Stability of bichromatic gravity waves on deep water

The stability of bichromatic gravity waves with small but finite amplitudes propagating in two directions on deep water is considered. Starting from the Zakharov equation, elementary quartet interactions are isolated and stability criteria are formulated. Results are illustrated for various combinations of bichromatic wave trains, from long-crested to standing waves. Two generic mechanisms operate: the first one is a modulational instability of one of the two components of the bichromatic wave train; the second mechanism is a modulation which couples both components of the wave train. However a third mechanism eventually comes into play: the resonant interaction of Phillips and Longuet-Higgins which leads initially to the linear growth of a third wave. When this latter is active, in particular for wave trains with wave vectors close together, it is shown by numerical integration that the long-time recurrence is destroyed.     

On steady three-dimensional deep water weakly nonlinear gravity waves

Various kinds of steady weakly nonlinear gravity waves are examined. Corrections to the linear phase speed and the direction of modulation are calculated.     

Parametric instability of gravity waves on the surface of deep water

We propose to investigate the characteristics of the parametric generation of gravity waves on the surface of a body of deep water. The threshold conditions for the onset of generation are determined, and the results are compared with the experimental data. The singularities of the excitation of parametric oscillations in a resonator are noted.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 182–185, May–June, 1972.The authors are grateful to V. N. Pshenichnikov for assisting with the experiments.     

Laboratory study on the evolution of waves parameters due to wave breaking in deep water

Understanding of the occurrence of the wave breaking, the process of the wave breaking and evolution of waves after they break in deep water is crucial to simulate the growth of wind wave in ocean. In this study, deep-water breaking waves with various spectral types, center frequencies and frequency bandwidths are generated in a wave flume based on energy focusing theory. The time series of the wave surface elevation along the flume are obtained by 22 wave probes mounted along the central line of the flume. The characteristics of deep-water wave breaking are analyzed using the spectrum analysis based on the Fast Fourier Transform (FFT). For small center frequency the maximum height of wave surface generated using the Pierson–Moskowitz (P–M) spectrum is produced and the impact of the frequency width is small in wave breaking zone. While the spectral type has a significant impact on the local wave steepness during breaking, the influence of center frequency and frequency width on the local wave steepness is very weak. The significant wave steepness changes significantly after wave breaking, but it remains stable in the upstream or the downstream of wave breaking zone. After wave breaking, the peak frequency remains stable, but the spectrally weighted wave frequency changes significantly. The relationship between the level of downshift and the incident wave steepness is approximately linear. By analyzing the energy spectra, it is found that the energy loses near high frequency of controlling frequencies range and increases near peak frequency during the wave breaking. After wave breaking, the total energy dissipates remarkably with increasing breaking intensity.     

Turbulent breaking of overturning gravity waves below a critical level

The interaction of an internal gravity wave with its evolving critical layer and the subsequent generation of turbulence by overturning waves are studied by three-dimensional numerical simulations. The simulation describes the flow of a stably stratified Boussinesq fluid between a bottom wavy surface and a top flat surface, both without friction and adiabatic. The amplitude of the surface wave amounts to about 0.03 of the layer depth. The horizontal flow velocity is negative near the lower surface, positive near the top surface with uniform shear and zero mean value. The bulk Richardson number is one. The flow over the wavy surface induces a standing gravity wave causing a critical layer at mid altitude. After a successful comparison of a two-dimensional version of the model with experimental observations (Thorpe [21]), results obtained with two different models of viscosity are discussed: a direct numerical simulation (DNS) with constant viscosity and a large-eddy simulation (LES) where the subgrid scales are modelled by a stability-dependent first-order closure. Both simulations are similar in the build-up of a primary overturning roll and show the expected early stage of the interaction between wave and critical level. Afterwards, the flows become nonlinear and evolve differently in both cases: the flow structure in the DNS consists of coherent smaller-scale secondary rolls with increasing vertical depth. On the other hand, in the LES the convectively unstable primary roll collapses into three-dimensional turbulence. The results show that convectively overturning regions are always formed but the details of breaking and the resulting structure of the mixed layer depend on the effective Reynolds number of the flow. With sufficient viscous damping, three-dimensional turbulent convective instabilities are more easily suppressed than two-dimensional laminar overturning.     

Standing vortex waves in deep water

Standing vibrations of the free surface of an infinitely deep fluid exist in the presence of a weak (of the order of the wave amplitude) shear flow. As a result of the interaction with the flow the waves acquire a vorticity proportional to the cube of their amplitude.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 158–161, May–June, 1996.     

