On the highest non-breaking wave in a group: fully nonlinear water wave breathers versus weakly nonlinear theory

2013 ◽  
Vol 735 ◽  
pp. 203-248 ◽  
Author(s):  
Alexey V. Slunyaev ◽  
Victor I. Shrira
Keyword(s):  
Water Waves ◽  
Nonlinear Models ◽  
Wave Train ◽  
Initial Perturbation ◽  
Practical Interest ◽  
Breaking Wave ◽  
Wave Group ◽  
Nls Equation ◽  
Fully Nonlinear ◽  
Weakly Nonlinear

AbstractIn nature, water waves usually propagate in groups and the open question about the characteristics of the highest possible wave in a group is of significant theoretical and practical interest. We examine the problem of the highest non-breaking wave in a wave group by direct numerical simulations of the exact Euler equations. The main aim of the study is twofold: (i) to describe the highest wave in a group in fully nonlinear setting and find its dependence on parameters; (ii) to examine correspondence between the exact breather solutions of weakly nonlinear analytic theory based on the integrable nonlinear Schrödinger (NLS) equation and their strongly nonlinear analogues. In contrast to weakly nonlinear models the very notion of the highest wave is ill-defined: the maximal crest elevation, the maximal trough-to-crest height and the deepest trough all occur at close but different moments; correspondingly, we have to speak about distinctively different extreme waves. In the simulations small initial perturbation of a uniform wave train were prescribed in a way ensuring that the initial perturbation excites a single breather-type modulation. The ensuing growth results in higher wave magnitudes and takes longer time to develop compared with the NLS theory. The maxima of crest elevation noticeably exceed their weakly nonlinear analogues. The wave with the highest crest differs significantly from the unmodulated wave: the local wavelength contracts considerably, the crest becomes noticeably higher; the vicinity of the crest of such an extreme wave is close to that of the limiting Stokes periodic wave. Thus, the shape of the maximal crest wave is almost universal, i.e. it practically does not depend on the way the wave group evolved, or even whether there was initially more than one group. The evolution of a single NLS breather has been shown to have a qualitatively similar but quantitatively quite different analogue in the fully nonlinear setting. The one-to-one mapping of the NLS breather solutions onto fully nonlinear ones has been constructed. The fully nonlinear breathers are found to be robust, which provides grounds for applying the results for developing short-term deterministic forecasting of rogue waves.

Fully nonlinear internal waves in a two-fluid system

1999 ◽  
Vol 396 ◽  
pp. 1-36 ◽  
Author(s):  
WOOYOUNG CHOI ◽  
ROBERTO CAMASSA
Keyword(s):  
Solitary Wave ◽  
Euler Equations ◽  
Deep Water ◽  
Large Amplitude ◽  
Nonlinear Models ◽  
Weak Nonlinearity ◽  
Fully Nonlinear ◽  
Weakly Nonlinear ◽  
The Waves ◽  
Two Fluid

Model equations that govern the evolution of internal gravity waves at the interface of two immiscible inviscid fluids are derived. These models follow from the original Euler equations under the sole assumption that the waves are long compared to the undisturbed thickness of one of the fluid layers. No smallness assumption on the wave amplitude is made. Both shallow and deep water configurations are considered, depending on whether the waves are assumed to be long with respect to the total undisturbed thickness of the fluids or long with respect to just one of the two layers, respectively. The removal of the traditional weak nonlinearity assumption is aimed at improving the agreement with the dynamics of Euler equations for large-amplitude waves. This is obtained without compromising much of the simplicity of the previously known weakly nonlinear models. Compared to these, the fully nonlinear models' most prominent feature is the presence of additional nonlinear dispersive terms, which coexist with the typical linear dispersive terms of the weakly nonlinear models. The fully nonlinear models contain the Korteweg–de Vries (KdV) equation and the Intermediate Long Wave (ILW) equation, for shallow and deep water configurations respectively, as special cases in the limit of weak nonlinearity and unidirectional wave propagation. In particular, for a solitary wave of given amplitude, the new models show that the characteristic wavelength is larger and the wave speed is smaller than their counterparts for solitary wave solutions of the weakly nonlinear equations. These features are compared and found in overall good agreement with available experimental data for solitary waves of large amplitude in two-fluid systems.

