Occurrence Of Rogue Waves Research Articles

Effect of shoaling length on rogue wave occurrence

The impact of shoaling on linear water waves is well known, but it has only been recently found to significantly amplify both the intensity and frequency of rogue waves in nonlinear irregular wave trains atop coastal shoals. At least qualitatively, this effect has been partially attributed to the ‘rapid’ nature of the shoaling process, i.e. shoaling occurs over a distance far shorter than that required for waves to modulate themselves and adapt to the reduced water depth. Through a theoretical model and highly accurate nonlinear simulations, we disentangle the respective effects of the length and angle of a shoal's slope. We investigate the effects of the shoaling process rapidness on the evolution of key statistical and spectral sea-state parameters. We let the wave field evolve over a slope with constant angle in all cases while we vary the slope length. Our results indicate that the non-equilibrium dynamics is not affected by the slope length, because further extending the slope length does not influence the magnitude of the statistical and spectral measures as long as the non-equilibrium dynamics dominates the wave evolution. Thus, the shoaling effect on rogue waves is deduced to be mainly driven by the slope magnitude rather than the slope length.

An Empirical Model for Skillful Rogue Wave Forecasts

Abstract One of the leading goals of rogue wave research is to develop a robust rogue wave warning system to mitigate the danger they pose. One such system has been developed by the European Centre for Medium-Range Weather Forecasts (ECMWF), called the freak wave warning system (FWWS), based on nonlinear wave effects. The FWWS predicts maximum expected wave envelope height as a risk parameter for forecasts. Recently, a data-driven alternative has been proposed by Häfner et al., which was distilled from a neural network using wave buoy observations. However, it has yet to be evaluated by a spectral wave model for application to operational wave forecasting. The data-driven, learned model emphasizes bandwidth-controlled linear superposition as the predominant mechanism in crest-to-trough rogue wave generation, while nonlinear effects are a secondary term. The present work evaluates the performance of the empirical model using output from an ECMWF global wave hindcast. We find that the prediction models based on bandwidth effects have the highest log likelihood scores, with the empirical model outperforming all other tested models. In contrast, the expected maximum envelope wave height from the FWWS does not predict the occurrence of rogue waves. These results indicate that the empirical model with wave model input is a skillful predictor and should be considered for operational implementation to improve rogue wave forecasting. Significance Statement Rogue waves are unexpectedly large and unpredictable waves. Encounters with rogue waves can result in damage to marine vessels and offshore infrastructures. Research is devoted to developing systems to predict the risk of rogue wave events. A novel predictive model has been shown to perform well in predicting rogue wave occurrences based on wave buoy observations. This model is a symbolic expression distilled from an artificial neural network that incorporates known rogue wave dynamics and that can be evaluated alongside traditional risk estimates with minor adjustments to operational forecasting systems. This study evaluates the newly proposed empirical model in a forecasting setting. Our work demonstrates the efficacy of the empirical model for operational forecasting purposes.

Prediction of freak waves from buoy measurements

Freak or rogue waves are a danger to ships, offshore infrastructure, and other maritime equipment. Reliable rogue wave forecasts could mitigate this risk for operations at sea. While the occurrence of oceanic rogue waves at sea is generally acknowledged, reliable rogue wave forecasts are unavailable. In this paper, the authors seek to overcome this shortcoming by demonstrating how rogue waves can be predicted from field measurements. An extensive buoy data set consisting of billions of waves is utilized to parameterize neural networks. This network is trained to distinguish waves prior to an extreme wave from waves which are not followed by an extreme wave. With this approach, three out of four rogue waves are correctly predicted 1 min ahead of time. When the advance warning time is extended to 5 min, it is found that the ratio of accurate predictions is reduced to seven out of ten rogue waves. Another strength of the trained neural networks is their capabilities to extrapolate. This aspect is verified by obtaining forecasts for a buoy location that is not included in the networks’ training set. Furthermore, the performance of the trained neural network carries over to realistic scenarios where rogue waves are extremely rare.

