C.C. Mei, The applied dynamics of ocean surface waves, Wiley, New York (1983) xvii + 740 pp., £ stg. 66.75.

A theoretical model is developed for wave heights and set-up in a surf zone. In the time-averaged equations of energy and momentum the energy flux, radiation stress and energy dissipation are determined by simple approximations which include the effect of the surface roller in the breaker and the actual shape of the waves. To the first approximation the roller represents a volume of water moving with the wave speed. It significantly changes both energy flux and radiation stress of the surf zone waves. The equations of energy and momentum are solved simultaneously to give the wave height variation and the set-up. Comparison with measurements shows good agreement. Also the transitions immediately after breaking are analyzed and shown to be in accordance with the above-mentioned ideas and results.     

Wave heights and set-up in a surf zone

A theoretical model is developed for wave heights and set-up in a surf zone. In the time-averaged equations of energy and momentum the energy flux, radiation stress and energy dissipation are determined by simple approximations which include the effect of the surface roller in the breaker and the actual shape of the waves. To the first approximation the roller represents a volume of water moving with the wave speed. It significantly changes both energy flux and radiation stress of the surf zone waves. The equations of energy and momentum are solved simultaneously to give the wave height variation and the set-up. Comparison with measurements shows good agreement. Also the transitions immediately after breaking are analyzed and shown to be in accordance with the above-mentioned ideas and results.     

近岸海浪几种数值计算模型的比较

除了统计模型，目前可用于近岸海浪计算的主要模型基本上可以分为Boussinesq方程、缓坡方程、能量平衡方程三大类。本文在对这三类模型作简要介绍的基础上，对它们用于近岸波浪计算各自存在的优缺点作了较为详细的比较，然后综合这三类模型的优点，根据目前我们所掌握的一些先进的海浪计算方法，提出一种比较实用的计算模型。     

Internal generation of waves: Delta source function method and source term addition method

In this study, we investigate two internal wave generation methods in numerical modeling of time-dependent equations for water wave propagation, i.e., delta source function method and source term addition method, the latter of which has been called the line source method in literatures. We derive delta source functions for the Boussinesq-type equations and extended mild-slope equations. By applying the fractional step splitting method, we show that the delta source function method is equivalent to the source term addition method employing the energy velocity. This suggests that the energy velocity should be used rather than the phase velocity for the transport of incident wave energy in the source term addition method. Finally, the performance of the delta source function method is verified by accurately generating nonlinear cnoidal waves as well as linear waves for horizontally one-dimensional cases.     

Improvements on Mean Free Wave Surface Modeling

Some new results of the modeling of mean free surface of waves or wave set-up are presented. The stream funetion wave theory is applied to incident short waves. The limiting wave steepness is adopted as the wave breaker indcx in the calculation of wave breaking dissipation. The model is based on Roelvink (1993), but the numerical techniques used in the solution are based on the Weighted-Average Flux (WAF) method (Watson et al. , 1992), with Time-Operator-Split-ting (TOS) used for the treatment of the source terms. This method allows a small number of eomputational points to be used, and is particularly efficient in modeling wave set-up. The short wave (or incident primary wave) energy equation is solved by use of a traditional Lax-Wendroff technique. The present model is found to be satisfactory compared with the measurements conducted by Stive (1983).     

High order Hamiltonian water wave models with wave-breaking mechanism

Based on the Hamiltonian formulation of water waves, using Hamiltonian consistent modelling methods, we derive higher order Hamiltonian equations by Taylor expansions of the potential and the vertical velocity around the still water level. The polynomial expansion in wave height is mixed with pseudo-differential operators that preserve the exact dispersion relation. The consistent approximate equations have inherited the Hamiltonian structure and give exact conservation of the approximate energy. In order to deal with breaking waves, we extend the eddy-viscosity model of Kennedy et al. (2000) to be applicable for fully dispersive equations. As breaking trigger mechanism we use a kinematic criterion based on the quotient of horizontal fluid velocity at the crest and the crest speed. The performance is illustrated by comparing simulations with experimental data for an irregular breaking wave with a peak period of 12 s above deep water and for a bathymetry induced periodic wave plunging breaker over a trapezoidal bar. The comparisons show that the higher order models perform quite well; the extension with the breaking wave mechanism improves the simulations significantly.     

Generation of waves in Boussinesq models using a source function method

A method for generating waves in Boussinesq-type wave models is described. The method employs a source term added to the governing equations, either in the form of a mass source in the continuity equation or an applied pressure forcing in the momentum equations. Assuming linearity, we derive a transfer function which relates source amplitude to surface wave characteristics. We then test the model for generation of desired incident waves, including regular and random waves, for both one and two dimensions. We also compare some model results with analytical solution and available experiment data.     

