Shoaling Surface Gravity Waves Cause a Force and a Torque on the Bottom

Freely propagating surface gravity waves are observed to slow down and to stop at a beach when the bottom has a relatively gentle upward slope toward the shore and the frequency range of the waves covers the most energetic wind waves (sea and swell). Essentially no wave reflection can be seen and the measured reflected energy is very small compared to that transmitted shoreward. One consequence of this is that the flux of the wave’s linear momentum decreases in the direction of wave propagation, which is equivalent to a time rate of change of the momentum. It takes a force to cause the time rate of change of the momentum. Therefore, the bottom exerts a force on the waves in order to decrease the momentum flux. By Newton’s third law (action equals reaction) the waves then impart an equal but opposite force to the bottom. In shallow (but finite) water depths the wave force per unit bottom area is calculated, for normal angle of incidence to the beach, to be directly proportional to the square of the wave amplitude and to the bottom slope and inversely proportional to the mean depth; it is independent of the wave frequency. Constants of proportionality are: 1/4, the fluid density and the acceleration of gravity. Swell attenuation near coasts and some characteristics of sand movement in the near-shore region are not inconsistent with the algebraic structure of the wave force formula. Since the force has a depth variation which is significantly faster than that of the dimensions of the particle orbits in the vertical direction, the bottom induces a torque on the fluid particles that decreases the angular momentum flux of the waves. By an extension of Newton’s third law, the waves also exert an equal but opposite torque on the bottom. And because the bottom force on the waves exists over a horizontal distance, it does work on the waves and decreases their energy flux. Thus, theoretically, the fluxes of energy, angular and linear momentum are not conserved for shoaling surface gravity waves. Mass flux, associated with the Stokes drift, is assumed to be conserved, and the wave frequency is constant for a steady medium.     

未破碎变浅随机海浪的波高统计分布

利用青岛海洋大学海洋实验室现代化的大型水槽,设计进行了多种海浪强度下,由深水传入近岸不同坡度水底上的变浅随机海浪的模型实验。依据实验资料统计分析结果表明：对近岸变浅随机海浪而言,其波高分布不再符合Ｒａｙｌｅｉｇｈ分布,与Глуховский的经验分布也有差异,它不仅与参量Ｈ＊＝Ｈ／ｄ有关,且与表征深水海浪的波形特征量Ｈ０／Ｔ２０有关,并获得其经验关系,为实验应用变浅随机海浪的波高统计分布提供了可能。     

A nonlinear model for nonbreaking shoaling random waves

ＡｎｏｎｌｉｎｅａｒｍｏｄｅｌｆｏｒｎｏｎｂｒｅａｋｉｎｇｓｈｏａｌｉｎｇｒａｎｄｏｍｗａｖｅｓＬｉｕＸｉｎ＇ａｎ,ＨｕａｎｇＰｅｉｊｉ,ＣｈｅｎＸｕｅｙｉｎｇ,ＨｕＺｅｊｉａｎ（ＲｅｃｅｉｖｅｄＯｃｔｏｂｅｒ１５,１９９６,ａｃｃｅｐｔｅｄＡｕｇｕ...     

未破碎变浅随机海浪的非线性模式

利用实测和实验室数据分析了未破碎变浅海浪波面高度分布三阶矩(或波面偏度)相对于Hs/d的空间演化,利用本文的经验关系,在二阶近似下,给出了以Hs/d为参量的非线性波面表示模式,并推导了功率谱、二阶谱以及波面高度分布函数等特征量.文中还讨论了波面偏度和峰度的相关关系.     

Momentum balance in shoaling gravity waves: Comment on ’shoaling surface gravity waves cause a force and a torque on the bottom’ by K. E. Kenyon

In a recent paper, Kenyon (2004) proposed that the wave-induced energy flux is generally not conserved, and that shoaling waves cause a mean force and torque on the bottom. That force was equated to the divergence of the wave momentum flux estimated from the assumption that the wave-induced mass flux is conserved. This assumption and conclusions are contrary to a wide body of observations and theory. Most importantly, waves propagate in water, so that the momentum balance generally involves the mean water flow. Although the expression for the non-hydrostatic bottom force given by Kenyon is not supported by observations, a consistent review of existing theory shows that a smaller mean wave-induced force must be present in cases with bottom friction or wave reflection. That force exactly balances the change in wave momentum flux due to bottom friction and the exchange of wave momentum between incident and reflected wave components. The remainder of the wave momentum flux divergence, due to shoaling or wave breaking, is compensated by the mean flow, with a balance involving hydrostatic pressure forces that arise from a change in mean surface elevation that is very well verified by observations.     

