Observation of a giant nonlinear wave-packet on the surface of the ocean

In many physical systems such as ocean waves, nonlinear optics, plasma physics etc., extreme events and rare fluctuations of a wave field have been widely observed and discussed. In the field of oceanography and naval architecture, their understanding is fundamental for a correct design of platforms and ships, and for performing safe operations at sea. Here, we report a measurement of an impressive and unique wave packet recorded in the Bay of Biscay in the North-East of the Atlantic Ocean. An analysis of the spatial extension of the packet that includes three large waves reveals that it extents for more than 1 km, with individual crests moving faster than 100 km/h. The central and largest wave in the packet was 27.8 m high in a sea with significant wave height of 11 m. A detailed analysis of the data using the nonlinear Fourier analysis reveals that the wave packet is characterized by a non trivial nonlinear content. This observation opens a new paradigm which requires new understanding of the dynamics of ocean waves and, more in general, of nonlinear and dispersive waves.


Results
Here, we report on a related phenomenon, i.e. the existence of a giant wave packet on the ocean surface, i.e. a group of coherent waves that emerge from a random sea state, see Fig. 1a: the central wave in the packet, the largest one, has a height of almost 27.8 m. To highlight the difference between a typical rogue wave and the giant wave-packet in Fig. 1b, we show a zoom of the measurements in the Gulf of Biscay and the well known Draupner wave 12,17 , (a shift in time has been applied in order to have the crest of the largest waves both at time t = 0 ): while the crest of the Draupner wave is slightly larger, the different size and volume of water involved in the two events is evident. Using the method described in 18 , the study of the meteocean conditions during which such object has been recorded and the understanding of its content in terms of the Nonlinear Fourier analysis are the subject of this article. The winter 2013-2014 has been the most active one of the last decade in the North Atlantic, the 4-6 January 2014 storm representing the most severe event of the season. The storm initiated on 3 January with an intensifying low pressure system off Newfoundland. In the following day the storm grew very rapidly, surface pressure dropping at 934 hPa, satisfying the condition of explosive cyclogenesis. Moving rapidly East, the strong winds, measured up to 28 m/s, led to exceptionally large waves, the peak conditions (according to the analysis of the European Centre for Medium-Range Weather Forecasts, ECMWF) being close to 20 m significant wave height. While the heart of the storm moved towards Ireland where the largest coastal waves took place, the cyclonic winds led also to a strong south-easterly moving swell that reached the coasts of the Iberian peninsula on 6 January. Heavy damage was reported along the Portuguese coast. When the swell reached the north-west corner of the peninsula, a south-westerly wind was locally blowing (Fig. 2). At 12 UTC the buoy of Villano-Sisargas (Cabo Vilán), located at 43.50 • N and 9.21 • W, close to the north-west corner of the Iberian peninsula, where the water depth h is 383 m, recorded the giant wave packet. The local wave conditions were characterized by a dominant South-East directed swell plus a superimposed relatively minor wind sea directed to 60 • N. The mean directions plotted in the panels of Fig. 2 represent the weighted combination of the two respective energies.
The largest wave measured is of 27.8 m in a significant wave height height, H s , of 11.1 m, the latter calculated as 4 times the standard deviation of the 1600 s time series shown in Fig. 1 (sampling rate 1.28 Hz). According to   19 . In the present case the distance (in time) between the two local minima around the largest wave is 19.5 s which, according to the linear dispersion relation, corresponds to a length of 593 m. Besides measuring the surface elevation, the position of the buoy in time has been recorded, its trajectory following, in deep water conditions, to some extent the local wave kinematics. In Fig. 1c the horizontal displacement of the buoy is shown. The colours in the plot indicate the vertical displacement; dark dots connected with a line indicate the time interval during which the large wave passed. For perfectly unidirectional waves, the motion of the buoy would be on a line inclined of some angle, depending on the direction of propagation of the waves. Interestingly, the Figure confirms that all waves in the wave packet travel in the same direction. The frequency-direction spectrum has been computed using the buoy 2-D horizontal displacement and heave implemented in the Extended Maximum Likelihood Method 18 , see Fig. 3. The spectrum displays two peaks with a small angle in between them. The higher frequency peak is related to the wind sea while the other more energetic peak corresponds to the swell.

