Wave breaking is a complex physical process about which open questions remain. For some applications, it is critical to include breaking effects in phase-resolved envelope-based wave models such as the non-linear Schrödinger. A promising approach is to use machine learning to capture breaking effects. In the present paper we develop the machine learning architecture to model breaking developed by Eeltink et al. (2022) further, potentially enabling more detailed breaking physics to be captured. We show that this model can be trained on focused wave groups but can also capture breaking in random waves and modulated plane waves. Analysis of the model suggests that the machine learning has broken the problem into two—one part which detects whether the wave is breaking and another which captures the subsequent behaviour, consistent with the way human scientists routinely understand the breaking problem.

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Wave breaking is the main mechanism that dissipates energy from ocean waves by wind. Its effects on the frequency spectrum cause a downshift of the spectral peak and dissipation of the total energy of the spectrum. Various reduced-form wave breaking models have been developed to capture wave breaking in envelope-based wave evolution equations for perturbed plane-wave systems, but their applicability to waves with a continuous spectrum has not been examined. In this paper we perform modified nonlinear Schrödinger equation simulations to study four existing wave breaking models and compare the results with new experimental data for breaking unidirectional wave groups. We first compare the different wave breaking models for perturbed plane-wave simulations and then examine their potential extension to waves with a continuous spectrum. We find that most existing models are able to model breaking in perturbed plane waves, but none produce the correct spectral dissipation for focused wave groups. We propose a modification to the breaking model by Kato and Oikawa [J. Phys. Soc. Jpn. 64, 4660 (1995)0031-901510.1143/JPSJ.64.4660] in order to model breaking in focused wave groups. The modified model incorporates both a breaking criterion, which activates and deactivates the dissipation term proposed by Kato and Oikawa, and a heuristic spectral weighting function that is obtained by fitting to experimental data. The modified model also predicts breaking in perturbed plane waves well.

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An increased number of rogue waves, relative to standard distributions, can be induced by unidirectional waves passing over abrupt decreases in water depth. We investigate this phenomenon in a more general setting of multidirectional waves. We examine the influence of the directionality on the occurrence probability of rogue waves using laboratory experiments and fully nonlinear potential flow simulations. Based on the analysis of the statistics of random waves, we find that directional spreading reduces the formation probability of rogue waves relative to unidirectional seas. Nevertheless, for typical values of directional spreading in the ocean (15∘–30∘), our numerical results suggest that there is still a significant enhancement to the number of rogue waves just beyond the top of a depth discontinuity.@en

Axisymmetric standing waves occur across a wide range of free surface flows. When these waves reach a critical height (steepness), wave breaking and jet formation occur. For travelling surface gravity waves, wave breaking is generally considered to limit wave height and reversible wave motion. In the ocean, the behaviour of directionally spread waves lies between the limits of purely travelling (two dimensions) and axisymmetric (three dimensions). Hence, understanding wave breaking and jet formation on axisymmetric surface gravity waves is an important step in understanding extreme and breaking waves in the ocean. We examine an example of axisymmetric wave breaking and jet formation colloquially known as the 'spike wave', created in the FloWave circular wave tank at the University of Edinburgh, UK. We generate this spike wave with maximum crest amplitudes of 0.15-6.0 m (0.024-0.98 when made non-dimensional by characteristic radius), with wave breaking occurring for crest amplitudes greater than 1.0 m (0.16 non-dimensionalised). Unlike two-dimensional travelling waves, wave breaking does not limit maximum crest amplitude, and our measurements approximately follow the jet height scaling proposed by Ghabache et al. (J. Fluid Mech., vol. 761, 2014, pp. 206-219) for cavity collapse. The spike wave is predominantly created by linear dispersive focusing. A trough forms, then collapses producing a jet, which is sensitive to the trough's shape. The evolution of the jets that form in our experiments is predicted well by the hyperbolic jet model proposed by Longuet-Higgins (J. Fluid Mech., vol. 127, 1983, pp. 103-121), previously applied to jets forming on bubbles.

