On the modelling of huge water waves called rogue waves

2007 ◽  
pp. 113-134
Author(s):  
Christian Kharif
Keyword(s):  
Water Waves ◽  
Rogue Waves

scholarly journals Super Rogue Waves: Observation of a Higher-Order Breather in Water Waves

2012 ◽  
Vol 2 (1) ◽  
Author(s):  
A. Chabchoub ◽  
N. Hoffmann ◽  
M. Onorato ◽  
N. Akhmediev
Keyword(s):  
Water Waves ◽  
Rogue Waves ◽  
Higher Order

Breather wave, rogue wave and lump wave solutions for a (3+1)-dimensional generalized Kadomtsev–Petviashvili equation in fluid

2018 ◽  
Vol 32 (20) ◽  
pp. 1850223 ◽  
Author(s):  
Ming-Zhen Li ◽  
Bo Tian ◽  
Yan Sun ◽  
Xiao-Yu Wu ◽  
Chen-Rong Zhang
Keyword(s):  
Wave Velocity ◽  
Water Waves ◽  
Rogue Wave ◽  
Rogue Waves ◽  
Dispersion Effect ◽  
Wave Solutions ◽  
Weak Perturbation ◽  
Petviashvili Equation ◽  
Nonlinearity Effect ◽  
Lump Wave

Under investigation in this paper is a (3[Formula: see text]+[Formula: see text]1)-dimensional generalized Kadomtsev–Petviashvili equation, which describes the long water waves and small-amplitude surface waves with the weak nonlinearity, weak dispersion and weak perturbation in a fluid. Via the Hirota method and symbolic computation, the lump wave, breather wave and rogue wave solutions are obtained. We graphically present the lump waves under the influence of the dispersion effect, nonlinearity effect, disturbed wave velocity effects and perturbed effects: Decreasing value of the dispersion effect can lead to the range of the lump wave decreases, but has no effect on the amplitude. When the value of the nonlinearity effect or disturbed wave velocity effects increases respectively, lump wave’s amplitude decreases but lump wave’s location keeps unchanged. Amplitudes of the lump waves are independent of the perturbed effects. Breather waves and rogue waves are displayed: Rogue waves emerge when the periods of the breather waves go to the infinity. When the value of the dispersion effect decreases, range of the rogue wave increases. When the value of the nonlinearity effect or disturbed wave velocity effects decreases respectively, rogue wave’s amplitude decreases. Value changes of the perturbed effects cannot influence the rogue wave.

Characteristics of solitary waves, quasiperiodic solutions, homoclinic breather solutions and rogue waves in the generalized variable-coefficient forced Kadomtsev–Petviashvili equation

2017 ◽  
Vol 31 (36) ◽  
pp. 1750350 ◽  
Author(s):  
Xue-Wei Yan ◽  
Shou-Fu Tian ◽  
Min-Jie Dong ◽  
Li Zou
Keyword(s):  
Solitary Waves ◽  
Water Waves ◽  
Variable Coefficient ◽  
Wave Equations ◽  
Direct Method ◽  
Dynamical Behavior ◽  
Rogue Waves ◽  
Nonlinear Wave Equations ◽  
Quasiperiodic Solutions ◽  
Petviashvili Equation

In this paper, the generalized variable-coefficient forced Kadomtsev–Petviashvili (gvcfKP) equation is investigated, which can be used to characterize the water waves of long wavelength relating to nonlinear restoring forces. Using a dependent variable transformation and combining the Bell’s polynomials, we accurately derive the bilinear expression for the gvcfKP equation. By virtue of bilinear expression, its solitary waves are computed in a very direct method. By using the Riemann theta function, we derive the quasiperiodic solutions for the equation under some limitation factors. Besides, an effective way can be used to calculate its homoclinic breather waves and rogue waves, respectively, by using an extended homoclinic test function. We hope that our results can help enrich the dynamical behavior of the nonlinear wave equations with variable-coefficient.

scholarly journals Super compact equation for water waves

2017 ◽  
Vol 828 ◽  
pp. 661-679 ◽  
Author(s):  
A. I. Dyachenko ◽  
D. I. Kachulin ◽  
V. E. Zakharov
Keyword(s):  
Wave Equation ◽  
Water Waves ◽  
Wave Breaking ◽  
Water Wave ◽  
Rogue Waves ◽  
Resonant Interaction ◽  
Zakharov Equation ◽  
Advection Term ◽  
New Equation ◽  
The Many

