Statistical estimates of characteristics of long-wave run-up on a beach

A run-up of irregular long sea waves on a beach with a constant slope is studied within the framework of the nonlinear shallow-water theory. This problem was solved earlier for deterministic waves, both periodic and pulse ones, using the approach based on the Legendre transform. Within this approach, it is possible to get an exact solution for the displacement of a moving shoreline in the case of irregular-wave run-up as well. It is used to determine statistical moments of run-up characteristics. It is shown that nonlinearity in a run-up wave does not affect the velocity moments of the shoreline motion but influences the moments of mobile shoreline displacement. In particular, the randomness of a wave field yields an increase in the average water level on the shore and decrease in standard deviation. The asymmetry calculated through the third moment is positive and increases with the amplitude growth. The kurtosis calculated through the fourth moment turns out to be positive at small amplitudes and negative at large ones. All this points to the advantage of the wave run-up on the shore as compared to a backwash at least for small-amplitude waves, even if an incident wave is a Gaussian stationary process with a zero mean. The probability of wave breaking during run-up and the applicability limits for the derived equations are discussed.     

Non-linear effects of the statistical model of shallow-water wind waves

This study deals with the non-linear effects of shallow-water wind waves. It is assumed that wind waves are an ergodic, random process, quasi-normal, stationary in time and homogeneous in space. The probability density function of sea surface oscillations is approximated by Gram-Charlier series in a modified form due to Edgeworth. Based on the above assumptions and on the fundamental statistical and stochastical laws the author has obtained the following characteristics: the probability density functions of wave heights, wave phases and velocity of the point mean wave height and mean wave length. The derived formulae are compared by the experimental data of the international expedition ‘Lubiatowo-74’. The analysis carried out has shown that real waves and the Gaussian model contradict considerably.     

Simulation of nearshore wave processes by a depth-integrated non-hydrostatic finite element model

This paper presents CCHE2D-NHWAVE, a depth-integrated non-hydrostatic finite element model for simulating nearshore wave processes. The governing equations are a depth-integrated vertical momentum equation and the shallow water equations including extra non-hydrostatic pressure terms, which enable the model to simulate relatively short wave motions, where both frequency dispersion and nonlinear effects play important roles. A special type of finite element method, which was previously developed for a well-validated depth-integrated free surface flow model CCHE2D, is used to solve the governing equations on a partially staggered grid using a pressure projection method. To resolve discontinuous flows, involving breaking waves and hydraulic jumps, a momentum conservation advection scheme is developed based on the partially staggered grid. In addition, a simple and efficient wetting and drying algorithm is implemented to deal with the moving shoreline. The model is first verified by analytical solutions, and then validated by a series of laboratory experiments. The comparison shows that the developed wave model without the use of any empirical parameters is capable of accurately simulating a wide range of nearshore wave processes, including propagation, breaking, and run-up of nonlinear dispersive waves and transformation and inundation of tsunami waves.     

A Numerical Investigation of the Reduction of Solitary Wave Runup by A Row of Vertical Slotted Piles

To improve the current understanding of the reduction of tsunami-like solitary wave runup by the pile breakwater on a sloping beach, we developed a 3D numerical wave tank based on the CFD tool OpenFOAM in this study. The Navier Stokes equations were applied to solve the two-phase incompressible flow, combined with an LES model to solve the turbulence and a VOF method to capture the free surface. The adopted model was firstly validated with existing empirical formulas for solitary wave runup on the slope without the pile structure. It is then validated using our new laboratory observations of the free surface elevation, the velocity and the pressure around a row of vertical slotted piles subjected to solitary waves, as well as the wave runup on the slope behind the piles. Subsequently, a set of numerical simulations were implemented to analyze the wave reflection, the wave transmission, and the shoreline runup with various offshore wave heights, offshore water depths, adjacent pile spaces and beach slopes. Finally, an improved empirical equation accounting for the maximum wave runup on the slope was proposed by taking the presence of the pile breakwater into consideration.     