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When making directional surface gravity waves in a wave tank or when initialising numerical simulations of the ocean, the wave spectrum is often curtailed suppressing higher frequencies and wavenumbers. We consider the impact of doing this by numerically simulating two seminal experiments, those of Onorato et al. (J. Fluid Mech., vol. 627, 2009, pp. 235–257, R2) and Latheef & Swan (Proc. R. Soc. A, vol. 469, no. 2152, 2013, p. 20120696). We simulate waves using a fully nonlinear potential-flow model. We find that curtailing the spectrum can have a significant impact on the subsequent evolution. In particular, for cases where the spectrum has been curtailed, the nonlinear physics produces significantly more extreme or rogue waves than are observed in the case where the full spectral tail was included in the initial conditions, and this difference persists over tens of periods after the waves are initialised. This suggests that sea states that are ‘out of equilibrium’ (i.e. with their tails removed) can produce a greater number of rogue waves. We show this can also have an impact on predicted loads on offshore infrastructure.@en

Abrupt changes in water depth are known to lead to abnormal free-surface wave statistics. The present study considers whether this translates into abnormal loads on offshore infrastructure. A fully non-linear numerical model is used which is carefully validated against experiments. The wave kinematics from the numerical model are used as input to a simple wave loading model. We find enhanced overturning moments, an increase of approximately 20%, occur over a distance of a few wavelengths after an abrupt depth transition. We observe similar results for 1:1 and 1:3 slopes. This increase does not occur in linear simulations.

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We perform simulations of random seas based on narrow-banded spectra with directional spreading. Our wavefields are spatially homogeneous and nonstationary in time. We truncate the spectral tail for the initial conditions at different cutoff wavenumbers to assess the impact of the spectral tail on the kurtosis and spectral evolution. We consider two cases based on truncation of the wavenumber tail at |k|/k_p = 2.4 and |k|/k_p = 6. Our simulations indicate that the peak kurtosis value increases if the tail is truncated at |k|/k_p = 2.4 rather than |k|/k_p = 6. For the case with a wavenumber cutoff at |k|/k_p = 2.4, augmented kurtosis is accompanied by comparatively more aggressive spectral changes including redevelopment of the spectral tail. Similar trends are observed for the case with a wavenumber cutoff at |k|/k_p = 6, but the spectral changes are less substantial. Thus, the spectral tail appears to play an important role in a form of spectral equilibrium that reduces spectral changes and decreases the peak kurtosis value. Our findings suggest that care should be taken when truncating the spectral tail for the purpose of simulations/experiments. We also find that the equation of Fedele (2015, “On the Kurtosis of Deep-Water Gravity Waves,” J. Fluid Mech., 782, pp. 25–36) provides an excellent estimate of the peak kurtosis value. However, the bandwidth parameter must account for the spectral tail to provide accurate estimates of the peak kurtosis.

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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.

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Abrupt depth transitions (ADTs) have been shown to induce the release of bound waves into free waves, which results in spatially inhomogeneous wave fields atop ADTs. Herein, we examine the role of free-wave release in the generation and spatial distribution of higher-harmonic wave components and in the onset of wave breaking for very steep periodic waves upon interaction with an ADT. We utilise a Smoothed Particle Hydrodynamics (SPH) model, making use of its ability to automatically capture breaking and overturning surfaces. We validate the model against experiments. The SPH model is found to accurately reproduce the phase-resolved harmonic components up to the sixth harmonic, particularly in the vicinity of the ADT. For the cases studied, we conclude that second-order free waves released at the ADT, and their interaction with the linear and second-order bound waves (beating), drive higher-order bound-wave components, which show spatial variation in amplitude as a result. For wave amplitudes smaller than the breaking threshold, this second-order beating phenomenon can be used to predict the locations where peak values of surface elevation are located, whilst also predicting the breaking location for wave amplitudes at the breaking threshold. Beyond this threshold, the contributions of the second-order and higher harmonics (second-harmonic amplitudes are up to 60% and sixth-harmonic up to 10% of the incident amplitude) cause breaking to occur nearer to the ADT, and hence the wave breaking onset location is confined to the region between the ADT and the first anti-node location of the second-order components. Counter-intuitively, we find that, at the point of breaking, steeper incident waves are found to display reduced non-linearity as a result of breaking nearer to the ADT.@en

