Effects of three-wave interactions in the gravity-capillary range of wind waves

The formation of the spectrum of short wind waves from the gravity-capillary and capillary ranges under the effect of three-wave interactions is considered. In order to determine the spectrum, the kinetic equation for wave packets is integrated to the point where the solution is established. Three-wave interactions are described by a collision integral without introducing any additional assumptions simplifying the problem. This calculation procedure reproduces the Zakharov-Filonenko theoretical spectra, which correspond to the cases of energy equipartition and the inertial range. It is shown that the main role of three-wave interactions lies in the energy transfer from the range of short gravity waves to waves with shorter wavelengths. This transfer is accomplished both locally in the Fourier space and as a result of interactions between short and long waves. Its characteristic features are the formation of a dip on the curvature spectrum in the region of a minimum phase velocity of waves and the formation of a secondary peak in the capillary range. The dip is filled and disappears as the wind speed increases. Taking into account the interaction between short and long waves increases the spectrum in the capillary range several times, and the balance between energy input from long waves and viscous dissipation is established in the capillary range. The energy sink caused by three-wave interactions, viscous dissipation, and wind forcing cannot give the stability of the spectrum of short gravity waves.     

On the nonlinear interactions in triads of edge waves on the sea shelf

We study nonlinear three-wave interactions between edge waves propagating in the same direction over the shelf step. The conditions of synchronism are determined and the coefficient of interaction is computed for the cases where the waves of the five lowest modes participate in the interaction. The space-time dynamics is studied by analyzing, as an example, a single triad of edge waves. The possibility of interaction of edge waves in the regions with actual topography is demonstrated. __________ Translated from Morskoi Gidrofizicheskii Zhurnal, No. 3, pp. 3–19, May–June, 2008.     

Disintegration of linear edge waves

It is demonstrated that offshore wavenumbers of edge waves change from imaginary wavenumbers in deep water to real wavenumbers in shallow water. This finding indicates that edge waves in the offshore direction exist as evanescent waves in deep water and as propagating waves in shallow water. Since evanescent waves can stably exist in a limited region while propagating waves cannot, energy should be released from nearshore regions. In the present study, the instability region is predicted based on both the full water wave solution and the shallow-water wave approximation.     

边缘波研究进展

由于折射作用,在波浪近岸传播过程中会出现一种特殊的、沿着岸线传播的波浪,这种波浪被称为边缘波。边缘波平行于岸线传播,其振幅在岸线处最大,在远离岸线的方向,其振幅呈指数型减小,它们的能量基本被限制在离海岸一波长的距离之内,因此边缘波对近岸地区工程、地貌等有着重要影响。本文对边缘波的研究历史、研究进展进行了阐述,主要介绍了以下几个方面：（1）基于不同控制方程、不同地形上的边缘波理论;（2）实际观测到的边缘波特性;（3）物理模型试验中边缘波的造波方式以及观测到的边缘波特性;（4）数值模拟方法在边缘波研究中的应用。最后,展望了边缘波在未来的研究趋势。     

剪切流中界面波的二阶Stokes波解

考虑了均匀剪切流场中强非线性界面波,建立了基于任意水深处速度而不是通常所用的平均速度为速度变量的模型,分析了其色散关系,并求得了各层速度的二阶渐近解和界面内波波面位移的二阶Stokes解,揭示了波流之间的非线性相互作用和界面波解之间的非线性相互作用。     

Energy properties and shoaling of higher-order stokes waves on a current

The energy density, the energy flux, the set-down, the radiation stress, and some wave energy velocities have been derived correct to fourth order in wave steepness for waves on a vorticity-free current. The energy flux and the set-down have been used for shoaling predictions for finite amplitude waves with and without a net volume flux. The results with a zero volume flux are compared with more accurate shoaling predictions showing rather good accordance, except for large steepnesses. This also applies to the deep water wave energy transport velocity. The results with a net volume flux show that the steepness of the waves reduces the influence of this flux on the wave evolution. Some problems in connection with the orders in Stokes waves are discussed, among others concerning the dispersion relation and the orders of integral properties. Bed shear and accompanying dissipation is neglected.     

Interpretation of edge waves observations on Shikotan Island

The observations of short-period edge waves and accompanying morphological features of the coast line in the Dimitrov Bay on Shikotan Island on August 11, 2005, are presented. The wave field characteristics and form of the coastline are interpreted within the frameworks of the exact solutions of three-dimensional nonlinear equations for the waves propagating over a flat sloping bottom.     

A numerical experiment on nearshore circulation in standing edge waves

The unsteady shallow-water vorticity equation dominating nearshore flow on a gently sloping plane beach has been solved by using the implicit finite difference technique under the assumption of constant viscosity over the flow field. The result of computation showed that pairs of nearshore circulation cells are generated through the nonlinear effect of flow in the boundary layer formed by the run-up movement of a standing edge wave along a shoreline and the paired cell has the spacing of half a wavelength of the edge wave. When the leaky-mode standing wave of Lamb with the same wave period as the edge wave and the wave crest parallel to a shoreline was superposed on the edge wave field, the alongshore spacing of circulation cell doubled and seaward flow in the cell concentrated in the narrow zone like a ‘rip current’. Although no effect of breaking waves is considered in the computation, such a mechanism may also generate some kinds of nearshore circulation systems observed in a sea.     