Mathematicians and physicists have long been interested in the subject of water waves. The problems formulated in this subject can be considered fundamental, but many questions remain unanswered. For instance, a satisfactory analytic theory of such a common and important phenomenon as wave breaking has yet to be developed. Our knowledge of the formation of rogue waves is also fairly poor despite the many efforts devoted to this subject. One of the most important tasks of the theory of water waves is the construction of simplified mathematical models that are applicable to the description of these complex events under the assumption of weak nonlinearity. The Zakharov equation, as well as the nonlinear Schrödinger equation (NLSE) and the Dysthe equation (which are actually its simplifications), are among them. In this article, we derive a new modification of the Zakharov equation based on the assumption of unidirectionality (the assumption that all waves propagate in the same direction). To derive the new equation, we use the Hamiltonian form of the Euler equation for an ideal fluid and perform a very specific canonical transformation. This transformation is possible due to the ‘miraculous’ cancellation of the non-trivial four-wave resonant interaction in the one-dimensional wave field. The obtained equation is remarkably simple. We call the equation the ‘super compact water wave equation’. This equation includes a nonlinear wave term (à la NLSE) together with an advection term that can describe the initial stage of wave breaking. The NLSE and the Dysthe equations (DystheProc. R. Soc. Lond.A, vol. 369, 1979, pp. 105–114) can be easily derived from the super compact equation. This equation is also suitable for analytical studies as well as for numerical simulation. Moreover, this equation also allows one to derive a spatial version of the water wave equation that describes experiments in flumes and canals.

scholarly journals Field Measurements of Rogue Water Waves

2014 ◽  
Vol 44 (9) ◽  
pp. 2317-2335 ◽  
Author(s):  
Marios Christou ◽  
Kevin Ewans
Keyword(s):  
Quality Control ◽  
Wave Height ◽  
Water Waves ◽  
Normal Population ◽  
Single Point ◽  
Field Measurements ◽  
Rogue Waves ◽  
Control Procedure ◽  
Sea State ◽  
The North

Abstract This paper concerns the collation, quality control, and analysis of single-point field measurements from fixed sensors mounted on offshore platforms. In total, the quality-controlled database contains 122 million individual waves, of which 3649 are rogue waves. Geographically, the majority of the field measurements were recorded in the North Sea, with supplementary data from the Gulf of Mexico, the South China Sea, and the North West shelf of Australia. The significant wave height ranged from 0.12 to 15.4 m, the peak period ranged from 1 to 24.7 s, the maximum crest height was 18.5 m, and the maximum recorded wave height was 25.5 m. This paper will describe the offshore installations, instrumentation, and the strict quality control procedure employed to ensure a reliable dataset. An examination of sea state parameters, environmental conditions, and local characteristics is performed to gain an insight into the behavior of rogue waves. Evidence is provided to demonstrate that rogue waves are not governed by sea state parameters. Rather, the results are consistent with rogue waves being merely extraordinary and rare events of the normal population caused by dispersive focusing.

On Signatures and Features of Modulational Instability in Ocean Waves

2019 ◽  
Author(s):  
Alexander V. Babanin
Keyword(s):  
Water Waves ◽  
Modulational Instability ◽  
Ocean Waves ◽  
Rogue Waves ◽  
Spectral Energy ◽  
Nonlinear Superposition ◽  
Relative Significance ◽  
Wave Fields ◽  
Ice Breakup ◽  
Surface Water Waves

Abstract Modulational instability of nonlinear waves in dispersive environments is known across a broad range of physical media, from nonlinear optics to waves in plasmas. Since it was discovered for the surface water waves in the early 60s, it was found responsible for, or able to contribute to the topics of breaking and rogue waves, swell, ice breakup, wave-current interactions and perhaps even spray production. Since the early days, however, the argument continues on whether the modulational instability, which is essentially a one-dimensional phenomenon, is active in directional wave fields (that is whether the realistic directional spectra are narrow enough to maintain such nonlinear behaviours). Here we discuss the distinct features of the evolution of nonlinear surface gravity waves, which should be attributed as signatures to this instability in oceanic wind-generated wave fields. These include: wave-breaking threshold in terms of average steepness; upshifting of the spectral energy prior to breaking; oscillations of wave asymmetry and skewness; energy loss from the carrier waves in the course of the breaking. We will also refer to the linear/nonlinear superposition of waves which is often considered a counterpart (or competing) mechanism responsible for breaking or rogue waves in the ocean. We argue that both mechanisms are physically possible and the question of in situ abnormal waves is a problem of their relative significance in specific circumstances.

scholarly journals Dynamical and statistical explanations of observed occurrence rates of rogue waves

2011 ◽  
Vol 11 (5) ◽  
pp. 1437-1446 ◽  
Author(s):  
J. Gemmrich ◽  
C. Garrett
Keyword(s):  
Standard Model ◽  
Water Waves ◽  
Rogue Waves ◽  
Value Theory ◽  
Occurrence Rate ◽  
Exceedance Probability ◽  
Extreme Wave ◽  
Straight Line ◽  
The Standard Model ◽  
The Waves

Abstract. Extreme surface waves occur in the tail of the probability distribution. Their occurrence rate can be displayed effectively by plotting ln(–ln P), where P is the probability of the wave or crest height exceeding a particular value, against the logarithm of that value. A Weibull distribution of the exceedance probability, as proposed in a standard model, then becomes a straight line. Earlier North Sea data from an oil platform suggest a curved plot, with a higher occurrence rate of extreme wave and crest heights than predicted by the standard model. The curvature is not accounted for by second order corrections, non-stationarity, or Benjamin-Feir instability, though all of these do lead to an increase in the exceedance probability. Simulations for deep water waves suggest that, if the waves are steep, the curvature may be explained by including up to fourth order Stokes corrections. Finally, the use of extreme value theory in fitting exceedance probabilities is shown to be inappropriate, as its application requires that not just N, but also lnN, be large, where N is the number of waves in a data block. This is unlikely to be adequately satisfied.

