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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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

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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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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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.

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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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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.

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