Methods
The results shown in Fig. 1c,d, the reconstruction of the storm in Fig. 2 and the energetic swell observable in the directional spectrum, allow us to assume that locally the dynamics can be approximated at least for short times to one dimensional propagation. Within this framework, a model that describes to the leading order in dispersion and nonlinearity the dynamics of surface gravity waves in infinite water depth as is the Nonlinear Schrödinger equation, written as an evolution equation in space, which reads: with A = A(x, t) represents the complex envelope of the surface elevation that changes in space x and in time t. k 0 and ω 0 are the wave number of the carrier wave and its associated frequency, respectively. They can be estimated directly from a time series taking its Fourier spectrum and considering the frequency corresponding to its peak.
For what follows the angular frequency ω 0 will be taken to be 0.314 s −1 ( f 0 = ω 0 /(2π) = 0.05 Hz); with this value of the angular frequency, the wavenumber k 0 = ω 2 0 /g has the numerical value k 0 = 10 −2 m −1 and k 0 h = 3.83 . The finite water depth has drastic effects on the dynamics: the NLS equation is still in a focussing regime, since k 0 h > 1.36 ; the numerical values of the coefficients are ν = 1.01 , γ = 0.72 , κ = 1.09 The surface elevation, η(x, t) is related to the complex envelope at leading order as follows: The NLS equation is integrable through the Inverse Scattering Transform 20 , IST, which can be considered as a generalization of the standard Fourier Transform which solves linear wave problems. Just like the Fourier analysis, the IST can be used to analyze data. Indeed, starting from the pioneering works in 21,22 , a nonlinear time series analysis tool has been developed and used for different applications [23][24][25][26][27][28] . Here we analyze the time series by solving the nonlinear spectral problem associated with Eq. (1): Indeed, the main tool for visualizing the results from a nonlinear spectral analysis is the so called -plane. can have a real and imaginary part, the latter forming the discrete spectrum associated with the soliton content of A 0 . As prescribed in 24,25 , the field A 0 used in Eq. (4) is obtained by limiting the experimental signal over a time interval of finite extent in order to establish locally the nonlinear contents of the data. For each window, the spectral problem in Eq. (4) has been solved using zero (ZBC) 25 and periodic boundary (PBC) conditions 22,29,30 In Fig. 4 we show the results of our analysis by plotting on the left the time series (the window of time over which the analysis has been performed is shaded) and on the central and right columns the nonlinear analysis performed with ZBC and PBC, respectively. Using ZBC, two discrete eigenvalues located above the real axis are obtained when the analysis is performed on a window that contains the large wave packet. These discrete eigenvalues are not found if the nonlinear spectral analysis is performed at other positions. The analysis has also been performed on wider windows confirming the same results. Similar results are obtained if PBC are used: instead of discrete eigenvalues, two large spines are observed in correspondence of the large wave packet. Those spines are a signature of nonlinear modes that can be interpreted in the framework of finite gap theory of periodic IST 29,30 . The results are also robust to small changes in the numerical values of the parameters related to the carrier wave: changing the numerical value of ω 0 by a 10 % does not change quantitatively the results obtained by nonlinear spectral analysis. Therefore, the main result obtained within the nonlinear analysis is that the wave packet recorded by the buoy has some nonlinear content, a result that is also comforted by the calculation of the nonlinear length and dispersive length defined as: where T is the width of the packet in time which is of the order of 50 s. �|A 0 | 2 � is a measure of the wave packet amplitude and can be estimated to be 14 m (half the total height). Substituting the numerical values, we obtain L nlin ∼ 7 km that is shorter than the linear dispersive length L dis ∼ 14.64 km. We emphasise that the analysis is based on the assumption that the evolution is ruled by the NLS equation at least for short distances. www.nature.com/scientificreports/ It is well known that solitons are unstable to transverse perturbations 31,32 which cannot be avoided in any natural environment; therefore, such structure will eventually disappear. The conservative dynamics of the considered wave packet is influenced by three physical phenomena: the dispersion, the nonlinearity and the diffraction. Based on the 2D+1 NLS equation, the diffraction length can be estimated as: where Y corresponds to the width of the wave packet in transverse to the direction of propagation scale. For what concerns the transverse scale, there is no direct measurement; however, to be very conservative, we estimate that the transverse size of the wave packet is at least of 1 km; note that the wavelength in the direction of propagation is almost 600 m. The diffractive effects, due to the finite crest, take place on a scale of 20 km. The nonlinear spatial scale is the shortest one. assuming that the packet is travelling with its group velocity, i.e. ≃ 56 km/h, its life time is about of 20 min.

Discussion
In this paper we have reported on the analysis of a unique time series of surface elevation recorded in the Biscay Bay which shows a giant wave packet whose maximum height is almost 28 m, i.e. one of the largest wave ever measured. Besides its height, the group extends spatially for almost 2 km in the propagation direction. Despite a detailed analysis of the storm that took place during the day of the recording, the condition in which such extraordinary packets are formed is still far from being understood. The analysis reveals that a wind sea is superimposed to a large swell travelling from North-West; the latter appears to be, at least from the analysis www.nature.com/scientificreports/ of the movement of the buoy, quite unidirectional. The leading order model that can be used to describe locally the dynamics of the wave packet is the Nonlinear Schrödinger equation; we exploit the integrability property of the latter equation and use the IST to establish the nonlinear contents of the wave packet. The analysis, using both PBC and ZBC, reveals that the giant wave packet is characterized by a nonlinear content. We mention that the latter result is based on the assumption that the dynamics is ruled, at least for short time and space, by the NLS equation. Effects of a finite transverse front have to be taken into account, but unfortunately, as most of the times it happens for ocean waves, no spatial measurements are available. A rough estimate of the life-time of the giant wave packet, based on a simple dimensional argument on two-dimensional version of the NLS equation, highlights that the wave packet may have survived at least for 20 min.