Quasi-three-dimensional simulation of crescent-shaped waves

2020 ◽  
Author(s):  
Alexander Dosaev ◽  
Yuliya Troitskaya
Keyword(s):  
Water Waves ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Rogue Waves ◽  
Boundary Integral ◽  
Main Direction ◽  
Two Dimensional ◽  
Dimensional Model ◽  
Three Dimensional Model ◽  
Wave Interactions

<p>Many features of nonlinear water wave dynamics can be explained within the assumption that the motion of fluid is strictly potential. At the same time, numerically solving exact equations of motion for a three-dimensional potential flow with a free surface (by means of, for example, boundary integral method) is still often considered too computationally expensive, and further simplifications are made, usually implying limitations on wave steepness. A quasi-three-dimensional model, put forward by V. P. Ruban [1], represents another approach at reducing computational cost. It is, in its essence, a two-dimensional model, formulated using conformal mapping of the flow domain, augmented by three-dimensional corrections. The model assumes narrow directional distribution of the wave field and is exact for two-dimensional waves. It was successfully applied by its author to study a nonlinear stage of of Benjamin-Feir instability and rogue waves formation.</p><p>The main aim of the present work is to explore the behaviour of the quasi-three-dimensional model outside the formal limits of its applicability. From the practical point of view, it is important that the model operates robustly even in the presence of waves propagating at large angles to the main direction (although we do not attempt to accurately describe their dynamics). We investigate linear stability of Stokes waves to three-dimensional perturbations and suggest a modification to the original model to eliminate a spurious zone of instability in the vicinity of the perpendicular direction on the perturbation wavenumber plane. We show that the quasi-three-dimensional model yields a qualitatively correct description of the instability zone generated by resonant 5-wave interactions. The values of the increment are reasonably close to those obtained from the exact equations of motion [2], despite the fact that the corresponding modes of instability consist of harmonics that are relatively far from the main direction. Resonant 5-wave interactions are known to manifest themselves in the formation of the so-called “horse-shoe” or “crescent-shaped” wave patterns, and the quasi-three-dimensional model exhibits a plausible dynamics leading to formation of crescent-shaped waves.</p><p>This research was supported by RFBR (grant No. 20-05-00322).</p><p>[1] Ruban, V. P. (2010). Conformal variables in the numerical simulations of long-crested rogue waves. <em>The European Physical Journal Special Topics</em>, <em>185</em>(1), 17-33.</p><p>[2] McLean, J. W. (1982). Instabilities of finite-amplitude water waves. <em>Journal of Fluid Mechanics</em>, <em>114</em>, 315-330.</p>

Rogue Waves and Lump Solitons of the (3+1)-Dimensional Generalized B-type Kadomtsev–Petviashvili Equation for Water Waves

2017 ◽  
Vol 68 (6) ◽  
pp. 693 ◽  
Author(s):  
Yan Sun ◽  
Bo Tian ◽  
Lei Liu ◽  
Han-Peng Chai ◽  
Yu-Qiang Yuan
Keyword(s):  
Water Waves ◽  
Rogue Waves ◽  
Petviashvili Equation

scholarly journals Evidence of Landau Damping in a Fluid Coupled with an Anharmonic Lattice

2018 ◽  
Author(s):  
Andrei Ludu ◽  
Eric Padilla ◽  
M. A. Qaayum Mazumder
Keyword(s):  
Drag Force ◽  
Water Waves ◽  
Stokes Equations ◽  
Collisionless Plasma ◽  
Rogue Waves ◽  
Landau Damping ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Anharmonic Lattice ◽  
Specific Frequency

The Landau damping effect was observed in collisionless plasma, as a microscopic resonant mechanism between electromagnetic radiation and the collective modes. In this paper we demonstrate the occurrence of the Landau damping at macroscopic scale in the interaction between water waves and anharmonic lattice of magnetic buoys. By coupling the Navier-Stokes equations for incompressible fluid with the nonlinear dynamics of an anharmonic magnetic lattice we obtain a resonant transfer of momentum and energy between the two systems. The velocity of the flow is obtained in the Stokes approximation with Basset type of drag force. The dynamics of the buoys is calculated in the surfactant approximation for a specific frequency, then we use Fourier analysis to obtain the general time variable interaction. After involving an integral Dirichlet transform we obtain the time dependent expression of the drag force, the interaction waves-lattice with a new term in the form of a Caputo fractional derivative. We compare the results of the model with experiments performed in a wave tank with free floating magnetic buoys under the action of small amplitude gravitational waves. This configuration can be applied in studies for the attenuation with resonant damping of rogue waves, storms or tsunamis.

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