Experimental Study of Wave Breaking on Gentle Slope

—An experimental study of regular wave and irregular wave breaking is performed on a gentleslope of 1:200.In the experiment,asymmetry of wave profile is analyzed to determine its effect on wavebreaker indices and to explain the difference between Goda and Nelson about the breaker indices of regu-lar waves on very mild slopes.The study shows that the breaker index of irregular waves is under less influ-ence of bottom slope i,relative water depth d/L_0 and the asymmetry of wave profile than that of regularwaves.The breaker index of regular waves from Goda may be used in the case of irregular waves, whilethe coefficient A should be 0.15.The ratio of irregular wavelength to the length calculated by linear wavetheory is 0.74.Analysis is also made on the waveheight damping coefficient of regular waves after break-ing and on the breaking probability of large irregular waves.     

Numerical Prediction of Regular Wave Breaking on Very Gentle Slopes

Regular wave deformation and breaking on very gende slopes is calculated by Mixed-Eulerian-Lagrangian procedure. The velocity potentials and their normal derivatives on the boundary are calculated through the mixed 0-1 boundary element method. The wave elevation and the potentials of time-stepping integration are detertnined by the 2nd-order Taylor expansion at the nodes of free surface boundary elements. During calculation the x-coordinates of the free surface element nodes are supposed to remain unchanged, i.e. the partial derivatives of wave elevation and potentials with respect to x are considered as zero. The numerical results of asymmetric parameters of breaking waves are verified by experimental study. It is shown that when the wave asymmetry is weak, the maximum horizontal velocity of water particales occurs at the wave peak and, the average ratio of this maximum velocity to wave celerity is 0.96. However, when the wave asymmetry is strong, the maximum horizontal velocity of water particles occu     

浅水破碎波对直立圆柱作用力的试验研究

分析了桩柱破波力的构成特性,认为破碎波与桩柱作用过程中在不完全绕流区将出现由于附连质量迅速变化引起的冲击力和柱前后波面高度差引起的附加压差力;影响破碎波对直立圆柱作用力的因素应该包括确定破波特性的水底坡度i、入射深水波陡H_0/L_0及代表圆柱对破波作用影响的相对柱径D/H_0。通过从i=1/100到1/15五种底坡上桩柱破波力的系统试验,探索了桩柱破波力的作用特性,归纳了计算桩柱破波力的经验公式。     

Examination of empirical formulas for wave shoaling and breaking on steep slopes

Estimation of the wave height transformation of shoaling and breaking is essential for the nearshore hydrodynamics and the design of coastal structures. Many empirical formulas have been well recognized to the wave height transformation, but most of them were only applicable for gentle slopes. This paper reports the experimental results of wave shoaling and breaking over the steep slopes to examine the applicability of the previous empirical formulas. Two steep bottom slopes of 1/3 and 1/5, and one gentle slope of 1/10 were conducted in the present experiments. Experimental results show that the shoaling distance of steep slopes become short and the surface waves may be partially reflected from the steep bottom, thus the estimation of wave shoaling using the well-known previous formula did not conform completely to the experimental results. The previous empirical formulas for the wave breaking criteria were also examined, and the modified equations to the steep beaches were proposed in this work. A numerical model was finally adopted to calculate the wave height transformation in the surf zone by introducing the modified breaking index.     

Breaking of Solitary Waves on Uniform Slopes

The propagation,shoaling and breaking of solitary waves on mild slopes are simulated byboundary element method.In this paper,the criterion of breaking solitary waves on mild slopes is discussed.The criterion is that the ratio of horizontal velocity of water particles on the wave crest to wave celerity equalsone.However,the case that the ratio of horizontal velocity of water particles on the wave crest to wave ce-lerity is below one but the front face of wave profile becomes vertical is also considered as a breaking criteri-on.According to the above criteria,the breaking index for slopes 1:10 to 1:25 is studied.The result is com-pared to other researchers'.The deformation of solitary waves on slopes is discussed and the distribution offluid velocities at breaking is shown.     

Wave Breaking Phenomena of Irregular Waves Combined with Opposing Current

- Experimental study and theoretical analysis show that the critical value of relative wave height (H/ d)b given by Goda and the critical wave steepness (H/ L)b given by Michell and Miche can be adopted as the spilling breaking indices of regular waves. According to the same principle, a systematic theoretical analysis and experiment of irregular waves have been done by the authors in order to solve the breaking problem of irregular waves. It is indicated that the authors' method for determining wave breaking of regular waves can also be used for irregular waves.     

