3.1. Rogue wave amplification is symmetric with regard to current direction

To characterise the tail of the wave height distribution, we focus on the exceedance probability $\mathbb{P}(H\gt \alpha H_s)$ , $\alpha$ being the normalised wave height. For rogue waves, $\mathbb{P}(\alpha = 2)$ is enhanced by moderate currents of either direction, whether following or opposing (figure 3). The limited number of rogue waves (278 events in the entire sample after quality control), however, limits the resolution of the binning, preventing us from accurately determining the current speed associated with maximum rogue wave amplification. This limited number also explains the width of the shaded 95 % confidence ranges in figure 3, which were calculated by assuming independent draws for each wave and consequently using a binomial analysis (Jeffreys Reference Jeffreys1961). As displayed in table 2, the two bins with smaller current in either direction ( $U/c_g = \pm 0.06, \pm 0.13$ ) display significantly higher exceedance probabilities than the rest condition ( $p \lt 0.05$ ). In contrast, the corresponding pairs of bins with the same velocity magnitude but opposing directions ( $U/c_g = -0.06$ and 0.06, respectively, as well as $U/c_g = -0.13$ and 0.13) are respectively at the very edge of significance and non-significant. This analysis confirms the amplification of the rogue wave probability by a current, and the symmetry of this amplification with regard to the current direction, at least for moderate current velocities. The limited number of events prevents the results for the outermost bin ( $U/c_g = -0.19$ and $U/c_g = 0.21$ ) from reaching statistical significance.

Figure 3.

Exceedance probability of rogue waves ( $\alpha = 2.0$ ) in the southern North Sea with low-resolution binning of the normalised tidal current $U/c_{g}$ . The error band is computed from the 95 % Jeffreys confidence interval. The numbers of waves and rogue waves corresponding to each point are displayed in table 1.

Table 1.

Numbers of waves and rogue waves associated with each point of figure 3.

Figure 4.

(a) Amplification of the observed rogue wave exceedance probability by opposing and following currents in comparison with laboratory measurements of Ducrozet et al. (Reference Ducrozet, Abdolahpour, Nelli and Toffoli2021), (3.1). Confidence intervals on observations are not shown as they are the same as in figure 3. (b) Exceedance probability for rest conditions as expected by Longuet-Higgins (Reference Longuet-Higgins1980) and for the bin of highest amplification ( $-0.12 \lt U/c_{{g}} \lt -0.16$ ), as a function of the threshold $\alpha$ .

Table 2.

Hypergeometric $p$ -value calculated from the Fisher exact test for relevant pairs of data points of figure 3.

Figure 4 compares the rogue wave probability amplification observed in our data with Toffoli et al.’s (Reference Toffoli, Waseda, Houtani, Cavaleri, Greaves and Onorato2015) experimental results, further described by Ducrozet et al. (Reference Ducrozet, Abdolahpour, Nelli and Toffoli2021), as a function of the relative velocity of the tidal current (panel a), and of the considered threshold $\alpha$ (panel b). The experiments are one-dimensional (1 + 1, unidimensional in space and time). In contrast, in our data, waves show a significant directional spread, introducing a second spatial dimension (2 + 1). Therefore, the exceedance probabilities in rest conditions are different, so that we compared the amplification ratios with regard to the latter

(3.1) \begin{equation} \frac {\mathbb{P}_{U/c_g}^{(2+1)}(\alpha )}{\mathbb{P}_{U/c_g=0}^{(2+1)}(\alpha )} \quad \textrm {and} \quad \frac {\mathbb{P}_{U/c_g}^{(1+1)}(\alpha )}{\mathbb{P}_{U/c_g=0}^{(1+1)}(\alpha )} \quad . \end{equation}

The match between field observations and flume experiments is remarkable, especially when keeping in mind the wide range of conditions and the directional spread present in our dataset.

Figure 5.

(a) Amplification of the exceedance probability of extreme waves as a function of the current velocity, for several values of the threshold $\alpha$ and (b) exceedance probability as a function of $\alpha$ for bins with rest conditions ( $U/c_g = 0$ ), maximal amplification ( $U/c_g = -0.13$ ) and of expected wave breaking conditions ( $U/c_g = -0.19$ ). To maximise legibility, confidence intervals in panel (a) observations are not shown, as panel (b) provides their magnitude for the rest conditions and the peak of the opposing current. Furthermore, the confidence intervals for $\alpha = 2.0$ are the same as in figure 3.

3.2. Influence of the threshold $\alpha$ on the exceedance probabilities

Hitherto, we have focused on the exceedance probability for a value of $\alpha = 2$ , corresponding to the common definition of rogue waves. As displayed in figure 5, the choice of $\alpha$ does not qualitatively impact our main finding, as long as $\alpha$ is sufficiently high. The difference between the rest and opposing current conditions is significant from a statistical point of view ( $p \lt 0.05$ , see table 3) for $1.7 \leqslant \alpha \leqslant 2.0$ . The ratio between the two curves of figure 5 begins to increase further for $\alpha \geqslant 1.9$ , which also corresponds to the regime in which amplification by following and opposing currents is observed, up to a normalised current speed $|U/c_g|\approx 0.13$ . In addition, we also compare it with the curve (cyan) for the bin with the highest values of the opposing current $|U/c_g|\approx 0.19$ where wave breaking is expected to appear. The amplification tends to increase for increasing values of $\alpha$ , where a smaller fraction of the wave height distribution is included, enhancing the weight of the tail of the probability distribution function. However, as the definition of extreme events becomes more stringent for higher $\alpha$ , such events become sparser, leading to the loss of statistical significance for $\alpha \geqslant 2.1$ .

