We analysed wave elevation data measured by a surface-following buoy off the coast of the island of Norderney in the German Bight in the time period between 2011 and 2016. The predominant wave propagation direction during this time period was southeast (Fig. ). The measurement buoy was deployed at a nominal water depth of h=10 m, which was assumed to be constant for the following analyses. In reality, the water depth off the coast of Norderney is not constant, as the bathymetry at the location is spatially highly variable with strong gradients (Fig. ). The bed slope perpendicular to the wave direction varies between 1 : 500 (offshore direction) and 1 : 200 (onshore direction). As the buoy is restricted only by its mooring, there is the possibility that it will move horizontally. The actual water depth h below the horizontally moving buoy may then be subject to rapid changes. In addition, the tidal range at the site is about 2.5 m , which further causes the water depth to vary.

The wave data were measured at a frequency of 1.28 Hz and are available as a set of time series (samples) of 30 min length. To exclude low-energy sea states in the following, only samples with a significant wave height Hs above the long-term 70th percentile of the significant wave height, Hs,70=1.29 m, were included in the analysis. Here, the significant wave height Hs is defined as the mean of the highest 30 % of the wave heights in a 30 min sample. Hs,70 was calculated from the significant wave heights Hs of all 30 min samples during the 6 years of available measurement data. On the one hand, this excludes possible measurement uncertainties caused by short waves that are only described by a few points; on the other hand, it includes only rogue waves of heights relevant for offshore activities. As the KdV equation for shallow water was to be applied to the data, only samples satisfying shallow-water conditions in terms of the validity of the KdV equation were included in the study. The definition of shallow water depths for the applicability of the KdV equation is different from the commonly used definition of shallow water in the engineering context, kh<π/10 . As explained in Sect. 1, the shallow-water condition used in this study was 1hL<0.22orkh≤1.36, with water depth h and wavelength L. The wavelength was calculated as 2L=Tp⋅c from the peak period Tp=fp-1 of each sample, where fp is the peak frequency in the linear fast Fourier transform (FFT) spectrum of the sample, and the linear phase speed c=gh, where g is gravity. Following Eqs. () and (), the condition for the peak period may be written as 3Tp>h0.22⋅c.

Thus, for a water depth of h=10 m, the peak period had to satisfy the condition Tp>4.6 s in order for a sample to classify for shallow depth conditions in which the KdV equation is valid. We based the shallow-water condition on the peak period Tp of the entire sample to assume that shallow-water wave properties as described by the KdV equation strongly contribute to the wave processes in the sample. Nevertheless, it was additionally ensured that each of the individual rogue waves (or the highest wave in each sample that did not contain a rogue wave) satisfied the depth conditions required for the applicability of the KdV equation, based on its period Tmax. Of all the selected samples above Hs,70, the required shallow depth conditions applied in more than 98 % of cases and were, thus, the dominant condition in these samples. The 2 % of the samples not satisfying the condition of shallow depth were discarded and not considered in the analysis. In the considered samples, kh ranged between 0.38 and 1.36.

Rogue waves are commonly defined as waves with an individual height H from crest to trough of 4H≥2.0Hs and/or waves with a crest height C above still water level of 5C≥1.25Hs.

In a previous study based on measurement data from the southern North Sea , we found that the rogue wave frequency significantly deviated from the Forristall distribution for wave heights larger than 2.3Hs. Therefore, in the present study we further define “extreme rogue waves” by a more strict height criterion of 6H≥2.3Hs.

For the definition of a wave, the zero-upcrossing method was used.

The measured time series were subdivided into five categories:

Non-rogue samples comprise measurement samples that did not include any rogue wave.

Examples of each time series category are shown in Fig. . Table  shows the number of samples and the percentage of samples in each category.

