Nonlinear Wave Groups Research Articles

Soliton groups and extreme wave occurrence in simulated directional sea waves

The evolution of nonlinear wave groups that can be associated with long-lived soliton-type structures is analyzed, based on the data of numerical simulation of irregular deep-water gravity waves with spectra typical to the ocean and different directional spreading. A procedure of the windowed Inverse Scattering Transform, which reveals wave sequences related to envelope solitons of the nonlinear Schrödinger equation, is proposed and applied to the simulated two-dimensional surfaces. The soliton content of waves with different directional spreading is studied in order to estimate its dynamical role, including characteristic lifetimes. Statistical features of the solitonic part of the water surface are analyzed and compared with the wave field on average. It is shown that intense wave patterns that persist for tens of wave periods can emerge in stochastic fields of relatively long-crested waves. They correspond to regions of locally enhanced on average waves with reduced kurtosis. This eventually leads to realization of locally extreme wave conditions compared to the general background. Although intense soliton-like groups may be detected in short-crested irregular waves as well, they possess much shorter lateral sizes, quickly disperse, and do not influence the local statistical wave properties.

Frequency downshifting in decaying wavetrains on the ocean surface covered by ice floes

We study analytically and numerically a frequency downshifting due to power-type frequency-dependent decay of surface waves in the ocean covered by ice floes. The downshifting is obtained both within the linear model and within the nonlinear Schrödinger (NLS) equation augmented by viscous terms for the initial condition in the form of an NLS envelope soliton. It is shown that the frequency-dependent dissipation produces a more substantial downshifting when the spectrum is relatively wide. As a result, the nonlinear adiabatic scenario of wavetrain evolution provides a downshifting remarkably smaller in magnitude than in the linear regime. Meanwhile, interactions between nonlinear wavegroups lead to spectral broadening and, thus, result in fast substantial frequency downshifts. Analytic estimates are obtained for an arbitrary power n of the dependence of a dissipation rate on frequency ∼ωn. The developed theory is validated by the numerical modeling of the generalized NLS equation with dissipative terms. Estimates of frequency downshift are given for oceanic waves of realistic parameters.

A study of non-equilibrium wave groups in two-phase flow in high-contrast porous media with relative permeability hysteresis

Non-equilibrium models are applicable in various physical situations, including phase transitions, hysteresis, and chemical reactions, among others. To model the dynamics of such phenomena, partial differential equations are employed with source terms, as seen in Eq. (1.1). This work focuses on situations where we connect states in equilibrium while allowing for non-equilibrium times. In our model, we model non-equilibrium through relaxation. The source term, in this case, acts as an attractor, and the model’s dynamics allow only a small-time far from equilibrium. We use this theory to model a two-equation system in a two-phase fluid in porous media, where different permeabilities of the flow lead to hysteresis phenomena. We show several examples and study the Riemann solution to gain a better understanding of the solution. We investigate the interplay between nonlinear wave groups of water–oil saturation with different permeabilities in the non-wetting phase and the high-contrast heterogeneity of the porous medium in two-space dimensions. This phenomenon is described by a coupled set of time-dependent partial differential equations of hyperbolic–parabolic–elliptic mixed type. We present and discuss a new projection method with relaxation to obtain 1D-analytical Riemann solutions in an idealized homogenized medium, and we use this solution to validate our numerical method. The primary components of obtaining these solutions are shocks, rarefactions, and bifurcations. We obtain 2D-numerical solutions using an operator splitting approach based on physics, highlighting the discretization methods used. We provide robust analytical–numerical examples to verify the theory and illustrate the approach’s capabilities. In particular, we discuss the 1D-2D behavior of wave groups in situations driven by water–oil infiltration problems of permeable reservoir rocks.

Partitioning the Extreme Wave Spectrum of Hurricane Wilma to Improve the Design of Wave Energy Converters

Analysis of the omnidirectional energy spectrum from storm wave measurements provides valuable parameters for understanding the specific local conditions that wave energy converters would have to withstand. Partitioning the energy spectrum also helps to identify wave groups with low directional spread propagating in the direction of the dominant waves of the more energetic wave systems. This paper analyzes the partition of the Hurricane Wilma energy spectrum using single-point measurements obtained in shallow water. Hurricane Wilma generated simultaneous crossing wave systems with different significant wave heights and steepnesses. The maximum estimated significant height among the wave groups was 5.5 m. The corresponding height of the partitions and the omnidirectional energy spectrum were 11.0 m (swell) and 12 m, respectively. While linear superposition was the main mechanism responsible for driving the wave groups, at times, modulational instability produced nonlinear wave groups. This is a new finding, since modulational instability is usually considered an open-sea phenomenon. For shorelines with multidirectional wave groups, submerged and semi-submerged devices should be designed to account for changes in wave direction and wave height, although under extreme hurricane conditions, energy harvesting might have to be sacrificed for the benefit of device integrity.

