations have been carried out for a number of cases by gradually increasing the amplitude until the wave overturning occurs as shown in figure 10. Various mesh sizes were used to ensure that these results were robust. Our simulation for the largest non-breaking wave has a time history similar to that measured at Draupner as shown in figure 11, in which both time histories have had the low-frequency components removed, and the signals match well around the large peak although the amplitude of the main crest is slightly too small. However, any increase in amplitude or steepness over this results in wave overturning. This result is consistent with the physical wave-tank experiments of Clauss & Klein (2009), whose attempt to reproduce the Draupner wave uni-directionally resulted in a scaled crest height slightly smaller than that measured. They found that this wave was on the point of breaking.

After low-pass filtering, the simulation in figure 11, we see the expected set-down, shown in figure 12 together with the second-order theoretical result based on a linearized profile and a spreading of 20^{°} (although the actual spreading will be slightly smaller than this owing to the nonlinear changes to the group shape (Gibbs & Taylor 2005; Adcock 2009)). The figure confirms that, while there is a small mismatch, the second-order theory is clearly still applicable in this case. More importantly, this figure implies that in the following sea state, the long wave (or the difference wave) components produce a set-down beneath an energetic group rather than set-up.

### (b) Modelling the Draupner wave as a wave in a crossing sea

We have also carried out fully nonlinear modelling of the two wave groups which are crossing in order to look at if the set-up can appear as predicted by the second-order theory in §4. We assume that each wave group has a NewWave-type profile and that the two mean wave directions are separated by 120^{°} and that each group has an r.m.s. spreading about its mean direction of 20^{°}. We start our simulations 60 s before linear focus, but in this case, we have modified the initial conditions so that both groups focus at the same spatial point. In the crossing sea case, the waves with different amplitudes have also been investigated and shown that they do not break, even for the cases higher and steeper than the Draupner wave. The case with a profile very close to that of the Draupner wave is shown in figure 13, where the crest of the numerical results is very close to the crest of the field-measured data (18.7 m). The corresponding low pass-filtered time history of the simulation (at 0.034 Hz) is given in figure 14, together with the low pass-filtered data of the Draupner wave. Although there is inevitably some difference between them owing to lack of information in the field data, this comparison shows that in the crossing wave case, the fully nonlinear simulation leads to local long wave set-up, again confirming the prediction of the second-order theory discussed in §4.

## 6. Discussion of analytical and numerical results

In §§4 and 5, we have investigated the causes of the low-frequency set-up under the Draupner wave. The numerical modelling demonstrates that for very steep, moderately broadbanded waves, second-order theory gives a good model for the long waves in a record. Thus, we can conclude that a ‘set-up’ will only occur under a large wave if its energy is not confined to a narrow directional spectrum.

Some discussion needs to be made as to the assumptions we have used in the reconstruction of the wave. Various alternative scenarios could be imagined where, consistent with second-order theory, a set-up results. However, given the evidence of the ECMWF hindcast, the most probable scenario is the wave was caused by two wave packets colliding whose mean wave directions were separated by more than 90^{°}. We cannot, of course, be accurate in giving this angle exactly, or in reconstructing the distribution of energy between the two wave packets.

Finally, we remark on our choice of spreading function for the two wave systems. We have used a simple frequency-independent Gaussian form. Other more complex choices are possible, a well-validated frequency-dependent model is given by Ewans (1998). However, our predictions of set-up/set-down for the Draupner wave only work when the two wave trains are separated by more than 90^{°}, well beyond the range of spreading consistent with the Ewans’ form. Almost any reasonable choice for the spreading forms for the two individual trains will give comparable results for the set-up only if they are separated in mean direction by at least 90^{°}.

## 7. Forces on the structure

The forces owing to this wave on the northern platform of the two at the Draupner field were significantly lower than would have been expected for a wave of this size (Hansteen *et al.* 2003; Haver 2004). Given that the height of the Draupner wave is roughly the 50–100 year design condition for that part of the North Sea, the observed peak horizontal force on the structure, as recorded by strain gauges on the suction caisson foundations and lower structural members, was roughly half of that expected for a wave of this height.

