THE EVENT.

We refer to the event of 1 January 1995 when, during stormy conditions, a particularly high, and apparently isolated, wave was recorded at the Draupner oil platform in the north-central North Sea. Being the first such recorded event, it triggered a sequence of studies (e.g., Dysthe et al. 2008; Fedele et al. 2016) that tried to analyze, and possibly explain, the 25.6-m recorded individual wave height with 18.5-m crest elevation under a significant wave height H_s close to 12 m. Two complementary approaches have been used. In the first approach, scholars tried to simulate as best as possible the meteorological and wave conditions to provide, together with the original time record, the background for the second, more theoretical, approach based on the nonlinear spectral instability or extreme statistics of nonlinear crest and wave height.

A basic problem with hindcasting the storm of 1 January 1995 is its rather unusual conditions, with a small polar low descending rapidly along the east coast of the North Sea, superimposed on a large-scale preexisting storm with wind and waves from a different direction. While the existing cross-sea conditions were identified [see the study by Adcock et al. (2011)], the available data did not lead to a rational explanation, generally ending with the conclusion of an exceptional, statistically rare, wave event.

In a parallel paper, Cavaleri et al. (2016) describe a new high-resolution hindcast of the storm obtained with the new version of the European Centre for Medium-Range Weather Forecasts (ECMWF) global coupled atmosphere–ocean–wave modeling system. As in the now operational model (since 8 March 2016), in the hindcast the resolution was 9 and 14 km for the atmospheric and wave models, respectively. The first one turned out to be especially crucial in obtaining a previously unrecognized picture of the situation summarized, for our present interest, in the following points: 1) on 1 January, large waves were moving from northwest to southeast with about H_s = 8 m; 2) the small, but very intense, polar low was descending rapidly from north to south (speed about 15 m s⁻¹), continuously creating an increasing system of large waves with a large background swell also moving with the low (but not forced by local wind), whose energy evidence strongly suggests it was fed by nonlinear interactions with the dominant waves of the polar low; and 3) this reached the Draupner latitude, slightly to the east of the platform, at 1500 UTC, at the time the large wave was recorded. The hindcast 2D wave spectrum is shown in Fig. 1 where, beside the bulk energy, the lobe of what Cavaleri et al. (2016) call “dynamical swell” (an actively energy receiving wave system, but not from wind) is evident at frequency 0.055 Hz and direction 180°.

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The modeled frequency–direction spectrum at the Draupner platform position at 1500 UTC 1 Jan 1995. Spectral energy density isolines are shown at 20 m² s rad⁻¹ intervals. The “dynamical swell” lobe is evident at about frequency 0.055 Hz and flow direction 180°.

Citation: Bulletin of the American Meteorological Society 98, 4; 10.1175/BAMS-D-15-00300.1

Starting from the full information derived from the hindcast, in this paper we do an intensive and detailed analysis of the nonlinear wave conditions at the Draupner platform and how these evolved during the day (see “Analysis of the event” section). We follow two approaches for extreme wave conditions. In the first one, an innovative piece of theory, we approach the problem with the analysis of the envelope of a wave train. The second approach deals with the space–time (ST) analysis of extreme sea surface elevations, which are connected to the spatial evolution of wave groups at the apex of their development stage (focusing point). We then reverse the problem and estimate the sea surface area and time necessary to consider the Draupner wave crest height (18.5 m = 1.55 H_s) as the expected extreme event. In the discussion in the “Two further points” section, we put forward two more arguments. The first one concerns the shape of the Draupner wave, and, on the basis of open-sea 3D (2D space + time) measurements, we show it to be typical of a high wave passing close to the focusing point of a large ST group. The final argument deals with the concept of “encounter probability” explaining why we consider this more appropriate than the commonly used “occurrence” or “happening” probability. We summarize our findings in the “Summary” section.

ANALYSIS OF THE EVENT.

The envelope approach.

