Rogue Waves in the Southern North Sea—The Role of Modulational Instability

Ina Teutsch; Ralf Weisse

doi:10.1175/JPO-D-22-0059.1

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Journal of Physical Oceanography

Abstract
- Significance Statement
1. Introduction
2. Data and methods
- a. Data
- b. Methods
3. Results
- a. Spectral bandwidth
- b. Directional spreading
4. Discussion
5. Conclusions
Acknowledgments.
Data availability statement.
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Fig. 1.

Measurement sites considered in the study. Red circles: radar stations; blue squares: wave buoys.

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Fig. 2.

Bandwidth parameters ε, ν, and Q_p, calculated from the available data at all measurement stations.

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Fig. 3.

Benjamin–Feir index, calculated from the available data at all measurement stations.

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Fig. 4.

Data from station L9, originally sampled at 4 Hz, and the same data subsampled at 2 and 1 Hz.

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Fig. 5.

Change in spectral shape with a change in sampling frequency.

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Fig. 6.

Bandwidth parameters ε, ν, and Q_p, calculated from the available data at all measurement stations. The 4-Hz time series measured at radar stations have been subsampled at f_s = 2 Hz for comparability.

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Fig. 7.

Benjamin–Feir index, calculated from the available data at all measurement stations. The 4-Hz time series measured at radar stations have been subsampled at f_s = 2 Hz for comparability.

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Fig. 8.

The 2D Benjamin–Feir index according to Mori et al. (2011), compared with the BFI according to Eq. (1) at the buoy stations.

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Fig. 9.

Broadness parameter ε, calculated from the different time series categories. Distributions are shown as box-and-whisker plots (box: interquartile range; whiskers: 1.5 times the interquartile range; horizontal line inside the box: median; dots: data outside the whiskers).

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Fig. 10.

Broadness parameter ε, calculated at the radar stations, comparing nonrogue to (a) height and extreme rogue samples and (b) crest rogue samples.

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Fig. 11.

As in Fig. 10, but calculated at the buoy stations.

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Fig. 12.

Result of the Monte Carlo technique for the broadness parameter ε. Results from (left) height and extreme rogue samples and (right) crest rogue samples are compared to results from 10 000 samples, drawn from the population of nonrogue time series. For these samples, the mean and the standard deviations of the distributions are plotted (Campbell 2022).

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Fig. 13.

As in Fig. 12, but for the narrowness parameter ν.

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Fig. 14.

As in Fig. 12, but for the peakedness parameter Q_p.

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Fig. 15.

As in Fig. 12, but for the BFI.

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Fig. 16.

As in Fig. 12, but for the directional spreading.

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Fig. 17.

Comparison of the bandwidth-directional spreading pairs calculated from nonrogue and from rogue wave samples. The bandwidth is quantified in terms of the parameters ε and ν. Color coding: density of nonrogue samples. Black dots: rogue wave samples, as defined in the legend to each panel.

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Fig. 18.

As in Fig. 12, but for the combination parameter R.

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Fig. 19.

Comparison of the bandwidth parameters ε and ν with rogue wave occurrence frequencies from Teutsch et al. (2020).

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Editorial Type:: Article

Article Type:: Research Article

Rogue Waves in the Southern North Sea—The Role of Modulational Instability

Ina Teutsch

Ina Teutsch^aHelmholtz-Zentrum Hereon, Geesthacht, Germany

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and

Ralf Weisse

Ralf Weisse^aHelmholtz-Zentrum Hereon, Geesthacht, Germany

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Online Publication:: 27 Dec 2022

Print Publication:: 01 Jan 2023

DOI:: https://doi.org/10.1175/JPO-D-22-0059.1

Page(s):: 269–286

Received:: 11 Mar 2022

Accepted:: 08 Sep 2022

Published Online:: 27 Dec 2022

Displayed acceptance dates for articles published prior to 2023 are approximate to within a week. If needed, exact acceptance dates can be obtained by emailing amsjol@ametsoc.org.

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Abstract

The role of the modulational instability for rogue wave generation in the ocean is still under debate. We investigated a continuous dataset, consisting of buoy and radar wave elevation data of different frequency resolutions, from eight measurement stations in the southern North Sea. For periods with rogue waves, we evaluated the presence of conditions for the modulational instability to work, that is, a narrow-banded wave spectrum in both frequency and angular direction. We found rogue waves that exceed twice the significant wave height indeed occur at slightly lower frequency bandwidths than usual. For rogue waves that are defined only by high crests, this was, however, not the case. The results were dependent on the measurement frequency. The directional spreading of the buoy spectra yielded no information on the presence of a rogue wave. In general, all spectra estimated from the dataset were found to be broad in frequency and angular direction, while the Benjamin–Feir index yielded no indication on a high nonlinearity of the sea states. These are unfavorable conditions for the evolution of a rogue wave through modulational instability. We conclude that the modulational instability did not play a substantial role in the formation of the rogue waves identified in our dataset from the southern North Sea.

Significance Statement

This work investigates whether rogue waves measured in the southern North Sea may have been generated by a modulational instability. The latter is a nonlinear mechanism of wave energy focusing that has been proven mathematically and confirmed in laboratory experiments. However, it is still unclear whether this mechanism is responsible for rogue wave generation under realistic ocean conditions. The modulational instability primarily arises when waves have similar frequencies and directions. In our data, these conditions were not satisfied. This finding leads to the insight that the modulational instability is not the most probable mechanism to generate rogue waves in our dataset.

This article is licensed under a Creative Commons Attribution 4.0 license (http://creativecommons.org/licenses/by/4.0/).

Corresponding author: Ina Teutsch, ina.teutsch@hereon.de

Abstract

Significance Statement

This article is licensed under a Creative Commons Attribution 4.0 license (http://creativecommons.org/licenses/by/4.0/).