Nonlinear random wave fields within a Boussinesq system

The dynamics of nonlinear random fields is important for understanding wave turbulence. In this work, we use a Boussinesq system to examine the distinctions between unidirectional and bidirectional waves. Our study demonstrates that in both scenarios, the wave spectra reach a stationary state. Moreover, we show that the occurrence of rogue waves is more probable in the unidirectional case. In the unidirectional case, the probability distribution of wave crests exceeds the one predicted by the Rayleigh distribution once the spectra reach the stationary state. Conversely, in the bidirectional case, the opposite trend is observed. The discovery of various types of rogue waves, including massive wave trains commonly known in the literature as “two sisters” and “three sisters” are found.

Brief communication: Implications of outstanding solitons for the occurrence of rogue waves at two additional sites in the North Sea

Brief communication: Implications of outstanding solitons for the occurrence of rogue waves at two additional sites in the North Sea

The Expected Dynamics for the Extreme Wind and Wave Conditions at the Mouths of the Danube River in Connection with the Navigation Hazards

The entrance in the Sulina channel in the Black Sea is the target area of this study. This represents the southern gate of the seventh Pan-European transport corridor, and it is usually subjected to high navigation traffic. The main objective of the work is to provide a more comprehensive picture concerning the past and future expected dynamics of the environmental matrix in this coastal area, including especially the extreme wind and wave conditions in connection with the possible navigation risks. The methodology considered assumes analyses performed at three different levels. First, an analysis of some in situ measurements at the zero-kilometer point of the Danube is carried out for the 15-year period of 2009–2023. Together with the maximum wind speed and the maximum value of the wind gusts, the water level variation was analyzed at this point. As a second step, the analysis is based on wind speed data provided by regional climate models. Two periods, each spanning 30 years, are considered. These are the recent past (1976–2005), when comparisons with ERA5 reanalysis data were also performed, and the near future (2041–2070), when two different models and three climate scenarios were considered. The focus was on the extreme wind speed values, performing comparisons between the past and future expected extreme winds. Finally, the third analysis is related to the wave conditions. Thus, using as a forcing factor each of the wind fields that was previously analyzed, simulations employing a spectral wave model were carried out. The wave modeling system was focused using three different computational domains with increasing resolution towards the target area, and the nearshore wave conditions were evaluated. The results show that both the extreme wind and wave conditions are expected to slightly increase in the future. Especially in the wintertime, strong wind fields are often expected in this area, with wind gusts exceeding more than 70% of the hourly average wind velocity. With regard to the waves, due to the complex nearshore phenomena, considerable enhancements in terms of significant wave heights are induced, and there is also an elevated risk of the occurrence of rogue waves. This work is still ongoing, and taking into account the high navigation risks highlighted, the next step would be to elaborate the risk assessment of severe shipping conditions, particularly related to the likelihood or probability of adverse conditions with the potential of generating hazardous situations in this coastal environment.

Predicting the occurrence of rogue waves in short-crested mixed seas

ABSTRACT In this paper we have for the first time proposed a novel transformed Gaussian process method for the estimation of rogue wave occurrence in a short crested mixed sea. The saddle point approximation of a specific joint density is utilised to further improve the computational efficiency of the proposed method. The proposed method has been applied to forecast the rogue wave occurrence in four short-crested mixed sea states, two with directional wave spectra based on measured surface elevation data at the coast of Yura, and another two with typical directional bimodal wave spectra. It is shown in these four cases that the proposed method can offer more accurate forecasting results than those obtained from the linear simulation method, from the transformation based on Hermite polynomials and from using the Rayleigh distribution model. It is also demonstrated that the proposed method is more efficient than the nonlinear simulation method.