Abrupt depth transitions (ADTs) have recently been identified as potential causes of 'rogue' ocean waves. When stationary and (close-to-) normally distributed waves travel into shallower water over an ADT, distinct spatially localized peaks in the probability of extreme waves occur. These peaks have been predicted numerically, observed experimentally, but not explained theoretically. Providing this theoretical explanation using a leading-order-physics-based statistical model, we show, by comparing to new experiments and numerical simulations, that the peaks arise from the interaction between linear free and second-order bound waves, also present in the absence of the ADT, and new second-order free waves generated due to the ADT.@en

We simulate focusing surface gravity wave groups with directional spreading using the modified nonlinear Schrödinger (MNLS) equation and compare the results with a fully-nonlinear potential flow code, OceanWave3D. We alter the direction and characteristic wavenumber of the MNLS carrier wave, to assess the impact on the simulation results. Both a truncated (fifth-order) and exact version of the linear dispersion operator are used for the MNLS equation. The wave groups are based on the theory of quasi-determinism and a narrow-banded Gaussian spectrum. We find that the truncated and exact dispersion operators both perform well if: (1) the direction of the carrier wave aligns with the direction of wave group propagation; (2) the characteristic wavenumber of the carrier wave coincides with the initial spectral peak. However, the MNLS simulations based on the exact linear dispersion operator perform significantly better if the direction of the carrier wave does not align with the wave group direction or if the characteristic wavenumber does not coincide with the initial spectral peak. We also perform finite-depth simulations with the MNLS equation for dimensionless depths (k_pd) between 1.36 and 5.59, incorporating depth into the boundary conditions as well as the dispersion operator, and compare the results with those of fully-nonlinear potential flow code to assess the finite-depth limitations of the MNLS.

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Recent experimental and numerical studies of surface gravity waves propagating over a sloping bottom have shown that an increase in the probability of extreme waves can be triggered by depth variations in sufficiently shallow waters. This phenomenon is studied here by means of a boundary element method with fast multipole acceleration to solve the fully nonlinear water wave equations. We focus on the case of a random, unidirectional wave field with prescribed statistical properties propagating over a submerged slope and consider different depth variations, including a step. Validation is provided by comparing with experiments by Trulsen, Zeng, and Gramstad [Phys. Fluids 24, 097101 (2012)PHFLE61070-663110.1063/1.4748346]. Strongly non-Gaussian statistics are observed in a region localized near the depth transition, beyond which the statistics settle rapidly to the steady statistical state of finite-depth random wave fields. Using a harmonic separation technique, we show that the second-order terms are responsible for the change in the statistical properties near the depth transition. We explore in detail the effects of peak frequency, significant wave height, the inclination of the slope, and the depth of the shallower water side on the kurtosis, skewness, and the excess probability of the crest height, including their spatial distributions.

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We have investigated steep three-dimensional surface gravity wave groups formed by dispersive focusing using a fully nonlinear potential flow solver. We find that third-order resonant interactions result in rapid energy transfers to higher wavenumbers and reduced directional spreading during focusing, followed by spectral broadening during defocusing, forming steep wave groups with augmented kinematics and a prolonged lifespan. If the wave group is initially narrow-banded, quasi-degenerate interactions arise, characterised by energy transfers along the resonance angle, of the Phillips 'figure-of-eight' loop. Spectral broadening due to the quasi-degenerate interactions facilitates non-degenerate interactions, characterised by oblique energy transfers at approximately to the spectral peak. We consider the influence of steepness, finite depth, directional spreading and the high-wavenumber tail on spectral evolution. Steepness is found to augment both the quasi-degenerate and non-degenerate interactions similarly. However, a reduction in depth is found to weaken the quasi-degenerate interactions more severely than the non-degenerate interactions. We observe that increased directional spreading reduces spectral evolution, partially because wave groups with more spreading focus for a shorter duration due to linear dispersion. However, we also find that directional spreading reduces the peak rates of energy transfer. Inclusion of the high-wavenumber tail of the Joint North Sea Wave Project spectrum further reduces rates of energy transfer compared with a Gaussian wavenumber spectrum. Thus, directional spreading and the high-wavenumber tail may be integral to a form of spectral equilibrium that reduces rapid energy transfers during a steep wave event.@en