Statistical and Dynamical Characteristics of Extreme Wave Crests Assessed with Field Measurements from the North Sea

Abstract Wave crests of unexpected height and steepness pose a danger to activities at sea, and long-term field measurements provide important clues for understanding the environmental conditions that are conducive to their generation and behavior. We present a novel dataset of high-frequency laser altimeter measurements of the sea surface elevation gathered over a period of 18 years from 2003 to 2020 on an offshore platform in the central North Sea. Our analysis of crest height distributions in the dataset shows that mature, high sea states with high spectral steepness and narrow directional spreading exhibit crest height statistics that significantly deviate from standard second-order models. Conversely, crest heights in developing sea states with similarly high steepness but wide directional spread correspond well to second-order theory adjusted for broad frequency bandwidth. The long-term point time series measurements are complemented with space–time stereo video observations from the same location, collected during five separate storm events during the 2019/20 winter season. An examination of the crest dynamics of the space–time extreme wave crests in the stereo video dataset reveals that the crest speeds exhibit a slowdown localized around the moment of maximum crest elevation, in line with prevailing theory on nonlinear wave group dynamics. Extending on previously published observations focused on breaking crests, our results are consistent for both breaking and nonbreaking extreme crests. We show that wave crest steepness estimated from time series using the linear dispersion relation may overestimate the geometrically measured crest steepness by up to 25% if the crest speed slowdown is not taken into account. Significance Statement Better understanding of the statistics and dynamical behavior of extreme ocean surface wave crests is crucial for improving the safety of various operations at sea. Our study provides new, long-term field evidence of the combined effects of wave field steepness and directionality on the statistical distributions of crest heights in storm conditions. Moreover, we show that the dynamical characteristics of extreme wave crests are well described by recently identified nonlinear wave group dynamics. This finding has implications, for example, for wave force calculations and the treatment of wave breaking in numerical wave models.

Experiments on uni-directional and nonlinear wave group shoaling

Experiments on uni-directional and nonlinear wave group shoaling

Introducing envelope soliton solutions for wave–structure investigations

This paper explores the applicability of envelope soliton solutions of the non-linear Schrödinger-type equation as an innovative design wave for wave–structure interaction investigations. The major benefits of envelope solitons are the potential to generate non-linear wave groups of certain frequency and (physically possible) wave steepness as well as the availability of an analytical solution. Furthermore, the fact that envelope solitons are characterized by a limiting frequency bandwidth enables the tailor-made generation of critical wave conditions for task-specific wave–structure investigations. For the evaluation of the applicability of envelope solitons, wave–structure investigations with a chemical tanker were performed in a set of envelope solitons at certain frequencies and wave steepness. The applicability of envelope solitons as design waves as well as their impact on the chemical tanker are compared to classical design wave concepts and tailored extreme wave events. For the extreme wave events, the Peregrine breather solution and a real world abnormal wave reproduction, the famous ‘New Year Wave’, were selected. The investigations comprised model tests as well as numerical simulations. Altogether, this paper shows that envelope solitons are excellent design waves for extreme response investigations suitable for model tests as well as numerical investigations.

Directional Coherent Wave Group From an Assimilated Non-linear Wavefield

The presence of coherent wave groups in the ocean has been so far postulated but still lacks evidence other than the indication from the radar images. Here, we attempt to reconstruct a wave field to monitor the evolution of a directional wave group based on a phase resolving two-dimensional non-linear wave model constrained by the stereo images of the ocean surface. The reconstructed wave field of around 20 wavelength squared revealed a coherent wave group compact in both propagating and transverse directions. The envelope of the wave group seems to be oriented obliquely to the propagation direction, somewhat resembling the directional soliton that was theoretically predicted and experimentally and numerically reproduced recently. A comparison with a constrained linear wave model demonstrated the coherence of the non-linear wave group that propagates for tens of wavelengths. The study elaborates a possibility of a spatially coherent short crested wave group in the directional sea.