The vKdV-NLFT was applied to the data in order to obtain the discrete soliton spectrum of each time series. The KdV equation was introduced by . It describes the evolution of weakly nonlinear and dispersive progressive unidirectional free-surface waves in shallow water with constant depth. For the analysis of space series (fixed at one point in time), the space-like KdV equation (sKdV) is given e.g. in as 7ut+cux+αuux+βuxxx=0, in which u=u(x,t) is a free-surface space series, developing in space x and time t. The subscripts x and t denote partial derivatives, c is the phase speed in shallow water, and α=(3c)(2h)-1 and β=(ch2)/6 are constants, depending on the phase speed c and the water depth h. Equation () can be adapted to the analysis of time series (fixed at one point in space, such as buoy measurements). For the case of a free-surface elevation time series u(x0,t) (see e.g. Fig. ) at location x0, the spatial evolution is then described by the time-like KdV equation (tKdV) 8ux+c′ut+α′uut+β′uttt=0, in which c′=c-1=(gh)-1, α′=-α(c2)-1 and β′=-β(c4)-1. For our application of the KdV-NLFT, we assumed initial conditions with vanishing boundaries, i.e. 9lim⁡t→±∞u(x0,t)=0 sufficiently fast. As we were mainly interested in the soliton part of the nonlinear spectrum and solitons are not periodic, we preferred vanishing to periodic boundary conditions. For vanishing boundary conditions, the initial wave packet develops into a train of solitons followed by an oscillatory trail that vanishes over time e.g.. The soliton spectrum therefore completely describes the behaviour of the wave train in the far field. The surface elevation in the far field is then described by 10u(x,t)≈∑n=1Nũnsech2ωnt-knx-ϕn, i.e. as the linear superposition of independent solitons after the oscillatory waves have dampened out, with ũn and kn uniquely determined by ωn, h0 and g Eq. 2.20a​​​​​​​. The nonlinear spectrum of the vKdV-NLFT consists of a discrete spectrum representing solitons and a continuous spectrum representing oscillatory waves. We applied the vKdV-NLFT by using the interface to a development version (commit 681191c) of the FNFT software library . Figure  shows an example of a measured time series, its linear FFT spectrum, the nonlinear continuous spectrum and the discrete nonlinear soliton spectrum. In this paper, only the discrete soliton spectrum will be discussed further. Each of the solitons in the discrete spectrum would be a physical soliton if the signal is propagated according to the KdV equation with vanishing boundary conditions. After sufficiently long propagation, the solitons will separate and their characteristic shapes become clearly visible. For visualisation of the role of solitons in the time series, Fig. a shows the soliton train that was obtained by nonlinear superposition of the solitons (considering their interactions but neglecting the continuous spectrum) using the algorithm from . Although a soliton does not cross the still water level, a mathematical definition of the angular frequency can be obtained from the soliton solution of the tKdV Eq. 12 as follows: 11Ω=2π⋅F=3Ag4h2. As this equation relates the frequency F to the amplitude A of the soliton, the frequency sorts the solitons in the spectrum by their amplitude. Following the convention in , the solitons in the discrete spectrum (Fig. d) are displayed on a negative frequency axis. The vKdV-NLFT was applied to all 15 156 samples listed in Table .

The aim of the study was to explore the role of the individual solitons in the generation of rogue waves. The following procedure was used to check whether individual solitons in the NLFT spectrum could be associated with the recorded rogue waves. First, the KdV-NLFT of the original time series was computed. Following this, all free-surface elevations between the two zero-upcrossings of a rogue wave (or the largest wave for non-rogue wave samples) were scaled down to 80 % (Fig. ). The KdV-NLFT was then repeated for the modified time series, which resulted in a new soliton spectrum. We monitored which of the solitons had changed in amplitude A (and, therefore, in frequency F), due to the change in wave height of the modified rogue wave. These solitons were assumed to have the same position in the time series as the rogue/maximum wave.

Solitons were attributed to specific rogue waves, following the procedure described in Sect. . In each case, we found that the amplitude of one large soliton significantly decreased for a reduced rogue wave (or maximum wave) height. Furthermore, slight changes in amplitudes were observed in the group of smaller solitons. As amplitude A and frequency F are related according to Eq. () for solitons, the reduction in amplitude corresponded to a simultaneous shift in frequency, which can be seen in the soliton spectrum (Fig. ). The reduced solitons can be regarded as being associated with the rogue wave in the time series, while the other solitons in the spectrum maintained their amplitudes. The solitons with constant amplitudes can be regarded as not being associated with the rogue wave. We refer to the amplitudes of the l=1…n solitons associated with the rogue wave as ASn, with AS1 denoting the largest attributed soliton. Although often the case, the largest soliton attributed to the rogue wave was not necessarily the largest soliton in the spectrum (Fig. ).