Corresponding author: Ina Teutsch, ina.teutsch@hereon.de

Keywords: Oceanic waves; Buoy observations; Radars/Radar observations; Spectral analysis/models/distribution; Statistics; Time series

1. Introduction

In numerical and physical experiments, nonlinear focusing has been identified as a possible mechanism for rogue wave generation. Its role for the formation of rogue waves in the real ocean, however, remains unclear. Nonlinear focusing was first described by Benjamin and Feir (1967): due to a modulational instability, a uniform wave train in deep water may dissolve into groups, which subsequently produce one large wave that is growing at the expense of the surrounding waves. Ten years later, Lake et al. (1977) demonstrated this so-called Benjamin–Feir instability to work in wave tank experiments. Onorato et al. (2004) showed—also in a wave flume—that as the instability develops, the rogue wave occurrence frequency increases. The Benjamin–Feir instability has therefore been proposed as a possible explanation for the formation of rogue waves (Janssen 2003).

Numerical experiments have shown a dependency of rogue wave occurrence on the ratio of wave steepness to spectral bandwidth (Alber and Saffman 1978; Onorato et al. 2001). This ratio quantifies the importance of nonlinear interactions, relative to that of dispersion in deep water, similar to the role of the Ursell number in shallow water. This deep-water Ursell number was later called a “Benjamin–Feir index” (BFI) by Janssen (2003):

BFI = \frac{\sqrt{2} ϵ}{Δ ω / ω_{0}} .

(1)

Here,

\sqrt{2} ϵ

represents steepness defined by the slope parameter

ϵ = k_{0} \sqrt{m_{0}}

. Here, k₀ is the mean wavenumber estimated from the mean frequency f₀ in the dispersion relation in deep water, k₀ = (2πf₀)²/g with gravity g, and m₀ is the 0th moment of the frequency spectrum. The term Δω denotes the frequency bandwidth, while ω₀ is the angular frequency at k₀. For BFI ≪ 1, a sea state may be described by the linear superposition of sinusoidal waves, while for BFI ≥ 1, nonlinear interactions are expected to dominate the evolution of the wave train (Onorato et al. 2001). The “BFI has been suggested as an indicator for the probability of occurrence of freak waves in the sense that large BFI means larger probability of freak waves” (Gramstad and Trulsen 2007).

Alber (1978) showed analytically that the requirement for a Benjamin–Feir-type instability to occur in a random wave field in deep water, is a sufficiently low bandwidth and small directional spreading of the wave spectrum. This has been substantiated numerically (Yuen and Ferguson 1978) and experimentally (Stansberg 1995). Waseda et al. (2009) showed in wave tank experiments that narrow-banded conditions favor an increased rogue wave occurrence.

Several authors have discussed the effect of broadening of the frequency spectrum on the distribution of wave heights and crests (Tayfun 1983; Naess 1985; Karmpadakis et al. 2020). Accounting for the spectral bandwidth in wave height and crest distributions influences the estimate of the significant wave height and the prediction of the largest wave in a wave train (Naess 1985), which both are important figures for the investigation of rogue waves. A number of parameters have been defined to describe the bandwidth of a spectrum in the frequency domain. Cartwright and Longuet-Higgins (1956) were the first to take the broadness of the frequency spectrum into account in the development of a distribution of the maximum surface elevation η_max in a time series. They introduced the parameter

ε = \sqrt{\frac{m_{0} m_{4} - m_{2}^{2}}{m_{0} m_{4}}} : 0 < ε < 1,

(2)

with the spectral moments

m_{n} = \int_{0}^{\infty} f^{n} S (f) d f

(3)

as a measure of the root-mean-square width of the (nondirectional) energy spectrum S(f). A wave spectrum is considered narrow banded if ε approaches zero (Cartwright and Longuet-Higgins 1956). In that case, the individual waves in the considered time series have similar frequencies and the wave energy is concentrated near the peak frequency (Cattrell et al. 2018). For an infinitely narrow spectrum with ε = 0, η_max is Rayleigh distributed (Cartwright and Longuet-Higgins 1956). In the case of a broad-banded spectrum, in which the wave energy is distributed over a wide range of frequencies and ε → 1, η_max is Gaussian distributed (Cartwright and Longuet-Higgins 1956).

When describing a probability density function of the wave period, Longuet-Higgins (1975) additionally introduced the narrowness parameter

ν = \sqrt{\frac{m_{0} m_{2}}{m_{1}^{2}} - 1} : 0 < ν < 1

(4)

as a measure of the bandwidth. For a narrow spectrum, ν ≈ 0.5 ε (Longuet-Higgins 1975). Typical values for wave conditions during a storm are ν ≈ 0.3–0.5 (Cattrell et al. 2018).

Some authors have stated a disadvantage of the parameters ε and ν, which is their sensitivity to high-frequency noise in the spectrum, represented by the higher spectral moments (e.g., Vandever et al. 2008; Janssen and Bidlot 2009). This becomes problematic at low sampling rates like the buoy and radar sampling rates of f_s = 1.28 and 2 Hz, as investigated in this study (Häfner et al. 2021a). It has also been criticized that ε for a spectrum of real sea waves describes the resolution of the data sampling rather than the spectral width (Goda 1988a,b). Rye (1977) found that Goda’s peakedness parameter (Goda 1970)

Q_{p} = \frac{2}{m_{0}^{2}} \int_{0}^{\infty} f {[\int_{0}^{2 π} S (f, Θ) d Θ]}^{2} d f,

(5)

as opposed to ε and ν, is independent of the high-frequency cutoff f_c. The Q_p is to some extent related to the spectral width parameter ε, but not by a simple function. While a small ε is associated with a large value of Q_p, the value of ε ≈ 0.7 is associated with a variety of Q_p values (Goda 1970). Due to the dependency of Eq. (5) on the square of the frequency spectrum, peaks in the spectrum are emphasized. This means that the weight is transferred to wave components with a higher contribution to the total wave energy (Janssen and Bidlot 2009). Values of Q_p range between 2 for fully developed wind seas and >4 for swell (Saulnier et al. 2011).

