Application of a nonlinear frequency domain wave–current interaction model to shallow water recurrence effects in random waves

The effect of ambient currents on nearshore nonlinear wave–wave energy transfer in random waves is studied with the use of a nonlinear frequency domain wave–current interaction model. We focus on the phenomenon of wave recurrence as a classical nonlinear phenomenon whose characteristics are well established for systems truncated to small numbers of frequency modes. The model used for this study is first extended to enhance accuracy; comparisons of permanent form solutions to analytical forms confirm the model accuracy. Application of the model to a highly truncated system confirmed the model’s consistency with published results for both positive (following) and negative (adverse) currents. Propagation of random wave spectra over a flat bottom was performed with the model, with the intent of determining the prevalence of recurrence between the spectral peak and its harmonics. For spectra of moderate Ursell number, it was found that positive currents extended the length scale of recurrence relative to the case with no currents; conversely, negative currents reduced the recurrence lengths. However, beyond a propagation distance of ≈40 wavelengths of the spectral peak, recurrence becomes almost completely damped as the spectra becomes broad and the spectral energies equilibrate. For spectra of high Ursell number, in contrast, recurrence is almost immediately damped, suggesting that the nonlinearity is sufficient to allow immediate spectral broadening and equilibration and overwhelming any preferential interactions among the spectral peak and its harmonics, regardless of current magnitude or direction.     

Separation of incident and reflected waves in wave–current flumes

A technique is developed to separate the incident and reflected waves propagating on a known current in a laboratory wave–current flume by analyzing wave records measured at two or more locations using a least squares method. It can be applied to both regular and irregular waves. To examine its performance, numerical tests are made for waves propagating on quiescent or flowing water. In some cases, to represent the signal noise and measurement error, white noise is superimposed on the numerically generated wave signal. For all the cases, good agreement is observed between target and estimation.     

Exact asymmetric slope distributions in stochastic Gauss–Lagrange ocean waves

The stochastic Lagrange wave model is a realistic alternative to the Gaussian linear wave model, which has been successfully used in ocean engineering for more than half a century. This paper presents exact slope distributions and other characteristic distributions at level crossings for symmetric and asymmetric Lagrange space and time waves. These distributions are given as expectations in a multivariate normal distribution, and they have to be evaluated by simulation or numerical integration. Interesting characteristic variables are: slopes obtained by asynchronous sampling in space or time, slopes in space or time, and horizontal particle velocity, when waves are observed when the water level crosses a predetermined level.     

Modelling linear wave transformation induced by dissipative structures—Random waves

The relevant theory is presented and numerical results are compared with the analytical solution for the interaction of non-breaking waves with an array of vertical porous circular cylinders on a horizontal bed. The extension to the cases of unidirectional and multidirectional waves is obtained by means of a transfer function. The influence of the mechanical properties of porous structures and wave irregularity on wave transformation is analysed. Results for unidirectional and multidirectional wave spectra are compared to those obtained for regular waves. The model presented reproduces well the analytical results and provides a tool for analysing several engineering problems.     

A function to determine wavelength from deep into shallow water based on the length of the cnoidal wave at breaking—Reply to discussion by T.S. Hedges

The equations of Hedges [Hedges, T.S., 2009. Discussion of “A function to determine wavelength from deep into shallow water based on the length of the cnoidal wave at breaking” by J.P. Le Roux, Coastal Eng.], although yielding similar wavelengths, are not consistent with the fact that the horizontal water particle velocity in the wave crest should equal the wave celerity at breaking over a nearly horizontal bottom.     

Euler–Lagrange equations for the spectral element shallow water system

We present the derivation of the discrete Euler–Lagrange equations for an inverse spectral element ocean model based on the shallow water equations. We show that the discrete Euler–Lagrange equations can be obtained from the continuous Euler–Lagrange equations by using a correct combination of the weak and the strong forms of derivatives in the Galerkin integrals, and by changing the order with which elemental assembly and mass averaging are applied in the forward and in the adjoint systems. Our derivation can be extended to obtain an adjoint for any Galerkin finite element and spectral element system.We begin the derivations using a linear wave equation in one dimension. We then apply our technique to a two-dimensional shallow water ocean model and test it on a classic double-gyre problem. The spectral element forward and adjoint ocean models can be used in a variety of inverse applications, ranging from traditional data assimilation and parameter estimation, to the less traditional model sensitivity and stability analyses, and ensemble prediction. Here the Euler–Lagrange equations are solved by an indirect representer algorithm.     