The 3-wave resonant interaction model: spectra and instabilities of plane waves

The three wave resonant interaction model (3WRI) is a non-dispersive system with quadratic coupling between the components that finds application in many areas, including nonlinear optics, fluids and plasma physics. Using its integrability, and in particular its Lax Pair representation, we carry out the linear stability analysis of the plane wave solutions interacting under resonant conditions when they are perturbed via localised perturbations. A topological classification of the so-called stability spectra is provided with respect to the physical parameters appearing both in the system itself and in its plane wave solution. Alongside the stability spectra, we compute the corresponding gain function, from which we deduce that this system is linearly unstable for any generic choice of the physical parameters. In addition to stability spectra of the same kind observed in the system of two coupled nonlinear Schrödinger equations, whose non-vanishing gain functions detect the occurrence of the modulational instability, the stability spectra of the 3WRI system possess new topological components, whose associated gain functions are different from those characterising the modulational instability. By drawing on a recent link between modulational instability and the occurrence of rogue waves, we speculate that linear instability of baseband-type can be a necessary condition for the onset of rogue wave types in the 3WRI system, thus providing a tool to predict the subsequent nonlinear evolution of the perturbation.

Non-homogeneous approximation for the kurtosis evolution of shoaling rogue waves

Bathymetric changes have been experimentally shown to affect the occurrence of rogue waves. We recently derived a non-homogeneous correction to the spectral analysis, allowing us to describe the evolution of the rogue wave probability over a shoal. Here, we extend this work to the evolution of the excess kurtosis of the surface elevation, that plays a central role in estimating rare event probabilities. Furthermore, we provide an upper bound to the excess kurtosis. In intermediate and deep water regimes, a shoal does not affect wave steepness nor bandwidth significantly, so that the vertical asymmetry between crests and troughs, the excess kurtosis and the exceedance probability of wave height stay rather constant. In contrast, in shallower water, a sharp increase in wave steepness increases the vertical asymmetry, resulting in a growth of both the tail of the exceedance probability and the excess kurtosis.

Contribution of solitons to enhanced rogue wave occurrence in shallow depths: a case study in the southern North Sea

Abstract. 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.

Long-term rogue wave occurrence probability from historical wave data on a spatial scale relevant for spar-type floating wind turbines

Very large waves, such as rogue waves, can be dangerous for ships and offshore structures. There is no consensus on the theoretical occurrence probability of these rogues and ocean measurements containing rogue waves are rare. This paper addresses the long-term occurrence probability of rogue waves using time-extreme (TE) and space–time extreme (STE) statistical wave models, evaluated with a significant amount of historical directional wave data from Monterey Bay, California, USA. A novel understanding of these models is obtained by exploring the area size for which the STE models estimate a higher rogue wave occurrence probability compared to the TE models. For four rogue wave and crest heights the return periods until a 95% rogue wave occurrence probability are estimated. To illustrate an application of this statistical analysis, the risk of rogue waves to a spar-type floating wind turbine is evaluated. Based on both the dynamic response and the long-term occurrence probability of the rogue wave, this research presents a new and advanced approach for a risk analysis of rogue waves to floating offshore structures.

Nonlinear Schrödinger waves in a disordered potential: Branched flow, spectrum diffusion, and rogue waves.

The problem of nonlinear Schrödinger (NLS) waves in a disordered potential arises in many physical occasions, such as hydrodynamics, optics, and cold atoms. It provides a paradigm for studying the interaction between nonlinearity and random effect, but the current results are far from perfect. In this paper, we systematically simulate the turbulent waves for the focusing NLS equation with dynamical (time-dependent) random potentials, where the enhanced branching structures evolve into branched soliton flows as the nonlinearity increases. In this process, the occurrence of rogue waves for short times results from the interplay of linear random focusing and modulation instability. While the nonlinear spectral analysis reveals that for longer times, it is due to a self-organization of larger solitons competing with breakup of intermediate solitons. On the other hand, we found that the strong nonlinearity can significantly increase the width of the linear (Fourier) spectrum for several time scales, but its spreading rate becomes suppressed, which has a dependence on the correlation length of the potential. We hope that our findings will facilitate a deeper understanding of the nonlinear waves interacting with disordered media.