Uncertainty affects estimates of the power potential of tidal currents, resulting in large ranges in values reported for sites such as the Pentland Firth, UK. Kreitmair et al. (2019, R. Soc. open sci. 6, 180941. (doi:10.1098/rsos.191127)) have examined the effect of uncertainty in bottom friction on tidal power estimates by considering idealized theoretical models. The present paper considers the role of bottom friction uncertainty in a realistic numerical model of the Pentland Firth spanned by different fence configurations. We find that uncertainty in removable power estimates resulting from bed roughness uncertainty depends on the case considered, with relative uncertainty between 2% (for a fully spanned channel with small values of mean roughness and input uncertainty) and 44% (for an asymmetrically confined channel with high values of bed roughness and input uncertainty). Relative uncertainty in power estimates is generally smaller than (input) relative uncertainty in bottom friction by a factor of between 0.2 and 0.7, except for low turbine deployments and very high mean values of friction. This paper makes a start at quantifying uncertainty in tidal stream power estimates, and motivates further work for proper characterization of the resource, accounting for uncertainty inherent in resource modelling.

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We study the evolution of unidirectional water waves from a randomly forced input condition with uncorrelated Fourier components. We examine the kurtosis of the linearised free surface as a convenient proxy for the probability of a rogue wave. We repeat the laboratory experiments of Onorato et al. (Phys. Rev. E, vol. 70, 2004, 067302), both experimentally and numerically, and extend the parameter space in our numerical simulations. We consider numerical simulations based on the modified nonlinear Schrödinger equation and the fully nonlinear water wave equations, which are in good agreement. For low steepness, existing analytical models based on the nonlinear Schrödinger equation (NLS) are found to be accurate. For cases which are steep or have very narrow bandwidths, these analytical models over-predict the rate at which excess kurtosis develops. In these steep cases, the kurtosis in both our experiments and numerical simulations peaks before returning to an equilibrium level. Such transient maxima are not predicted by NLS-based analytical models. Above a certain threshold of steepness, the steady-state value of kurtosis is primarily dependent on the spectral bandwidth. We also examine how the average shape of extreme events is modified by nonlinearity over the evolution distance, showing significant asymmetry during the initial evolution, which is greatly reduced once the spectrum has reached equilibrium. The locations of the maxima in asymmetry coincide approximately with the locations of the maxima in kurtosis.

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The relative contributions of free waves and bound waves to the formation of an extreme wave event remains a topic of interest in offshore engineering. A variety of methods have been proposed for identifying and removing the bound wave components. The method of “phase separation” or “phase manipulation” repeats simulations/experiments of a wave field with an offset in the initial phase of the wave components and relies upon summation of the resulting wave fields to isolate the bound harmonics, following from a Stokes expansion in steepness; the method has proven effective in isolating bound harmonics but requires that all cases be repeated. Alternatively, the bound harmonics can be removed using a three-dimensional fast Fourier transform (3D-FFT) of the wave field. However, the Fourier transform requires periodicity in the signal and assumes homogeneity in space and stationarity in time, producing spurious modes otherwise. We compare the phase separation and 3D-FFT approaches for a steep, focusing wave group in deep water using the numerical simulation tool, OceanWave3D, and discuss the effectiveness of both methods.@en

Many ocean engineering problems involve bound harmonics which are slaved to some underlying assumed close to linear time series. When analyzing signals we often want to remove the bound harmonics so as to "linearise" the data or to extract individual bound harmonic components so that they may be studied. For even moderately broadbanded systems filtering in the frequency domain is not sufficient to separate components as they overlap in frequency. One way to overcome this difficulty is to use input signals with the same linear envelope but with different phases and then use simple addition and subtraction of the resulting signals to extract different harmonics. This approach has been established for the analysis of wave groups. In this paper we examine whether this approach can be used on random time series as well. We analyse random wave time series of wave elevation from the towing tank in Shanghai Jiao Tong University and force measurements on a cylinder taken in the Kelvin tank at the University of Strathclyde.