Calculation of a capsizing rate of a ship in stochastic beam seas

This study introduces a method of calculating a capsizing rate of a ship. The phenomenon ‘capsizing’ is described as a jump of local equilibrium point from that near the upright position of a ship to what describes the upside-down attitude of the capsized ship; the rate of occurrence of such jumps was calculated. The potential function corresponding to the roll restoring moment have two potential wells located at the roll displacement angle 0 and 180°, respectively. A nonlinear Fokker–Planck equation for the joint probability density function of roll angle and velocity was solved. The excitation to the ship was assumed to be a combination of a regular harmonic wave and a white noise process.     

Calibration and verification of a spectral wind–wave model for Lake Okeechobee

A spectral wind wave model SWAN (Simulation WAves Nearshore) that represents the generation, propagation and dissipation of waves was applied to Lake Okeechobee. This model includes the effects of refraction, shoaling, and blocking in wave propagation. It accounts for wave dissipation by whitecapping, bottom friction, and depth-induced wave breaking. The wave–wave interaction effect also is included in this model. Measurements of wind and wave heights were made at different stations and different time periods in Lake Okeechobee. Significant wave height values were computed from the recorded data. The correlation between wind stress and significant wave height also was analyzed. A 6-day simulation using 1989 data was conducted for model calibration. Another 6-day simulation using 1996 data was conducted for model verification. The simulated significant wave heights were found to agree reasonably well with measured significant wave heights for calibration and verification periods. Agreement between observed and simulated values was based on graphical comparisons, mean, absolute and root mean square errors, and correlation coefficient. Comparisons showed that the model reproduced both general observed trends and short term fluctuations.     

Validation of the three-wave quasi-kinetic approximation for the spectral evolution in shallow water

This paper aims at validating the three-wave quasi-kinetic approximation for the spectral evolution of weakly nonlinear gravity waves in shallow water. The problem is investigated using a one-dimensional numerical wave propagation model, formulated in the spectral representation. This model includes both a nonlinear triad interactions term and a wave breaking dissipation term. Some numerical tests were carried out in order to show the importance of using the triad nonlinear term in wave propagation spectral models, particularly to describe both behavior of the spectral integral parameters and of the spectral shape evolution in shallow water depth. Furthermore; a comparison against different set of experimental observations was carried out. Comparing the numerical results with the experimental observations made it possible to show the modeling efficiency of the three-wave quasi-kinetic approximation.     

Nonlinear progressive waves in water of finite depth — an analytic approximation

An analytical solution using homotopy analysis method is developed to describe the nonlinear progressive waves in water of finite depth. The velocity potential of the wave is expressed by Fourier series and the nonlinear free surface boundary conditions are satisfied by continuous mapping. Unlike the perturbation method, the present approach is not dependent on small parameters. Thus solutions are possible for steep waves. Furthermore, a significant improvement of the convergence rate and region is achieved by applying Homotopy-Padé Approximants. The calculated wave characteristics of the present solution agree well with previous numerical and experimental results.     

Effective boundary element method for the interaction of oblique waves with long prismatic structures in water of finite depth

An effective boundary element method (BEM) is presented for the interaction between oblique waves and long prismatic structures in water of finite depth. The Green's function used here is the basic Green's function that does not satisfy any boundary condition. Therefore, the discretized elements for the computation must be placed on all the boundaries. To improve the computational efficiency and accuracy, a modified method for treatment of the open boundary conditions and a direct analytical approach for the singularity integrals in the boundary integral equation are adopted. The present BEM method is applied to the calculation of hydrodynamic coefficients and wave exciting forces for long horizontal rectangular and circular structures. The performance of the present method is demonstrated by comparisons of results with those generated by other analytical and numerical methods.     