Internal gravity waves: parametric instability and deep ocean mixing

The Boussinesq approximation provides a convenient framework to describe the dynamics of stably-stratified fluids. A fundamental motion in these fluids consists of internal gravity waves, whatever the strength of the stratification. These waves may be unstable through parametric instability, which results in turbulence and mixing. After a brief review of the main properties of internal gravity waves, we show how the parametric instability of a monochromatic internal gravity wave organizes itself in space and time, using energetics arguments and a simple kinematic model. We provide an example, in the deep ocean, where such instability is likely to occur, as estimates of mixing from in situ measurements suggest. We eventually discuss the fundamental role of internal gravity wave mixing in the maintenance of the abyssal thermal stratification. To cite this article: C. Staquet, C. R. Mecanique 335 (2007).     

Traveling gravity water waves in two and three dimensions

This paper discusses the bifurcation theory for the equations for traveling surface water waves, based on the formulation of Zakharov [58] and of Craig and Sulem [15] in terms of integro-differential equations on the free surface. This theory recovers the well-known picture of bifurcation curves of Stokes progressive wavetrains in two-dimensions, with the bifurcation parameter being the phase velocity of the solution. In three dimensions the phase velocity is a two-dimensional vector, and the resulting bifurcation equations describe two-dimensional bifurcation surfaces, with multiple intersections at simple bifurcation points. The integro-differential formulation on the free surface is posed in terms of the Dirichlet–Neumann operator for the fluid domain. This lends itself naturally to numerical computations through the fast Fourier transform and surface spectral methods, which has been implemented in Nicholls [32]. We present a perturbation analysis of the resulting bifurcation surfaces for the three-dimensional problem, some analytic results for these bifurcation problems, and numerical solutions of the surface water waves problem, based on a numerical continuation method which uses the spectral formulation of the problem in surface variables. Our numerical results address the problem in both two and three dimensions, and for both the shallow and deep water cases. In particular we describe the formation of steep hexagonal traveling wave patterns in the three-dimensional shallow water regime, and their transition to rolling waves, on high aspect ratio rectangular patterns as the depth increases to infinity.     

Instability of periodic capillary-gravity waves on deep water

Pure gravity waves of finite amplitude in infinitely deep water are unstable to small disturbances in the form of modulated side-bands (Benjamin-Feir instability). These disturbances will undergo unbounded magnification if the parameter Ω associated with the frequency of the side-bands lies in the range $\text{0<Ω<√2 ka}$, where k and a are the wavenumber and amplitude of the basic wavetrain. The present paper is devoted to studying the situation for capillary-gravity waves. It is shown that the corresponding range of Ω, for which instability can arise, is smaller, see eq. 4.17, than for the pure gravity wave situation. Even pure capillary waves admit instabilities.     

Surface waves in deep water due to arbitrary periodic surface pressure

Summary The expressions for the velocity potential and the surface elevation are obtained for the three dimensional motion on surface waves generated by a surface pressure F(x, y)e^{i t}, t>0, on the surface of a deep sea. The wave integrals are asymptotically evaluated for large distances and times by the method of stationary phase. By a direct passage to the limit t, the steady state solution of the problem is evaluated in exact terms. The results are illustrated for a class of physically plansible pressure distributions on the surface. Some features of the wave motions are discussed.

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Sommario Si ottengono le espressioni per il potenziale della velocità e la elevazione della superficie nel moto tridimensionale di onde superficiali generate da una pressione superficiale F(x, y)e^{i t}, t>0, sulla superficie di un mare profondo. Gli integrali di onde sono calcolati asintoticamente per grandi distanze e tempi con il metodo della fase stazionaria. Con un passaggio diretto al limite t, si calcola in termini esatti la soluzione di regime del problema. I risultati vengono illustrati per una classe di distribuzioni di pressione superficiale fisicamente plausibile, Si discutono alcune caratteristiche dei moti ondosi.

     

Occurrence of standing surface gravity waves modulation in shallow water

Arising of modulations of surface gravity waves in a shallow water resonator under harmonic forcing is investigated in laboratory experiments. Different types of modulations are found, when the wave amplitude exceeds a certain threshold. Bifurcation diagram on the plane “amplitude of excitation – frequency of excitation” is determined. Numerical simulations of the Euler equations within the frameworks of the High-Order Spectral Method are performed with the purpose of reproducing the modulational regimes observed in the laboratory experiments. The simulations allowed us to determine physical mechanisms responsible for the occurrence of modulated waves.     

Uniform approximation to surface gravity waves around a breakwater on water of variable depth

The linear surface gravity wavefield around a breakwater and on water of varying depth is described by a uniform asymptotic representation. The scattering of not necessarily irrotational wave packets generated by sources at arbitrary distance from the breakwater can be treated with the technique here expounded.     

New families of pure gravity waves in water of infinite depth

Nonlinear periodic gravity waves propagating at a constant velocity at the surface of a fluid of infinite depth are considered. The fluid is assumed to be inviscid and incompressible and the flow to be irrotational. It is known that there are both regular waves (for which all the crests are at the same height) and irregular waves (for which not all the crests are at the same height). We show numerically the existence of new branches of irregular waves which bifurcate from the branch of regular waves. Our results suggest there are an infinite number of such branches. In addition we found additional new branches of irregular waves which bifurcate from the previously calculated branches of irregular waves.     

