An explicit approximation to the wavelength of nonlinear waves

An explicit and concise approximation to the wavelength in which the effect of nonlinearity is involved and presented in terms of wave height, wave period, water depth and gravitational acceleration. The present approximation is in a rational form of which Fenton and Mckee's (1990, Coastal Engng 14, 499–513) approximation is reserved in the numerator and the wave steepness is involved in the denominator. The rational form of this approximation can be converted to an alternative form of a power-series polynomial which indicates that the wavelength increases with wave height and decreases with water depth. If the determined coefficients in the present approximation are fixed, the approximating formula can provide a good agreement with the wavelengths numerically obtained by Rienecker and Fenton's (1981, J. Fluid Mech. 104, 119–137) Fourier series method, but has large deviations when waves of small amplitude are in deep water or all waves are in shallow water. The present approximation with variable coefficients can provide excellent predictions of the wavelengths for both long and short waves even, for high waves.     

An explicit approximation to the wavelength of nonlinear waves

An explicit and concise approximation to the wavelength in which the effect of nonlinearity is involved and presented in terms of wave height, wave period, water depth and gravitational acceleration. The present approximation is in a rational form of which Fenton and Mckee's (1990, Coastal Engng 14, 499–513) approximation is reserved in the numerator and the wave steepness is involved in the denominator. The rational form of this approximation can be converted to an alternative form of a power-series polynomial which indicates that the wavelength increases with wave height and decreases with water depth. If the determined coefficients in the present approximation are fixed, the approximating formula can provide a good agreement with the wavelengths numerically obtained by Rienecker and Fenton's (1981, J. Fluid Mech. 104, 119–137) Fourier series method, but has large deviations when waves of small amplitude are in deep water or all waves are in shallow water. The present approximation with variable coefficients can provide excellent predictions of the wavelengths for both long and short waves even, for high waves.     

非传统近似下海洋内波的一类WKB近似解

在非传统近似(即,包含地转水平分量在内的完整地转效应)条件下,用 WKB(Wentzel-Kramers-Brillouin)方法得到了密度连续分层海洋内波的一类 WKB近似解.为了检验所得到的 WKB 近似解的有效性,对WKB解各垂向速度模态与基于三点中心差分格式及QR算法的数值计算结果进行了详细比对,结果表明,当浮...     

稳恒水波的Fourier近似解研究

A computational method for steady water waves is presented on the basis of potential theory in the physical plane with spatial variables as independent quantities. The finite Fourier series are applied to approximating the free surface and potential function. A set of nonlinear algebraic equations for the Fourier coefficients are derived from the free surface kinetic and dynamic boundary conditions. These algebraic equations are numerically solved through Newton＇s iterative method, and the iterative stability is further improved by a relaxation technology. The integral properties of steady water waves are numerically analyzed, showing that （1） the set-up and the set-down are both non-monotonic quantities with the wave steepness, and （2） the Fourier spectrum of the free surface is broader than that of the potential function. The latter further leads us to explore a modification for the present method by approximating the free surface and potential function through different Fourier series, with the truncation of the former higher than that of the latter. Numerical tests show that this modification is effective, and can notably reduce the errors of the free surface boundary conditions.     

On nonlinear sea waves and the induced mean flow

The nonlinear modulation of water wave groups is investigated and the interaction equations with induced flows are obtained. The analysis is performed up to the third order of the wave steepness a by treating it as a small parameter in the singular perturbation technique by means of the Krylov-Bogoliubov-Mitropolski method. The equation which governs the development of the wave envelope is found by a modification of the ordinary nonlinear Schroedinger equation for the case of uniform depth. The equations governing the behavior of the induced mean flow are examined by deriving the second order flow when the form of the modulated wave train is prescribed. The present theory can describe the mean flow caused by the radiation stress. Some applications containing the monochromatic wave instability are given to confirm the theoretical results.An outline of this paper was presented at The Ocean Surface Symposium (Sendai, 1984).     