Gravity torques for surface waves

The effects of the gravity torques acting on the angular momentum of surface gravity waves are calculated theoretically. For short crested waves the gravity torque is caused by the force of gravity on the orbiting fluid particles acting down the slopes of the crests and troughs and in the direction parallel to the crests and troughs. The gravity torque tries to rotate the angular momentum vectors, and thus the waves themselves, counterclockwise in the horizontal plane, as viewed from above, in both hemispheres. The amount of rotation per unit time is computed to be significant assuming reasonable values for the along-crest and trough slopes for waves in a storm area. The gravity torque has a frequency which is double the frequency of the waves. For long crested waves the gravity torque acts in the vertical plane of the orbit and tries to decelerate the particles when they rise and accelerate them when they fall. By disrupting the horizontal cyclostrophic balance of forces on the fluid particles (centrifugal force versus pressure force) the gravity torque accounts qualitatively for the three characteristics of breaking waves: that they break at the surface, that they break at the crest, and that the crest breaks in the direction of wave propagation.     

Internal solitary waves shoaling onto a shelf: Comparisons of weakly-nonlinear and fully nonlinear models for hyperbolic-tangent stratifications

In this study the evolution of internal solitary waves shoaling onto a shelf is considered. The results of high resolution two-dimensional numerical simulations of the incompressible Euler equations are compared with the predictions of several weakly-nonlinear shoaling models of the Korteweg–de Vries family including the Gardner equation and the cubic regularized long wave (or Benjamin–Bona–Mahoney) equation. Wave models in both physical x–t space and in s–x space are considered where s is a commonly used characteristic time variable. The effects of rotation, background currents and damping are ignored. The Boussinesq and rigid lid approximations are also used. The shoaling internal solitary waves generally fission into several waves. Reflected waves are negligible in the cases considered here. Several hyperbolic tangent stratifications are considered with and without a critical point. Among the equations in x–t space the cubic regularized long wave equation gives the best predictions. The Gardner equation in s–x space gives the best predictions of the shape of the leading waves on the shelf, but for many stratifications it predicts a propagation speed that is too large.     

未破碎变浅随机海浪的波面高度概率分布

利用青岛海洋大学物理海洋实验室现代化的大型水槽,设计进行了多种海浪强度下,由深水传入近岸不同坡度水底上的变浅随机海浪的实验.依据实验资料分析结果表明,对变浅非正态海浪过程而言,其波面高度分布取Gram-Charlier级数前3项,所得结果与实验分布符合良好.该分布中σ、λ₃、λ₄3个参量是测点水深和波浪强度的函数,并获得了与无因次参量Hs/d之间的经验关系,为预测变浅随机海浪的波面高度分布提供了可能.     

Burial and scour around short cylinder under progressive shoaling waves

The results of a laboratory experimental program aimed at better understanding the scour around and burial of heavy cylindrical objects under oscillating flow on a sandy bed are described. This study was motivated by its application to the dynamics of isolated cobbles/mines on a sandy floor under nonlinear progressive waves, such as that occur in shallow coastal waters beyond the wave-breaking region. In the experiments, nonlinear progressive waves were generated in a long wave tank of rectangular cross-section with a bottom slope. Model mines (short cylinders) were placed on the sandy bottom and the temporal evolution of the bed profile and the velocity field in the near field of the object were observed. Experiments were conducted at relatively high Reynolds numbers for a range of flow conditions, which can be characterized by the Keulegan–Carpenter number and Shields parameter. Depending on the values of these parameters, four different scour regimes around the cylinder including periodical burial of cylinder under migrating sand ripples were observed; they were classified as: (i) no scour/burial, (ii) initial scour, (iii) expanded scour, and (iv) periodic burial cases. A scour regime diagram was developed and the demarcation criteria between different regimes were deduced. Semi-empirical formulae that permit estimation of the scour depth with time, the equilibrium maximum scour depth and length, and conditions necessary for the burial of the cylinder as a function of main external parameters are also proposed.     