Table 3.

Hypergeometric $p$ -value calculated from Fisher’s exact test for differences between wave height exceedance probabilities at rest and in opposing current conditions (red and blue curves of figure 5 b).

3.3. Wave crest statistics

To check if our analysis is specific to wave height statistics, we performed the same work based on wave crest distributions (figure 6). In this case, the exceedance probability $\mathbb{P}(\beta , U)$ corresponds to crest heights $H_c$ exceeding $\beta H_s$ . The rest conditions and the bin that maximises the crest-to-trough height probability are compared with both the Rayleigh distribution for crests (Longuet-Higgins Reference Longuet-Higgins1952) and its nonlinear counterpart (Tayfun Reference Tayfun1980). The latter distribution is plotted for a steepness typical of the rest conditions ( $U/c_g = 0$ ) with $\varepsilon = 0.06$ . The wave crest corresponds to 55 %–65 % of the total wave height (Mendes et al. Reference Mendes, Scotti and Stansell2021). Thus, we expect similar behaviours for $\mathbb{P}(\beta , U)$ and $\mathbb{P}(\alpha , U)$ with $\beta \simeq 0.6 \alpha$ .

Indeed, the ratio between the exceedance probabilities of wave crests in rest and in opposing current conditions (figure 6) visually rises beyond $\beta = 0.6 \times 1.9 \simeq 1.15$ . The case $\alpha = 1.9$ corresponds to the point where the similar ratio for wave heights starts to increase (figure 5 b). However, the amplification of the exceedance probabilities for wave crests looses its statistical significance as compared with wave heights beyond $\beta = 1.7$ . This can be understood by considering that the Rayleigh distribution for wave crests evolves as

(3.2) \begin{equation} {\exp {(-8\beta ^2)} \approx \exp {(-2.9\alpha ^2)},} \end{equation}

as compared with $\exp {(-2\alpha ^2)}$ for wave heights (Longuet-Higgins Reference Longuet-Higgins1952). This faster decay of the wave crest distribution results in sparser events, which undermine the statistical significance. Consequently, wave heights appear more relevant than wave crests for assessing heavy-tailed wave distributions.

We also note that the exceedance probabilities for both the rest and the opposing current conditions drastically deviate from the Rayleigh distribution up to a level similar to the nonlinear distribution introduced by Tayfun (Reference Tayfun1980). This deviation confirms the nonlinearity of the statistical distribution induced by currents, although it is more marked for wave crests than for wave heights.

Figure 6.

Exceedance probability of rogue wave crests ( $\beta = H_c/H_s$ ) for the entire North Sea data. The error band is computed from the 95 % Jeffreys confidence interval.

Table 4.

Hypergeometric $p$ -value calculated from the Fisher exact test on the difference between wave crest height exceedance probabilities in rest and in opposing current conditions (red and blue curves of figure 6).

3.4. Influence of nonlinearity

We further investigate the behaviour in different ranges of wave steepness. Figure 7 displays the rogue wave amplification probability as a function of the current speed, for three ranges of steepness, corresponding to linear ( $\varepsilon \leqslant 0.05$ ), second-order ( $0.05 \leqslant \varepsilon \leqslant 0.9$ ) and higher-order ( $\varepsilon \geqslant 0.9$ ) regimes, respectively. The amplification of the rogue wave probability observed in figure 3 mainly originates from the second-order steepness range. In contrast, the amplification is weaker in the linear regime. More surprisingly, it vanishes in the higher-order regime. In the latter case, the onset of wave breaking associated with the large steepness likely levels off the wave height distribution, limiting the occurrence of rogue waves. Thus, the steepness regime is relevant for rogue wave occurrence.

3.5. Robustness assessment: definition of depth and rest conditions

Figure 7.

Effect of steepness on the rogue wave amplification by tidal currents for different linearity ranges.

Figure 8.

Exceedance probability $P_\alpha$ for $\alpha = 2.0$ as a function of the normalised current velocity $U/c_g$ measured at $h=-5.5$ m and taken as an average in the range $-22.5\,\textrm {m} \lt h \lt -7.5\,\textrm {m}$ . Confidence intervals are not shown since they are the same as in figure 3.