Oblique water entry of a wedge into waves with gravity effect

The hydrodynamic problem of a two dimensional wedge entering waves with gravity effect is analysed based on the incompressible velocity potential theory. The problem is solved through the boundary element method in the time domain. The stretched coordinate system in the spatial domain, which is based on the ratio of the Cartesian system in the physic space to the vertical distance the wedge has travelled into the water, is adopted based on the consideration that the decay of the effect of the impact away from the body is proportional to this ratio. The solution is sought for the total potential which includes both the incident and disturbed potentials, and decays towards the incident potential away from the body. A separate treatment at initial stage is used, in which the solution for the disturbed potential is sought to avoid the very large incident potential amplified by dividing the small travelled vertical distance of the wedge. The auxiliary function method is used to calculate the pressure on the body surface. Detailed results through the free surface elevation and the pressure distribution are provided to show the effect of the gravity and the wave, and their physical implications are discussed.     

Actio-reactio symmetry breaking model fitted to the new tower-mine gravity experiments

Summary In the present paper the gravity acceleration in the neighbourhood of the earth-atmosphere interface is analyzed. The solution of the actio-reactio equations (see [7, 8]) are compared with the data measured by Holding et al. [4] in mines and Eckehardt et al. [6] on a tower.

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Anpassung eines die actio/reactio-symmetrie verletzenden modells an gravitationsexperimente im bergwerk und turm

Übersicht Analysiert wird die Gravitationsbeschleunigung in der Nähe der Grenze Erde/Atmosphäre. Die Lösung der Actio/Reactio-Gleichungen (vgl. [7, 8]) wird mit den Daten, die von Holding et al. [4] und von Eckehardt et al. [6] im Bergwerk bzw. auf einem Turm gemessen wurden, verglichen.

     

Rotationally symmetric internal gravity waves

Many mathematical models formulated in terms of non-linear differential equations can successfully be treated and solved by Lie group methods. Lie group analysis is especially valuable in investigating non-linear differential equations, for its algorithms act here as reliably as for linear cases. The aim of this article is to provide the group theoretical modeling of internal waves in the ocean. The approach is based on a new concept of conservation laws that is utilized to systematically derive the conservation laws of non-linear equations describing propagation of internal waves in the ocean. It was shown in our previous publication that uni-directional internal wave beams can be obtained as invariant solutions of non-linear equations of motion. The main goal of the present publication is to thoroughly analyze another physically significant exact solution, namely the rotationally symmetric solution and the energy carried by this solution. It is shown that the rotationally symmetric solution and its energy are presented by means of a bounded oscillating function.     

Governing equations of envelopes created by nearly bichromatic waves on deep water

In this paper, the author derives the modified Schrödinger equation that governs the envelope created by nearly bichromatic waves, which are defined by the waves whose energy is almost concentrated in two closely approached wavenumbers. The stability of the solution of the modified Schrödinger equation for nearly bichromatic waves on deep water is discussed and the fact that the Benjamin–Feir instability occurs in a condition is shown. Moreover, the solutions of the modified Schrödinger equation for nearly bichromatic waves on deep water are obtained and, in a special case, the solution becomes the standing wave solution is shown.     

Selection of a staggered grid for inertia‐gravity waves in shallow water

The problem of accuracy in propagating inertia‐gravity waves on Arakawa grids is investigated. It is shown that the sole analysis of spatial discretization and the recommendation of the B‐grid for coarse resolution models and C‐grid for high resolution models must be re‐analysed when time discretization is taken into account as well. For a chosen time discretization, a coarse C‐grid is shown for example, to increase precision when using larger time‐steps (up to the stability limit) whereas the precision of the B‐grid decreases. Here, an analysis of error for different grids in function of the space–time resolutions and computational costs is presented and some recommendations on the choice of the particular staggered grid for a given application are outlined. Copyright © 2002 John Wiley & Sons, Ltd.     

Electromagnetic scattering from large steady breaking waves

A submerged hydrofoil generated large steady breaking waves of 0.3 m and 0.4 m height in a circulating water channel. We measured water fraction in the breakers with conductivity probes. We observed the radar cross-section of the breakers at X-band with a pulsed step-frequency instrumentation radar with high spatial resolution in the downstream direction. The normalized radar cross-section increases with increasing elevation angle of observation for both vertical and horizontal polarization. This variation is consistent with a simple interpretation of the breaking wave as a diffuse (Lambertian) surface. However, the observed sizes and shapes of fluid elements in the breakers clearly show that construction of a theory for electromagnetic scattering from first principles will be challenging. We also obtained the velocity spectrum of the scattering features within the breakers. This spectrum indicates that slower moving small liquid elements rather than the faster moving large disturbances are responsible for most of the electromagnetic scattering. Received: 27 January 1999/Accepted: 7 August 2000