Note on the propagation of a shallow water wave in water of variable depth

In this technical note, the phenomena of non-linear water-wave propagation above a seabed with variable depth is re-examined. The conventional Korteweg-de Vries (KdV) equation is re-derived for the general case of variable water depth. In the new form of KdV equation, the seabed bottom function is included. Two different bottom profiles are considered in this study; case 1: $b^{\prime }\left(x^{\prime }\right)=c\epsilon \sin \lambda x^{\prime }$ and case 2: $b^{\prime }\left(x^{\prime }\right)=c\epsilon {\mathrm{e}}^{-\lambda {\left(x^{\prime }-x_0^{\prime }\right)}^2}$. The effects of three bottom profile parameters, c, λ and ε on the wave profile are examined. Numerical results indicate that both ε and λ affect the wave profile significantly in case 1, while ε significantly affects the wave profile in case 2.     

Numerical and experimental study on seakeeping performance of ship in finite water depth

Vessels operating in shallow waters require careful observation of the finite-depth effect. In present study, a Rankine source method that includes the shallow water effect and double body steady flow effect is developed in frequency domain. In order to verify present numerical methods, two experiments were carried out respectively to measure the wave loads and free motions for ship advancing with forward speed in head regular waves. Numerical results are systematically compared with experiments and other solutions using the double body basis flow approach, the Neumann-Kelvin approach with simplified m-terms, and linearized free surface boundary conditions with double-body m-terms. Furthermore, the influence of water depths on added mass and damping coefficients, wave excitation forces, motions and unsteady wave patterns are deeply investigated. It is found that finite-depth effect is important and unsteady wave pattern in shallow water is dependent on both of the Brard number τ and depth Froude number F_h.     

Solution of an unstable axisymmetric Cauchy–Poisson problem of dispersive water waves for a spectrum with compact support

 It has been known that the axisymmetric Cauchy–Poisson problem for dispersive water waves is well posed in the sense of stability. Thereby time evolution solutions of wave propagation depend continuously on initial conditions. However, in this paper, it is demonstrated that the axisymmetric Cauchy–Poisson problem is ill posed in the sense of stability for a certain class of initial conditions, so that the propagating solutions do not depend continuously on the initial conditions. In order to overcome the difficulty of the discontinuity, Landweber–Fridman's regularization, famous and well known in applied mathematics, are introduced and investigated to learn whether it is applicable to the present axisymmetric wave propagation problem. From the numerical experiments, it is shown that stable and accurate solutions are realized by the regularization, so that it can be applicable to the determination of the ill-posed Cauchy–Poisson problem.     

浅水极限波浪几何特征的实验研究

该文通过物理模型实验,对浅水区域内的波浪在破碎前极限状态下的几何特征进行了研究。实验基于JONSWAP谱对不规则波浪进行模拟,通过对波群中出现的单体极限波浪进行捕捉并对波形进行测量而得到研究样本。为了考察底坡因素对极限波浪几何特征的影响,实验共考虑了3组大小分别为β=1/15、1/30以及1/45的地形坡度。统计结果表明,在实验所采用的坡度范围内,当地波高与水深对近岸极限波浪的影响最为显著,随着水深与波高因素变化,极限波浪的几何特征也出现明显的改变。坡度因素对极限波陡和偏度的影响很小,可以被忽略,但是对不对称度参数的影响相对比较明显,坡度越陡,不对称程度越剧烈。最后,通过参数化,本文给出了极限波浪几何特征变化的经验公式。     

Nonlinear wave profiles of wave–wave interaction in a finite water depth by fixed point approach

In this paper, a superposition of two periodic wave profiles in a finite water depth was investigated. This paper is focused on the improvement of a wave profile on the linear superposition of two waves. This improvement was realized by introducing an iterative method, which was based on a fixed point approach. Application of the fixed point approach to the wave superposition made it possible to obtain a wave profile of wave–wave interaction. The improved result of the wave profile was in good agreement with that of the nonlinear perturbation solution of the second order. It was interesting that the improved result revealed the higher-order nonlinear frequencies for two interacting Stokes waves while Dalzell's solution by a perturbation method could not predict them.     