Ion acoustic wave group dynamics near critical non-thermal parameter in non-thermal plasmas

Weakly nonlinear ion acoustic wave group dynamics at critical non-thermal parameter is investigated in the frame work of Gardner equation in non-thermal plasmas. Also the wave modulation due to self-interactions near the critical parameter is investigated by a nonlinear Schrodinger equation. The non-thermal parameter modifies the wave dispersion and the parameter space of wave modulation. At the boundary of this parameter space, the wave group dynamics is shown to be governed by a modified nonlinear Schrodinger equation. These equations are solved numerically with the help of a pseudo-spectral scheme and the solutions predict a wide class of nonlinear structures related to wave and wave group dynamics of the weakly nonlinear wave. The numerical solutions and the growth rate of the modified nonlinear Schrodinger equation confirm that the ion acoustic wave is modulationally unstable near the marginal state. The results are discussed in the context of astrophysical observations in auroral region.

Analysis of higher harmonics in a focused water wave group by a nonlinear potential flow model

This work analyses the higher harmonic wave elevations of focused wave groups based on the assumption of a Stokes-type nonlinear structure. A fully nonlinear potential flow model is employed to generate nonlinear wave groups by the NewWave theory, which represents an extreme event in a random sea state. We present a methodology to generate high-quality nonlinear wave groups of a narrow-banded wave spectrum in a numerical wave tank. A phase-manipulation approach is employed to accurately extract the higher harmonic elevations. The elevation spectra show clean separation of the first four harmonics. Comparisons with the experimental data show remarkably good agreements for the higher harmonics. We confirm the Stokes-type underlying nonlinear structure of the harmonic elevations in focused wave groups. This is found by simulating wave groups with varying wave steepness and calculating the corresponding elevation coefficients of the higher harmonics. The harmonic coefficients are found almost constant against varying steepness. An implication of the Stokes-type structure for the nonlinear wave elevations is that, it allows us to estimate the higher harmonics based only on the linear component. This is successfully demonstrated by reconstructing a nonlinear focused wave group using the linear NewWave model and the coefficients at its higher harmonics.

On the near resonances of collinear steady-state wave groups in finite water depth

In this paper, the role nearly resonant components play in collinear steady-state wave groups in finite water depth is investigated theoretically. Fully nonlinear water wave equations are solved when the resonance conditions are nearly satisfied. Convergent solutions are obtained for steady-state wave groups with multiple near resonances. Spectra analysis shows that, as water depth decreases from deep water to finite water depth, the four-wave nearly resonant interactions play a dominant role in nonlinear steady-state wave groups. Meanwhile the dispersion relation changes with the water depth and some low and high frequency components appear in the spectrum due to the three-wave nearly resonant interactions. Comparison between the linear and nonlinear wave groups are made in finite water depth when the mean rates of mass, momentum and energy fluxes are almost conserved. It is found that wave energy in nonlinear wave groups is redistributed as more components join the resonance when the nonlinearity increases. Besides, both the wave group steepness and magnitude of water particle horizontal velocities near the crests and troughs in nonlinear wave groups increase more rapidly with the nonlinearity. The significance of near resonances in finite water depth for collinear steady-state wave groups is demonstrated.

Finite-amplitude steady-state wave groups with multiple near-resonances in finite water depth

Finite-amplitude wave groups with multiple near-resonances are investigated to extend the existing results due to Liu et al. (J. Fluid Mech., vol. 835, 2018, pp. 624–653) from steady-state wave groups in deep water to steady-state wave groups in finite water depth. The slow convergence rate of the series solution in the homotopy analysis method and extra unpredictable high-frequency components in finite water depth make it hard to obtain finite-amplitude wave groups accurately. To overcome these difficulties, a solution procedure that combines the homotopy analysis method-based analytical approach and Galerkin method-based numerical approaches has been used. For weakly nonlinear wave groups, the continuum of steady-state resonance from deep water to finite water depth is established. As nonlinearity increases, the frequency bands broaden and more steady-state wave groups are obtained. Finite-amplitude wave groups with steepness no less than $0.20$ are obtained and the resonant sets configuration of steady-state wave groups are analysed in different water depths. For waves in deep water, the majority of non-trivial components appear around the primary ones due to four-wave, six-wave, eight-wave or even ten-wave resonant interactions. The dominant role of four-wave resonant interactions for steady-state wave groups in deep water is demonstrated. For waves in finite water depth, additional non-trivial high-frequency components appear in the spectra due to three-wave, four-wave, five-wave or even six-wave resonant interactions with the components around the primary ones. The amplitude of these high-frequency components increases further as the water depth decreases. Resonances composed by components only around the primary ones are suppressed while resonances composed by components around the primary ones and from the high-frequency domain are enhanced. The spectrum of steady-state resonant wave groups changes with the water depth and the significant role of three-wave resonant interactions in finite water depth is demonstrated.