离岸流灾害水动力学过程的数值模拟研究

离岸流是近岸流的重要组成部分,当波浪受到特殊海滩地形的影响,会形成一股沿着离岸方向运动的高速水流,能够迅速将人带离海岸,对海滨安全造成威胁。为了深入探究离岸流的形成机理及水动力学特性,本文基于二阶Stokes波浪理论,采用了更为光滑的变截面沙坝模型,通过流体体积法捕捉自由液面,对离岸流进行三维数值模拟探究。本文重点分析了离岸流产生时流场的瞬时速度、时均速度、压强等不同参量的分布规律,结果显示在沙坝和海岸线之间,有一对方向相反的水循环体系;对比不同流层离岸流的速度,了解到波浪与离岸流的耦合作用;并探究了入射波波高对离岸流强度及分布区域的影响,深化了对离岸流水动力学过程的认识。     

Surf zone dynamics simulated by a Boussinesq type model. Part II: surf beat and swash oscillations for wave groups and irregular waves

This is the second of three papers on the modelling of various types of surf zone phenomena. In the first paper the general model was described and it was applied to study cross-shore motion of regular waves in the surf zone. In this paper, part II, we consider the cross-shore motion of wave groups and irregular waves with emphasis on shoaling, breaking and runup as well as the generation of surf beats. These phenomena are investigated numerically by using a time-domain Boussinesq type model, which resolves the primary wave motion as well as the long waves. As compared with the classical Boussinesq equations, the equations adopted here allow for improved linear dispersion characteristics and wave breaking is modelled by using a roller concept for spilling breakers. The swash zone is included by incorporating a moving shoreline boundary condition and radiation of short and long period waves from the offshore boundary is allowed by the use of absorbing sponge layers. Mutual interaction between short waves and long waves is inherent in the model. This allows, for example, for a general exchange of energy between triads rather than a simple one-way forcing of bound waves and for a substantial modification of bore celerities in the swash zone due to the presence of long waves. The model study is based mainly on incident bichromatic wave groups considering a range of mean frequencies, group frequencies, modulation rates, sea bed slopes and surf similarity parameters. Additionally, two cases of incident irregular waves are studied. The model results presented include transformation of surface elevations during shoaling, breaking and runup and the resulting shoreline oscillations. The low frequency motion induced by the primary-wave groups is determined at the shoreline and outside the surf zone by low-pass filtering and subsequent division into incident bound and free components and reflected free components. The model results are compared with laboratory experiments from the literature and the agreement is generally found to be very good. Finally the paper includes special details from the breaker model: time and space trajectories of surface rollers revealing the breakpoint oscillation and the speed of bores; envelopes of low-pass filtered radiation stress and surface elevation; sensitivity of surf beat to group frequency, modulation rate and bottom slope is investigated. Part III of this work (Sørensen et al., 1998) presents nearshore circulations induced by the breaking of unidirectional and multi-directional waves.     

Surf zone dynamics simulated by a Boussinesq type model. Part I. Model description and cross-shore motion of regular waves

This is the first of three papers on the modelling of various types of surf zone phenomena. In this first paper, part I, the model is presented and its basic features are studied for the case of regular waves. The model is based on two-dimensional equations of the Boussinesq type and it features improved linear dispersion characteristics, possibility of wave breaking, and a moving boundary at the shoreline. The moving shoreline is treated numerically by replacing the solid beach by a permeable beach characterized by an extremely small porosity. Run-up of nonbreaking waves is verified against the analytical solution for nonlinear shallow water waves. The inclusion of wave breaking is based on the surface roller concept for spilling breakers using a geometrical determination of the instantaneous roller thickness at each point and modelling the effect of wave breaking by an additional convective momentum term. This is a function of the local wave celerity, which is determined interactively. The model is applied to cross-shore motions of regular waves including various types of breaking on plane sloping beaches and over submerged bars. Model results comprise time series of surface elevations and the spatial variation of phase-averaged quantities such as the wave height, the crest and trough elevations, the mean water level, and the depth-averaged undertow. Comparisons with physical experiments are presented. The phaseaveraged balance of the individual terms in the momentum and energy equation is determined by time-integration and quantities such as the cross-sectional roller area, the radiation stress, the energy flux and the energy dissipation are studied and discussed with reference to conventional phase-averaged wave models. The companion papers present cross-shore motions of breaking irregular waves, swash oscillations and surf beats (part II) and nearshore circulations induced by breaking of unidirectional and multidirectional waves (part III).     

