Formation Of Extreme Waves Research Articles

Laboratory observation of nonlinear wave shapes due to spatial varying opposing currents

Physical experiments were conducted to examine the influence of adverse currents on the propagation of shallower water waves and their impact on the evolution of nonlinear wave shapes. Irregular waves, characterized by varying initial peak periods and amplitudes, were generated in a physical wave flume equipped with a bottom slope of 1/20. Three groups of spatially varying opposing currents were generated in the flume and interacted with the aforementioned wave trains. Experimental results confirm that strong opposing currents can significantly intensify local wave nonlinearity, thereby further influencing the deformation characteristics of waves. In contrast, a weak opposing current (without wave-blocking) has a negligible effect. Bicoherence analysis revealed that the degree of phase coupling among triads of waves increased with an elevated current velocity before wave-blocking. Nevertheless, during partial wave-blocking, the degree of phase coupling was seemingly weakened by an increase in opposing current. Moreover, the influence of a co-existing current in the formation of extreme waves was recognized. The study confirms that adverse currents notably affect extreme wave steepness and extreme wave skewness, with a negligible impact on extreme wave asymmetry. As a result, empirical formulas modified by current effects are presented, describing certain key nonlinear wave shapes as functions of the local Ursell number.

A Study of Extreme Water Waves Using a Hierarchy of Models Based on Potential-Flow Theory

The formation of extreme waves arising from the interaction of three line-solitons with equal far-field amplitudes is examined through a hierarchy of water-wave models. The Kadomtsev–Petviashvili equation (KPE) is first used to prove analytically that its exact three-soliton solution has a ninefold maximum amplification that is achieved in the absence of spatial divergence. Reproducing this ninefold maximum paves the way for simulations based on both the Benney–Luke equations (BLE) and more advanced potential-flow equations (PFE). To preserve (for the sake of computations) the region of interaction, exact KPE solutions on an infinite domain are used to yield initial conditions that seed the BLE and PFE models within a periodic domain. The above strategies are realised by directly implementing the corresponding time-discretised variational principles within the finite-element environment Firedrake, one aim being automation of the generation of the algebraically cumbersome weak formulations. In the case of three-soliton interactions, it is found numerically that an amplification factor in the interval circa (7.6, 9) can be achieved within the BLE framework, whereas in the PFE framework, this falls to circa 7.8.

Extreme wave formation past a submerged circular sill: The role of linear spatial focusing and resonant trapping

We investigate dispersive wave amplification past a submerged circular sill on an otherwise flat seabed. This phenomenon is important because it can generate large-amplitude waves near the sill, due to spatial focusing and resonant trapping in linear dispersive regime, endangering navigation. Based on the potential flow theory, the velocity potential is solved separately in the ocean region and in the sill region. Matching is then achieved by means of integral equations involving Galerkin expansion of the unknown velocity field at the border between the two regions. The model is successfully validated against known analytical expressions for long waves and a smoothed particle hydrodynamics numerical solution. Our results advance existing theories valid either for non-dispersive waves or for shallow submergence of the sill. We show that, for relatively short waves as compared to the ocean depth, the sill acts as a wave lens focusing energy behind it. Increasing the wavelength of the incident wave promotes transition from wave focusing behind the sill to partial trapping atop the sill. In intermediate water depth, the concurrence of focusing and partial trapping favors the emergence of extreme wave amplitudes that can exceed up to 6 times the amplitude of incident waves. We hypothesize that this phenomenon is the main cause of local peaks in skewness and kurtosis near a submerged circular shoal obtained in recent numerical simulations. Indications for further studies in the nonlinear regime are finally provided.

Excitation of an extreme wave by standing current

The statistics suggest that extreme waves cause more damage in shallow waters and at the coast than in the deep sea. In the linear theory of the formation of extreme waves, their existence is interpreted as a local superposition of surface monochromatic waves. The event of excitation of extreme waves can be understood as an increase in natural oscillations of the water basin. The conditions for the excitation and sustaining of natural oscillations are the proximity of the periods of exciting traveling waves to the period of traveling waves and the speed of movement of the exciting current to the phase speed of propagation of traveling waves of the reservoir. Examples of stimulating natural oscillations are presented. We determined the range of expected periods of natural oscillations, which range from 30 seconds to 24 hours. Synchronously and in common-mode with the oscillations of standing waves between their antinodes, a "standing" current occurs with a measured speed of up to 11 km/h. We presented a hypothesis about the possibility of stimulating natural oscillations of water bodies by a standing current, which changes its direction due to the movement of the water surface from the trough of the wave to its crest, and back. A model of stimulating oscillations by the waves with a constant period and currents with constant and variable speeds has been developed.