A narrow and peaked frequency spectrum indicates regular wave conditions and more pronounced wave groups than a broad-banded spectrum. The bandwidth parameters described above are thus also a measure of wave groupiness (Holthuijsen 2007). High Q_p values and low ε and ν values are therefore expected in swell-dominated sea states or wave fields including wave groups.

Janssen and Bidlot (2009) showed that under the assumption of a Gaussian shape of the frequency spectrum, the BFI may also be expressed in terms of Q_p. Then,

Δ ω / ω_{0} = \frac{1}{Q_{p} \sqrt{π}}, and

(6)

BFI = k_{0} \sqrt{m_{0}} Q_{p} \sqrt{2 π} .

(7)

Casas-Prat et al. (2009) successfully applied this formulation to buoy measurement data.

Besides a narrow bandwidth, another prerequisite for the modulational instability to occur is a narrow directional spreading. Numerical calculations have shown that unidirectional seas may result in much higher waves than predicted by second-order theory (Gibson and Swan 2006). This has been interpreted as an increase in the probability for rogue wave occurrences due to the modulational instability in unidirectional waves (Gramstad and Trulsen 2007). A broadening of the spectrum on the contrary leads to a reduction in rogue wave occurrence probability (Onorato et al. 2002; Waseda 2006; Janssen and Bidlot 2009). Latheef and Swan (2013) reported from a laboratory study that higher-order nonlinear effects can be important in directionally spread seas as well, provided the waves are sufficiently steep and not too short crested. In Onorato et al.’s (2009) laboratory experiments in a 3D wave basin, waves with a directional spreading larger than σ_Θ = 15° showed an almost Gaussian distribution, thus, a rogue wave occurrence frequency close to the expectations of second-order theory. In the real ocean, this would correspond, for example, to prevailing swell conditions. In pure wind-wave conditions, a high BFI does not necessarily indicate the presence of a modulational instability (Waseda 2006).

Gramstad and Trulsen (2007) stated that the probability of large waves depends on both the spectral bandwidth and the directional spreading, and they recommended including both parameters in the rogue wave probability prediction.

The bandwidth parameters described by Eqs. (2)–(5) did not take the directional spreading of waves into account. Janssen and Bidlot (2009) introduced a parameter to quantify the importance of the angular width, compared to the frequency width, which may be applied as a “measure of short-crestedness for the dominant waves” (Fedele 2015):

R = \frac{1}{2} \frac{σ_{Θ}^{2}}{ν^{2}}

(8)

with the spectral bandwidth ν [Eq. (4)], following Fedele et al. (2016), and the directional spreading

σ_{Θ} = \sqrt{2 (1 - C_{1})},

(9)

using the spectrally weighted, thus frequency-independent, averages of the first-order Fourier coefficients a₁ and a₂ with

C_{1} = \sqrt{a_{1}^{2} + b_{1}^{2}}

. Janssen and Bidlot (2009) found that when the directional spreading is larger than

\sqrt{2}

times the frequency width (corresponding to R = 1, according to Eq. (8), the sea state is defocusing and rogue waves occur less frequently than expected. They concluded that the occurrence probability of rogue waves is highest for almost unidirectional waves with a high BFI. Mori et al. (2011) pointed out that the parameter R follows from the assumption of a narrow-banded spectrum, in which both σ_Θ and ν are small. Since this assumption does not hold for most wind-wave data, the parameter should be seen as an approximation only.

Meanwhile, the validity of the modulational instability has been extended to intermediate water depths (0.8 < kh < 1.36) (e.g., Toffoli et al. 2013) It was confirmed numerically that the destabilization of a wave train due to oblique perturbations is possible in intermediate water (Fernandez et al. 2014). Toffoli et al. (2009) found that unlike in deep water (kh > 1.36), the deviation from Gaussian statistics in intermediate water is independent of the directional spread. Thus, in intermediate water, modulational instabilities may also occur for realistic directional spreading (Karmpadakis et al. 2019).

Dysthe et al. (2008) argued that the deep-water preconditions for a modulational instability are unlikely to occur in wind waves conditions. Based on hindcast data from the Sea of Japan, which include typhoon conditions, several authors found that narrow wave spectra that favor a modulational instability are possible in some parts of the ocean (Waseda et al. 2009; Tamura et al. 2009). A number of authors found, based on measurement studies, that the occurrence frequency of rogue waves was dependent on the spectral bandwidth in the underlying wave field. Karmpadakis et al. (2020) described, based on radar measurements in the North Sea, a reduction in wave heights with an increase in spectral bandwidth. Cattrell et al. (2018) investigated buoy data from the U.S. coast and found that the spectral bandwidth parameters of rogue seas displayed different probability distributions than those estimated from nonrogue seas. Christou and Ewans (2014) stated, based on radar and laser data from the North Sea and other locations, that the spectral bandwidth might be an indicator for distinguishing rogue waves from high nonrogue waves. Most recent findings by Häfner et al. (2021b), based on machine learning algorithms applied to buoy data from the U.S. coast, showed that the spectral bandwidth was much more informative about rogue wave probability than the BFI.

On the other hand, some authors have found that the rogue wave occurrence frequency was not or only weakly dependent on the spectral bandwidth (e.g., Stansell 2004). Goda (1970) wrote, supported by numerical experiments, that wave heights defined by the zero-upcrossing method practically followed the Rayleigh distribution, independently of the spectral bandwidth. When examining the broadness parameter ε, Christou and Ewans (2014) found little difference between rogue wave samples and the highest nonrogue samples.

Also concerning the directional spreading during rogue wave occurrence, investigations have come to different conclusions in different parts of the ocean. While Waseda et al. (2011) confirmed, based on radar measurement data from a platform in the North Sea, that on days with a high occurrence of rogue waves, the directional spreading of the wave spectrum was narrower than on other days, Christou and Ewans (2014) found no significant differences between nonrogue samples and rogue wave samples and concluded that the environmental conditions generating normal waves, were also able to form rogue waves.