On the approximation of equatorially-trapped waves in the numerical models of ocean dynanics

Finite-differential schemes (continous in time) are studied for the shallow water equations on the equatorial -plane on the grids B and C. The B-grid spectrum is shown to be symmetrical about fourgrid interval waves. It results in the wrong sign of the group velocity in the ddomain of wavelengths less than four intervals of the grid. An analytical solution is given for the differential analogues of Kelving and Yanai waves on grid C. A conclusion is drawn on the advantage of grid, C and the necessary filtration of the high-frequency band of the spectrum for grid B, if the grid intervals are 100 km or less.Translated by Mikhail M. Trufanov     

On the quasi-geostrophic dynamics of a stratified two-component medium: The ocean

The quasi-geostrophic dynamics of a stratified two-component medium (salty sea water) are described by formulating a closed system of equations containing the temperature and salinity among the sought field variables. The corresponding system consists of the conservation laws of two Lagrange invariants, namely, the quasi-geostrophic potential vorticity and some “thermodynamic” invariant playing the role of a passive tracer. The temperature and salinity fields determined from the values of these invariants are separated into the density and density-compensated parts. In this case, the density part participates immediately in the dynamics, while the density-compensated part is simply transferred by the geostrophic velocity filed with no contribution to the density field. The system thus formulated is used to describe a number of specific features in the dynamics of thermohaline disturbances in zonal geostrophic flows. These features include the sharp-ening of the spatial gradients of the density-compensated distributions in shear currents and breaking of the initial temperature disturbance into two (density and density-compensated) wave packets propagating with different velocities.     

On the nonlinear hydrodynamic forces for a ship advancing in waves

In this paper, using a second-order steady-state approach and a three-dimensional (3D) pulsating source distribution method derives the nonlinear hydrodynamic forces on a ship advancing in waves. The nonlinear hydrodynamic forces considered here consist of the mean lateral drifting force and the added resistance, which can be expressed as products of the ship-motion responses, the radiation potential, diffraction potential and the incident-wave potential. All related velocity potentials applied in the calculations are in 3D form. The Series 60 and Marine ship hulls are used for numerical calculations and the results are compared with existing experimental data and two-dimensional (2D) solutions. The comparisons show that the results obtained in the paper generally agree with experimental data well. It is also found that the nonlinear hydrodynamic forces obtained based on the present 3D source distribution methods are indeed improved in some calculations compared with the 2D method, especially for the mean lateral drifting force.     

Improved explicit approximation of linear dispersion relationship for gravity waves: A discussion

Recently, an accurate explicit approximation to linear dispersion relationship is proposed based on Eckart's explicit relationship (Beji, 2013). The author has nicely improved Eckart's explicit dispersion relationship by introducing an empirical correction function. The resulting expression is valid for the entire range of relative water depths and accurate to within 0.044%.In this discussion, the proposed expression by the author is simplified and improved to an accuracy of 0.019%. Moreover, a near exact solution with 0.001% accuracy is also given.     

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.     

Influence of quasi-geostrophic currents and inertial waves on the elution of fine sediments in the Southeast Baltic

Numerical simulation based on the Princeton Ocean Model (POM) was performed for a region of the Southeast Baltic in order to compare data on the spatial distribution of velocity and bottom sediments. Special attention was focused on the influence of western and northeastern winds, which generate intense quasi-geostrophic currents can may cause very high velocities in the near bottom layer, which results in the elution of bottom sediments and transport of their fine fractions. An abrupt change in wind velocity intensifies the effect of elution due to generation of inertial internal waves that penetrate into the bottom layer. The spatial distributions of the velocity in the surface and near bottom layers are compared with data on bottom sediments. It turned out that areas with the highest velocities that formed under the effect of western and northeastern winds in most cases coincide with areas where bottom sediments are represented by coarse-grain fractions of gravel and sands.     