If the two wave components of amplitudes 8 and 6 m were travelling at close to 90^{°} apart in direction, but in phase locally in time, the net horizontal velocity amplitude would be approximately proportional to vector (8,6), resulting in a combined velocity equivalent to a 10 m wave since . In contrast, if the wave system were close to collinear, the vector sum would be (14,0). Taking the Morison drag as proportional to velocity magnitude squared, this gives a peak force proportional to 10^{2} for the crossing system, compared with 14^{2} had the wave been collinear. Hence, we suspect that the unexpectedly low value of the total horizontal load on the structure, as observed by Hansteen *et al.* (2003), is at least consistent with the giant wave arising in a crossing sea state.

## 8. Other freak wave events in crossing seas

Most studies of freak waves investigate the dynamics of extreme waves in following seas (see reviews Kharif & Pelinovsky (2003) and Dysthe *et al.* (2008)). This is because nonlinear Benjamin-Feir-type instabilities only occur in seas which have a small directional spreading (Waseda *et al.* 2009). Recent studies by Onorato *et al.* (2006*b*) and Shukla *et al.* (2007) have looked at the nonlinear dynamics of wave groups in crossing seas, and Donelan & Magnusson (2005) looked at the statistics of waves in such seas, but at present, the understanding of these sea states has not been as thoroughly researched as that for following seas.

However, as well as the evidence presented in this paper, there is other evidence that unusual and dangerous waves do occur in crossing seas. Ferreira de Pinho *et al.* (2004) studied extreme waves in the Campos basin off Brazil. They only observed extreme waves (*H*/*H*_{s}>2.4) in sea states, which were highly directionally spread and can be assumed to be crossing seas. Rosenthal & Lehner (2008) also found evidence that freak waves occur in crossing sea states.

Toffoli *et al.* (2005) 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, the situation our analysis suggests was the case at Draupner.

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 large-wave 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. Similar accounts are given by the crews of the Gloucester dragger (Prybot 2007) and the RMS Etruria (Liu 2007).

## 9. Conclusions

The Draupner wave was an event which was extremely improbable in a standard second-order model of the sea state. There is no obvious physical reason why the amplitude of the wave should be larger than that expected in a model based on linear wave-group evolution. Certainly, Benjamin-Feir-type instabilities, as we currently understand these, would not cause this wave as these would be suppressed by the finite depth and the directional spreading of the sea state.

The Draupner wave has an unusual feature, that is the long wave (or low-frequency wave) components give a set-up under the giant wave. In this paper, we first apply second-order theory to show that the set-up of the long waves was the result of two wave packets whose mean wave direction was separated by around 90^{°} or more. This explanation is consistent with all the other evidence the authors have been able to find. These include the hindcast of the sea state and the measured forces on the structure. Secondly, we carry out fully nonlinear numerical modelling. This confirms that the set-up cannot occur in the following sea but would occur in a crossing sea. This modelling also shows that a wave as steep as the Draupner wave would break in a following sea, whereas it would not in a crossing sea. Therefore, our investigations in this paper lead to the conclusion that the set-up of the long wave components, and hence the giant Draupner wave, is the result of the two wave groups propagating at a large angle.

This paper does not explain why the wave was so large (although the set-up, as opposed to a set-down, does add 1 m to the crest elevation above mean sea level). However, we discuss other evidence that freak waves do occur in crossing sea-states, suggesting that there may be some physical process which occurs as two wave systems merge which causes abnormal waves.

## Acknowledgements

The authors thank Dr Sverre Haver (Statoil) for the Draupner time series. Jean Bidlot is thanked for doing the ECMWF hindcast. T.A.A.A. was funded by an EPSRC studentship and a ‘PhD plus’ fellowship. S.Y. was funded by Leverhulme Trust, UK.

- Received January 18, 2011.
- Accepted May 18, 2011.

- This journal is © 2011 The Royal Society