We extracted the amplitude of the largest attributed soliton AS1 for each time series and compared it to the rogue wave height H (for rogue waves according to any of the two height criteria, including double rogue waves; Fig. a) or the crest height C of the rogue wave (for rogue waves according to the crest criterion, including double rogue waves; Fig. b). A comparison of the soliton amplitude AS1 to the largest wave height Hmax and the largest crest height Cmax in non-rogue wave samples has been added for reference (Fig. c, d). The slopes of the linear regression curves express increasing AS1 with increasing H or Hmax and C or Cmax. For the analysed samples, the scatter of the data suggests an upper limit of AS1 of between 2 and 3 m. The goodness of fit of each curve to the data is given in terms of the coefficient of determination 12R2=1-SSresSStotal, in which SSres is the sum of squares of residuals with respect to the regression curve and SStotal is the sum of squares of residuals with respect to the average value of the data (and thus a measure of the variance). R2 indicates that the linear curves fit the results from height and extreme rogue samples better than the results from non-rogue, double and crest rogue samples. R2 is higher in Fig. a than in Fig. b–d.

Moreover, it is seen that the amplitude of the largest soliton is always smaller than the rogue wave crest/height itself. This is in agreement with results by , who identified solitons in measurement data from the Adriatic Sea by applying the NLFT with quasi-periodic boundary conditions to the KdV equation. Our investigation revealed that, in all cases, some smaller solitons were additionally associated with a rogue wave. Typical values of the amplitude of the second largest soliton AS2 are 20 %–30 % of AS1. The amplitude of the third largest attributed soliton AS3 is typically 10 %–20 % of AS1.

So far, the results show that high soliton amplitudes in the spectrum are associated with high absolute values of wave heights or crests. However, this does not necessarily imply that high solitons play a role in forming individual waves that are exceptional with respect to the surrounding wave field. To be able to compare different measurement samples, the soliton amplitudes AS1 were normalised by the significant wave height Hs of the corresponding sample. By relating the normalised soliton amplitudes to the different time series categories, the importance of solitons for the relative height of rogue or maximum waves was investigated (Fig. ). If solitons are to play a major role in the presence of rogue waves, their normalised amplitudes are expected to increase from non-rogue wave samples with H(Hs)-1<2.0 through height and double rogue waves (2.0≤H(Hs)-1<2.3) to extreme rogue waves (H(Hs)-1≥2.3). In fact, the median values of AS1(Hs)-1 are higher for rogue wave samples than for non-rogue wave samples, meaning that the distributions calculated from the rogue wave samples are shifted towards higher normalised soliton amplitudes with respect to the distribution calculated from non-rogue wave samples (Fig. ). Additionally, the rogue wave sample distributions, especially those calculated from crest and extreme rogue samples, show heavier tails. The differences in the distributions suggest that solitons play a role in rogue wave generation. It is striking that not only extreme rogue waves but also crest rogue waves had a tendency to be associated with higher solitons. This makes sense when recalling that a soliton is not an oscillating wave and, due to its shape, contributes more to wave crests than to wave heights. However, although differences in normalised soliton amplitudes AS1(Hs)-1 are present for the different categories, the distributions overlap and the positive trend with increasing relative wave height is not as pronounced as the positive trend of AS1 with increasing maximum wave height (as presented in Fig. ). This emphasises the relevance of the considered sea state for the soliton amplitude, in that large solitons are only found in high sea states. Large solitons correspond to high wave heights H and high crest heights C but not necessarily to high relative wave heights H(Hs)-1 or high relative crest heights C(Hs)-1.

As we were interested in the importance of nonlinear processes in rogue wave generation at the buoy location, we intended to quantify the nonlinearity of the rogue waves. In shallow water, the nonlinearity of waves can be described by the Ursell number . According to Eqs. 10.151 and 10.154, the Ursell number in its time-like form is given by 13Ur=332π2HL2h3=mK2(m)2π2, with the modulus m.