As mentioned in § 2, the water depth at the measurement location (30 m) is sufficiently small compared with the wavelength ( $\sim$ 100 m) for the waves to be influenced by the current over the full water column. We checked that our results are robust against the choice of the depth at which the current speed was measured, by comparing our results obtained by considering the current at $h=-5.5$ m with those obtained by considering $U/c_g$ averaged between $h = -7.5$ and $-22.5$ m. The behaviours are comparable, except in the rightmost bin corresponding to a fast following current (figure 8). However, as discussed above, the extremely large confidence interval on this bin limits the relevance of these differences. A similar check was performed by considering the current at $h = -15$ m (not shown), leading to the same conclusion. This relative independence of our results of the depth at which the current speed was measured, facilitates the comparison with experiments, where the vertical current profile is generally homogeneous (Ducrozet et al. Reference Ducrozet, Abdolahpour, Nelli and Toffoli2021; Zhang et al. Reference Zhang, Ma, Tan, Dong and Benoit2023). Finally, figure 9 displays the rogue wave amplification of probability for three definitions of the boundaries of the central bin, i.e. the rest condition. Obviously, the impact of this choice on our results is minimal, especially when comparing the differences between the curves with the typical width of the confidence interval of the data (see figure 3).

Figure 9.

Sensitivity to different definitions of the rest condition (legend) of the exceedance probability of rogue waves ( $\alpha = 2$ ). Confidence intervals are not shown since they are the same as in figure 3.

Figure 10.

Estimated mean excess kurtosis and its 95% confidencen intervals as a function of the tidal current.

3.6. Alternative metrics of enhanced rogue wave activity

The excess kurtosis is widely used as a proxy for measuring the degree of nonlinearity of ocean waves (Longuet-Higgins Reference Longuet-Higgins1963; Marthinsen Reference Marthinsen1992; Mori & Janssen Reference Mori and Janssen2006). Figure 10 displays the average excess kurtosis for each tidal velocity bin. The variations with the current are well below statistical significance (table 5). This can be understood by considering that excess kurtosis holds as a proxy of the wave nonlinearity only if the initially linear waves have a quasi-Gaussian distribution of the surface elevation (Longuet-Higgins Reference Longuet-Higgins1952). While this is the case in laboratory-controlled experiments (Zhang et al. Reference Zhang, Benoit, Kimmoun, Chabchoub and Hsu2019; Li et al. Reference Li, Draycott, Zheng, Lin, Adcock and Van Den Bremer2021), it is less trustworthy in real-ocean observations (Stansell Reference Stansell2004; Christou & Ewans Reference Christou and Ewans2014; Cattrell et al. Reference Cattrell, Srokosz, Moat and Marsh2018). The rest condition data of figure 5(b) illustrate the empirical sub-Gaussian exceedance probability at rest conditions $(U/c_g =0)$ . This dependence is well fitted by (Longuet-Higgins Reference Longuet-Higgins1980; Mendes & Scotti Reference Mendes and Scotti2020)

(3.3) \begin{equation} {\mathbb{P}_{\nu }(\alpha ) = e^{-2\nu _{\star }\alpha ^{2}},} \end{equation}

where

Table 5.

Hypergeometric $p$ -value calculated from the Fisher exact test for selected pairs of data points of figure 10.

Figure 11.

Response of the mean directional Benjamin–Feir index ( $\textit{BFI}_{2D}$ ), as computed from (37) of Mori et al. (Reference Mori, Onorato and Janssen2011), to the normalised tidal intensity. Error bands display data with plus or minus one standard deviation.

(3.4) \begin{equation} {\nu _{\star } = \left [ 1 - \left ( \frac {\pi ^2}{8} - \frac {1}{2} \right ) \nu ^{2} \right ]^{-1} \quad .} \end{equation}

This sub-Gaussian probability is likely due to the broadbanded spectrum in the North Sea data. If we use a Gram–Charlier expansion for the exceedance probability of the type $1 + ({\mu _{4}}/{2}) \alpha ^{2} ( \alpha ^{2} - 1 )$ for the ratio between non-Gaussian and Gaussian distributions (Mori & Yasuda Reference Mori and Yasuda2002), comparison with the broadbanded distribution of (3.3) leads to $\mu _4 \approx ( 1 - \pi ^2/4 ) 2\nu ^{2}/(\alpha ^{2} - 1) \sim - \nu ^{2}$ . As such, the reference kurtosis in rest conditions is sub-Gaussian (negative). Consequently, a threefold increase in the probability of exceedance with such rest conditions will increase the kurtosis by small fractions ( ${\lesssim}1/10$ ). In contrast, if the rest conditions had $\mu _4 = 0$ , its increase would be ${\gtrsim}1/3$ . This kurtosis reference bias weakens the kurtosis as a relevant metric in the conditions of our open sea data.

Table 6.

Hypergeometric $p$ -value calculated from the Fisher exact test for selected pairs of data points of figure 11. $p$ -values are very small in all cases, ergo we also highlight their $\chi ^2$ values in italics to allow for comparison.

Finally, BFI is also generally considered as a key indicator of the ability of rogue waves to develop and grow (Mori & Janssen Reference Mori and Janssen2006; Mori et al. Reference Mori, Onorato and Janssen2011). Its dependency on the tidal current relative to the wave direction (figure 11) displays a significant asymmetry between opposing and following currents (table 6). However, as also evidenced in figure 2(f), the values of the BFI are well below the value of $0.3$ that would be required for it to initiate the development of rogue waves.