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.

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.

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.

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.

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.

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.

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.

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.