​​​​​​​Different definitions of the Ursell number exist. A common definition is Eq. 11.109 U1=HL2h3=163K2k2, with K the complete elliptic integral of the first kind and with the modulus k. Comparison of Ur and U1 shows the Ursell numbers to differ by a factor of 3/(32π2). The moduli of Ur and U1 are related by m=k2. Thus, different Ursell number definitions will yield different thresholds for the separation of wave theories. In this study, we use the definition given in Eq. () and adjust the cited threshold values accordingly. (For consistency with U1, the wave amplitude a in the original equation of has been replaced by the wave height a=H/2.)

The Ursell number has been used to classify wave types. In , solitary-like waves are defined by a modulus of m>0.99. According to this classification and by applying U1, Ursell numbers Ur>0.559 are obtained for solitary-like waves. Waves with Ur≤0.559 are classified as oscillatory waves.

According to U1, the Ursell number is defined either by the modulus m or by height H and wavelength L of a single wave oscillation over depth h. Thus, we can calculate the Ursell number for the identified rogue waves using the H and L obtained by zero-upcrossing. In our case, the amplitudes of the largest attributed solitons show an almost linear positive trend with increasing Ursell number up until approximately Ur=0.5 (Fig. ). For our data, in which the bulk of waves are located below Ur=0.559, this means that most rogue waves are not classified as solitons. This is in agreement with several previous studies that have shown that rogue waves in shallow water, despite their large amplitudes, have very small ratios of nonlinearity to dispersion (Ursell numbers) and, thus, are almost linear . Another observation made from Fig.  is an upper limit in soliton amplitude between AS1=2.0 m and AS1=2.8 m, depending on the time series category, for Ursell numbers larger than approximately Ur=0.5. Referring to the classification given above, this implies that soliton amplitudes are limited for the most nonlinear waves, which are those satisfying solitary wave theory. A limit in soliton height as a result of breaking is expected at amplitudes of approximately A=8 m for a water depth of h=10 m, as the breaking criterion for solitary waves is Ah-1=0.78 or Ah-1=0.83 . Therefore, shallow-water wave breaking at the location of the buoy can be excluded. The reason for the limit in soliton amplitude at AS1=2.5 m to AS1=3 m could be limited energy input by wind (see , for soliton generation by wind) or a shoal in front of the measurement buoy causing the larger waves to break before they reach the buoy.

When investigating the attribution of solitons to rogue waves in Sect. , we found that the largest soliton in the nonlinear spectrum could be attributed to the rogue wave in the majority of cases. In addition, this soliton was often outstanding from the other solitons in the spectrum, with a much larger amplitude than the remaining solitons in the spectrum (see the example in Fig. ). Therefore, we were interested in whether the existence of an outstanding soliton in the nonlinear spectrum was typical of rogue wave samples off Norderney. We investigated this question statistically by comparing soliton spectra, calculated from vKdV-NLFT, for non-rogue wave samples and the four different categories of rogue wave samples. In fact, while all 15 156 considered time series yielded discrete spectra with a large number of solitons, we identified two characteristic classes of soliton spectra. The typical appearance of a soliton spectrum calculated from a time series without rogue waves was a cluster of solitons (Fig. ). On the contrary, in the majority of cases, soliton spectra calculated from time series including a rogue wave showed one outstanding soliton with an amplitude much larger than that of the remaining cluster of solitons in the spectrum (Fig. ).

To distinguish between clustered soliton spectra and those featuring an outstanding soliton, we compared the amplitudes of the largest soliton, A1, and the second largest soliton, A2, in the discrete spectrum. From the visual inspection of the spectra, we identified a threshold of the ratio A2(A1)-1, below which the largest soliton could be called outstanding: 14A2A1≤0.8.

Thus, a soliton spectrum had an outstanding soliton if the second largest soliton was at least 20 % smaller than the largest soliton in the spectrum. The choice of this threshold was further supported by the fact that the threshold A2(A1)-1=0.8 coincides with the median value of A2(A1)-1 for maximum wave heights just below the rogue wave criterion H(Hs)-1≥2.0 (Fig. ). This reveals that our threshold chosen for the distinction between clustered spectra and those featuring an outstanding soliton concurrently indicates a difference between the spectra calculated from non-rogue and those calculated from rogue wave time series.