Theoretical Model and Random Simulation for Nonlinear Interaction of Irregular Waves with Vertical Wall

According to the theoretical solutions for the nonlinear three-dimensional gravity surface waves and their interactions with vertical wall previously proposed by the lead author, in this paper an exact second-order random model of the unified wave motion process for nonlinear irregular waves and their interactions with vertical wall in uniform current is formulated, the corresponding theoretical nonlinear spectrum is derived, and the digital simulation model suitable to the use of the FFT (Fast Fourier Tansform) algorithm is also given. Simulations of wave surface, wave pressure, total wave pressure and its moment are performed. The probability properties and statistical characteristics of these realizations are tested, which include the verifications of normality for linear process and of non-normality for nonlinear process; the consistances of the theoretical spectra with simulated ones; the probability properties of apparent characterstics, such as amplitudes, periods, and extremes (maximum and minimum, positive and negative extremes). The statistical analysis and comparisons demonstrate that the proposed theoretical and computing models are realistic and effective, the estimated spectra are in good agreement with the theoretical ones, and the probability properties of the simulated waves are similar to those of the sea waves. At the same time, the simulating computation can be completed rapidly and easily.     

基于LPB1-2型声学测波仪资料的近岸浪特征分析

为研究近岸浪的形态特征,利用LPB1-2型声学测波仪在某海域所测得的海浪数据对近岸浪特征要素进行统计分析,利用最大"熵"原理推导出了波高的最大"熵"分布,研究了状态参量对波高分布和波高"熵"的影响。研究表明,该海域的波主要以波高0.3~0.7 m,周期2.5~5.5 s的波为主,表示离散程度的特征值平均差为0.267,日极差值主要在0.1~0.4 m区间内,最大值为1.0 m,分布较为集中,波高分布近似为正态分布。     

Probability distributions for maximum wave and crest heights

The paper discusses short- and long-term probability models of ocean waves. The Gaussian theory is reviewed, and nonlinear short-term probability distributions are derived from a narrow band second-order model. The nonlinearity has different impact on different measurement techniques, and this is further demonstrated for wave data from the WAVEMOD Crete measurement campaign and laser data from the North Sea. Finally, we give some examples on how the short-term statistics may be used to estimate the probability distributions for the maximum waves during individual storms as well as in a wave climate described by long-term distributions.     

Nonlinear effects in the process of propagation of internal waves in the presence of turbulence

In the Boussinesq approximation, we study the nonlinear effects observed in the process of propagation of internal waves in the presence of turbulence. The space damping factor of the waves is evaluated. The Stokes drift velocity and the Euler velocity of the mean current induced by waves due to the presence of nonlinearity are determined. It is shown that the principal contribution to the wave transfer is made by the horizontal velocity of the induced current. The Stokes drift is significant only near the bottom. The vertical component of the Stokes drift velocity obtained with regard for the turbulent viscosity is nonzero.     

Computation of solitary waves during propagation and runup on a slope

A numerical time-simulation algorithm for analysing highly nonlinear solitary waves interacting with plane gentle and steep slopes is described by employing a mixed Eulerian–Lagrangian method. The full nonlinear free surface conditions are considered here in a Lagrangian frame of reference without any analytical approximations, and thus the method is valid for very steep waves including overturning. It is found that the runup height is crucially dependent on the wave steepness and the slope of the plane. Pressures and forces exerted on impermeable walls of different inclinations (slopes) by progressive shallow water solitary waves are studied. Strong nonlinear features in the form of pronounced double peaks are visible in the time history of pressure and force signals with increasing heights of the oncoming solitary waves. The effect of nonlinearity is less pronounced as the inclination of the wall decreases with respect to the bottom surface.     

Wave Run-up on A Vertical Seawall Protected by An Offshore Submerged Barrier

Submerged barriers are constructed in coastal zones for shoreline or harbor protection or to prevent the beach erosion. In the present study, the wave run-up on a vertical seawall protected by a submerged barrier is analyzed. The physical configurations include a rigid barrier and a long channel of finite depth. For linear water waves, by matching the velocity along the barrier and along the gap, the systems of linear equations about the velocity potentials are obtained. The wave run-up is further analyzed for various settings of barrier height and distance between the barrier and the wall, i.e. the chamber length. For nonlinear waves and random sea waves, a numerical model is extended to investigate the effect parameters of the barrier on the wave run-up against the seawall. Not only the numerical simulations, but also the analytical results illustrate that the wave run-up on the seawall depends very much on the distance between the barrier and the vertical seawall.     