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High wind speeds generated during hurricanes result in the formation of extreme waves. Extreme waves by nature are steep meaning that linear wave theory alone is insufficient in understanding and predicting their occurrence. The complex, highly transient nature of the direction of wind and hence of waves generated during hurricanes affects this nonlinear behavior. Herein, we examine how this directionality can affect the second-order nonlinearity of extreme waves generated during hurricanes. This is achieved through both deterministic calculations and experiments based on the observations of Young (2006, "Directional Spectra of Hurricane Wind Waves," J. Geophys. Res. Oceans, 111(C8), epub). Our calculations show that interactions between the tail and peak of the spectrum can become significant when they travel in different directions, resulting in second-order difference components that exist in the linear range of frequencies. These calculations are generally supported by experimental observations, but we note the difficulty of generating and focusing the high-frequency tail of the spectrum experimentally. Bound second-order difference components or subharmonics typically exist as low frequency infra-gravity waves. Components that exist in the linear range of frequencies may be missed by conventional methods of processing field data where low-pass filtering is used and hence overlooked. In this note, we show that in idealized directional spreading conditions representative of a hurricane, failing to account for second-order difference components may lead to underestimation of extreme wave height.

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Freak or rogue waves are so called because of their unexpectedly large size relative to the population of smaller waves in which they occur. The 25.6 m high Draupner wave, observed in a sea state with a significant wave height of 12 m, was one of the first confirmed field measurements of a freak wave. The physical mechanisms that give rise to freak waves such as the Draupner wave are still contentious. Through physical experiments carried out in a circular wave tank, we attempt to recreate the freak wave measured at the Draupner platform and gain an understanding of the directional conditions capable of supporting such a large and steep wave. Herein, we recreate the full scaled crest amplitude and profile of the Draupner wave, including bound set-up. We find that the onset and type of wave breaking play a significant role and differ significantly for crossing and non-crossing waves. Crucially, breaking becomes less crest-amplitude limiting for sufficiently large crossing angles and involves the formation of near-vertical jets. In our experiments, we were only able to reproduce the scaled crest and total wave height of the wave measured at the Draupner platform for conditions where two wave systems cross at a large angle.

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For sufficiently directionally spread surface gravity wave groups, the set-down of the wave-averaged free surface, first described by Longuet-Higgins and Stewart (J. Fluid Mech. vol. 13, 1962, pp. 481-504), can turn into a set-up. Using a multiple-scale expansion for two crossing wave groups, we examine the structure and magnitude of this wave-averaged set-up, which is part of a crossing wave pattern that behaves as a modulated partial standing wave: in space, it consists of a rapidly varying standing-wave pattern slowly modulated by the product of the envelopes of the two groups; in time, it grows and decays on the slow time scale associated with the translation of the groups. Whether this crossing wave pattern actually enhances the surface elevation at the point of focus depends on the phases of the linear wave groups, unlike the set-down, which is always negative and inherits the spatial structure of the underlying envelope(s). We present detailed laboratory measurements of the wave-averaged free surface, examining both single wave groups, varying the degree of spreading from small to very large, and the interaction between two wave groups, varying both the degree of spreading and the crossing angle between the groups. In both cases, we find good agreement between the experiments, our simple expressions for the set-down and set-up, and existing second-order theory based on the component-by-component interaction of individual waves with different frequencies and directions. We predict and observe a set-up for wave groups with a Gaussian angular amplitude distribution with standard deviations of above ( for energy spectra), which is relatively large for realistic sea states, and for crossing sea states with angles of separation of and above, which are known to occur in the ocean.

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