Modulational instability and statistical properties of irregular waves in finite water depth

Modulational instability is one of the possible basic mechanisms for the generation of rogue waves in deep water. Under the condition of finite depth, this instability is reduced and disappears as k1h<1.363 (k1 is the wavenumber of the carrier wave and h is the water depth). In this study, we addressed this phenomenon through three approaches, theoretical analysis of the Zakharov’s equation and the nonlinear Schrödinger equation and direct numerical simulations of the Euler equation. Their results on the modulational instability over arbitrary depths, particularly the growth rates of sidebands, are analysed and compared. In addition, the consequences of the suppression of modulational instability for realistic ocean waves are illustrated by direct simulations of long-crested, irregular waves at finite water depth. The results show that many phenomena related to the third-order nonlinearity (i.e. the downshift of the spectral peak frequency, the enhancement of kurtosis of surface elevations and rouge wave occurrence probability) are suppressed with decreasing depth k1h and eventually cease to exist as k1h<1.363. Considering the dependence of wave evolution on water depth, two Benjamin–Feir indices for arbitrary depths derived from Zakharov’s equation and nonlinear Schrödinger equation are used as indicators to estimate the modulational instability effect on nonlinear wave fields. Our results confirm that both parameters provide reasonable predictions of the modulational instability and probability of rogue wave occurrence.

Numerical Study of Statistical Properties for Semiconductor Laser Chaos by Exploring the Injection Parameter Space

The statistical properties of semiconductor laser chaos are of great importance for practical applications. By exploring the injection parameter space of an optically injected semiconductor laser, the associated probability density functions (PDFs) are numerically investigated for generation of chaotic waveforms with rogue waves (RWs) or Gaussian distributions. First, generation of RWs are demonstrated by considering chaotic waveforms with long-tailed PDFs. The dynamical behaviors for RW generations are systematically investigated by exploring the entire injection parameter space, where a pattern is identified for parameter regions of nonzero RWs. The occurrence of RWs is also investigated for different injection parameters by evaluating the effects of laser bias current, spontaneous emission noise, and detection bandwidth. Alternatively, Gaussian PDFs are generated by bandwidth-limited detection of chaotic waveforms. The bandwidth-dependent statistical properties are investigated through tuning the injection parameters for different detection bandwidths. Generation of Gaussian PDFs are demonstrated for a large region of injection parameters and for a detection bandwidth as comparable to the laser relaxation resonance frequency. This numerical study reveals the importance of injection parameters for tuning the statistics associated with chaotic lasers, and confirms the feasibility of generating RWs and Gaussian PDFs on demand for potential applications in various fields.

Probabilistic assessment of rogue wave occurrence in directional wave fields

Probabilistic assessment of rogue wave occurrence in directional wave fields

The Rayleigh-Haring-Tayfun distribution of wave heights in deep water

Nearly every known exceedance probability distribution applied to rogue waves has shown disagreement with its peers, suggesting that the underlying phenomenon may be too complex to grasp within a single distribution. In recent studies, this view has been supported by the realisation that there is no distribution suitable for wide ranges of sea states, even though distributions can perform well in a limited and small number of conditions. Towards the better quantitative assessment of rogue wave occurrence, the present work focuses on the apparent discrepancy among the majority of previous works. Furthermore, on the grounds of the uneven distribution of rogue waves found in North Sea observations, a new exceedance probability distribution for rogue waves is introduced. The proposed distribution is a geometrical composition of some of the most popular simple models for waves while introducing additional algebraic structures. The model also obeys empirical physical bounds obtained from the analysis of nearly 350,000 waves from storms recorded in North sea and supports the qualitative analysis of rogue wave occurrence based on the interaction among three sea state parameters.