Impact of the four-wave quasi-resonance on freak wave shapes in the ocean

Two freak waves were observed a day apart in October 2009 at a 5000-m deep moored station in the northwest Pacific Ocean. As the typhoon passed by, the wave system transitioned within a day from a narrow and unimodal spectrum to the broad and bi-modal spectrum. The occurrence probability of a freak wave is known to increase due to a modulational instability; however, whether the modulational instability survives under a realistic directional sea state has not been conclusively determined yet. In this study, a phase-resolving wave model was used to obtain ensembles of realizations based on observed and simulated directional spectra. Unlike previous studies that focused only on the probability of freak wave occurrence, this study focuses on wave shape. It reveals that the front-to-rear asymmetry and crescent shape deformation of the crest are more pronounced for narrower spectrum and longer-lifetime freak waves; this distortion of wave shape and extended lifetime are both characteristics of nonlinear wave groups. This study also shows that the distribution of the lifetime of a freak wave depends on the sea state and that the number of nonlinear wave groups increases for a narrower spectrum. We therefore conjecture that both the four-wave quasi-resonance and dispersive focusing are responsible for freak wave generation, but their relative significance depends on the spectral broadness. Investigating the total kurtosis or occurrence probability alone is insufficient to unravel the underlying mechanisms of individual freak-wave generation.

Group-wave resonances in nonlinear dispersive media: The case of gravity water waves

The dynamics of coherent nonlinear wave groups is shown to be drastically different from the classical scenario of weakly nonlinear wave interactions. The coherent groups generate nonresonant (bound) waves which can be synchronized with other linear waves. By virtue of the revealed mechanism, the groups may emit waves with similar or different lengths, which propagate in the same or opposite direction.

Mass, momentum, and energy flux conservation between linear and nonlinear steady-state wave groups

This paper provides a mass, momentum, and energy flux conservation analysis between the linear and nonlinear steady-state wave groups. Convergent high-order solutions for nonlinear wave groups with multiple steady-state near resonances in deep water have been obtained by means of the homotopy analysis method. The small divisors associated with nearly resonant components are transformed to singularities that are originally caused by exact resonances by a piecewise auxiliary linear operator. Both two primary components and other nearly resonant ones are considered in the initial guess to search for finite amplitude wave groups. It is found that as nonlinearity of wave groups increases, more wave components appear in the spectrum due to the nearly resonant interactions. The nonlinear wave fields change from the initial bi-chromatic waves that contain only two nontrivial primary components into the steady-state resonant waves that contain both two primary components and other nearly resonant ones. The conservation of mean rates of mass, momentum, and energy fluxes is established between the nonlinear wave groups and linear waves that are combined by two primary components with the same frequencies as in nonlinear wave groups. Comparison of the linear and nonlinear wave fields shows that the nearly resonant components influence the wave field distribution significantly: the nonlinear free surfaces have more peaked crests, steeper troughs, and more flatten wave nodes, and the related velocities at the crests and troughs increase more rapidly with the nonlinearity. All of these findings are helpful to enrich and deepen our understanding about nonlinear wave groups.

Sharp-Crested Breaking Surface Waves Observed from a Ship-Based Stereo Video System

AbstractA new ship-based stereo video system is used to observe breaking ocean waves (i.e., whitecaps) as three-dimensional surfaces evolving in time. First, the stereo video measurements of all waves (breaking and nonbreaking) are shown to compare well with statistical parameters from traditional buoy measurements. Next, the breaking waves are detected based on the presence of whitecap foam, and the geometry of these waves is investigated. The stereo measurements show that the whitecaps are characterized by local extremes of surface slope, though the larger-scale, crest-to-trough steepness of these waves is unremarkable. Examination of 103 breaking wave profiles further demonstrates the pronounced increase in the local wave steepness near the breaking crest, as estimated using a Hilbert transform. These crests are found to closely resemble the sharp corner of the theoretical Stokes limiting wave. Results suggest that nonlinear wave group dynamics are a key mechanism for breaking, as the phase speed of the breaking waves is slower than predicted by the linear dispersion relation. The highly localized and transient steepness, along with the deviation from linear phase speed, explains the inability of conventional wave buoys to observe the detailed geometry of breaking waves.

