The effects of randomness on the stability of two-dimensional surface wavetrains

1978 ◽  
Vol 363 (1715) ◽  
pp. 525-546 ◽  
Keyword(s):  
Water Waves ◽  
Modulational Instability ◽  
Wave Spectrum ◽  
Random Wave ◽  
Length Scale ◽  
Wave System ◽  
Random Surface ◽  
Homogeneous Wave ◽  
Amplification Rate ◽  
The Stability

A simplified nonlinear spectral transport equation, for narrowband Gaussian random surface wavetrains, slowly varying in space and time, is derived fron the weakly nonlinear equations of Davey & Stewartson. The stability of an initially homogeneous wave spectrum, to small oblique wave perturbations is studied for a range of spectral bandwidths, resulting in an integral equation for the amplification rate of the disturbance. It is shown for random deep water waves that instability of the wavetrain can exist, as in the corresponding deterministic Benjamin-Feir (B-F) problem, provided that the normalized spectral bandwidth σ / k 0 is less than twice the root mean square wave slope, multiplied by a function of the perturbation wave angle ϕ = arctan ( m/l ). A further condition for instability is that the angle ϕ be less than 35.26°. It is demonstrated that the amplification rate, associated with the B-F type instability, diminishes and then vanishes as the correlation length scale of the random wave field ( ca . 1/ σ )is reduced to the order of the characteristic length scale for modulational instability of the wave system.

scholarly journals Soliton Turbulence in Approximate and Exact Models for Deep Water Waves

Fluids ◽  
2020 ◽  
Vol 5 (2) ◽  
pp. 67 ◽  
Author(s):  
Dmitry Kachulin ◽  
Alexander Dyachenko ◽  
Vladimir Zakharov
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Water Waves ◽  
Schrodinger Equation ◽  
Modulational Instability ◽  
Nonlinear Schrodinger Equation ◽  
Wave System ◽  
Zakharov Equation ◽  
Nonlinear Schrödinger ◽  
Homogeneous Wave

We investigate and compare soliton turbulence appearing as a result of modulational instability of the homogeneous wave train in three nonlinear models for surface gravity waves: the nonlinear Schrödinger equation, the super compact Zakharov equation, and the fully nonlinear equations written in conformal variables. We show that even at a low level of energy and average wave steepness, the wave dynamics in the nonlinear Schrödinger equation fundamentally differ from the dynamics in more accurate models. We study energy losses of wind waves due to their breaking for large values of total energy in the super compact Zakharov equation and in the exact equations and show that in both models, the wave system loses 50% of energy very slowly, during few days.

The Diffraction of Multidirectional Random Waves by Trapezoidal Submarine Pits

2003 ◽  
Author(s):  
Hong Sik Lee ◽  
A. Neil Williams ◽  
Sung Duk Kim
Keyword(s):  
Numerical Model ◽  
Wave Diffraction ◽  
Boundary Integral ◽  
Random Wave ◽  
Linear Wave ◽  
Wave System ◽  
Random Waves ◽  
Random Surface ◽  
Directional Spectrum ◽  
Integral Equation Approach

A numerical model is presented to predict the interaction of multidirectional random surface waves with one or more trapezoidal submarine pits. In the present formulation, each pit may have a different side slope, while the four side slopes at the interior edge of any given pit are assumed equal. The water depth in the fluid region exterior to the pits is taken to be uniform, and the solution method for a random wave system involves the superposition of linear-wave diffraction solutions based on a two-dimensional boundary integral equation approach. The incident wave conditions are specified using a discrete form of the Mitsuyasu directional spectrum. The results of the present numerical model have been compared with those of previous theoretical studies for regular and random wave diffraction by single or multiple rectangular pits. Reasonable agreement was obtained in all cases. Based on these comparisons it is concluded that the present numerical model is an accurate and efficient tool to predict the wave field around multiple submarine pits of trapezoidal section in many practical situations.