The starting point for the envelope approach is Janssen (2014), where the analysis of the sea surface elevation η time series is based on its envelope ρ, complemented with theoretical and experimental results from nonlinear optics. The latter is a convenient step because 1) nonlinear wind waves and nonlinear optics share part of the physics and most of the relevant equations, and 2) optics has the advantage of being able to explore experimentally the range of extremely low (order 10⁻⁶) exceedance probabilities. This opens the way to provide estimates of the probability of extremely rare (i.e., large) ocean waves.

For nonextreme values, that is, when the effects of nonlinearity are small, the probability density function (pdf) of the normalized envelope wave height h = 2ρ/4σ, with σ² being the variance of the sea surface elevation, can be obtained by a Taylor expansion of the logarithm of the Rayleigh generating function as

The related coefficients C3 and C4 are the third-order (skewness) and fourth-order (kurtosis) cumulants of the random envelope. These can be obtained from the wave spectra following Janssen (2014). However, this approach is not suitable for the largest wave heights, where the envelope wave height 2ρ can be higher than 2.5 times the significant wave height (in the Draupner case h is 3.1). Taking advantage of the similarity of the basic equations, the solution for high sea waves is provided by nonlinear optics, where experimental results (here the envelope, not the single waves, is measured) strongly suggest an exponential tail for the extreme height distribution. Indeed, the analysis by Janssen and Bidlot (2009, see their Fig. 8) also suggested that the statistics of the largest ocean waves tend to depart from the general wave height distribution and approach an exponential one.

In nonlinear optics the pdf p(E), with E = 2h² = ρ²/(2σ²) the power of the signal normalized with its average, can be approximated as (Montina et al. 2009)

where the normalization factor N and the coefficients c₁and c₂ follow in optics from a fit to the observations. Equation (2) allows us to derive the distributions p(h) and p(ρ), with a progressive transition from a Gaussian state to an exponential distribution for large E (Janssen 2015). Rather than fitting to the observations, the coefficients c₁ and c₂ have been determined by Janssen (2015) matching the approximate expression of the wave height probability at its upper range of validity. The correctness of (2) has been verified by extensive numerical experiments (Janssen 2015). Figure 2 shows how the exponential tail distribution fits well to the Monte Carlo results, diverging substantially for the highest h range (>2.5) from those also of the nonlinear envelope theory.

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The pdf p(h) of envelope wave height h at the Draupner platform position at 1500 UTC according to linear, nonlinear, and nonlinear exponential theories vs Monte Carlo numerical experiment results.

Citation: Bulletin of the American Meteorological Society 98, 4; 10.1175/BAMS-D-15-00300.1

The Euler characteristic approach.

There is theoretical and experimental evidence that high waves travel with wave groups (Kharif et al. 2009). At a single measurement point, the random nature of wave phases and the narrow spatial localization of high waves make it unlikely to capture 3D moving groups at the apex of their development, an apex that generally represents the highest crest or wave height that can occur anywhere within a sea surface area. Therefore, to obtain realistic statistics of the largest waves of 3D groups, a possible approach stems from the analysis of the extremes of multidimensional random fields. In this respect we follow the Euler characteristics (EC) approach (Adler 1981), adapted to Gaussian short-crested oceanic waves by Fedele (2012) and assessed with observed ST wave data (Benetazzo et al. 2012) by Fedele et al. (2013) and Benetazzo et al. (2015). The EC method provides a straightforward way to compute the exceedance distribution function of maximum sea surface elevations (η_max) over a statistically homogeneous and stationary ST sea region U. To account for second-order nonlinearities due to bound harmonics, Fedele et al. (2013) and Benetazzo et al. (2015) express the nonlinear wave crest amplitude using the Tayfun formulation (Tayfun 1980). Putting all this together, the final form of the ST exceedance probability of second-order nonlinear maximum crest heights is approximated, for large values of the normalized linear crest height z, as

where ξ is the nonlinear wave crest height related to z via the Tayfun’s quadratic equation (Tayfun 1980); σ is the standard deviation of the sea surface elevation field; and N₃, N₂, and N₁ are the average number of 3D, 2D, and 1D waves within U, along its boundaries and edges, respectively (Fedele 2012). At first, we consider the distribution (3) over a 2D spatial domain covering the wave model computational cell (14 × 14 km²) to analyze the conditions that supported the generation of a very high wave at the Draupner platform around 1500 UTC 1 January 1995. In this condition the 3D term vanishes (indeed, N₃ = 0 as the time span of U is equal to 0), and the dominant contribution to the probability of maxima is the 2D spatial term, N₂z in (3), which is scaled by the average number of 2D waves N₂ given by