南沙群岛礁概化地形上拍岸浪数值模拟与统计分析

南沙群岛珊瑚岛礁众多,大多数岛礁具有向海坡陡峭、外礁坪比较平缓的特征。将南沙群岛岛礁的迎浪向地形概化为陡坡和缓坡组成的双斜坡,采用FUNWAVE-TVD模式数值模拟概化地形上的波浪,根据模拟的破碎波高分析其拍岸浪特征。对拍岸浪数值模拟结果进行比较分析,向海坡的坡度对拍岸浪影响不大,外礁坪上拍岸浪高随地形坡度增大而略有增大;向海坡和外礁坪交界位置(即坡折点)水深对拍岸浪有比较明显的影响,拍岸浪高随坡折点水深增大而减小;拍岸浪高随入射波高和波周期增大而增大。利用大量的拍岸浪数值模拟数据对国内外5种统计模型进行检验,并且基于拍岸浪数值模拟数据建立了3种南沙群岛岛礁拍岸浪统计模型,计算结果显示这些模型适用性较好。     

Modeling of nonlinear wave propagation over fringing reefs

The applicability of existing nonlinear (triad) spectral models for steep slopes (0.1–0.2) characteristic of reef environments was investigated, using both deterministic (phase-resolving) and stochastic (phased-averaged) formulations. Model performance was tested using laboratory observations of unidirectional wave transformation over steep and smooth bathymetry profiles. The models, developed for mild slopes, were implemented with minimal modifications (the inclusion of breaking parametrizations and linear steep-slope corrections) required by laboratory data. The deterministic model produced typically more accurate predictions than the stochastic one, but the phase averaged formulation proved fast enough to allow for an inverse modeling search for the optimal breaking parametrization. The effects of the additional assumptions of the stochastic approach resulted in a slower than observed evolution of the infragravity band. Despite the challenge posed by the fast wave evolution and energetic breaking characteristic to the steep reef slopes, both formulations performed overall well, and should be considered as good provisional candidates for use in numerical investigation of wave–current interaction processes on steep reefs.     

规则波在缓坡上变形与破碎数值分析

通过在二维数值水槽内用边界元法直接求解Laplace方程,对规则波在缓坡上的变形及破碎进行了数值计算。分析了不同底坡及采用不同底摩阻系数时规则波的破碎特征,并对规则波破碎的极限坡度进行了研究。重点分析了规则波破碎时海底坡度、底摩阻系数及波形不对称性对破碎指标的影响。     

The Relative Trough Froude Number for initiation of wave breaking: Theory,experiments and numerical model confirmation

It is important to accurately locate the wave breaking region for the calculation of nearshore hydrodynamics. Energy from breaking waves drives hydrodynamic phenomena such as wave set-up, set-down, wave run-up, longshore currents, rip currents, and nearshore circulation. Numerous studies have been undertaken to describe when and where wave breakings occurs. Recent development of computer resources permits the use of phase-resolving numerical models for the study of wave propagation, transformation, and nearshore hydrodynamics. This requires new types of wave breaking criteria for the numerical model. The Relative Trough Froude Number (RTFN) is a new wave breaking criterion. This model is based on the moving hydraulic jump concept, therefore it satisfies properly posed boundary-value conditions. It has been experimentally proved that a critical RTFN at the initiation of wave breaking is consistent with and without the presence of an opposing current, but previous efforts did not investigate the theory for the critical value. This paper provides a theoretical analysis and a numerical analysis to demonstrate why the RTFN theory works as a wave breaking initiation (trigger) index. The theoretical analysis provides a universal constant for the initiation of wave breaking for all water depths assuming the Miche formula properly describes the wave breaking condition. A subroutine for wave breaking in a numerical model, FUNWAVE was modified to include the RTFN trigger. The numerical model was calibrated with data from wave tank experiments, and it was found that the critical condition is very close to the theoretical number, CTFN = 1.45. A second paper (in preparation) provides details of the theory and experiments for a second criterion for termination of wave breaking. The time scale for the establishment of the breaking region i.e., between the initiation position and termination position, depends upon the additional momentum present under turbulent condition within the breaking wave. This subject is not considered herein.     