Non-linear propagation of unidirectional wave fields over varying topography

The present study aims at investigating the non-linear triad interaction process affecting shoaling surface gravity wave fields. The triad interaction phenomenon being enhanced towards the shore, the domain of study is extended up to the surf zone. Three 1D non-linear wave models (one phase-resolving and two phase-averaged spectral models) have been implemented and compared to laboratory experiments performed in a wave flume. This set of models includes two existing models and a new one which has been developed in the frame of this work. The models include a breaking dissipation term based on the parametrical model of Battjes and Janssen [Battjes, J.A., Janssen, P.A.E.M., 1978. Energy loss and set-up due to breaking of random waves. Proc. 16th Int. Conf. Coastal Eng. (ASCE), Vol. 1, pp. 569–587.]. The investigations concern the evolution of variance spectra, spectral significant wave height and mean period over a barred bathymetric profile. In addition, the performances of the different models are analysed by computing the spectral source term for triad interactions. We found that all models are able to reproduce the main features of non-linear mechanisms affecting a wave field in the near-shore zone. The phase-resolving model gives the most accurate results for non-breaking situations. It correctly reproduces the non-linear coupling effect in decreasing water depths due to wave–wave interactions, as well as the harmonic release after a bar. However, the model is computationally time-consuming. The CPU time is considerably reduced using phase-averaged models. They give satisfactorily results on harmonic generation. However, they do not reproduce the release of harmonics as water depth increases. In breaking conditions, the variance spectra undergo significant changes under the combined effects of non-linear energy transfers and dissipation. The depth-induced wave breaking model included in the equations provides a good estimate of the energy decay in the surf zone.     

Shock-capturing Boussinesq-type model for nearshore wave processes

This paper describes the formulation and validation of a nearshore wave model for tropical coastal environment. The governing Boussinesq-type equations include the conservative form of the nonlinear shallow-water equations for shock capturing. A Riemann solver supplies the inter-cell flux and bathymetry source term, while a Godunov-type scheme integrates the evolution variables in time. The model handles wave breaking through momentum conservation with energy dissipation based on an eddy viscosity concept. The computed results show very good agreement with laboratory data for wave propagation over a submerged bar, wave breaking and runup on plane beaches as well as wave transformation over fringing reefs. The model accurately describes transition between supercritical and subcritical flows as well as development of dispersive waves in the processes.     

波浪增减水的实用数学模型及其数值模拟

提出了一种模拟近岸区波浪增减水的实用数学模型.首先采用考虑波能损失的抛物型缓坡方程数值模拟波浪破碎引起的波浪复振幅变化,接着根据计算得到的波浪复振幅,采用一种新的辐射应力公式计算辐射应力分量,然后采用深度平均方程计算波浪破碎产生的增减水.采用该模型对规则波和不规则波破碎引起的增减水进行了数值模拟,数值模拟结果与实验结果吻合良好,表明该模型可有效模拟近岸区由于波浪破碎引起的增减水.     

Cross-shore hydrodynamics within an unsaturated surf zone

This paper concerns the hydrodynamics induced by random waves incident on a steep beach. New experimental results are presented on surface elevation and kinematic probability density functions, cross-shore variation in wave heights, the fraction of broken waves and velocity moments. The surf zone is found to be unsaturated at incident wave frequencies, with a significant proportion of the incident wave energy remaining at the shoreline in the form of bores. Wave heights in both the outer and inner surf zones are best described by a full Rayleigh distribution [Thornton, E.B., Guza, R.T., 1983. Transformation of wave height distribution. J. Geophys. Res. 88, 5925–5938], rather than a truncated Rayleigh distribution as used by Battjes and Janssen (1978) [Battjes, J.A, Janssen, J.P., 1978. Energy loss and setup due to breaking of random waves. Proc. 16th Int. Conf. Coastal Eng. ASCE, New York, pp. 569–588]. A new parametric wave transformation model is outlined which provides explicit expressions for the fraction of broken waves and the energy dissipation rate within the surf zone. On steep beaches, the model appears to offer improved predictive capabilities over the original Battjes and Janssen model. Cross-shore variations in the velocity variance and velocity moments are best described using Linear Gaussian wave theory, with less than 20% of the velocity variance in the inner surf zone due to low frequency energy.     