Here X and Y are the side lengths of the region U (we assume the x axis is oriented as the mean direction of wave propagation and the y axis is orthogonal to x); L_x and L_y are characteristic mean wave lengths along the two axes (equal to 102 and 141 m, respectively, at 1500 UTC); and α_xy is a parameter representing the irregularity of the sea state. The smaller α²_xy is (always ≤1 and equal to 0.005 at 1500 UTC), the more irregular the sea surface elevation field and the larger the number N₂ will be.

The conditions that led to the Draupner wave.

We analyze the storm following the two approaches presented in the previous two subsections and report the results at the Draupner position. Starting from the hindcast spectra (Fig. 1) and following Janssen (2014), Fig. 3 shows the time evolution, during 1 January 1995, of the nonlinear parameters of the sea state, namely, skewness factor C₃ and kurtosis factor C₄. It is clear that both the factors increase substantially after 1000 UTC, reaching very large values when the polar low reaches the Draupner position around 1500 UTC. The exceedance probability of the envelope increases accordingly (Fig. 4). Along the Euler characteristic approach, we choose as the nonlinear threshold in (3) the observed normalized Draupner crest height ξ = 6.2 = 1.55 H_s/σ and derive the related probability. For the nonlinear ST extreme approach, we have used the measure of the wave steepness μ (Fig. 3) derived according to Fedele and Tayfun (2009).

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Evolution in time at the Draupner platform position of the nonlinear parameters skewness (C3), kurtosis (C4), and steepness (μ) evaluated from the local frequency–direction wave spectrum following Janssen (2014) and Fedele and Tayfun (2009).

Citation: Bulletin of the American Meteorological Society 98, 4; 10.1175/BAMS-D-15-00300.1

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Evolution in time of the wave conditions at the Draupner position and related probabilities. The estimates are in an area of 14 × 14 km² (one grid element of the wave model). (left) Significant wave height and expected maximal crest height, E{η_max}, according to the ST extreme nonlinear distribution. (right) Exceedance probability (Pr) of crest height > 1.55 H_s according to linear and nonlinear ST extreme model (ST-L and ST-NL, respectively), and to linear, nonlinear, and exponential tail envelope approaches (ENV-L, ENV-NL, and ENV-NL-ET, respectively). The time of the Draupner wave is marked (dashed red line) in all panels.

Citation: Bulletin of the American Meteorological Society 98, 4; 10.1175/BAMS-D-15-00300.1

All these results are summarized in Fig. 4, which we discuss in detail. The left panel shows the modeled evolution of the significant wave height reaching a maximum around the time the polar low reaches the Draupner area, plus, according to nonlinear ST extreme theory, the expected maximal crest height over a local area of 14 × 14 km² (corresponding to the single grid cell in our simulation).

The evolution on 1 January of the exceedance probability of ST crest height and envelope larger than 1.55 H_s (i.e., the observed Draupner crest height) according to the two different approaches is in the right panel. Starting with the linear ST probability, we see that this suggests very small probabilities (below 10⁻³), further decreasing after 1000 UTC when the large-scale storm (waves from northwest) develops, leading to longer waves. As a consequence, the number N₂ in (4) decreases, not supported by a concurrent reduction of . However, the nonlinear ST probability, much higher in the morning and decreasing at 1000 UTC, pushes up again, reaching its maximum slightly before 1500 UTC. A similar difference and evolution is found for the envelope approach. It is the combination of high probability and high significant wave height that leads to the expected large crest height. From a physical perspective, it is interesting to note how from 1000 to 1500 UTC H_s grew about 35%, with in turn only a limited (10%) reduction of the average wavenumber. The reason is the convergence in the Draupner area of two independent wave systems, the large-scale one from the northwest and the one from the north associated with the polar low, with similar characteristics, and hence an immediate increase of the local steepness. We have also estimated the area and time extent of the ST region U to consider the Draupner wave crest height as the expected event of the ST extreme nonlinear distribution. The time span was considered coincident with the length (1200 s) of the wave record at the platform. The spatial extent turns out to be 850 × 850 m².