As a result, a number of authors have come to the conclusion that the modulational instability was not the main reason for the formation of rogue waves in their measurement data (e.g., Cattrell et al. 2018). Fedele et al. (2016) found, based on a large collection of field data from various locations in Europe, that the main generation mechanism of rogue waves is the constructive interference of elementary waves (dispersive and directional focusing), enhanced by second-order bound nonlinearities and not the modulational instability. They concluded that rogue waves are likely to be rare occurrences of weakly nonlinear random seas.

In a previous study, we investigated a measurement dataset from the southern North Sea (Teutsch et al. 2020). We found an enhanced occurrence frequency of rogue waves compared to the Forristall distribution in the overall dataset, but especially when recorded by radar devices. The Forristall distribution was developed based on waves in the Gulf of Mexico during hurricane conditions (Forristall 1978). Although it was found to be a good fit to measurement data in other ocean regions (e.g., Casas-Prat et al. 2009; Waseda et al. 2011), it should be kept in mind that this empirical distribution accounts for nonlinearities present in the original buoy dataset. Nonlinearities might be different in the southern North Sea, which is especially relevant when investigating rogue waves. In the present study, we investigate the possibility that those rogue waves exceeding the expectations of random superposition, were generated by modulational instabilities. Even at stations showing a good agreement with the Forristall distribution, modulational instabilities might be present. These are not necessarily reflected by the wave statistics (Fernandez et al. 2016). Based on a field measurement dataset of approximately 123 000 samples from radar stations and 63 000 samples from wave buoys in the southern North Sea, our aim is to investigate whether the requirements for a modulational instability, that is, narrow bandwidth in both frequency and angular direction, is given during rogue wave occurrence. This is done in terms of the bandwidth parameters [Eqs. (2)–(5)], the directional spreading [Eq. (9)] and the combined parameter R for directional spreading and spectral bandwidth [Eq. (8)]. Finally, the BFI [Eq. (1)] in our dataset is evaluated and compared for time series with and without rogue wave occurrence.

The measurement area and the dataset, as well as the estimation method of the directional spectrum, are described in section 2. The results of the evaluation of the bandwidth parameters and the BFI are described in section 3. In section 4, our results are related to previous experimental studies, and differences in the findings from the two different measurement devices are discussed. From the results and discussions, we draw our conclusions in section 5.

2. Data and methods

a. Data

Surface elevation measurement data from 2011 to 2016 at eight measurement stations in the southern North Sea were investigated (Fig. 1). The data are part of a larger set that was introduced in an earlier study (Teutsch et al. 2020). At four of the stations, the data were provided by fixed radar devices, measuring the air gap to the sea surface at a sampling frequency of either f_s = 2 Hz or f_s = 4 Hz. The remaining four stations are equipped with surface-following buoys of type MkIII, measuring at a sampling rate of f_s = 1.28 Hz. The buoy data were delivered in samples of 30-min duration. The radar data, which were available as continuous time series, were split into 30-min samples accordingly. All samples were quality controlled according to the procedure described in Teutsch et al. (2020).

To exclude low-energy sea states, only samples with a significant wave height H_s above the long-term 70th percentile of the significant wave height at each station H_s_,70 were included in the analysis. The significant wave height H_s is here defined as the mean of the highest 30 % of the wave heights in a 30-min sample. The H_s_,70 was calculated from the significant wave heights H_s of all 30-min samples during the six years of available measurement data. On the one hand, this excludes possible measurement uncertainties caused by small waves that are only described by a few points, and on the other hand, it includes only rogue waves of heights relevant for offshore activities. The H_s_,70 is presented for each station in Table 1.

Table 1

Water depth h and long-term 70th percentile of the significant wave height H_s_,70 at all stations included in the analysis.

Unidirectional waves are expected to be modulationally unstable above a threshold of kh = 1.363, in which k is the wavenumber and h is the water depth (Benjamin 1967). Modulational instabilities have also been identified in intermediate water, but there their preconditions are not easily assessed, as they do not only occur in narrow directional spreading (Karmpadakis et al. 2019). Therefore, we considered only samples above kh = 1.363. In deep water, where k = 2π(L)⁻¹, the condition becomes h(L)⁻¹ > 0.22. Inserting

L = g {(2 π)}^{- 1} T_{p}^{2}

, leads to the condition for the peak period

T_{p} < \sqrt{\frac{π h}{0.11 g}}

(10)

with gravity g and the peak period

T_{p} = 1 f_{p}^{- 1}

of each sample, with f_p representing the peak frequency in the linear fast Fourier transform (FFT) spectrum of the sample. Table 2 shows the maximum accepted peak period for the samples at each station, according to Eq. (10). It is compared to the 99th percentile of peak periods T_p_,99 for samples above H_s_,70 at each station. The number of available samples above H_s_,70 is shown in Table 3. Rogue waves were identified and visually checked, as described in Teutsch et al. (2020).

Table 2

The 99th percentile of the peak period T_p_,99, in all samples above H_s_,70, compared to the peak period threshold T_p, below which samples are analyzed with respect to preconditions for a modulational instability.

Table 3

Number of available 30-min measurement samples with H_s above the long-term 70th percentile of the significant wave height H_s_,70. The data are classified according to the station and the sampling frequency f_s of the recording.

b. Methods

For all 30-min samples in Table 3, the spectral bandwidth was estimated from the FFT spectrum in terms of the broadness parameter ε [Eq. (2)], the narrowness parameter ν [Eq. (4)] and Goda’s peakedness parameter Q_p [Eq. (5)]. In addition, the BFI was calculated for each sample, according to Eq. (7). As described in Teutsch et al. (2020), each sample was assigned to one of the following categories:

“Nonrogue samples”—measurement samples that did not contain a rogue wave.
“Height rogue samples”—measurement samples that contained a rogue wave according to the height criterion (Haver and Andersen 2000)

2.3 H_{s} > H \geq 2.0 H_{s},

(11)

with H denoting the height of the rogue wave from crest to trough.