scholarly journals Water waves, nonlinear Schrödinger equations and their solutions

1983 ◽  
Vol 25 (1) ◽  
pp. 16-43 ◽  
Author(s):  
D. H. Peregrine
Keyword(s):  
Water Waves ◽  
Rational Solution ◽  
Three Dimensional ◽  
Length Scale ◽  
Nls Equation ◽  
Nonlinear Water Waves ◽  
Weakly Nonlinear ◽  
Nonlinear Schrödinger ◽  
Wave Induced ◽  
Modulation Length

AbstractEquations governing modulations of weakly nonlinear water waves are described. The modulations are coupled with wave-induced mean flows except in the case of water deeper than the modulation length scale. Equations suitable for water depths of the order the modulation length scale are deduced from those derived by Davey and Stewartson [5] and Dysthe [6]. A number of ases in which these equations reduce to a one dimensional nonlinear Schrödinger (NLS) equation are enumerated.Several analytical solutions of NLS equations are presented, with discussion of some of their implications for describing the propagation of water waves. Some of the solutions have not been presented in detail, or in convenient form before. One is new, a “rational” solution describing an “amplitude peak” which is isolated in space-time. Ma's [13] soli ton is particularly relevant to the recurrence of uniform wave trains in the experiment of Lake et al.[10].In further discussion it is pointed out that although water waves are unstable to three-dimensional disturbances, an effective description of weakly nonlinear two-dimensional waves would be a useful step towards describing ocean wave propagation.

scholarly journals Modelling nonlinear electrohydrodynamic surface waves over three-dimensional conducting fluids

2017 ◽  
Vol 473 (2200) ◽  
pp. 20160817 ◽  
Author(s):  
Zhan Wang
Keyword(s):  
Surface Tension ◽  
Free Surface ◽  
Electric Field ◽  
Nonlinear Models ◽  
Three Dimensional ◽  
Separation Distance ◽  
Nls Equation ◽  
Fully Nonlinear ◽  
Special Cases ◽  
Kp Equations

The evolution of the free surface of a three-dimensional conducting fluid in the presence of gravity, surface tension and vertical electric field due to parallel electrodes, is considered. Based on the analysis of the Dirichlet–Neumann operators, a series of fully nonlinear models is derived systematically from the Euler equations in the Hamiltonian framework without assumptions on competing length scales can therefore be applied to systems of arbitrary fluid depth and to disturbances with arbitrary wavelength. For special cases, well-known weakly nonlinear models in shallow and deep fluids can be generalized via introducing extra electric terms. It is shown that the electric field has a great impact on the physical system and can change the qualitative nature of the free surface: (i) when the separation distance between two electrodes is small compared with typical wavelength, the Boussinesq, Benney–Luke (BL) and Kadomtsev–Petviashvili (KP) equations with modified coefficients are obtained, and electric forces can turn KP-I to KP-II and vice versa; (ii) as the parallel electrodes are of large separation distance but the thickness of the liquid is much smaller than typical wavelength, we generalize the BL and KP equations by adding pseudo-differential operators resulting from the electric field; (iii) for a quasi-monochromatic plane wave in deep fluid, we derive the cubic nonlinear Schrödinger (NLS) equation, but its type (focusing or defocusing) is strongly influenced by the value of the electric parameter. For sufficient surface tension, numerical studies reveal that lump-type solutions exist in the aforementioned three regimes. Particularly, even when the associated NLS equation is defocusing for a wave train, lumps can exist in fully nonlinear models.