The effect of the Airy phase during the propagation of edge waves

Peculiarities of the edge wave evolution a great distance away from the source are considered. It is shown that waveguide dispersion may result in discrimination of the region in a wave train corresponding to the Airy phase. The effect of the discrimination of the Airy wave in the remote zone for the model of a step-like shelf is compared with real edge waves (using the marigrams of tsunami observed in the Sea of Japan in 1983).Translated by Mikhail M. Trufanov.     

Shelf waves with diurnal tidal frequency at the Greenland shelf edge

Tidal analysis was carried out on current measurements at a “cross-shelf” transect off Greenland at 71°N. The diurnal tides manifest themselves mainly as a barotropic continental shelf wave, travelling southward along the shelf slope. This follows from the amplitude distribution of the diurnal tidal components and from the rotation sense of the tidal ellipses at different cross-slope locations, as calculated with simple two-dimensional models. The well organized cross-slope pattern of the velocity amplitudes is absent in observations further north near 75°N. These observations suggest that the local vanishing of the group velocity, which is caused by topography, is of importance for the existence and local amplification of these continental shelf waves with diurnal tidal frequency.     

Bound waves and triad interactions in shallow water

Boussinesq type equations with improved linear dispersion characteristics are derived and applied to study wave-wave interaction in shallow water. Weakly nonlinear solutions are formulated in terms of Fourier series with constant or spatially varying coefficients for two purposes: to derive higher order boundary conditions for regular and irregular wave trains and to derive evolution equations on constant or variable water depth. Wave transformation of monochromatic, bichromatic and irregular waves is studied and comparison with measurements and direct time domain solutions shows good agreement. The improvement relative to classical models from the literature is discussed.     

Hydrodynamic interactions between two ships advancing in waves

In this paper the hydrodynamic problems between two moving ships in waves are analyzed using a three-dimensional potential-flow theory based on the source distribution technique. The potential is presented by a distribution of source over the ship hull. The corresponding Green functions and their derivatives can be easily solved numerically by using the series expansions of Telste and Noblesse's algorithm for the Cauchy principal value integral of unsteady flow. The numerical solution is evaluated by applying the present method to two pairs of models and compared with experimental data and strip theory. From the comparisons, it shows that the hydrodynamic interactions are generally important. In the resonance region, the hydrodynamic interaction calculated by the 3D method is more reasonable, which is not so significant as that by the 2D method. The technique developed here may serve as a more rigorous tool to analyze the related problems of two ships doing underway replenishment in waves.     

Water levels in a dual-basin harbour in response to infragravity and edge waves

Seiche modes in a compound harbour (an “Outer Harbour” connected both to the sea and to an “Inner Harbour”) were studied using water level data and a numerical model. A variety of harbour oscillations are present, with periods up to 67 min. Periods longer than 25 min exceed resonant modes of the harbour. This paper addresses the characteristics and causes of the open-basin modes. The dual harbour open-basin mode is modified by constriction at the connection between harbours, by partial reflection at the antinode, and by the geometry of the entrance. The single-harbour open-basin mode excites the dual harbour closed-basin mode, which has nearly the same period. This forcing moves the closed-basin antinode and slightly changes the modal period, but the coupling permits the amplitude to increase through the closed-basin resonance. The water level response to wind stress is weak, but significant residual currents can occur, which take the form of clockwise gyres in each basin. Energetic peaks in the water level spectrum at 26, 35, and 67 min are shown to correspond to possible edge waves on the local shelf. The work has practical implications to port design, e.g. towards minimisation of ship ranging while at anchor.     

Amplitude variations of edge waves on a shelf slowly varying in the alongshore direction

Within the framework of the linear shallow-water theory, the dynamics of edge waves over a shelf characterized by a cylindrical bottom relief is investigated under the assumption that shelf parameters vary slowly in the alongshore direction. An asymptotic theory and an energy approach are used to calculate the amplitude of the edge wave. In the analytic form, the results are obtained for shelves of three different profiles with parameters varying along the shore: an infinite linear profile, a concave exponential profile, and a stepwise profile.     