“Extreme rogue samples”—measurement samples that contained a rogue wave according to a more strict height criterion of

H \geq 2.3 H_{s} .

(12)

“Crest rogue samples”—measurement samples that contained a rogue wave according to the crest criterion (Haver and Andersen 2000)

C \geq 1.25 H_{s},

(13)

with C denoting the crest height of the rogue wave above still water level.

“Double rogue samples”—measurement samples that contained a rogue wave according to both the criteria defined in Eqs. (11) and (13). Double rogue samples were excluded in the groups of height and crest rogue samples.

Table 4 shows the number and percentage of available samples in each category.

Table 4

Number of samples in each category for the groups of radar and buoy stations.

The Datawell Waverider buoys provide information on their position on the water surface in heave, north and west directions. Based on the three-dimensional information, it is possible to estimate a directional wave spectrum for each 30-min sample. We calculated the wave spectrum from buoy data according to Huntley et al. (1977), who applies the iterated maximum likelihood method (IMLM) developed by Pawka (1983). According to Young (1994), a maximum likelihood method yields a broader spectrum with a lower peak than the real spectrum. However, as the author points out, it is “the analysis technique with the highest directional resolving capabilities” for wave buoy data (Young 1994). Based on the directional spectrum, the directional spreading of each sample, and subsequently the parameter R, which relates the directional spreading to the spectral bandwidth, were calculated [Eqs. (8) and (9)].

3. Results

a. Spectral bandwidth

The measurement data analyzed in this study, were recorded by different measurement instruments and at different sampling rates (Table 3). When comparing spectral bandwidth parameters and BFI at all stations, we observed that the parameter range differed with the measurement frequency f_s (Figs. 2 and 3). This became most obvious for the broadness parameter ε, whose median value showed to be approximately 30% higher in radar samples measured at f_s = 4 Hz than in buoy samples with f_s = 1.28 Hz (Fig. 2). To test the hypothesis that the investigated spectral parameters are sensitive to the measurement frequency, we subsampled the 4-Hz data at station L9 at 2 and at 1 Hz, respectively. The result demonstrates that the same time series yield different spectral parameters when sampled at different frequencies (Fig. 4). The changes are most pronounced for the broadness parameter ε. The reason for this dependency of the parameters on the sampling frequency is a change in spectral shape with the measurement frequency (Fig. 5). This results in a change in spectral moments (Table 5). A change in sampling frequency strongly affects those bandwidth parameters that are dependent on the higher moments of the spectrum [Eqs. (2) and (4)]. This issue has already been raised by Goda (1988a,b), who introduced a peakedness parameter that is independent of the higher moments of the frequency spectrum [Eq. (5)]. Indeed, the peakedness parameter Q_p is least of all bandwidth parameters affected by the measurement frequency (Fig. 2).

Table 5

Spectral moments m_n, bandwidth parameters, and BFI, calculated from the same time series, measured starting at 0000 LT 1 Jan 2016. It was originally sampled at f_s = 4 Hz and subsamples at 2 and 1 Hz.

Since our dataset consists of buoy and radar measurement data, which have been sampled at different frequencies, buoy and radar data will be treated separately throughout this study. Within the set of radar measurements, we subsampled all time series recorded at 4 Hz with f_s = 2 Hz. This way, we still retain measurement data at a high frequency, while enlarging the set of 2-Hz samples and obtaining bandwidth parameters in a range that is more comparable to the buoy data than the 4-Hz data. Figure 6 is a repetition of Fig. 2, but with all 4-Hz radar data subsampled at 2 Hz. The BFI for 2-Hz radar data and 1.28-Hz buoy data is shown in Fig. 7.

The differences between the respective stations are insignificant. Within the subset of radar measurement samples, one could infer an increased broadness of the frequency spectrum in terms of both ε and ν with a decrease in water depth, with L9 and Clipper situated in water depths much shallower than Leman (see Table 1). This trend, however, is not seen in the peakedness parameter Q_p, and it is not confirmed by the buoy stations, with the deepest stations being FN1 and LTH with a water depth of h = 30 m (Fig. 6). The BFI does not display any relation to the water depth in which the samples were recorded (Fig. 7).

Serio et al. (2005) suggested to include information on the water depth in the formulation of the BFI. We have recalculated the BFI according to their proposed equations and found no significant differences compared to the BFI calculated from Eq. (1). However, an extension of the BFI to directional effects, as proposed by Mori et al. (2011) [ ${BFI}_{2D} = \sqrt{{BFI}^{2} / (1 + 7.1 R)}$ ], led to a significant reduction of the BFI (Fig. 8). Since radar stations do not provide directional wave information, we proceed to compare the original BFI [Eq. (1)], which is available for the full dataset.

Figure 9 shows the broadness parameter ε, combined for each instrument category, but separated into the different sample categories defined in section 2. Figure 9 and Table 6 indicate that, compared to nonrogue samples, spectral bandwidth is slightly enhanced in crest rogue samples and slightly reduced in height and extreme rogue samples. The differences are somewhat more apparent when displayed in a histogram, comparing nonrogue to height and extreme rogue samples, and nonrogue to crest rogue samples (Figs. 10 and 11). Double rogue samples display a lower bandwidth than usual (Table 6), but they will not be treated further, as they belong to both height and crest rogue categories.

Table 6

Median values of the distributions of ε in the different categories, as shown in Fig. 9.

To test whether the difference between the ε distributions in the different sample categories is statistically significant, a Monte Carlo technique was applied to ε, and subsequently for all bandwidth parameters and the BFI, comparing the parameters calculated from nonrogue samples to height and extreme rogue samples on the one hand and to crest rogue samples on the other hand. An overview of the Monte Carlo calculations that were performed, is given in Table 7. The applied Monte Carlo technique is described in the following. In each execution, 10 000 samples of the length of the comparison population (height and extreme or crest rogue time series) were drawn with replacement from the population of nonrogue time series. For these samples, the respective parameters in Table 7 were calculated. The mean and standard deviation of the 10 000 parameter distributions is displayed and compared to the comparison population in Figs. 12–15. A significant difference between nonrogue time series and the comparison population is given if the distribution of the comparison population is located outside of two standard deviations of the distribution calculated from the nonrogue time series.

