Modulation Instability and Extreme Events Beyond Initial Three Wave Systems

2016 ◽  
Author(s):  
Amin Chabchoub ◽  
Takuji Waseda
Keyword(s):  
Numerical Simulations ◽  
Extreme Events ◽  
Water Waves ◽  
Modulation Instability ◽  
Stokes Wave ◽  
Side Band ◽  
Wave Basin ◽  
Stokes Waves ◽  
A Minor ◽  
Band Pair

One possible mechanism that models the dynamics of extreme events in the ocean is the modulation instability (MI). The latter has been discovered in the 60s and significant progress in understanding the physics of modulationally unstable deep-water waves has been achieved since then. The MI instability starts its dynamics from a minor periodic perturbation of a regular Stokes wave, which enhances in amplitude, generating therefore periodic large waves, within the specific range of modulation period. In the spectral domain the same process starts from in amplitude very small symmetric side-band pair, lying in the unstable range from the main carrier frequency peak, which then starts to grow while generating by their own a side-band cascade. We report a new type of periodically modulated and unstable Stokes waves which initial dynamics starts from more that one unique unstable side-band pair. Laboratory experiments have been conducted in a large water wave basin, while numerical simulations have been performed using the modified nonlinear Schrödinger equation and the boundary element method. Both, experiments and numerical simulations are in reasonable agreement. Furthermore, the validity, limitations and applicability of such models will be discussed in detail.

scholarly journals Dynamic-pressure distributions under Stokes waves with and without a current

2017 ◽  
Vol 376 (2111) ◽  
pp. 20170103 ◽  
Author(s):  
Motohiko Umeyama
Keyword(s):  
Water Waves ◽  
Dynamic Pressure ◽  
Water Pressure ◽  
Travelling Waves ◽  
Stokes Wave ◽  
Front Face ◽  
Nonlinear Water Waves ◽  
Near Surface ◽  
Stokes Waves ◽  
A Current

To investigate changes in the instability of Stokes waves prior to wave breaking in shallow water, pressure data were recorded vertically over the entire water depth, except in the near-surface layer (from 0 cm to −3 cm), in a recirculating channel. In addition, we checked the pressure asymmetry under several conditions. The phase-averaged dynamic-pressure values for the wave–current motion appear to increase compared with those for the wave-alone motion; however, they scatter in the experimental range. The measured vertical distributions of the dynamic pressure were plotted over one wave cycle and compared to the corresponding predictions on the basis of third-order Stokes wave theory. The dynamic-pressure pattern was not the same during the acceleration and deceleration periods. Spatially, the dynamic pressure varies according to the faces of the wave, i.e. the pressure on the front face is lower than that on the rear face. The direction of wave propagation with respect to the current directly influences the essential features of the resulting dynamic pressure. The results demonstrate that interactions between travelling waves and a current lead more quickly to asymmetry. This article is part of the theme issue ‘Nonlinear water waves’.

scholarly journals On slowly-varying Stokes waves

1970 ◽  
Vol 41 (4) ◽  
pp. 873-887 ◽  
Author(s):  
Vincent H. Chu ◽  
Chiang C. Mei
Keyword(s):  
Wave Train ◽  
Perturbation Technique ◽  
Stokes Wave ◽  
Monochromatic Wave ◽  
Side Band ◽  
Depth Variation ◽  
Instability Problem ◽  
Stokes Waves ◽  
Points Of View ◽  
Slow Modulation

A WKB-perturbation technique is applied to study the slow modulation of a Stokes wave train on the surface of water. It is found that new terms directly representing modulation rates must be included to extend the scope of validity of Whitham's theory based on an averaged Lagrangian. Two examples are discussed. In the first, a monochromatic wave normally incident on a mild beach is studied and the local rate of depth variation is found to affect the wave phase. In the second, the ‘side-band instability’ problem of Benjamin & Feir is discussed from both linear and non-linear points of view.

