The shallow waters off the coast of Norderney in the southern North Sea are characterised by a higher frequency of rogue wave occurrences than expected. Here, rogue waves refer to waves exceeding twice the significant wave height. The role of nonlinear processes in the generation of rogue waves at this location is currently unclear. Within the framework of the Korteweg–de Vries (KdV) equation, we investigated the discrete soliton spectra of measured time series at Norderney to determine differences between time series with and without rogue waves. For this purpose, we applied a nonlinear Fourier transform (NLFT) based on the Korteweg–de Vries equation with vanishing boundary conditions (vKdV-NLFT). At measurement sites where the propagation of waves can be described by the KdV equation, the solitons in the discrete nonlinear vKdV-NLFT spectrum correspond to physical solitons. We do not know whether this is the case at the considered measurement site. In this paper, we use the nonlinear spectrum to classify rogue wave and non-rogue wave time series. More specifically, we investigate if the discrete nonlinear spectra of measured time series with visible rogue waves differ from those without rogue waves. Whether or not the discrete part of the nonlinear spectrum corresponds to solitons with respect to the conditions at the measurement site is not relevant in this case, as we are not concerned with how these spectra change during propagation. For each time series containing a rogue wave, we were able to identify at least one soliton in the nonlinear spectrum that contributed to the occurrence of the rogue wave in that time series. The amplitudes of these solitons were found to be smaller than the crest height of the corresponding rogue wave, and interaction with the continuous wave spectrum is needed to fully explain the observed rogue wave. Time series with and without rogue waves showed different characteristic soliton spectra. In most of the spectra calculated from rogue wave time series, most of the solitons clustered around similar heights, but the largest soliton was outstanding, with an amplitude significantly larger than all other solitons. The presence of a clearly outstanding soliton in the spectrum was found to be an indicator pointing towards the enhanced probability of the occurrence of a rogue wave in the time series. Similarly, when the discrete spectrum appears as a cluster of solitons without the presence of a clearly outstanding soliton, the presence of a rogue wave in the observed time series is unlikely. These results suggest that soliton-like and nonlinear processes substantially contribute to the enhanced occurrence of rogue waves off Norderney.

Rogue waves are commonly defined as individual waves exceeding twice the significant wave height, where the significant wave height refers to the average height of the highest one-third of waves in a record. The occurrence of a rogue wave is a rare incident in the framework of a second-order process

Map of the German Bight, showing the location of the measurement buoy close to the island of Norderney.

In a previous study, we analysed measurement data from various stations in the southern North Sea

Nonlinear Fourier transforms are also known as scattering transforms in the literature.

(NLFT)To date, the nonlinear behaviour of deep-water rogue waves has received considerably more attention than that of shallow-water rogue waves. The evolution of the complex envelope of unidirectional wave trains in deep water can be described by the cubic nonlinear Schrödinger (NLS) equation

The form of the NLFT for the NLS equation (NLS-NLFT) depends on the boundary conditions. Initially, the NLS-NLFT was developed for vanishing boundary conditions, where localised wave packets with sufficient decay are considered

The vanishing NLS-NLFT, which detects envelope solitons and radiation in deep-water wave packets, has been applied to rogue waves as well. As pointed out by

The role of nonlinear processes with respect to rogue wave generation in shallow water has received considerably less attention than for deep water. Shallow-water wind waves substantially differ from deep-water wind waves; therefore, it is not appropriate to simply scale the deep-water nonlinear interaction to shallow-water waves

In shallow water, the wave evolution is described by the Korteweg–de Vries (KdV) equation

The KdV equation can again be solved using suitable NLFTs. The NLFT for the KdV equation (KdV-NLFT) with vanishing boundaries was found by

By numerically solving the KdV equation,

The nonlinear interaction of solitons in shallow water has been discussed with regard to its role in rogue wave generation.

At the moment, rogue wave occurrence in shallow water has not been sufficiently explained beyond second order. Moreover, almost all investigations in previous work have been based on theoretical considerations, numerical simulations or laboratory experiments. In this study, we instead leverage the vanishing KdV-NLFT to analyse the soliton spectrum of a large number of time series with and without rogue waves that have been measured off the coast of Norderney in the southern North Sea. For this location, wave height distributions based on linear superposition have been shown to underestimate rogue wave occurrence

The structure of the paper is outlined in the following. Section

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.

Bathymetry conditions at Norderney relative to NHN (Normalhöhennull), which represents the standard elevation zero of the German reference height system, and the position of the measurement buoy.

Mean directional wave spectrum for the time period from 2011 to 2016, obtained using the DIrectional WAve SPectra Toolbox (DIWASP;

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

Thus, for a water depth of

Rogue waves are commonly defined as waves with an individual height

In a previous study based on measurement data from the southern North Sea

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

The measured time series were subdivided into five categories:

Panels

Number of samples and total number of individual waves in the considered time series categories.

Examples of each time series category are shown in Fig.

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

Example of a time series including a rogue wave at approximately 820 s, and its corresponding FFT and NLFT spectra. The nonlinear spectra were calculated from vKdV-NLFT. The time series with

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.

Panel

Regular and irregular wave trains in very shallow water are known to often contain solitons, even without the presence of rogue waves

Solitons were attributed to specific rogue waves, following the procedure described in Sect.

Panel

Amplitude of the largest soliton attributed to the highest wave,

We extracted the amplitude of the largest attributed soliton

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

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

Amplitude of the highest soliton attributed to the rogue wave or maximum wave in the time series, normalised by the significant wave height, for the different categories of time series. Distributions are shown as box-and-whisker plots (box: interquartile range; whiskers: 1.5 times the interquartile range; horizontal line inside the box: median; red crosses: data outside the whiskers).