Experimental study of extreme waves based on nonlinear Schrödinger equation under background of a random sea

The Peregrine breather (PB) solution of the nonlinear Schrödinger equation is used to model ocean extreme waves in a water wave flume. Triangular spectral features of wave elevations are observed over the nonlinear evolution of the extreme waves, which can be applied for early detection of the formation of extreme waves. To model a more realistic sea state, a background random wave is superposed to the PB in this study. We examine the spectral features of the nonlinear wave evolution with random background waves by the spectral analysis. It is found that the wave elevations show similar triangular spectral features for extreme waves with a relatively mild background wave. Moreover, we find that the second harmonic elevations of the extreme waves also show triangular spectral features, suggesting the potential use of the second harmonic elevations (in addition to the first) for detection of the formation of extreme waves.

Experimental Evidence of Nonlinear Focusing in Standing Water Waves.

Nonlinear wave focusing originating from the universal modulation instability (MI) is responsible for the formation of strong wave localizations on the water surface and in nonlinear wave guides, such as optical Kerr media and plasma. Such extreme wave dynamics can be described by breather solutions of the nonlinear Schrödinger equation (NLSE) like by way of example the famed doubly-localized Peregrine breathers (PB), which typify particular cases of MI. On the other hand, it has been suggested that the MI relevance weakens when the wave field becomes broadband or directional. Here, we provide experimental evidence of nonlinear and distinct PB-type focusing in standing water waves describing the scenario of two counterpropagating wave trains. The collected collinear wave measurements are in excellent agreement with the hydrodynamic coupled NLSE (CNLSE) and suggest that MI can undisturbedly prevail during the interplay of several wave systems and emphasize the potential role of exact NLSE solutions in extreme wave formation beyond the formal narrow band and unidirectional limits. Our work may inspire further experimental investigations in various nonlinear wave guides governed by CNLSE frameworks as well as theoretical progress to predict strong wave coherence in directional fields.

On the extreme value statistics of spatio-temporal maximum sea waves under cyclone winds

Principles of the spatio-temporal statistics are used to investigate the characteristics of short-term/range extreme sea waves and related sea-state parameters under cyclone winds (northern hemisphere). We base our analysis upon consistent stereo-imaging observations of the 3D (2D space + time) sea surface elevation field, and spectral wave model results in the Northwestern Pacific during tropical storm Kong-rey (2018). The focus is on the extreme value analysis of individual maximum sea surface elevations (crest heights) and maximum crest-to-trough wave heights. Results highlight the sea areas around the storm centre where the spatio-temporal highest waves are more likely, and, via scale analysis, the principal mechanisms responsible for the occurrence of extreme conditions in bimodal (composed of wind-sea and swell) and short-crested storm seas. We find that individual waves are the highest to the north-east of the translating cyclone, where sea states are more energetic. However, in the south/south-west of the centre, where opposing wind-sea/swell sea states dominate, directional spread and bound nonlinear interactions are enhanced. In this area, more extreme waves may occur, having the maximum crest and wave heights mean values in excess of 1.3 and 2.1 times the significant wave height, respectively. This set of results provides insights into the role of the dispersive and directional focusing enhanced by nonlinearities up to the second order as an effective mechanism for the formation of extreme waves under cyclone winds. To examine what physical mechanism is behind the generation of extreme waves in different regions around the cyclone, we also explore for comparison areas where nonlinear four-wave interactions are more likely to occur.

ЗБУДЖЕННЯ ХВИЛІ-УБИВЦІ ВЛАСНИМИ КОЛИВАННЯМИ ВОДНОГО БАСЕЙНУ

In linear theory the formation of extreme waves their existence is interpreted as a local superposition of surface monochromatic waves. Natural water areas are resonators that have their own set of natural oscillations – standing waves of stable spatial structure and fixed period. In the spectra of waves of many water bodies of World Ocean observed double high waves, this is explained by the tidal-seiche resonance. During a storm, the energy of natural oscillations increases ten times the background energy, during a tsunami it can increase up to three orders of magnitude. Examples of the effects of natural oscillations on the coast are given, and it is reported about the increased probability of the occurrence on the coast freak waves. Additionally, it is noted that natural oscillations in water mass are a normal state for any body of water at any time of its existence. The corresponding indices of the water fluctuations of the water basins are given. The events of extreme waves during the accidents at DniproHES (Zaporizhia) on August 18, 1941, and the Kurenivsky dam (Kyiv) on March 13, 1961, are presented. The excitement of the freak wave can be interpreted as enhancing the natural oscillations of the water basin, represented by standing waves of stable spatial structure, fixed period and high probability of waves in the water body. This does not contradict the linear theory of the resonant formation of abnormally high waves. The purpose of the article is to investigate possible sources of the excitement of freak waves, the results are proposed to be implemented in the development of countermeasures to the destructive process. However, the waves carry out both destructive and creative work. A task is presented, which involves the development of measures to stimulate extreme waves. This will increase electricity generation. Affiliation of dam-break waves to freak waves can be doubtful. However, they formally correspond to the classical condition of double exceeding the significant wave height. Most water basins are integral anthropogenic sites. The variability of both natural and anthropogenic environments forces the overriding of systematization and definition. It is proposed to attribute extreme waves of dam-break waves to freak waves.