Phenomena on Rogue Waves and Rational Solution of Korteweg-de Vries Equation

2013 ◽  
Author(s):  
Ni Song ◽  
Wei Zhang ◽  
Qian Wang
Keyword(s):  
Shallow Water ◽  
Water Waves ◽  
Kdv Equation ◽  
Rational Solution ◽  
Rogue Waves ◽  
Nls Equation ◽  
Engineering Structures ◽  
Weakly Nonlinear ◽  
Nonlinear Mechanism ◽  
De Vries

An appropriate nonlinear mechanism may create the rogue waves. Perhaps the simplest mechanism, which is able to create considerate changes in the wave amplitude, is the nonlinear interaction of shallow-water solitons. The most well-known examples of such structure are Korteweg-de Vries (KdV) solitons. The Korteweg-de Vries (KdV) equation, which describes the shallow water waves, is a basic weakly dispersive and weakly nonlinear model. Basing on the homogeneous balanced method, we achieve the general rational solution of a classical KdV equation. Numerical simulations of the solution allow us to explain rare and unexpected appearance of the rogue waves. We compare the rogue waves with the ones generated by the nonlinear Schrödinger (NLS) equation which can describe deep water wave trains. The numerical results illustrate that the amplitude of the KdV equation is higher than the one of the NLS equation, which may causes more serious damage of engineering structures in the ocean. This nonlinear mechanism will provide a theoretical guidance in the ocean and physics.

scholarly journals Laboratory measurements of deep-water breaking waves

1990 ◽  
Vol 331 (1622) ◽  
pp. 735-800 ◽  
Keyword(s):  
Deep Water ◽  
Water Waves ◽  
Phase Speed ◽  
Stationary Flow ◽  
Breaking Waves ◽  
Return Flow ◽  
Breaking Wave ◽  
Wave Group ◽  
Wave Channel ◽  
Wave Heights

The results of laboratory experiments on unsteady deep-water breaking waves are reported. The experiments exploit the dispersion of deep-water waves to generate a single breaking wave group. The direct effects of breaking are then confined to a finite region in the wave channel and the influence of breaking on the evolution of the wave field can be examined by measuring fluxes into and out of the breaking region. This technique was used by us in a preliminary series of measurements. The loss of excess momentum flux and energy flux from the wave group was measured and found to range from 10% for single spilling events to as much as 25% for plunging breakers. Mixing due to breaking was studied by photographing the evolution of a dye patch as it was mixed into the water column. It was found that the maximum depth of the dye cloud grew linearly in time for one to two wave periods, and then followed a t 1/4 power law (t is the time from breaking) over a range of breaking intensities and scales. The dyed region reached depths of two to three wave heights and horizontal lengths of approximately one wavelength within five wave periods of breaking. A detailed velocity survey of the breaking region was made and ensemble averages taken of the non-stationary flow. Mean surface currents in the range 0.02-0.03 C (C is the characteristic phase speed) were generated and took as many as 60 wave periods to decay to 0.005 C. A deeper return flow due to momentum lost from the forced long wave was measured. Together these flows gave a rotational region of approximately one wavelength. Turbulent root mean square velocities of approximately 0.02 C were measured near the surface and were still significant at depths of three to four wave heights. More than 90 % of the energy lost from the waves was dissipated within four wave periods. Subsequently measured kinetic energy in the residual flow was found to have a t -1 dependence. Correlation of all the above measurements with the amplitude, bandwidth and phase of the wave group was found to be good, as was scaling of the results with the centre frequency of the group,. Local measures of the breaking wave were not found to correlate well with the dynamical measurements.