Modulation instability in nonlinear chiral fiber

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Demissie Jobir Gelmecha ◽  
Ram Sewak Singh
Keyword(s):  
Theoretical Model ◽  
Numerical Simulations ◽  
Pulse Propagation ◽  
Modulation Instability ◽  
Constitutive Relations ◽  
Gain Spectrum ◽  
Circularly Polarized ◽  
Left Handed ◽  
Simulation Results ◽  
The Difference

AbstractIn this paper, the rigorous derivations of generalized coupled chiral nonlinear Schrödinger equations (CCNLSEs) and their modulation instability analysis have been explored theoretically and computationally. With the consideration of Maxwell’s equations and Post’s constitutive relations, a generalized CCNLSE has been derived, which describes the evolution of left-handed circularly polarized (LCP) and right-handed circularly polarized (RCP) components propagating through single-core nonlinear chiral fiber. The analysis of modulation instability in nonlinear chiral fiber has been investigated starting from CCNLSEs. Based on a theoretical model and numerical simulations, the difference on the modulation instability gain spectrum in LCP and RCP components through chiral fiber has been analyzed by considering loss and chirality into account. The obtained simulation results have shown that the loss distorts the sidebands of the modulation instability gain spectrum, while chirality modulates the gain for LCP and RCP components in a different manner. This suggests that adjusting chirality strength may control the loss, and nonlinearity simultaneously provides stable modulated pulse propagation.

On the modulation of water waves on shear flows

1976 ◽  
Vol 347 (1651) ◽  
pp. 537-546 ◽  
Keyword(s):  
Water Waves ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Vries Equation ◽  
Harmonic Wave ◽  
Short Wave ◽  
Long Wave ◽  
Stokes Waves ◽  
The Stability ◽  
Wave Limit

The method of multiple scales is used to examine the slow modulation of a harmonic wave moving over the surface of a two dimensional channel. The flow is assumed inviscid and incompressible, but the basic flow takes the form of an arbitrary shear. The appropriate nonlinear Schrödinger equation is derived with coefficients that depend, in a complicated way, on the shear. It is shown that this equation agrees with previous work for the case of no shear; it also agrees in the long wave limit with the appropriate short wave limit of the Korteweg-de Vries equation, the shear being arbitrary. Finally, it is remarked that the stability of Stokes waves over any shear can be examined by using the results derived here.

scholarly journals “Extraordinary” modulation instability in optics and hydrodynamics

2021 ◽  
Vol 118 (14) ◽  
pp. e2019348118
Author(s):  
Guillaume Vanderhaegen ◽  
Corentin Naveau ◽  
Pascal Szriftgiser ◽  
Alexandre Kudlinski ◽  
Matteo Conforti ◽  
...  
Keyword(s):  
Wave Propagation ◽  
Stability Analysis ◽  
Linear Stability ◽  
Nonlinear Theory ◽  
Linear Stability Analysis ◽  
Water Waves ◽  
Modulation Instability ◽  
Nonlinear Effects ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Theory

The classical theory of modulation instability (MI) attributed to Bespalov–Talanov in optics and Benjamin–Feir for water waves is just a linear approximation of nonlinear effects and has limitations that have been corrected using the exact weakly nonlinear theory of wave propagation. We report results of experiments in both optics and hydrodynamics, which are in excellent agreement with nonlinear theory. These observations clearly demonstrate that MI has a wider band of unstable frequencies than predicted by the linear stability analysis. The range of areas where the nonlinear theory of MI can be applied is actually much larger than considered here.

scholarly journals Using machine learning to predict extreme events in complex systems

2019 ◽  
Vol 117 (1) ◽  
pp. 52-59 ◽  
Author(s):  
Di Qi ◽  
Andrew J. Majda
Keyword(s):  
Dynamical Systems ◽  
Learning Strategies ◽  
Extreme Events ◽  
Water Waves ◽  
Extreme Values ◽  
Training Data ◽  
Partition Functions ◽  
Grand Challenge ◽  
Natural Systems ◽  
Using Data

Extreme events and the related anomalous statistics are ubiquitously observed in many natural systems, and the development of efficient methods to understand and accurately predict such representative features remains a grand challenge. Here, we investigate the skill of deep learning strategies in the prediction of extreme events in complex turbulent dynamical systems. Deep neural networks have been successfully applied to many imaging processing problems involving big data, and have recently shown potential for the study of dynamical systems. We propose to use a densely connected mixed-scale network model to capture the extreme events appearing in a truncated Korteweg–de Vries (tKdV) statistical framework, which creates anomalous skewed distributions consistent with recent laboratory experiments for shallow water waves across an abrupt depth change, where a remarkable statistical phase transition is generated by varying the inverse temperature parameter in the corresponding Gibbs invariant measures. The neural network is trained using data without knowing the explicit model dynamics, and the training data are only drawn from the near-Gaussian regime of the tKdV model solutions without the occurrence of large extreme values. A relative entropy loss function, together with empirical partition functions, is proposed for measuring the accuracy of the network output where the dominant structures in the turbulent field are emphasized. The optimized network is shown to gain uniformly high skill in accurately predicting the solutions in a wide variety of statistical regimes, including highly skewed extreme events. The technique is promising to be further applied to other complicated high-dimensional systems.