Table 7

Parameters and stations for which a Monte Carlo technique was performed: nonrogue samples were compared with height/extreme rogue samples and with crest rogue samples, respectively.

From the result of the Monte Carlo technique for the broadness parameter ε, it can be inferred that the frequency spectra calculated from height and extreme rogue samples at the investigated radar and buoy stations were significantly more narrow than the frequency spectra calculated from time series without rogue waves (Fig. 12). It can be concluded that in our case, height and extreme rogue waves typically occurred in slightly more narrow sea states than observed under nonrogue conditions. For crest rogue samples, Fig. 12 shows a different result: these did not differ significantly from nonrogue time series.

The standard deviation in the 10 000 samples drawn from radar data, is very small, due to a narrow ε distribution in radar samples, as seen in Fig. 10.

The result of the Monte Carlo technique for the narrowness parameter ν is slightly different. A significantly more narrow frequency spectrum for height and extreme rogue samples can only be identified at the buoy stations (Fig. 13). The distribution of ν for height and extreme rogue samples at radar stations is situated within two standard deviations of the ν distribution calculated from 10 000 nonrogue sample realizations. For crest rogue samples at both radar and buoy stations, the result remains that the distribution of ν is not significantly different from the ν values in nonrogue samples.

In terms of the peakedness parameter Q_p, for which a mean value slightly above Q_p = 2 is typical at all stations, no significant difference between nonrogue and rogue wave samples was identified (Fig. 14). Although the peakedness/groupiness of the wave spectrum appears slightly lower for crest rogue samples than usual and slightly higher for extreme rogue samples than usual, the curves of the comparison populations are within two standard deviations of the distribution from nonrogue samples. The same result applies to the BFI. Under the hypothesis that the modulational instability caused the rogue waves measured at our stations, a higher BFI would be expected in rogue wave samples. However, this does not seem to be the case (Fig. 15). Although within two standard deviations, low BFI values (BFI ≈ 0.2) seem to be more unusual in height and extreme rogue samples than in nonrogue samples.

b. Directional spreading

The directional spreading of the wave energy in each sample is only available at the buoy stations, which provide three-dimensional information on their location, but not at the radar stations with only one-dimensional information on the air gap. Since a narrow directional spreading represents a requirement for the process of the modulational instability to occur, we investigated the directional spreading at all buoy stations and compared it for time series with and without rogue waves. Following the reasoning in the previous section, this was done by means of a Monte Carlo technique, in which a sample of 10 000 time series was randomly drawn from the population of nonrogue time series. Figure 16 shows that the directional spreading in the measured rogue wave samples was not significantly different from the directional spreading in samples without rogue waves. Thus, the condition of a specifically low directional spreading in rogue wave samples as a prerequisite for the modulational instability to operate, was not present in our buoy data.

Finally, the combination of spectral bandwidth and directional spreading was examined. This was done because the favoring condition for the formation of a modulational instability has been formulated as a narrow spectrum in both direction and frequency (e.g., Alber 1978; Stansberg 1995). Figure 17 shows bandwidth-directional spreading pairs for all buoy samples. It shows that nonrogue samples, as well as the plotted rogue wave samples, cluster around medium bandwidth and directional spreading values. No specific accumulation of rogue wave samples at low bandwidth and directional spreading is seen, which would point toward favorable conditions for the modulational instability.

Additionally, the Monte Carlo technique was repeated for parameter R [Eq. (8)] in buoy measurement samples, which quantifies the importance of the directional spreading with respect to the spectral bandwidth. Although it appears that the maximum in the rogue wave distributions is shifted toward a higher R value, compared to nonrogue samples, the comparison with the standard deviation reveals that there are no significant differences apparent between rogue wave samples and nonrogue samples (Fig. 18). We conclude that rogue waves in our buoy dataset did not occur in sea states that are narrow-banded in frequency and angular direction. This indicates that the modulational instability is an unlikely mechanism for the formation of these rogue waves.

4. Discussion

Wave measurement time series from the southern North Sea that were evaluated in the present study, generally showed a wide spectral bandwidth, which is an unlikely condition for nonlinear focusing to occur in deep water (Alber 1978; Fedele et al. 2016). A similar observation of unfavorable conditions for the modulational instability was made by Fedele et al. (2016), who presented sea state parameters for three historical rogue wave occurrences in the North Sea. Due to a high bandwidth parameter ν and a large directional spreading σ_Θ, they came to the conclusion that these rogue waves were probably not the result of nonlinear focusing. This conclusion is transferable to our dataset, which displays high spectral bandwidth and directional spreading values for a substantially larger number of rogue waves in the North Sea.