Investigation of nonlinear interactions in ocean swell waves usingthe reduced bispectrum

In this paper, a two-dimensional (2-D) reduced bicoherence function which can be calculated via the reduced bispectrum is introduced. The statistical and computational aspects of the 2-D reduced bicoherence function are compared with those of the conventional bicoherence function. It is shown that the proposed technique is less noise-sensitive than the conventional bicoherence estimate. The use of the 2-D reduced bicoherence function to investigate nonlinear interactions in ocean swell waves is also demonstrated. As the 2-D reduced bicoherence can be efficiently calculated, it is proposed to use this novel technique in oceanographic field experiments that require real-time data processing and interpretation     

Inverse modeling of discrete interaction approximations for nonlinear interactions in wind waves

In the last two decades, the Discrete Interaction Approximation (DIA) has been the only economically feasible parameterization for nonlinear wave–wave interactions in operational wind wave models. Its major drawback is its limited accuracy. Several improvements to the DIA have been suggested recently. The present study summarizes these improvements and suggests some new modifications to the DIA. Using inverse modeling techniques, where the potential of various DIAs is assessed by optimal fitting to exact solutions, a comprehensive comparison of the potential of several such improvements is made. An in depth analysis of the behavior of DIAs in full wave models will be the subject of a second study, to be reported elsewhere. The major findings of this study are that: (i) An expanded definition of the representative quadruplet with additional degrees of freedom is necessary for an accurate representation of the exact interactions; (ii) Slowly varying the free parameters in such a DIA as a function of the spectral frequency f results in a (mostly qualitative) improvement; (iii) A DIA with expanded quadruplet definition and with four representative quadruplets is found to reproduce the exact source term accurately; (iv) Adding additional tunable constants to the equation for the strength of the interactions has little impact on the quality of the DIA.     

Resonance excitation of internal waves by fluctuations of the atmospheric pressure and the tangential wind stress

Resonance excitation of internal waves by running waves of the atmospheric pressure and the tangential wind stress in a continuously stratified rotating ocean is studied in a linear statement. A comparative analysis of the rates of the increase in amplitude of resonance-excited waves for these external disturbances is carried out numerically for the case of the average distribution of the Väisälä-Brunt frequency in the Caribbean Sea.Translated by Mikkail M. Trufanov.     

Stabilization of a linearly unstable wave participating in a three-wave resonant interaction

An analytical study of the influence of three-wave resonant interactions on the evolution of unstable wave disturbances is presented in the Kelvin-Helmholtz model. These results may be of interest in analyzing the dynamics of disturbances at the ocean-atmosphere interface and in two-layer flows which arise in the ocean and are characterized by large gradients of flow velocity at the boundary of layers. In the case under consideration, the instability arises when eigenfrequencies coincide in the framework of a single mode and the instability is algebraic. The amplitudes of the two other interacting stable waves are assumed to be small compared to the amplitude of the third, unstable, mode. The system of amplitude equations for this case is investigated using the WKB method. As a result, we obtain the formulas coupling the solutions for the time before and after a transition through a singular point, where the amplitude of the linearly unstable wave has a local minimum. These formulas give the rule of transformation of the parameter that characterizes a phase shift between fast and slow modes and determines the behavior of the system. It is shown that, in a transition through a singular point, this parameter changes randomly. As long as the parameter is positive, the amplitude of the linearly unstable wave remains limited and oscillates stochastically. In a transition of the parameter through zero, we exit the stabilization region and have an infinite growth of amplitude. The transition into the instability region is random. However, if the time interval where the amplitude remains limited is large enough, the scenario of the behavior of the system we have obtained can be treated as the partial stabilization of instability. The results make it possible for us to investigate the stochasticity caused by the nonlinear interaction of gravity-capillary waves in a two-layer model of a shear flow. These results are also of interest in analyzing secondary flows in laboratory facilities modeling the ocean and atmospheric processes.     

Observation of edge waves trapped on the continental shelf in the vicinity of Makurazaki Harbor, Kyushu, Japan

Sea-bottom pressure gauges were used to measure sea levels at two points on the shelf off the southern coast of Satsuma Peninsula, Kyushu, Japan. Spectral analysis of the observed records and the tide-gauge record of Makurazaki Harbor revealed several predominant common peaks. At the same time, the eigenmodes for the trapped waves on the shelf and inside Makurazaki Bay were obtained numerically using a two-dimensional model, and the periods and the spatial distribution of amplitudes of the proper modes were obtained. A comparison of the calculated modes with the periods and phase patterns of the observed peaks clarified that peaks with periods of 19.5, 16, 13.3, and 12.2 minutes in the shelf region were the modes of standing-edge waves, and the peak with the period of 16 minutes in Makurazaki Harbor was the fundamental mode of the harbor. Among the modes of standing-edge waves, the mode of the period 16 minutes on the shelf had nearly the same period as that of the fundamental mode of Makurazaki Harbor. An analysis of changes of spectral densities of these two modes confirmed that the fundamental mode of the Makurazaki Harbor was induced by this standing-wave mode.     

Simulation of the nonlinear dynamic interactions between waves, a submerged breakwater and the seabed

This study employed direct numerical simulation to simulate the fully nonlinear interaction between the water waves, the submerged breakwater, and the seabed under differing wave conditions. In the numerical simulation, the laminar flow condition in the seabed was applied to evaluate the more exact fluid resistance acting on the porous media. Varying incident wave conditions were applied to the flow field resulting from the wave–structure–seabed interaction, and the variation in the pore water pressure beneath the submerged breakwater was investigated along the cross-section of the submerged breakwater. Structural safety and scouring were also considered on the basis of the numerical results for the flow field around the structure and the variation of the pore water pressure.