Experimental study of breathers and rogue waves generated by random waves over non-uniform bathymetry

We present experimental evidence of formation and persistence of localized waves, breathers, and solitons, occurring in a random sea state and uniformly traveling over non-uniform bathymetry. Recent studies suggest connections between breather dynamics and irregular sea states and between extreme wave formation and breathers, random sea states, or non-uniform bathymetry individually. In this paper, we investigate the joint connection between these phenomena, and we found that breathers and deep-water solitons can persist in more complex environments. Three different sets of significant heights have been generated within a Joint North Sea Wave Observation Project wave spectrum, and the wave heights were recorded with gauges in a wave tank. Statistical analysis was applied to the experimental data, including the space and time distribution of kurtosis, skewness, Benjamin–Feir index, moving Fourier spectra, and probability distribution of wave heights. Stable wave packages formed out of the random wave field and traveling over shoals, valleys, and slopes were compared with exact solutions of the nonlinear Schrödinger equation with a good match, demonstrating that these localized waves have the same structure as deep-water breathers. We identify the formation of rogue waves at moments and over regions where the kurtosis and skewness have local maxima. These results provide insights for understanding of the robustness of Peregrine and higher-order Akhmediev breathers, Kuznetsov–Ma solitons, and rogue waves, and their occurrence in realistic oceanic conditions, and may motivate analogous studies in other fields of physics to identify limitations of exact weakly nonlinear models in non-homogeneous media.

Statistics of Extreme Waves in Coastal Waters: Large Scale Experiments and Advanced Numerical Simulations

The formation mechanism of extreme waves in the coastal areas is still an open contemporary problem in fluid mechanics and ocean engineering. Previous studies have shown that the transition of water depth from a deeper to a shallower zone increases the occurrence probability of large waves. Indeed, more efforts are required to improve the understanding of extreme wave statistics variations in such conditions. To achieve this goal, large scale experiments of unidirectional irregular waves propagating over a variable bottom profile considering different transition water depths were performed. The validation of two highly nonlinear numerical models was performed for one representative case. The collected data were examined and interpreted by using spectral or bispectral analysis as well as statistical analysis. The higher probability of occurrence of large waves was confirmed by the statistical distributions built from the measured free surface elevation time series as well as by the local maximum values of skewness and kurtosis around the end of the slope. Strong second-order nonlinear effects were highlighted as waves propagate into the shallower region. A significant amount of wave energy was transmitted to low-frequency modes. Based on the experimental data, we conclude that the formation of extreme waves is mainly related to the second-order effect, which is also responsible for the generation of long waves. It is shown that higher-order nonlinearities are negligible in these sets of experiments. Several existing models for wave height distributions were compared and analysed. It appears that the generalised Boccotti’s distribution can predict the exceedance of large wave heights with good confidence.

Experimental investigation of weakly three-dimensional nonlinear wave interactions

Abstract The properties of surface waves coalescing and spreading directionally is presented where two identical wave trains propagate at an angle to each other and hence interact in an “X” configuration ranging from near-linear to violent breaking. An event of this type is believed to be one of the foundations of the formation of extreme waves in the open sea. The study involved measuring the wave surface elevation along the channels and hence both frequency spectra and energy were calculated. Comparisons between these data confirm that the directional spread has a significant effect on the nonlinear wave–wave interactions, especially for the wave profile and energy loss due to breaking. In general, the greater the approach angle of the two wave trains, the more cross-channel variation there is following breaking and the larger the energy loss during breaking. It is believed that the observed results will provide a helpful preliminary study of full three-dimensional interactions.

A Note on the Second-Order Contribution to Extreme Waves Generated During Hurricanes

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.

Extreme wave formation in unidirectional sea due to stochastic wave phase dynamics

The authors consider a stochastic model based on the interaction and phase coupling amongst wave components that are modified envelope soliton solutions to the nonlinear Schrödinger equation. A probabilistic study is carried out and the resulting findings are compared with ocean wave field observations and laboratory experimental results. The wave height probability distribution obtained from the model is found to match well with prior data in the large wave height region. From the eigenvalue spectrum obtained through the Inverse Scattering Transform, it is revealed that the deep-water wave groups move at a speed different from the linear group speed, which justifies the inclusion of phase correction to the envelope solitary wave components. It is determined that phase synchronization amongst elementary solitary wave components can be critical for the formation of extreme waves in unidirectional sea states.