Energy Content Analysis of Closed Basin Wave Simulation Using Linear and Nonlinear Fourier Analysis

2018 ◽  
Author(s):  
Ali Mohtat ◽  
Solomon C. Yim ◽  
Alfred R. Osborne
Keyword(s):  
Fourier Analysis ◽  
Energy Content ◽  
Wave Train ◽  
Stokes Wave ◽  
State University ◽  
Linear Spectrum ◽  
Wave Basin ◽  
Significant Difference ◽  
Nonlinear Components ◽  
Oregon State University

This study focuses on the computation and analysis of the energy content of a wave train and the influence of nonlinear components, such as nonlinear wave profile as in Stokes wave and phased locked breathers, on the content. To this end, an overview of a state-of-the-art nonlinear Fourier analysis tools for the nonlinear Schrödinger equation is presented. Experimental measurements from a set of performance tests of the directional wave basin at Oregon State University were analyzed using this tool and the energy contents, both from the linear spectrum and nonlinear spectrum, were calculated. The deviation of the energy content from linear analysis and its relationship to the level of nonlinearity of the wave train is investigated. The Benjamin-Feir parameter presents the degree of nonlinearity of the wave train. An increasing energy deviation was observed for increasing nonlinearity of the wave field. Spatial evolution of such behavior is also investigated. It was confirmed that the significant difference from the linear energy is due to increase in the nonlinear components and the more distance the wave train could travel (without substantial dissipation) the more erratic and more significant energy deviations were observed.

scholarly journals A Study on the Minimization of Mooring Load in Fish-Cage Mooring Systems with a Damping Buoy

2020 ◽  
Vol 8 (10) ◽  
pp. 814
Author(s):  
Gun-Ho Lee ◽  
Bong-Jin Cha ◽  
Hyun-young Kim
Keyword(s):  
Numerical Simulations ◽  
Wave Theory ◽  
Line Length ◽  
Stokes Wave ◽  
Mooring Line ◽  
Spring Model ◽  
Mass Spring Model ◽  
Wave Conditions ◽  
Mass Spring ◽  
Fifth Order

This study established the conditions in which mooring load is minimized in a fish cage that includes a damping buoy in specific wave conditions. To derive these conditions, numerical simulations of various mooring contexts were conducted on a fish cage (1/15 scale) using a simplified mass-spring model and fifth-order Stokes wave theory. The simulation conditions were as follows: (1) bridle-line length of 0.8–3.2 m; (2) buoyancy of 2.894–20.513 N for the damping buoy; and (3) mooring-rope thickness of 0.002–0.004 m. The wave conditions were 0.333 m in height and 1.291–2.324 s of arrival period. Consequently, the mooring tensions tended to decrease with decreasing mooring line thickness and increasing bridle-line length and buoyancy of the buoy. Accordingly, it was assumed to be advantageous to minimize the mooring tension by designing a thin mooring line and long bridle line and for the buoyancy of the buoy to be as large as possible. This approach shows a valuable technique because it can contribute to the improvement of the mooring stability of the fish cage by establishing a method that can be used to minimize the load on the mooring line.

scholarly journals Two-dimensional modulation instability of wind waves

2019 ◽  
Vol 5 (4) ◽  
pp. 413-417 ◽  
Author(s):  
Roger Grimshaw
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Water Waves ◽  
Schrodinger Equation ◽  
Modulation Instability ◽  
Wind Waves ◽  
Nonlinear Schrodinger Equation ◽  
Wind Forcing ◽  
Nonlinear Schrödinger ◽  
Peregrine Breather

Abstract It is widely known that deep-water waves are modulationally unstable and that this can be modelled by a nonlinear Schrödinger equation. In this paper, we extend the previous studies of the effect of wind forcing on this instability to water waves in finite depth and in two horizontal space dimensions. The principal finding is that the instability is enhanced and becomes super-exponential and that the domain of instability in the modulation wavenumber space is enlarged. Since the outcome of modulation instability is expected to be the generation of rogue waves, represented within the framework of the nonlinear Schrödinger equation as a Peregrine breather, we also examine the effect of wind forcing on a Peregrine breather. We find that the breather amplitude will grow at twice the rate of a linear instability.

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