A coupled-mode model for the refraction–diffraction of linear waves over steep three-dimensional bathymetry

A consistent coupled-mode model recently developed by Athanassoulis and Belibassakis [1], is generalized in 2+1 dimensions and applied to the diffraction of small-amplitude water waves from localized three-dimensional scatterers lying over a parallel-contour bathymetry. The wave field is decomposed into an incident field, carrying out the effects of the background bathymetry, and a diffraction field, with forcing restricted on the surface of the localized scatterer(s). The vertical distribution of the wave potential is represented by a uniformly convergent local-mode series containing, except of the ususal propagating and evanescent modes, an additional mode, accounting for the sloping bottom boundary condition. By applying a variational principle, the problem is reduced to a coupled-mode system of differential equations in the horizontal space. To treat the unbounded domain, the Berenger perfectly matched layer model is optimized and used as an absorbing boundary condition. Computed results are compared with other simpler models and verified against experimental data. The inclusion of the sloping-bottom mode in the representation substantially accelerates its convergence, and thus, a few modes are enough to obtain accurately the wave potential and velocity up to and including the boundaries, even in steep bathymetry regions. The present method provides high-quality information concerning the pressure and the tangential velocity at the bottom, useful for the study of oscillating bottom boundary layer, sea-bed movement and sediment transport studies.     

On the modeling of wave propagation on non-uniform currents and depth

By transforming two different time-dependent hyperbolic mild slope equations with dissipation term for wave propagation on non-uniform currents into wave-action conservation equation and eikonal equation, respectively, shown are the different effects of dissipation term on the eikonal equation in the two different mild slope equations. The performances of intrinsic frequency and wave number are also discussed. Thus the suitable mathematical model is chosen in which the wave number vector and intrinsic frequency are expressed both more rigorously and completely. By using the perturbation method, an extended evolution equation, which is of time-dependent parabolic type, is developed from the time-dependent hyperbolic mild slope equation which exists in the suitable mathematical model, and solved by using the alternating direction implicit (ADI) method. Presented is the numerical model for wave propagation and transformation on non-uniform currents in water of slowly varying topography. From the comparisons of the numerical solutions with the theoretical solutions of two examples of wave propagation, respectively, the results show that the numerical solutions are in good agreement with the exact ones. Calculating the interactions between incident wave and current on a sloping beach [Arthur, R.S., 1950. Refraction of shallow water waves. The combined effects of currents and underwater topography. EOS Transactions, August 31, 549–552], the differences of wave number vector between refraction and combined refraction–diffraction of waves are discussed quantitatively, while the effects of different methods of calculating wave number vector on numerical results are shown.     

A 3-D model based on Navier–Stokes equations for regular and irregular water wave propagation

A spatial fixed σ-coordinate is used to transform the Navier–Stokes equations from the sea bed to the still water level. In the fixed σ-coordinate system only a very small number of vertical grid points are required for the numerical model. The time step for using the spatial fixed σ-coordinate is efficiently larger than that of using a time dependent σ-coordinate, as there is substantial truncation error involved in the time dependent σ-coordinate transformation. There is no need to carry out the σ-coordinate transformation at each time step, which can reduce computational times. It is important that wave breaking can be potentially modeled in the fixed σ-coordinate system, but in a time-dependent σ-coordinate system the wave breaking cannot be modeled. A projection method is used to separate advection and diffusion terms from the pressure terms in Navier–Stokes equations. The pressure variable is further separated into hydrostatic and hydrodynamic pressures so that the computer rounding errors can be largely avoided. In order to reduce computational time of solving the hydrodynamic pressure equation, at every time step the initial pressure is extrapolated in time domain using computed pressures from previous time steps, and then corrected in spatial domain using a multigrid method. For each time step, only a few of iterations (typically six iterations) are required for solving the pressure equation. The model is tested against available experimental data for regular and irregular waves and good agreement between calculation results and the measured data has been achieved.     

地形与流对水平无旋浅水波的影响

本文在水平无旋及Boussinesq假设之下,导出了水面变化与水平速度场耦合方程组,以及相应的压力与垂向速度解析表示式。通过数值方法求得水面变化及某一特定水深的水平速度分布之后,压力分布及其余水深的速度分布即可由简单计算得到。由色散关系式可知,不同水深的长波色散关系在O(ε)近似之下是等同的。粘性的存在会使波高随时间的增加而衰减,但粘性与底斜率的耦合又可能使波高增长,形成不稳定;计算及分析说明,当同向流增大时,波速增大,波长增大,波高会增加;而水深减少,使波速减少,波长缩短,振幅增加。