Long-time behaviour of a random inhomogeneous field of weakly nonlinear surface gravity waves

1983 ◽  
Vol 133 ◽  
pp. 113-132 ◽  
Author(s):  
Peter A. E. M. Janssen
Keyword(s):  
Wave Spectrum ◽  
Inhomogeneous Field ◽  
Time Behaviour ◽  
Phase Mixing ◽  
Weakly Nonlinear ◽  
Large Bandwidth ◽  
Long Time Behaviour ◽  
Homogeneous Wave ◽  
Long Time ◽  
The Stability

In this paper we investigate nonlinear interactions of narrowband, Gaussian-random, inhomogeneous wavetrains. Alber studied the stability of a homogeneous wave spectrum as a function of the width σ of the spectrum. For vanishing bandwidth the deterministic results of Benjamin & Feir on the instability of a uniform wavetrain were rediscovered whereas a homogeneous wave spectrum was found to be stable if the bandwidth is sufficiently large. Clearly, a threshold for instability is present, and in this paper we intend to study the long-time behaviour of a slightly unstable modulation by means of a multiple-timescale technique. Two interesting cases are found. For small but finite bandwidth – the amplitude of the unstable modulation shows initially an overshoot, followed by an oscillation around the time-asymptotic value of the amplitude. This oscillation damps owing to phase mixing except for vanishing bandwidth because then the well-known Fermi–Pasta–Ulam recurrence is found. For large bandwidth, however, no overshoot is found since the damping is overwhelming. In both cases the instability is quenched because of a broadening of the spectrum.

scholarly journals Evolution of statistically inhomogeneous degenerate water wave quartets

2017 ◽  
Vol 376 (2111) ◽  
pp. 20170101 ◽  
Author(s):  
R. Stuhlmeier ◽  
M. Stiassnie
Keyword(s):  
Water Waves ◽  
Wave Interaction ◽  
Theme Issue ◽  
Nonlinear Water Waves ◽  
Random Surface ◽  
Freak Wave ◽  
Long Time Behaviour ◽  
Long Time ◽  
Spatio Temporal ◽  
The Stability

A discretized equation for the evolution of random surface wave fields on deep water is derived from Zakharov's equation, allowing for a general treatment of the stability and long-time behaviour of broad-banded sea states. It is investigated for the simple case of degenerate four-wave interaction, and the instability of statistically homogeneous states to small inhomogeneous disturbances is demonstrated. Furthermore, the long-time evolution is studied for several cases and shown to lead to a complex spatio-temporal energy distribution. The possible impact of this evolution on the statistics of freak wave occurrence is explored. This article is part of the theme issue ‘Nonlinear water waves’.

Modulational Instability in JONSWAP Sea States Using the Alber Equation

2017 ◽  
Author(s):  
Odin Gramstad
Keyword(s):  
Stability Analysis ◽  
Nonlinear Evolution ◽  
Modulational Instability ◽  
Wave Spectrum ◽  
Rogue Waves ◽  
Systems Of Nonlinear Equations ◽  
Optimal Parameters ◽  
Weakly Nonlinear ◽  
Marquardt Algorithm ◽  
The Stability

An investigation of the instability of JONSWAP wave fields is carried out within the framework of the Alber equation [1]. The Alber equation describes the weakly nonlinear evolution of an inhomogeneous wave spectrum, and by linear stability analysis of this equation [1–3] the instability of an arbitrary wave spectrum subject to inhomogeneous perturbation is investigated. We are solving the equations for instability using a numerical method based on the Levenberg-Marquardt algorithm for solving systems of nonlinear equations, as implemented in the FORTRAN library MINPACK. Results from previous works addressing related topics [4, 5] are verified and refined, providing new results for the stability of JONSWAP wave spectra. Based on the results of the instability analysis we propose more optimal parameters for parameterizing the effects of modulational instability and probability of rogue waves in JONSWAP sea states. The results from the stability analysis of the Alber equation as well as the proposed parameters for parameterizing the effect of modulational instability are verified and tested by performing phase-resolving numerical simulations with the Higher Order Spectral Method [6, 7].