Wind and Current Effects on Extreme Wave Formation and Breaking

AbstractWind and current effects on the evolution of a two-dimensional dispersive focusing wave group are investigated using a two-phase flow model. A Navier–Stokes solver is combined with the Smagorinsky subgrid-scale stress model and volume of fluid (VOF) air–water interface capturing scheme. Model predictions compare well with the experimental data with and without wind. It was found that the following and opposing winds shift the focus point downstream and upstream, respectively. The shift of focus point is mainly due to the action of wind-driven current instead of direct wind forcing. Under strong following/opposing wind forcing, there appears a slight increase/decrease of the extreme wave height at the focus point and an asymmetric/symmetric behavior in the wave focusing and defocusing processes. Under a weak following wind, however, the extreme wave height decreases with increasing wind speed because of the dominant effect of the wind-driven current over direct wind forcing. The vertical shear of the wind-driven current plays an important role in determining the location of and the extreme wave height at the focus point under wind actions. Furthermore, it was found that the thin surface layer current is a better representation of the wind-driven current for its role in wind influences on waves than the depth-uniform current used by previous studies. Airflow structure above a breaking wave group and its link to the energy flux from wind to wave as well as wind influence on breaking are also examined. The flow structure in the presence of a following wind is similar to that over a backward-facing step, while that in the presence of an opposing wind is similar to that over an airfoil at high angles of attack. Both primary and secondary vortices are observed over the breaking wave with and without wind of either direction. Airflow separates over the steep crest and causes a pressure drop in the lee of the crest. The resulting form drag may directly affect the extreme wave height. The wave breaking location and intensity are modified by the following and opposing wind in a different fashion.

Ocean rogue waves and their phase space dynamics in the limit of a linear interference model.

We reanalyse the probability for formation of extreme waves using the simple model of linear interference of a finite number of elementary waves with fixed amplitude and random phase fluctuations. Under these model assumptions no rogue waves appear when less than 10 elementary waves interfere with each other. Above this threshold rogue wave formation becomes increasingly likely, with appearance frequencies that may even exceed long-term observations by an order of magnitude. For estimation of the effective number of interfering waves, we suggest the Grassberger-Procaccia dimensional analysis of individual time series. For the ocean system, it is further shown that the resulting phase space dimension may vary, such that the threshold for rogue wave formation is not always reached. Time series analysis as well as the appearance of particular focusing wind conditions may enable an effective forecast of such rogue-wave prone situations. In particular, extracting the dimension from ocean time series allows much more specific estimation of the rogue wave probability.

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.

Polarization modulation instability in a Manakov fiber system

The Manakov model is the simplest multicomponent model of nonlinear wave theory: It describes elementary stable soliton propagation and multisoliton solutions, and it applies to nonlinear optics, hydrodynamics, and Bose-Einstein condensates. It is also of fundamental interest as an asymptotic model in the context of the widely used wavelength-division-multiplexed optical fiber transmission systems. However, although its physical relevance was confirmed by the experimental observation of Manakov (vector) solitons in a planar waveguide in 1996, there have in fact been no quantitative experiments confirming its validity for nonlinear dynamics other than soliton formation. Here, we report experiments in optical fiber that provide evidence of passband and baseband polarization modulation instabilities in a defocusing Manakov system. In the spontaneous regime, we also reveal a unique saturation effect as the pump power increases. We anticipate that such observations may impact the application of this minimal model to describe and understand more complicated phenomena in nature, such as the formation of extreme waves in multicomponent systems.

Seasonal characteristics of waves in the southeastern part of the Baltic Sea in 2008–2009

Presented are the characteristics of waves in the southeastern part of the Baltic Sea obtained from the results of continuous instrumental observations in 2008–2009 on the offshore oil-and-gas platform. Discussed are the conditions and prerequisites for the formation of extreme waves.

Deep-Water Waves: on the Nonlinear Schrödinger Equation and its Solutions

We present a brief discussion on the nonlinear Schrodinger equation for modelling the propagation of the deep-water wavetrains and a discussion on its doubly-localized breather solutions, that can be con- nected to the sudden formation of extreme waves, also known as rogue

Linear and nonlinear rogue wave statistics in the presence of random currents*** This invited article is one of a series published throughout 2011 in Nonlinearity on the topic of Rogue Waves.

We review recent progress in modelling the probability distribution of wave heights in the deep ocean as a function of a small number of parameters describing the local sea state. Both linear and nonlinear mechanisms of rogue wave formation are considered. First, we show that when the average wave steepness is small and nonlinear wave effects are subleading, the wave height distribution is well explained by a single ‘freak index’ parameter, which describes the strength of (linear) wave scattering by random currents relative to the angular spread of the incoming random sea. When the average steepness is large, the wave height distribution takes a very similar functional form, but the key variables determining the probability distribution are the steepness, and the angular and frequency spread of the incoming waves. Finally, even greater probability of extreme wave formation is predicted when linear and nonlinear effects are acting together.