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。     

Simplified Method and Improved Solution for Deriving Fifth Order Stokes Wave

As the solution of the two equations for determining the existing fifth order Stokes wave de-rived by Skjelbreia is complex and tedious,the two equations are simplified into one equation for deter-mining d/L,i.e.,f(H,T,d/L)=0.According to this simplified method,three cases of the solution forthe Skjelbreia equations have been found:one accurate solution;more than one accurate solution and noaccurate solution(but there exists the optimum approximate solution in the area of satisfying Skjelbreiaequations).As to the case of more than one accurate solution,the reasonable solution can be judged fromthe method of variational principle,by means of which an optimum solution improved from the solutionof Skjelbreia equations in the area of satisfying the original mathematical equations of non-vortex andnonlinear wave theory,i.e.,the optimum fifth order Stokes wave,is given.     

Wave Transformation and Breaking in Diagonal Opposing Current

- When waves propagate into diagonal opposing current from non-current area, not only the wave parameters but also the direction of wave propagation will change, that is, wave refraction will occur. The authors have calculated the changes of wave parameters, including wave refraction, by Linear Wave Theory, and have also done systematic study on wave transformation and breaking in opposing current by means of experimental analysis and theoretical calculation. In order to know the effect of wave refraction, computation is done in this paper about wave transformation and breaking on gentle slopes in diagonal opposing current.     

Wave Breaker Indices in Finite Water Depth

Based on the analysis and comparison of wave breaker indices defined by geometric, kinetic as well as dynamic stabilities and verified by observation, the value a, which is equal to H / Lthkd by Miche's result and may be modified by Goda's results, is" suggested as the wave breaking criteria. The applicable values of a for pure waves or wave-current co-existing field are given in this paper. They are smaller than Miche's result (0.142), and they have been verified by model tests.     

Wave modelling in the vicinity of submerged breakwaters

This paper describes a simple method for modelling wave breaking over submerged structures, with the view of using such modelling approach in a coastal area morphodynamic modelling system.A dominant mechanism for dissipating wave energy over a submerged breakwater is depth-limited wave breaking. Available models for energy dissipation due to wave breaking are developed for beaches (gentle slopes) and require further modifications to model wave breaking over submerged breakwaters.In this paper, wave breaking is split into two parts, namely: 1) depth-limited breaking modelled using Battjes and Janssen's (1978) theory [Battjes, J.A. and Jannsen, J.P.F.M. (1978). Energy loss and setup due to breaking of random waves. Proceedings of the 16th Int. Conf. Coast. Eng., Hamburg, Germany, pp. 569-587.] and 2) steepness limited breaking modelled using an integrated form of the Hasselmann's whitecapping dissipation term, commonly used in fully spectral wind–wave models. The parameter γ₂, governing the maximum wave height at incipient breaking (H_max = γ₂d) is used as calibration factor to tune numerical model results to selected laboratory measurements. It is found that γ₂ varies mainly with the relative submergence depth (ratio of submergence depth at breakwater crest to significant wave height), and a simple relationship is proposed. It is shown that the transmission coefficients obtained using this approach compare favourably with those calculated using published empirical expressions.     

The study on the bottom friction and the breaking coefficient for typhoon waves in radial sand ridges—the Lanshayang Channel as an example

Owing to the interactions among the complex terrain, bottom materials, and the complicate hydrodynamics, typhoon waves show special characteristics as big waves appeared at the high water level(HWL) and small waves emerged at low and middle water levels(LWL and MWL) in radial sand ridges(RSR). It is assumed that the mud damping, sandy bed friction and wave breaking effects have a great influence on the typhoon wave propagation in this area. Under the low wave energy, a mud layer will form and transport into the shallow area, thus the mud damping effects dominate at the LWL and the MWL. And high Collins coefficient(c around 1) can be applied to computing the damping effects at the LWL and the MWL. But under the high wave energy, the bottom sediment will be stirred and suspended, and then the damping effects disappear at the HWL. Thus the varying Collins coefficient with the water level method(VCWL) is implemented into the SWAN to model the typhoon wave process in the Lanshayang Channel(LSYC) of the RSR, the observed wave data under "Winnie"("9711") typhoon was used as validation. The results show that the typhoon wave in the RSR area is able to be simulated by the VCWL method concisely, and a constant wave breaking coefficient(γ) equaling 0.78 is better for the RSR where wide tidal flats and gentle bed slopes exist.     