Dynamics of air-sea planetary boundary layer under the geostrophic momentum approximation condition

In this paper, the air-sea planetary boundary is divided into three layers. With the aid of geostrophic momentum approximation, wind and current profiles, surface wind, surface wind stress and Ekman pumpinii in the atmosphere as well as in the ocean affected by the atmospheric baroclini-city, stratification and nonlinear effects are investigated systematically for an ocean of infinite depth. Meanwhile, the characteristics of the air-sea interaction is analyzed.     

A model for the propagation of nonlinear surface waves over viscous muds

The effect of a thin viscous fluid–mud layer on nearshore nonlinear wave–wave interactions is studied using a parabolic frequency-domain nonlinear wave model, modified to incorporate a bottom dissipation mechanism based on a viscous boundary layer approach. The boundary-layer formulation allows for explicit calculation of the mud-induced wave damping rate. The model performed well in tests based on laboratory data. Numerical tests show that damping of high frequency waves occurs, mediated by “difference” nonlinear interactions. Simulations of 2-dimensional wave propagation over a mud “patch” of finite extent show that the wave dissipation causes significant downwave diffraction effects.     

On pressure forcing techniques for numerical modelling of unidirectional nonlinear water waves

A method of incorporating pressure forcing into a nonlinear potential flow wave model is presented. A semi-analytical pseudo-spectral method is used to calculate dynamic response of a water body exposed to evolving local pressure distribution. Surface slope coherent and slope proportional pressure functions are directly applied through a pressure term appearing in the dynamic free-surface boundary condition of a formulated initial boundary-value problem. First, a monochromatic pressure distribution is used to generate steady regular waves of permanent form. The pressure-induced wave motion exhibits stable harmonic structure for deepwater, transitional water and shallow water waves. In the next step, a more complex pressure system is used to initiate multi-component wave propagation. It is demonstrated that the proposed method provides well-posed initial conditions for studying various water wave scenarios within a framework of nonlinear potential flow solutions.     

非线性波浪时域计算的三维耦合模型

将计算区域Ω划分为内域Ω₁和外域Ω₂(Ω₂=Ω-Ω₁),外域控制方程采用改进线性频散特性的二维Boussinesq方程,用预报一校正法数值求解;结构物附近的内域控制方程为三维Navier-Stokes方程,由VOF方法数值求解。通过在外域和内域相匹配的交界面上设置合适的速度和波面边界条件,建立了三维非线性波浪时域计算的耦合模型。模拟试验表明:(1)耦合模型数值波浪水池可以产生稳定的、重复性较好的波动过程;(2)用耦合模型数值波浪水池求解较大浅水区域上的非线性波浪数值计算问题可以取得较高的计算效率,同时又能得出结构物附近的复杂流场。     

A second approximation to the time-mean lagrangian drift beneath a series of progressive gravity waves

A fourth-order solution is derived for the mean drift induced by a steady train of waves in water of constant depth. New measurements are carried out of the drift in the body of the fluid and the drift velocity gradient at the free surface. Comparison of theory and experiment shows significantly better agreement with the present fourth-order solution than with the previous second-order solution of Longuet-Higgins, M.S., 1953 [Phil. Trans. R. Soc.245, 535–581]. In particular, the present solution reproduces the observed tendency of the surface drift velocity to rise in shallow water and to level-off in very deep water.     

Study of nonlinear currents induced by coincident surface waves in liquid

This is an experimental study of the mixing induced by coincident surface waves in a liquid. The main mechanism leading to the emergence of mixing was shown to be the middle currents generated by coincident waves. The regime of these currents strongly depends on the amplitude of surface waves. For waves of near-critical amplitudes, an intense turbulization of middle currents is observed. Patterns of the velocity field were obtained using the Particle Imaging Velocimetry (PIV) technique for different amplitudes of surface waves. The results obtained can be used to estimate mixing in the near-surface oceanic layer.