Although a ubiquitous natural phenomenon, the onset and subsequent process of surface wave breaking are not fully understood. Breaking affects how steep waves become and drives air–sea exchanges1. Most seminal and state-of-the-art research on breaking is underpinned by the assumption of two-dimensionality, although ocean waves are three dimensional. We present experimental results that assess how three-dimensionality affects breaking, without putting limits on the direction of travel of the waves. We show that the breaking-onset steepness of the most directionally spread case is double that of its unidirectional counterpart. We identify three breaking regimes. As directional spreading increases, horizontally overturning ‘travelling-wave breaking’ (I), which forms the basis of two-dimensional breaking, is replaced by vertically jetting ‘standing-wave breaking’ (II). In between, ‘travelling-standing-wave breaking’ (III) is characterized by the formation of vertical jets along a fast-moving crest. The mechanisms in each regime determine how breaking limits steepness and affects subsequent air–sea exchanges. Unlike in two dimensions, three-dimensional wave-breaking onset does not limit how steep waves may become, and we produce directionally spread waves 80% steeper than at breaking onset and four times steeper than equivalent two-dimensional waves at their breaking onset. Our observations challenge the validity of state-of-the-art methods used to calculate energy dissipation and to design offshore structures in highly directionally spread seas.@en

Correction to: Naturehttps://doi.org/10.1038/s41586-024-07886-z Published online 14 September 2024 In the version of the article initially published, there was a typographical error where in the Fig. 5 title, now reading “For 3D waves, breaking onset does not limit crest height,” the word “not” was missing. The error has been corrected in the HTML and PDF versions of the article.

@en

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

Deep-water surface wave breaking affects the transfer of mass, momentum, energy and heat between the air and sea. Understanding when and how the onset of wave breaking will occur remains a challenge. The mechanisms that form unforced steep waves, i.e. nonlinearity or dispersion, are thought to have a strong influence on the onset of wave breaking. In two dimensions and in deep water, spectral bandwidth is the main factor that affects the roles these mechanism play. Existing studies, in which the relationship between spectral bandwidth and wave breaking onset is investigated, present varied and sometimes conflicting results. We perform potential-flow simulations of two-dimensional focused wave groups on deep water to better understand this relationship, with the aim of reconciling existing studies. We show that the way in which steepness is defined may be the main source of confusion in the literature. Locally defined steepness at breaking onset reduces as a function of bandwidth, and globally defined (spectral) steepness increases. The relationship between global breaking onset steepness and spectral shape (using the parameters bandwidth and spectral skewness) is too complex to parameterise in a general way. However, we find that the local surface slope of maximally steep non-breaking waves, of all spectral bandwidths and shapes that we simulate, approaches a limit of 1/tan(π/3)≈0.5774. This slope-based threshold is simple to measure and may be used as an alternative to existing kinematic breaking onset thresholds. There is a potential link between slope-based and kinematic breaking onset thresholds, which future work should seek to better understand.@en

Ocean wave breaking is a difficult-to-model oceanographic process, which has implications for extreme wave statistics, the dissipation of wave energy, and air–sea interaction. Numerical methods capable of reliably simulating real-world directionally spread breaking waves are useful for investigating the physics of wave breaking and for the design of offshore structures and floating bodies. Smoothed particle hydrodynamics is capable of modelling highly steep and overturning free surfaces, which makes it a promising method for simulating breaking waves. This paper investigates the effect of smoothing length on simulated wave breaking in both following and crossing seas. To do so, we reproduce numerically the experiments of highly directionally spread breaking waves in McAllister et al. (J Fluid Mech 860:767–786, 2019. https://doi.org/10.1017/jfm.2018.886) using a range of normalised smoothing lengths: h/ d_p= 1.4 , 1.7, 2.0, 2.3, with h smoothing length and d_p particle spacing. The smallest smoothing length we use appears to adversely affect the fidelity of the simulated surface elevation, so that the tallest wave crest observed in experiments is not fully reproduced (coefficient of determination r²≈ 0.7). For smoothing lengths h/ d_p= 1.7 , 2.0, and 2.3, the experiments are well reproduced (r²≥ 0.88); in these simulations smoothing length predominantly affects the spatial extent and duration of breaking. Qualitative and quantitative comparison of our simulations shows that values of h/ d_p in the range 1.7 - 2 best reproduce the wave breaking phenomena observed in experiments.

@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.

@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.

@en

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

Wave breaking in the ocean affects the height of extreme waves, energy dissipation, and interaction between the atmosphere and upper ocean. Numerical modelling is a critical step in understanding the physics of wave breaking and offers insight that is hard to gain from field data or experiments. High-fidelity numerical modelling of three-dimensional breaking waves is extremely challenging. Conventional grid-based numerical methods struggle to model the steep and double-valued free surfaces that occur during wave breaking. The Smoothed Particle Hydrodynamics (SPH) method does not fall prey to these issues. Herein, we examine the SPH method's ability to model highly directionally spread overturning breaking waves by numerically reproducing the experiments presented in McAllister et al. (2019). We find that the SPH method reproduces the experimental observations well; when comparing experimental and numerical measurements we achieve coefficient of determination values of 0.92−0.95, with some smaller-scale features less well reproduced owing to finite resolution. We also examine aspects of the simulated wave's geometry and kinematics and find that existing breaking criteria are difficult to apply in highly directionally spread conditions.

@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

Freak waves, abnormally large waves, that occur in the open-ocean can cause significant damage to offshore structures and vessels. In this paper, we attempt to numerically reproduce the experiments of McAllister et al., (2019, J. Fluid Mech. [1]), to investigate the potential properties of the Draupner freak wave [2] in more detail. We use a Smoothed Particle Hydrodynamics (SPH) method to solve the full-3D Navier-Stokes equations. This Lagrangian method is able to recreate wave breaking, and has the potential to fully reproduce these experiments with the aim of providing further insight into properties of the waves created such as their kinematics and geometry. We compare time histories of water surface elevation produced numerically using four different particle sizes with experimentally-obtained data. We find good agreement in the time domain, with r2 (coefficient of determination) values between experimental and numerical data of over 0.94 the error in maximum wave height was less than 5 % for the finest particle size (over 100 million particles). We also numerically reproduce wave breaking observed in the experiments, where jet formation and breaking phenomena are qualitatively similar in appearance.

@en

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.

@en