Finite-amplitude analysis of some Boussinesq-type equations

Recent progress in formulating Boussinesq-type equations includes improved features of linear dispersion and higher-order nonlinearity. Nonlinear characteristics of these equations have been previously analysed on the assumption of weak nonlinearity, being therefore limited to moderate wave height. The present work addresses this aspect for an important class of wave problems, namely, regular waves of permanent form on a constant depth. Using a numerical procedure which is valid up to the maximum wave height, permanent-form waves admitted by three sets of advanced Boussinesq-type equations are analysed. Further, the characteristics of each set of the Boussinesq-type equations are discussed in the light of those from the potential theory satisfying the exact free-surface conditions. Phase velocity, amplitude dispersion, harmonic amplitudes (namely, second and third) and skewness of surface profile are shown over a two-parameter space of frequency and wave height.     

Nonlinear effects on the distribution of crest-to-trough wave heights

The statistical distribution of the crest-to-trough heights of narrowband nonlinear sea waves is derived in a semi-closed form. A quantitative comparison of the resulting density and exceedance probability distributions with those of the linear theory is given. It is shown that the nonlinearity of waves, even with steepnesses typical of extreme sea states, has an insignificant influence on the distribution of crest-to-trough heights.     

Effects of spectrum band width on the distribution of wave heights and periods

The statistical distribution of the crest-to-trough heights of narrowband nonlinear sea waves is derived in a semi-closed form. A quantitative comparison of the resulting density and exceedance probability distributions with those of the linear theory is given. It is shown that the nonlinearity of waves, even with steepnesses typical of extreme sea states, has an insignificant influence on the distribution of crest-to-trough heights.     

Stem waves along vertical wall due to random wave incidence

This study investigates stem waves, propagating along a vertical wall, due to obliquely incident random waves through laboratory experiments and numerical simulations. Attention is paid to the difference or similarity between the stem waves due to periodic waves and random waves, the nonlinear and linear characteristics, and the effect of wave breaking on the evolution of stem waves. The following were found from this study: as the incident angle of waves become large or the nonlinearity of the incident waves become small, the significant stem wave height, normalized by the incident significant wave height, becomes large. This tendency is the same as that generated by the Stokes waves or cnoidal waves. However, regardless of the nonlinearity of incident waves, the width of stem waves is almost the same. This is a different point between the stem waves due to periodic and random waves. The wave breaking suppresses the growth of the stem waves.     

Effect of Nonlinearity on Groupiness of Random Wave Fields

Theoretical studies so far on random wave groups have all been in linear ways.Methods to sim-ulate random wave groups,an important subject in ocean engineering,also employ relationship resulting froma Gaussian process.Many filed measurements have shown that the real sea surfase displacement deviatessomewhat from Gaussian distribution.Tayfun et al.have further depicted in theory that the envelope spectralpeak frequency is constantly zero for a Gaussian process which means that the groupiness factors will be con-stants,too.In this paper,the effect of nonlinearity on groupiness of a random wave field is examined via thetheoretical results derived by Tayfun et al.from an expression of amplitude-modulated Stokes waves.Whenthe surface displacement is treated as a non-Gaussian process,it is found that the group height factors GF_1and GF_2 proposed by Zhao et al.and Yu et al.,respectively,depend on a nonlinearity factor as well as aspectrum-bandwidth factor,deferring from the case of a Gaussion process.Compariso     

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。     

Analysis of the extreme wave elevation due to second-order diffraction around a vertical cylinder

Statistical analysis of nonlinear random waves is important in coastal and ocean engineering. One approach for modeling nonlinear waves is second-order random wave theory, which involves sum- and difference-frequency interactions between wave components. The probability distribution of the non-Gaussian surface elevation can be solved using a technique developed by Kac and Siegert [21]. The wave field can be significantly modified by wave diffraction due to a structure, and the nonlinear diffracted wave elevation can be of interest in certain applications, such as the airgap prediction for an offshore structure. This paper investigates the wave statistics due to second-order diffraction, motivated by the scarcity of prior research. The crossing rate approach is used to evaluate the extreme wave elevation over a specified duration. The application is a bottom-supported cylindrical structure, for which semi-analytical solutions for the second-order transfer functions are available. A new efficient statistical method is developed to allow the distribution of the diffracted wave elevation to be obtained exactly, accounting for the statistical dependency between the linear, sum-frequency and difference-frequency components. Moreover, refinements are proposed to improve the efficiency for computing the free surface integral. The case study yields insights into the problem. In particular, the second-order nonlinearity is found to significantly amplify the extreme wave elevation, especially in the upstream region; conversely, the extreme elevation at an oblique location downstream is attenuated due to sheltering effects. The statistical dependency between the linear and sum-frequency components is also shown to be important for the extreme wave statistics.