As a final note on this point, Onorato et al. (2006), then applied by Cavaleri et al. (2012) on the Louis Majesty accident, indicated that two similar systems crossing at an angle close to, but less than, 60° (see Fig. 1) would be prone to instability and hence favor the appearance of particularly high waves. This too may have had a role in favoring the appearance of single particularly high waves.

TWO FURTHER POINTS.

One of the apparently anomalous characteristics of the Draupner wave was its high crest, 18.5 m, compared to the 25.6-m wave height and 12-m significant wave height. Indeed, at first look the wave appears as coming out of the blue (Fig. 5), surrounded, before and after in the original record, by a sequence of relatively small waves. Walker et al. (2004) have shown that nonlinearities up to fifth order are necessary to describe the crest-to-trough asymmetry of the Draupner shape. However, Benetazzo et al. (2015) have a quite different approach. Exploiting 3D (2D space + time) wave measurements (Benetazzo et al. 2012) in the open sea, for each point where a particularly high wave (with crest height exceeding 1.25H_s) had appeared, they plot the normalized (with respect to the H_s and the peak period T_p) time series. These profiles (23 of them) are summarized in Fig. 5 (right panel) in the “mean” profile and its uncertainty bands. It is clear that all these “particularly high crests” have, once normalized, almost the same profile. This is also the case of the Draupner wave, also superimposed in Fig. 5, and almost coincident with the “mean” profile. This strongly suggests that the Draupner wave was not so exceptional after all, simply reproducing on a larger scale what should be expected in those conditions as the highest crest of a dynamical space–time wave group was passing close to the rig. The order of magnitude of the related probability is given by the extent of the area, 850 × 850 m² (see the previous section), where the crest height of 1.55H_s appears as the expected value.

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(left) Temporal profile of the sea surface elevation (η) recorded at the platform and its envelope (ρ). The Draupner wave crest is shown around 264 s, when the local wave energy E is about 20 times the average energy. (right) Normalized temporal profile (blue solid line) of the Draupner wave compared to the mean profile (black solid line) of waves with crest height > 1.25 H_s observed at the near-focusing position of ST groups (following Benetazzo et al. 2015). The uncertainty band of observations is bordered by the dashed black lines.

Citation: Bulletin of the American Meteorological Society 98, 4; 10.1175/BAMS-D-15-00300.1

We close stressing a concept exposed by Cavaleri et al. (2016) and concerning the probability of coming across a particularly large wave. In the literature this is often referred to as “occurrence probability.” However, this expression may be misleading as its words imply the occurrence of the event but implicitly refer to a specific position. Cavaleri et al. (2016) propose to use in this sense the expression “encounter probability,” thus implying the encounter with the wave, that is, considering the probability of the event not only in time, but also in space. The implications for the difference between a buoy (a single point) and a rig or a ship (extended in space) are obvious.

SUMMARY.

Starting from Cavaleri et al. (2016), who analyzed the meteo-oceanographic conditions during the Draupner wave event, our main findings are summarized as follows.

1. Given the local wave conditions, the probability of the Draupner wave has been evaluated with both a new approach based on the wave envelope theory, supported by the corresponding nonlinear optic results, and the EC approach of ST wave maxima.
2. Both approaches confirm the substantial high occurrence probability of the Draupner wave, which exceeded 15% over an area of 14 × 14 km² (corresponding to the single grid cell in the simulations).
3. Once quantified in space and time, the Draupner wave appears as the expected event in the time span of the 20 min and an area estimated to be less than 1 km².
4. The Draupner wave temporal profile appears to be typical of a large space–time nonlinear wave group close to its focusing point. This supports the evidence that rogue waves can be generated by constructive interference (in frequency and direction) of elementary nonlinear waves.