Implementation and modification of a three-dimensional radiation stress formulation for surf zone and rip-current applications

Regional Ocean Modeling System (ROMS v 3.0), a three-dimensional numerical ocean model, was previously enhanced for shallow water applications by including wave-induced radiation stress forcing provided through coupling to wave propagation models (SWAN, REF/DIF). This enhancement made it suitable for surf zone applications as demonstrated using examples of obliquely incident waves on a planar beach and rip current formation in longshore bar trough morphology (Haas and Warner, 2009). In this contribution, we present an update to the coupled model which implements a wave roller model and also a modified method of the radiation stress term based on Mellor (2008, 2011a,b,in press) that includes a vertical distribution which better simulates non-conservative (i.e., wave breaking) processes and appears to be more appropriate for sigma coordinates in very shallow waters where wave breaking conditions dominate. The improvements of the modified model are shown through simulations of several cases that include: (a) obliquely incident spectral waves on a planar beach; (b) obliquely incident spectral waves on a natural barred beach (DUCK'94 experiment); (c) alongshore variable offshore wave forcing on a planar beach; (d) alongshore varying bathymetry with constant offshore wave forcing; and (e) nearshore barred morphology with rip-channels. Quantitative and qualitative comparisons to previous analytical, numerical, laboratory studies and field measurements show that the modified model replicates surf zone recirculation patterns (onshore drift at the surface and undertow at the bottom) more accurately than previous formulations based on radiation stress (Haas and Warner, 2009). The results of the model and test cases are further explored for identifying the forces operating in rip current development and the potential implication for sediment transport and rip channel development. Also, model analysis showed that rip current strength is higher when waves approach at angles of 5° to 10° in comparison to normally incident waves.     

波浪在珊瑚礁地形上破碎特性试验研究

对波浪在珊瑚礁地形上的传播特性进行了物理试验研究,将珊瑚礁地形简化为坡度为1∶5的陡坡(向海坡)加较长水平礁坪段的地形,对规则波和不规则波在该地形条件下的波浪破碎及波高沿程衰减进行了研究。结果表明,波高较小时,波浪破碎发生在礁坪上,但随着入射波高的增大,破碎位置逐渐向来浪方向移动,直至在向海坡段破碎。对于在礁坪上破碎的波浪,相对水深db/L0一定的条件下,破碎波高与入射波陡H0/L0相关,且变化趋势受相对水深db/L0的影响。同时给出了该地形条件下波浪破碎指标以及礁坪段破碎后沿程波高的计算公式。     

Control of pelagic sediment distribution by internal waves of tidal period: Possible interpretation of data from the southern East Pacific Rise

Sediment depositional patterns were observed on acoustic-reflection profiles at 36 and 42°S across the East Pacific Rise, near 100°W longitude. The sediment thickness as a function of distance from the crest shows a remarkable linearity on the east side of the rise, where the bottom topography is unusually subdued. The rate of sedimentation is 3.8 m/m.y at 42°S and 7.2 m/m.y. at 36°S. Disturbance to the even sedimentation appears to be correlated with topographic features more than 300 m high, and may therefore be associated with the interference between the barotropic tides and the topography. The group velocity of internal waves of semi-daily period is 15 cm sec⁻¹ in this area for a vertical wave number of 300 m⁻¹, and the characteristic slopes at 9° to the horizontal. The waves travel faster than the flow velocity of the tides and at an angle less than the slopes associated with the larger topographic features. Therefore a typical tidal velocity of 3 cm sec⁻¹ can be magnified substantially before the particle velocity approaches the group velocity and breaking occurs. Less magnification is possible near smaller topography because the group velocity is proportional to wavelength for internal waves of constant period. The tidal flow is magnified most near the boundary where the internal waves are reflected, and the higher velocities should cause settling sediment particles to remain in suspension locally. Thick boundary layers caused by breaking and mixing can shield the smaller-scale topography from the tidal motion.     

沙坝海岸沿岸流速度剖面特征研究

通过对两个坡度沙坝地形沿岸流实验测量和基于能量方程的沿岸流数值模拟,研究了沙坝海岸平均沿岸流速度剖面的双峰剖面特征,重点分析了第二个峰值的特征和两峰值的比值。综合考虑入射波高、入射波类型和坡度对波生沿岸流垂直岸线速度剖面的影响。结果表明,平均沿岸流速度剖面出现双峰剖面特征:第一峰值发生在沙坝向岸侧面的中部,第二个峰值发生在靠近岸线处;同一坡度情况两个峰值的位置和比值,不受入射波类型、入射波高的影响。数值模型中包括了侧混、底摩擦和水滚等因素,其数值模拟结果和实验值拟合较好,并讨论了有无侧混和水滚对速度剖面的影响。