Hydraulic Stability of Interlocking Concrete Unit-V (ICU-V)

2014 ◽  
Vol 606 ◽  
pp. 211-215
Author(s):  
Nur Diyana Md Noor ◽  
Ahmad Mustafa Hashim
Keyword(s):  
Wave Spectrum ◽  
Ratio Method ◽  
Coastal Protection ◽  
Random Wave ◽  
Wave Condition ◽  
Damage Level ◽  
Protection Scheme ◽  
Damage Progression ◽  
The Stability ◽  
Hydraulic Stability

Continuous research is conducted to improve the available coastal protection scheme, including to develop an innovative concrete armor unit with interlocking capability. Interlocking Concrete Unit-V (ICU-V) has light weight characteristic and developed specifically for mild wave condition. This paper discusses the performance of ICU-V represented by the Stability coefficient,KDand results of the damage progression investigation under different wave conditions. These were achieved by conducting a 2-D physical model investigation by using JONSWAP random wave spectrum. The damage ratio method was used to assess the damage progression. The optimumKDobtained was 12 with highest damage level of 0.4%, which is comparable with available armor units

scholarly journals Modulational instability in the full-dispersion Camassa–Holm equation

2017 ◽  
Vol 473 (2203) ◽  
pp. 20170153 ◽  
Author(s):  
Vera Mikyoung Hur ◽  
Ashish Kumar Pandey
Keyword(s):  
Surface Tension ◽  
Water Waves ◽  
Modulational Instability ◽  
Travelling Wave ◽  
Wave Number ◽  
Stokes Wave ◽  
Holm Equation ◽  
Stability And Instability ◽  
The Stability ◽  
Whitham Equation

We determine the stability and instability of a sufficiently small and periodic travelling wave to long-wavelength perturbations, for a nonlinear dispersive equation which extends a Camassa–Holm equation to include all the dispersion of water waves and the Whitham equation to include nonlinearities of medium-amplitude waves. In the absence of the effects of surface tension, the result qualitatively agrees with the Benjamin–Feir instability of a Stokes wave. In the presence of the effects of surface tension, it qualitatively agrees with those from formal asymptotic expansions of the physical problem and improves upon that for the Whitham equation, predicting the critical wave number at the strong surface tension limit. We discuss the modulational stability and instability in the Camassa–Holm equation and other related models.

On stability of parallel flow of an incompressible fluid of variable density and viscosity

1962 ◽  
Vol 58 (4) ◽  
pp. 646-661 ◽  
Author(s):  
P. G. Drazin
Keyword(s):  
Incompressible Fluid ◽  
Water Waves ◽  
Salt Water ◽  
Reynolds Numbers ◽  
Variable Density ◽  
Stability Equation ◽  
Wave Disturbances ◽  
Long Wave ◽  
Vertical Variations ◽  
The Stability

ABSTRACTSome aspects of generation of water waves by wind and of turbulence in a heterogeneous fluid may be described by the theory of hydrodynamic stability. The technical difficulties of these problems of instability have led to obscurities in the literature, some of which are elucidated in this paper. The stability equation for a basic steady parallel horizontal flow under the influence of gravity is derived carefully, the undisturbed fluid having vertical variations of density and viscosity. Methods of solution of the equation for large Reynolds numbers and for long-wave disturbances are described. These methods are applied to simple models of wind blowing over water and of fresh water flowing over salt water.

scholarly journals Evolution of a narrow-band spectrum of random surface gravity waves

2003 ◽  
Vol 478 ◽  
pp. 1-10 ◽  
Author(s):  
KRISTIAN B. DYSTHE ◽  
KARSTEN TRULSEN ◽  
HARALD E. KROGSTAD ◽  
HERVÉ SOCQUET-JUGLARD
Keyword(s):  
Three Dimensional ◽  
Low Frequency ◽  
Three Dimensions ◽  
Band Spectrum ◽  
Nls Equation ◽  
Random Surface ◽  
Narrow Bandwidth ◽  
Quasi Stationary State ◽  
The Stability ◽  
Three Dimensional Simulations

Numerical simulations of the evolution of gravity wave spectra of fairly narrow bandwidth have been performed both for two and three dimensions. Simulations using the nonlinear Schrödinger (NLS) equation approximately verify the stability criteria of Alber (1978) in the two-dimensional but not in the three-dimensional case. Using a modified NLS equation (Trulsen et al. 2000) the spectra ‘relax’ towards a quasi-stationary state on a timescale (ε2ω0)−1. In this state the low-frequency face is steepened and the spectral peak is downshifted. The three-dimensional simulations show a power-law behaviour ω−4 on the high-frequency side of the (angularly integrated) spectrum.

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.

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