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

Different definitions of the Ursell number exist. A common definition is

The Ursell number has been used to classify wave types.
In

According to

When investigating the attribution of solitons to rogue waves in Sect.

Example of

To distinguish between clustered soliton spectra and those featuring an outstanding soliton, we compared the amplitudes of the largest soliton,

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

Distribution of the ratio between the second largest and the largest soliton in the discrete spectrum calculated from non-rogue wave time series.

Share of samples in each category showing an outstanding soliton or a clustered soliton spectrum, respectively.

Equation (

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.

Contingency table of forecast–event pairs. The letters used in the table denote the following:

In Fig.

Box plots of the ratio between the second largest soliton (

Figure

Ratio between the second largest soliton (

We investigated discrete nonlinear soliton spectra obtained by the application of the vKdV-NLFT to time series measured by a surface-following buoy off the coast of the island of Norderney in the southern North Sea. The impulse to investigate the data at this specific site using nonlinear methods was given by a previous study

Throughout the study, indications were found that, although solitons play a role in the presence of rogue waves at Norderney, the soliton spectrum alone does not yield a satisfactory explanation of the formation of extreme waves/ crests. A first hint is given in Fig.

The bathymetry below the measurement buoy at Norderney is characterised by a strong decrease in water depth. Non-Gaussian wave characteristics as a result of decreasing water depth have already been described by studies such as

The solutions of the KdV equation for a given free-surface elevation time series strongly depend on the water depth (see Eq.

Share of samples in each category showing an outstanding soliton in the soliton spectrum, for the respective water depth adopted in the NLFT calculation. Note that the shallow-depth criterion in Eq. (

Share of samples in each category showing an outstanding soliton, for the approximately 10 % of samples with the lowest directional spreading.

The KdV equation is only valid for unidirectional waves. Although

In our study, we applied the vKdV-NLFT as a trace method for (extreme) rogue waves and demonstrated, for the first time, that certain distinctive patterns in the NLFT spectrum of real-world time series indicate extreme rogue waves. The method may provide further information on possibly dangerous time series in future applications. Further research is required on the applicability of the KdV equation to our data, which cannot be validated on the basis of single-point measurements. If wave propagation at Norderney is well described by KdV theory, the NLFT spectrum is approximately constant during propagation. The method may then identify time series with the potential of forming extreme rogue waves. Moreover, even if the KdV equation does not describe the propagation well, we still consider the NLFT a more appropriate transform than the linear FFT, which is often applied even if waves are nonlinear. Similar to the FFT in the linear case, our method should be treated as a signal transform

We would like to put an emphasis on the limitation of our suggested definition of an outstanding soliton (Eq.

Due to the limited recording frequency of the wave buoy, one might question the correct assignment of time series to the different categories (Table

Our result that rogue wave samples have a higher probability of showing an outstanding soliton in the nonlinear spectrum compared with non-rogue wave samples becomes most obvious in the categories of double and extreme rogue samples. In these categories, differences from non-rogue wave samples are visible not only in the percentage of outstanding solitons but also in the magnitude of the amplitude gap between the first and second solitons in the spectrum. Height rogue waves, on the contrary, do not seem to differ very much from high waves in non-rogue wave samples, both in terms of the gap between first and second soliton in the spectrum and in the height of the solitons associated with the maximum wave. The fact that differences between time series with and without rogue waves become apparent only in some of the chosen categories raises questions regarding whether the choice of rogue wave definitions is reasonable for the considered location. The rogue wave definitions serving as a basis to this study were introduced by

Rogue wave occurrence recorded off the coast of the island of Norderney is not sufficiently explained by the Forristall distribution of wave heights. We investigated the role of solitons as components of the discrete vKdV-NLFT spectrum in the enhanced rogue wave occurrence. Our main results for this specific measurement site are as follows.

Each measured rogue wave could be associated with at least one soliton in the NLFT spectrum.

The soliton heights were always smaller than those of the rogue waves. Samples with rogue waves were more likely to contain an outstanding soliton in the NLFT spectrum than samples without rogue waves.

The soliton spectrum analysis is a good indicator of extreme rogue waves in the corresponding time series.

The presence of a strongly outstanding soliton, with a ratio between the second largest and the largest soliton in the nonlinear spectrum of

Conversely, the absence of an outstanding soliton in the spectrum is a strong indicator for the absence of an extreme rogue wave of

We conclude that nonlinear processes are important in the generation of rogue waves at this specific site and may explain the enhanced occurrence of such waves beyond common wave height distributions. Rogue waves at Norderney are likely to be a result of the interaction of solitons with the underlying field of oscillatory waves. The nature of this interaction should be subject to further research.

The FNFT Software library is available at

The underlying wave buoy data are the property of the Lower Saxony Water Management, Coastal Defence and Nature Conservation Agency (NLWKN). They can be obtained upon request from the agency (

All authors contributed to the idea and scope of the paper. IT performed the analyses and wrote the manuscript. MB, RW and SW provided help with data analysis, discussed the results and contributed to writing the paper. RW supervised the work.

The contact author has declared that none of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The buoy data were kindly provided by the Lower Saxony Water Management, Coastal Defence and Nature Conservation Agency (NLWKN).

This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 Research and Innovation programme (grant agreement no. 716669). Ina Teutsch received funding for this work from the Federal Maritime and Hydrographic Agency (BSH).The article processing charges for this open-access publication were covered by the Helmholtz-Zentrum Hereon.

This paper was edited by Ira Didenkulova and reviewed by two anonymous referees.

^{®}: User Manual, Centre for Water Research, University of Western Australia, Tech. Rep., Research Report WP-1601-DJ (V1.1), 2002.