Theoretical considerations and experimental results clearly confirm an increased rogue wave occurrence in narrow bandwidth and directional spreading (Waseda et al. 2009). Observations of field measurement data, however, have come to contradictory conclusions. In our data, the spectral bandwidth was significantly lower than usual in height and extreme rogue wave samples, when quantified by the broadness parameter ε [Eq. (2)]. This result is basically consistent with a finding by Christou and Ewans (2014), who discovered, mainly based on radar data from the North Sea, recorded at f_s = 2 or 4 Hz, that rogue wave samples are more narrow-banded than the highest samples of nonrogue waves, and concluded that the spectral bandwidth might be an indicator for distinguishing time series with rogue waves from time series with high nonrogue waves. Christou and Ewans (2014) did not distinguish between height and crest rogue waves, but given that in our data, the sample of height rogue samples is approximately 5 times larger than that of crest rogue samples (Table 4), our results are still comparable to theirs. One major difference in the result, however, is that Christou and Ewans (2014) based their observation on the narrowness parameter ν. In terms of the broadness parameter ε, where we detected the strongest deviations, Christou and Ewans (2014) found little difference between rogue wave samples and the highest nonrogue samples. The dataset of Christou and Ewans (2014) included data from the same radar stations as we used in our study, however, for a different time period. Another difference to our dataset is the use of additional data from other regions from a large range of water depths (7–1311 m). The differences in the investigated data could be the reason for our discrepancies from their study. At the buoy stations, we also recognized differences between nonrogue and rogue wave samples in the narrowness parameter ν. Also, Karmpadakis et al. (2020) described a clear influence of the spectral bandwidth ν on the wave height distribution in radar measurements in the North Sea. They found wave heights to decrease with increasing bandwidth. Most of their time series were recorded at f_s = 2–4 Hz. Also, the dataset of Karmpadakis et al. (2020) includes the radar stations we used in the present study, with the time series starting somewhat earlier and continuing until February 2017. In addition to the difference in measurement period, it should be noted that Karmpadakis et al. (2020) did not exclude any H_s values, which makes our study not exactly comparable to theirs. However, Karmpadakis et al. (2020) distinguish between deep and shallow water in their results. Most recent findings by Häfner et al. (2021b) revealed, based on machine learning algorithms applied to Coastal Data Information Program (CDIP) buoy data from the U.S. coast with H_s > 1 m, that the spectral bandwidth (ν and Q_p) provides some information on the probability of rogue waves according to the height criterion. Furthermore, their study revealed that bandwidth effects are not relevant for wave crests. Based on our measurement data, we can confirm these results: although we observed slightly higher bandwidths than usual in crest rogue wave time series, these differences showed to be statistically insignificant. The CDIP dataset includes data from water depths between approximately 10 and 4500 m. Häfner et al. (2021b) supposed that the influence of second-order nonlinearities on wave crests might be higher than estimated on the basis of buoy measurement data. Wave buoys tend to cancel second-order nonlinearities by their own Lagrangian movement and thus to overestimate the mean water level, which in turn leads to an underestimation of crest heights (Forristall 2000). In addition, wave buoys may be dragged through or slide away from short wave crests, which might result in missing the maximum amplitudes. Especially for strongly nonlinear waves, this leads to significant differences compared to radar devices (Longuet-Higgins 1986). These in turn may overestimate crests by misinterpreting spray, breaking waves, or even fog (Grønlie 2006). Forristall (2005) noted that there is no standard way to calibrate measurement instruments and that it is not possible to decide which instrument yields the “most correct” results. With our dataset consisting of both buoy and radar time series, we had the possibility to compare the results for wave crests based on both instruments and came to the conclusion that the influence of spectral bandwidth on crest rogue waves is similarly negligible in both types of time series. Also, Cattrell et al. (2018) stated an influence of the spectral bandwidth on rogue wave occurrence. However, they came to different conclusions. In their investigation of 80 of the 161 CDIP buoys that were available to Häfner et al. (2021b), the distributions of the spectral bandwidth parameters ε and ν indicated that the probability of observing a height rogue wave increased in seas with a higher bandwidth, while that of observing a crest rogue wave increased in seas with a narrow spectral bandwidth. As opposed to Häfner et al. (2021b), Cattrell et al. (2018) investigated the entire range of H_s values, which could lead to the difference in findings. Our data from the southern North Sea seem to resemble the results of Häfner et al. (2021b). Finally, Stansell (2004) claimed, based on a statistical analysis of storm waves at the North Alwyn platform in the North Sea, measured by laser altimeters (H_s > 3 m, h = 130 m, f_s = 5 Hz), that the occurrence probability of rogue waves is only weakly dependent on the spectral bandwidth ν, which confirms our results regarding the narrowness parameter ν at the radar stations.

It is possible that the results on the influence of spectral bandwidth on rogue wave occurrence are dependent on the sampling frequency of the measurement device: while buoy measurements at f_s = 1.28 Hz identified strong dependencies of at least height rogue wave occurrence on the spectral bandwidth (Häfner et al. 2021b; Cattrell et al. 2018), and these results were still valid (Karmpadakis et al. 2020) or less pronounced (Christou and Ewans 2014) in 2–4-Hz radar measurements, the influence of the spectral bandwidth was described as insignificant in a study based on 5-Hz laser measurements (Stansell 2004). We can confirm that in our dataset, the influence of spectral bandwidth on height rogue waves was more pronounced in the time series recorded at f_s = 1.25 Hz than in the 2 Hz and subsampled at 2 Hz, radar data. This leads to the question to what extent the influence of the spectral bandwidth on rogue wave occurrence is actually a physical mechanism and how much of the effect should be attributed to the measurement instrument. Measurement instruments with low sampling frequencies are known to underestimate the occurrence frequency and the heights of the largest waves (Stansell et al. 2002). A false estimation of the spectral bandwidth could be a consequence. For further investigations on this issue, it would be valuable to have a radar device and a wave buoy installed not only in the same area, but at exactly the same place and compare measurement results.

Other reasons for differences in measurement results could be the restriction to specific H_s ranges applied by some authors (e.g., Stansell 2004; Häfner et al. 2021b), or a different behavior of waves at shallow water stations, which were not considered separately in all studies. Although rogue waves may occur in all sea states (Teutsch et al. 2020), we have focused our study on rogue waves in high sea states with a great relevance for offshore activities. This choice may have an influence on the results and conclusions of our study. An alternative option would be to group time series by the metocean conditions during the recordings or by sea state characteristics like spectral shape or steepness, while including sea states of all heights. However, including low sea states introduces measurement uncertainties. Subdividing the dataset of samples above H_s_,70 into sea state groups reduces the size of the different groups in a way that results cannot be drawn on a reliable basis. Therefore, our results and conclusions are presented for all samples above H_s_,70.

In our data, the BFI as the commonly accepted indicator for the modulational instability, was not characteristic during rogue wave occurrence. This again confirms a result of Häfner et al. (2021b), who showed for their data that the BFI as a measure of nonlinear effects does not play a role for the prediction of rogue waves. Furthermore, it should be kept in mind that the formulation of the BFI was derived under the assumption of a narrow-banded Gaussian spectrum (Janssen and Bidlot 2009). With this in mind, it should be questioned whether the BFI is a valid indicator for the application to real ocean data. Waseda (2006) showed in tank experiments that the BFI was only informative on the non-Gaussianity of a sea state in very narrow-banded waves.