Modelling of the temporal and spatial evolutions of weakly nonlinear random directional waves with the modified nonlinear Schrödinger equations

The temporal and spatial evolutions of nonlinear wave group with an initial Gaussian envelope are theoretically studied under the governing of MNLS equations, demonstrating that the temporal and spatial versions of numerical model are not always consistent in the whole evolution process, particularly in the presence of strong nonlinearity. Moreover, a large set of numerical simulations, performed respectively by these two versions of numerical model, are systematically compared to mechanically generated waves with different initial directional spreading and Benjamin–Feir Index, mainly focusing on the evolution properties of surface elevations such as the coefficients of skewness and kurtosis, the probability density function, and the maximal surface elevation. On the whole, it can be argued that the statistical properties of both numerically simulated wave fields are basically consistent with the laboratory observations.

Simulation of evolution of gravity wave groups with moderate steepness

The evolution of nonlinear deep-water wave groups in one-horizontal dimension is studied. The strongly nonlinear Green–Naghdi (GN) models are used to simulate this phenomenon. There are different levels of the GN models depending on the different velocity assumption used for the vertical structure of the flow field, such as GN-1, GN-2, and so forth. In this work, we use both the GN-3 and GN-4 models to do the simulations. Calculations are done for three different numbers of waves per group (or packet) as the number of envelope solitons depends on the number of waves per group (N). The numerical results show that the GN-3 model can give the converged GN results for the cases tested here. We conduct a series of physical experiments to investigate the evolution of wave groups. We also use other’s experimental data and present laboratory measurements to compare the data with the predictions of the GN models and show that good agreement is obtained overall.

Nonlinear dynamics of wave-groups in random seas: unexpected walls of water in the open ocean

This paper investigates the size and structure of large waves on the open ocean. We investigate how nonlinear physics modifies waves relative to those predicted by a linear model. We run linear random simulations and extract extreme waves and the surrounding sea-state. For each extreme event, we propagate the waves back in time under linear evolution before propagating the wave-field forward using a nonlinear model. The differences between large linear and nonlinear wave-groups are then examined. The general trends are that under nonlinear evolution, relative to linear evolution, there is, on average, little or no extra amplitude in the nonlinear simulations; that there is an increase in the width of the crest of the wave-group and a contraction of large wave-groups in the mean wave direction; that large waves tend to move to the front of a wave-packet meaning that the locally largest wave is relatively bigger than the wave preceding it; and that nonlinearity can increase the duration of extreme wave events. In all these trends, there is considerable scatter, although the effects observed are clear. Our simulations show that nonlinearity does play an important part in the formation of extreme waves on deep water.

A general boundary-fitted 3D non-hydrostatic model for nonlinear focusing wave groups

This paper employs a three-dimensional (3D) non-hydrostatic model to simulate nonlinear focusing wave groups. The non-hydrostatic model utilizes an explicit projection method to solve the Navier–Stokes equations. To accurately simulate the steep free surface involved in focusing waves, the model is built upon a general boundary-fitted coordinate system. This grid system allows for a great adaptability of the vertical discretization and meanwhile maintains the boundary-fitted properties of better fitting the bed and free surface. The advantage of the general boundary-fitted model is first validated by two test cases of nonlinear waves, including nonlinear standing waves and two-dimensional (2D) focusing freak wave. Then, the model is applied to simulate 2D focusing waves in deep and intermediate-water depths and 3D focusing waves in deep-water depth. By comparing with experimental data, the model results well reproduce the main characteristics of 2D deep-water focusing waves and 2D intermediate-water focusing waves as well as 3D deep-water focusing waves, demonstrating the model׳s capability to resolve 2D or 3D focusing wave groups. Furthermore, in the test of 2D intermediate-water focusing waves, the downstream shifting of the focusing position and time is also studied numerically, which is not presented in the experiments.