Numerical Simulation of Wave Height and Wave Set-Up in Nearshore Regions

Based on the time dependent mild slope equation including the effect of wave energy dissipation, an expression for the energy dissipation factor is derived in conjunction with the wave energy balance equation, and then a practical method for the simulation of wave height and wave set-up in nearshore regions is presented. The variation of the complex wave amplitude is numerically simulated by use of the parabolic mild slope equation including the effect of wave energy dissipation due to wave breaking. The components of wave radiation stress are calculated subsequently by new expressions for them according to the obtained complex wave amplitude, and then the depth-averaged equation is applied to the calculation of wave set-up due to wave breaking. Numerical results are in good agreement with experimental data, showing that the expression for the energy dissipation factor is reasonable and that the new method is effective for the simulation of wave set-up due to wave breaking in nearshore regions.     

On the application of a Boussinesq model to river flows including shocks

SERR-1D is a 4th-order finite volume 1D Boussinesq model including wave breaking energy dissipation through extra diffusive-like terms. This model has been primarily conceived to compute wave propagation in coastal areas and has been validated for breaking and non-breaking waves propagating over uneven bathymetries (Cienfuegos et al., 2005, 2006a, b, 2007). The present paper aims at investigating the ability of SERR-1D to simulate challenging fluvial hydraulic applications such as sudden gate operation in open channels generating short waves, dam-break flows and a steady hydraulic jump over a bump. The performance of the absorbing-generating boundary condition implemented in SERR-1D is first analysed in the context of fluvial applications where relatively short waves must be evacuated from the computational domain without producing spurious reflection. Next, by comparing numerical results to analytical and experimental dam-break test cases we show that the model is able to reproduce the overall features of these flows, but that additional care should be paid to the representation of energy dissipation and front speed in order to accurately represent bore dynamics.     

Energy dissipation in waves breaking on gentle slopes

The flow field of waves breaking on a gently sloping beach is shown to closely resemble that of hydraulic jumps. This supports the use of the hydraulic jump formulation for the breaking wave energy dissipation. A correction to this formulation, which takes into account the effects of turbulent flow, is found to explain the observed discrepancies between the classical theoretical result and the experiments satisfactorily. These findings are used to propose a simple, semi-empirical model for the wave height decay which includes the set-up. The model is generalized to a wider range of wave conditions by analyzing published data.     

A nearshore wave breaking model

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海岸波浪多次破碎波能耗散模型

在坡度很缓(接近或小于1∶100)的海岸，波浪在向海岸传播的过程中，可能经历多次破碎，而在两次波浪破碎之间将伴随着波浪恢复(波浪恢复到不破碎状态)。在现有海岸波高计算模型中，波浪破碎是通过波能耗散来模拟的，但所采用的波能耗散模型都不能自动考虑波浪出现多次破碎的过程，特别精确模拟这一过程中出现的波浪恢复。本文提出了解决这一问题的新的波能耗散模型，模型的建立是通过在Dally模型中重新建立稳定波能、饱和波高水深比和波能耗散系数，并引入了波浪恢复的判断条件实现的。该模型的波能耗散在波浪恢复区的值很小故能描述波浪恢复区的波浪运动。与实验结果的对比表明，新模型可以适合缓坡情况波浪多次破碎的波高模拟，而且对不同坡度的平坡和沙坝海岸(1∶100～1∶10)的破碎波模拟都可以给出与实验结果符合的结果，并且可以自动识别多次波浪破碎的存在和波浪恢复的发生。     

一种近岸区波浪破碎模型

从波浪破碎的能量关系入手,以紊流能量方程为基础,考虑破碎区内单个波在不同破碎阶段所提供的紊动能量强度的变化过程,提出了一种波浪破碎模式.通过将这一模型引入Boussinesq方程中,初步建立了一种近岸区波浪变形数学模型,并用波浪水槽实验资料对模型模拟波高和平均水位的情况进行了初步验证,得到了良好的结果.     

潜堤上规则波辐射应力的数值研究

在近岸波浪相关研究中,辐射应力是波动在水体中引起的剩余动量流,是波浪运动的重要物理量。在波浪从深水逐渐传向浅水的过程中,波浪的非线性逐渐增强,甚至会发生破碎等剧烈变形,引起辐射应力的强烈变化,对次重力波生成等有重要贡献。应用OpenFOAM精细模拟波浪在潜堤上的传播,得出波浪运动的详细流场信息,计算了有波浪破碎情况下潜堤地形上波浪的辐射应力和波浪增减水情况。研究结果表明,在潜堤地形下,辐射应力值在堤前平底处受波幅变化影响较敏感,波浪完全破碎后其值在堤后坡面处呈增大趋势直至平底处趋于平稳。辐射应力沿程变化对波浪增减水的影响趋势与波浪增减水方程符合良好。