On Modelling Kinematics of Steep Irregular Seaway and Freak Waves

2008 ◽  
Author(s):  
Gu¨nther F. Clauss ◽  
Florian Stempinski ◽  
Robert Stu¨ck
Keyword(s):  
Water Waves ◽  
Wave Train ◽  
Breaking Waves ◽  
Breaking Wave ◽  
Coupling Mechanism ◽  
Fast Method ◽  
Freak Waves ◽  
Irregular Wave ◽  
Wave Kinematics ◽  
Stokes Waves

The realistic modelling of velocity and pressure fields in steep, irregular seaways is still a challenging task, especially when extreme events such as freak waves are under investigation. Conventional wave theories provide fast and reliable results while CFD-codes based on RANSE or potential theory are gaining more acceptance for simulating water waves even though they are still considerably time consuming. This paper presents an approach to approximate irregular wave trains with known surface elevations by interacting Stokes waves of up to third order. This is a fast method to determine the wave potential of wind generated waves for long lasting wave registrations with arbitrary origin. The technique is applied to a steep breaking wave package as well as to a realization of a wave train in a wave tank (scale 1:120) which contain a measured extreme wave sequence. Here, special attention is paid to the distinction between the kinematics of the wave crests in extremely high waves and their surrounding irregular wave field. The predicted wave kinematics are validated by experiments employing particle velocity measurements (by Laser Doppler Velocimetry) as well as by pressure recordings. Kinematics of breaking waves are not covered by concurrent analytical wave theories. To address this deficiency a coupling mechanism between a conventionally determined velocity field with a RANSE/VoF-method is applied.

scholarly journals The fully nonlinear stratified geostrophic adjustment problem

2016 ◽  
Author(s):  
A. Coutino ◽  
M. Stastna
Keyword(s):  
Wave Train ◽  
Initial Perturbation ◽  
Classical Problem ◽  
Point Of View ◽  
Rossby Number ◽  
Adjustment Problem ◽  
Inertial Oscillations ◽  
Number Range ◽  
Fully Nonlinear ◽  
Rotation Rates

Abstract. The study of adjustment to equilibrium is a classical problem in geophysical fluid dynamics. We consider the fully nonlinear, stratified adjustment problem from a numerical point of view. We present results of smooth dam break simulations based on experiments in the published literature. We focus on both the wave trains that propagate away from the nascent geostrophic state and the geostrophic state itself. For the Rossby number range considered the rank ordered solitary wave train of the non rotating adjustment problem breaks down into a leading packet-like disturbance and a trailing tail. We consider variations in Rossby number and demonstrate that the wave train emanates from the inertial oscillations of the geostrophic state itself, but that the precise phase of the oscillation that yields the wave train is Rossby number dependent. We quantify the strength of the inertial oscillations of the geostrophic state and find results that are in agreement with hydrostatic theory in the literature. We consider the effects of changes in the polarity of the initial perturbations and find that the leading wave packet never completely separates from the trailing tail in this case. Finally we demonstrate that both polarities yield a unique signature in the spectrum of the depth averaged kinetic energy. This signature is fundamentally different from that found in non-rotating cases, and allows for low rotation rates allows for the discrimination between the polarity of the initial perturbation.

scholarly journals Bound Coherent Structures Propagating on the Free Surface of Deep Water

Fluids ◽  
2021 ◽  
Vol 6 (3) ◽  
pp. 115
Author(s):  
Dmitry Kachulin ◽  
Sergey Dremov ◽  
Alexander Dyachenko
Keyword(s):  
Free Surface ◽  
Deep Water ◽  
Coherent Structures ◽  
Water Waves ◽  
Nonlinear Models ◽  
Ideal Incompressible Fluid ◽  
Equation Model ◽  
System Of Nonlinear Equations ◽  
Potential Flows ◽  
Deep Water Waves

This article presents a study of bound periodically oscillating coherent structures arising on the free surface of deep water. Such structures resemble the well known bi-soliton solution of the nonlinear Schrödinger equation. The research was carried out in the super-compact Dyachenko-Zakharov equation model for unidirectional deep water waves and the full system of nonlinear equations for potential flows of an ideal incompressible fluid written in conformal variables. The special numerical algorithm that includes a damping procedure of radiation and velocity adjusting was used for obtaining such bound structures. The results showed that in both nonlinear models for deep water waves after the damping is turned off, a periodically oscillating bound structure remains on the fluid surface and propagates stably over hundreds of thousands of characteristic wave periods without losing energy.

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