Predicting the occurrence of rogue waves in the presence of opposing currents with a high-order spectral method

This work concerns the study of surface water waves in the presence of a horizontally varying opposing current. We demonstrate experimentally and numerically that spatial inhomogeneity increases the occurrence probability of extreme and rogue waves in the course of unidirectional wave propagation. We clearly report the transition between Gaussian and strongly non-Gaussian properties. An existing experimental model of wave-current interaction is revisited by comparing it against numerical simulations from current-modified Euler equations. The latter substantiate that the underlying physics is associated with quasi-resonant nonlinear interactions that are triggered by the background current.

Modelling rogue waves in 1D wave trains with the JONSWAP spectrum, by means of the High Order Spectral Method and a fully nonlinear numerical model

Probability of rogue wave occurrence due to modulational instability in unidirectional JONSWAP sea states are investigated. This investigation has been conducted based on the quantitative indicators of Benjamin-Feir index (BFI), Π-number and kurtosis, which are used to represent instability in wave trains. Evolution of wavefields is simulated using the fully nonlinear phase-resolving numerical model of Chalikov-Sheinin (CS) and the high order spectral model of HOS-Ocean. Effects of the high-frequency end (tail) of the spectrum on modulational instability and rogue wave evolution are discussed considering four different tail lengths.According to our simulation results, indicators considered for both model results perform as proxy for the occurrence of extreme events. Therefore, it is possible to define a certain interval for the indicators, where the probability of rogue wave occurrence is the highest. Nonlinearity (described in terms of steepness) is the dominant parameter on probability of rogue wave occurrence since it is related to various active physical processes in sea states (both modulational instability and breaking). Rogue wave occurrence increases with increased nonlinearity until it is restricted again by high nonlinearity which causes waves to reach breaking onset earlier.

Real-world rogue wave probabilities

Rogue waves are dangerous ocean waves at least twice as high as the surrounding waves. Despite an abundance of studies conducting simulations or wave tank experiments, there is so far no reliable forecast for them. In this study, we use data mining and interpretable machine learning to analyze large amounts of observational data instead (more than 1 billion waves). This reveals how rogue wave occurrence depends on the sea state. We find that traditionally favored parameters such as surface elevation kurtosis, steepness, and Benjamin–Feir index are weak predictors for real-world rogue wave risk. In the studied regime, kurtosis is only informative within a single wave group, and is not useful for forecasting. Instead, crest-trough correlation is the dominating parameter in all studied conditions, water depths, and locations, explaining about a factor of 10 in rogue wave risk variation. For rogue crests, where bandwidth effects are unimportant, we find that skewness, steepness, and Ursell number are the strongest predictors, in line with second-order theory. Our results suggest that linear superposition in bandwidth-limited seas is the main pathway to “everyday” rogue waves, with nonlinear contributions providing a minor correction. This casts some doubt whether the common rogue wave definition as any wave exceeding a certain height threshold is meaningful in practice.

Anomalous wave statistics following sudden depth transitions: application of an alternative Boussinesq-type formulation

Recent studies of water waves propagating over sloping seabeds have shown that sudden transitions from deeper to shallower depths can produce significant increases in the skewness and kurtosis of the free surface elevation and hence in the probability of rogue wave occurrence. Gramstad et al. (Phys. Fluids 25 (12): 122103, 2013) have shown that the key physics underlying these increases can be captured by a weakly dispersive and weakly nonlinear Boussinesq-type model. In the present paper, a numerical model based on an alternative Boussinesq-type formulation is used to repeat these earlier simulations. Although qualitative agreement is achieved, the present model is found to be unable to reproduce accurately the findings of the earlier study. Model parameter tests are then used to demonstrate that the present Boussinesq-type formulation is not well-suited to modelling the propagation of waves over sudden depth transitions. The present study nonetheless provides useful insight into the complexity encountered when modelling this type of problem and outlines a number of promising avenues for further research.