Also in terms of the directional spreading, we could not identify any characteristic tendency during rogue wave occurrence, which agrees with findings by Christou and Ewans (2014) and Häfner et al. (2021b). We agree with Christou and Ewans (2014), who conclude from this observation that “the environmental conditions generating normal waves, are also able to form rogue waves.” In a recent study that evaluated numerical models in comparison with in situ data, it was confirmed that both linear and nonlinear effects are important for the formation of rogue waves and thus, rogue waves may also be expected in sea states with large directional spreading (Kirezci et al. 2021). On the other hand, Waseda et al. (2011) observed in a hindcast that on days with a high occurrence of height rogue waves, as identified by radar measurements at a platform in the North Sea, the directional spreading of the wave spectrum was narrower than on other days. In our combined analysis of directional spreading and spectral bandwidth, we found that our rogue waves did not occur in sea states that were narrow-banded in frequency and angular direction.

We ensured for all parameters in rogue wave samples, by reducing the height of each identified rogue wave by 50%, that the bandwidth parameters were not influenced by the presence of the rogue wave itself. In fact, this test showed that while the significant wave height of the samples decreased as a result of the reduction of the rogue wave height, the bandwidth parameters and the BFI did not change. Thus, the influence of one rogue wave on the spectral parameters of a 30-min time series was small, as opposed to the 20-min windows investigated by Stansell (2004). He found that the difference in spectral bandwidth disappeared when the rogue wave was removed. In our 30-min time series, it appeared that the information in the bandwidth on the possible presence of a rogue wave was contained within the wave spectrum, not in the rogue wave itself. Analyzing measurement data and comparing the results to previous measurement studies raises the issue of sampling variability: statistical results become less reliable due to limited data availability. In 30-min time series, the effect of sampling variability may dominate over nonlinear effects, which are then difficult to assess (Bitner-Gregersen et al. 2021). This effect is certainly important when studying rogue waves, which are located at the tail of the wave height distribution and therefore naturally represented by only few measurements. The uncertainty imposed by the limited number of data, may be counteracted by spatial measurements (Bitner-Gregersen et al. 2021). This possibility was not available to us, a problem of many measurement studies in the ocean. The analysis of longer records could be another solution, but samples longer than 30 min cannot be assumed to be stationary, which again poses problems to the statistical evaluation (Holthuijsen 2007). The use of 30-min samples is therefore a compromise between long enough records (to produce reliable statistical results) and short enough records (to reasonably assume stationary wave conditions). The measurement data from the eight stations in the southern North Sea have been part of an earlier study (Teutsch et al. 2020). In that study, the rogue wave frequency was documented with respect to the total number of measured waves at each station (although without the H_s_,70 restriction). When comparing the bandwidth parameters ε and ν, as presented in Fig. 6, with the rogue wave frequencies from the previous study, it is seen that a high rogue wave occurrence corresponds to a narrow frequency spectrum (Fig. 19).

The parameters Q_p and BFI did not show any correlation with the rogue wave frequency. The directional spreading could not be compared at all stations, since the radar measurements are one-dimensional. From Teutsch et al. (2020), it is noted that all buoy stations showed rogue wave frequencies below the expectation from second-order theory. Here, Janssen and Bidlot (2009) should be cited, who state that the sea state defocusses above a certain threshold of σ_Θ, which yields even fewer rogue waves than expected according to second-order theory. Having the thresholds of 14°–15° in wave tank experiments (Waseda 2006; Onorato et al. 2009) and 30° in the ocean (Waseda et al. 2009) in mind, this could indeed be the case in our measurement data.

5. Conclusions

In a dataset consisting of radar and buoy measurements from high sea states in the southern North Sea, we identified lower values of spectral bandwidth than usual during height rogue wave occurrence. Samples with rogue waves according to the crest criterion in turn could not be attributed to specific bandwidth conditions. The directional spreading did not give any indication on the occurrence of rogue waves of any kind, neither did the BFI as a commonly applied indicator for the modulational instability. We conclude that the majority of rogue waves in our dataset were probably not generated by a modulational instability, since high spectral bandwidth and directional spreading are unfavorable conditions for nonlinear focusing.

Acknowledgments.

Ina Teutsch received funding for this work from the Federal Maritime and Hydrographic Agency (BSH). The buoy data were kindly provided by BSH. The authors are grateful to Graham Feld and Shell for providing the radar data. The authors are grateful to Jens Möller and Christian Senet for their valuable input. The article processing charges for this publication were covered by a Research Centre of the Helmholtz Association.

Data availability statement.

The underlying wave buoy and radar data are the property of and were made available by the Federal Maritime and Hydrographic Agency, Germany, and Shell, United Kingdom, respectively. They can be obtained upon request from these organizations.

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Rogue Waves in the Southern North Sea—The Role of Modulational Instability

Abstract

Significance Statement

Abstract

Significance Statement

1. Introduction

2. Data and methods

a. Data

b. Methods

3. Results

a. Spectral bandwidth

b. Directional spreading

4. Discussion

5. Conclusions

Acknowledgments.

Data availability statement.

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GENERAL CONDITIONS

No. III

BAROMETRIC PRESSURE

VERIFICATIONS

MISCELLANEOUS PHENOMENA

Rogue Waves in the Southern North Sea—The Role of Modulational Instability

Abstract

Significance Statement

Abstract

Significance Statement

1. Introduction

2. Data and methods

a. Data

b. Methods

3. Results

a. Spectral bandwidth

b. Directional spreading

4. Discussion

5. Conclusions

Acknowledgments.

Data availability statement.

REFERENCES

Share Link

AMS Publications

Get Involved with AMS

Affiliate Sites

Follow Us

Contact Us