Equation () is valid for 30 min samples at the measurement site, which is the standard window size of measurement samples delivered by Datawell Waverider buoys. As the ratio between soliton amplitudes might be dependent on the window size, it is not clear if Eq. () would apply to time window sizes other than 30 min. The effect of a larger time window size will be discussed in Sect. . Table  shows the share of outstanding solitons and clustered soliton spectra in each of the categories defined in Sect. . It is seen that the typical appearance of the soliton spectrum for 30 min wave measurement samples off Norderney without rogue waves is a cluster of solitons (64 % of the samples); at the same time, it is not unlikely to obtain a soliton spectrum with one outstanding soliton from vKdV-NLFT (36 % of the samples). For 30 min rogue wave samples, in contrast, it is more likely to obtain a soliton spectrum with one outstanding soliton than a clustered soliton spectrum. This is true for height rogue samples (57 %), and it is even more pronounced for crest rogue samples (64 %), double rogue samples (72 %) and, finally, extreme rogue samples (87 %). The conclusion can be drawn that the absence of an outstanding soliton is a strong indicator of the absence of an extreme rogue wave. The differences between the four rogue wave categories, indicating that the presence of an outstanding soliton is not equally expressive for all types of rogue waves, may lead to the presumption that not all rogue waves found off Norderney can necessarily be explained by the same theory.

The question regarding whether inferences can be made from the time to the spectral domain or vice versa is answered by a contingency table (Fig. ). Here, all previously defined rogue wave categories are combined into one joint group of rogue wave samples. Two statements can be made based on the table. On the one hand, the probability that an NLFT spectrum calculated from a normal sample shows an outstanding soliton is 4986/13.984=36 %, whereas the probability that a spectrum calculated from a rogue wave sample shows an outstanding soliton is 726/1172=62 %. This indicates that, although not all rogue waves can necessarily be explained by the same theory, outstanding solitons occurred in connection with the majority of observed rogue waves off Norderney. While outstanding solitons play a role in 62 % of the cases in the combined group of rogue waves, the share differs between the rogue wave categories (Table ). On the other hand, although rogue waves are more likely to be observed when an outstanding soliton is present in the NLFT spectrum, the presence of an outstanding soliton alone is not a sufficient an indicator for the detection of rogue waves. The main difficulty is the imbalance in sample size between non-rogue wave and rogue wave samples.

In Fig. , the ratio between the amplitudes of the second largest and the largest soliton in the nonlinear spectrum, A2(A1)-1, is visualised in a box plot for each of the time series categories. A ratio above A2(A1)-1=0.8, meaning that the second largest soliton has a rather similar amplitude to the largest soliton, implies that the soliton spectrum is clustered (Eq. ). For non-rogue wave samples, this is the case for the bulk of time series. The median of the ratio A2(A1)-1 decreases from the leftmost to the rightmost category on the right axes in Fig. . For height rogue samples, the median of A2(A1)-1 is below the 80 % line, with the distribution extending above and below. For double and extreme rogue waves, the gap between the soliton amplitudes may become much larger than for height rogue waves. In some cases, the amplitude A2 amounts to less than 30 % of the amplitude A1. In all categories except extreme rogue samples, there are samples for which the first and second solitons are almost similar in amplitude (A2(A1)-1≈1). On the contrary, for all extreme rogue wave samples, A2 is below 93 % of A1. The large part of soliton spectra from extreme rogue samples shows an outstanding soliton.

Figure  presents the ratio A2(A1)-1 in a scatter plot with one data point for each individual time series. According to this representation, although the presence of an outstanding soliton with A2(A1)-1≤0.8 is not a useful indicator of whether a rogue wave is present in the time series or not, the presence of a rogue wave becomes much more likely when one soliton in the nonlinear spectrum is strongly outstanding with A2(A1)-1≤0.3: of all 23 samples satisfying A2(A1)-1≤0.3, only 4/23=17 % are non-rogue wave samples, whereas 19/23=83 % of the samples are rogue wave samples (1 height, 1 crest, 8 double and 9 extreme rogue samples).