季节性层结对孤立内波作用于圆柱载荷的影响

On the basis of Morison＇s empirical formula and modal separation method in estimating the force and torque exerted by internal solitary waves （ISWs） on a cylindrical pile, it is found that the loads exerted by the ISWs change largely in different seasons at the same site of the continental shelf in the South China Sea （SCS） even under the condition that the amplitudes of ISWs are the same. Thus, the effect of a seasonal water stratification variation on the force and torque exerted by the ISWs is investigated, and a three-parameter stratification model is employed. It is shown that the loads exerted by the ISWs depend largely on the wa- ter stratification. The stronger the water stratification, the larger the force and the torque; when the depth where the maximum thermocline appears is deepened, the force decreases but the torque increases; when the width of the thermocline is narrowed, the force increases but the torque decreases. The seasonal varia- tion of the force and the torque exerted by the ISWs in four seasons in the SCS is thus explained. Key words： internal solitary waves, force, torque, water stratification, South China Sea     

Scour below marine pipelines in shoaling conditions for random waves

This paper provides an approach by which the scour depth below pipelines in shoaling conditions beneath non-breaking and breaking random waves can be derived. Here the scour depth formula in shoaling conditions for regular non-breaking and breaking waves with normal incidence to the pipeline presented by Cevik and Yüksel [Cevik, E. and Yüksel, Y., (1999). Scour under submarine pipelines in waves in shoaling conditions. ASCE J. Waterw., Port, Coast. Ocean Eng., 125 (1), 9–19.] combined with the wave height distribution including shoaling and breaking waves presented by Mendez et al. [Mendez, F.J., Losada, I.J. and Medina, R., (2004). Transformation model of wave height distribution on planar beaches. Coast. Eng. 50 (3), 97–115.] are used. Moreover, the approach is based on describing the wave motion as a stationary Gaussian narrow-band random process. An example of calculation is also presented.     

A study of Coriolis effects on long waves propagating in an unbounded ocean, channel and basin

The effects of Coriolis force on long waves have been discussed based on gravity waves propagating in an unbounded ocean, channel and basin. In case of ocean, results show that the Coriolis effect will be significant and negligible, when the wave period is comparable to 2π/f and much shorter, respectively. Results also show in a channel, the wave amplitude and water particle velocity decrease exponentially in the positive y direction in the northern hemisphere (where f is positive). Moreover, in a basin, the Cotidal lines have been found as curves and rotate counterclockwise around the origin.     

热带气旋下海浪对大气向海洋输入的动量和能量的影响

海浪不仅决定着海洋表面的粗糙度,由热带气旋引起的海浪,还通过其发展演化控制着大部分的海气之间的动量和能量传递。本文采用热带气旋观测数据IBTrACS和海浪模式WW III的模拟结果探究了热带气旋下海浪对大气向海洋输入的动量和能量的影响。结果发现,近30 a热带气旋的强度约每10 a增加 1 m/s,但移速没有明显变化。热带气旋的强度越大,从大气输入到海浪和从海浪输入到海流中的动量之差和能量之差也越大。由于热带气旋的风场和海浪场都有较强的不对称性,海气动量差和能量差也表现出非均匀分布：动量差较大的区域在热带气旋移动方向的后方,能量差的最大值则分布在右后象限,且二者均为左前方比较小。逆波龄与动量差和能量差呈高度正相关,相关系数约为0.95,说明波越年轻吸收的动量和能量越多。气旋移速越快逆波龄越大,且热带气旋移动速度与动量差和能量差呈正相关,相关系数在0.8以上。因此,海浪影响着大气向海洋输入的动量和能量的分布和大小,在以后关于海洋边界动力学和热力学的研究中,考虑海浪的演化可能会使结果更加准确。     

多向不规则波浪的确定性模拟

波浪波动时间过程及波列的模拟,对于开展实际波浪对于工程建筑物的作用具有重要的意义。本文采用线性叠加的单叠加模型,建立了多向不规则波浪的确定性模拟方法。基于理论模拟的规则波、单向不规则波和多向不规则波,验证了波浪确定性模拟方法的有效性。定性地对比分析了模拟波列和已知波列的一致性;定量地研究了模拟波浪在空间范围rr/L_s的误差分布情况(rr表示指定位置与给定位置的空间距离,L_s为有效波长)。并且建议,采用本文方法进行波浪确定性模拟时,最佳的浪高仪间距应小于0.12L_s。     

Field observations of waves generated by passing ships: A note

This short contribution reports the results of a field study on the nearshore characteristics of waves generated by both conventional and high speed passenger ferries. The field observations took place in the late summer of 2005, at a beach close to the port of Mytilene (Island of Lesbos, Greece), and involved the visual observation of ship waves, using digital video recordings and image processing techniques. The results showed that passage of the fast ferry was associated with a longer, more complex and energetic nearshore event; this event not only did include higher nearshore waves (up to 0.74 m) and was organised in different wave packets, but it was also an order of magnitude longer (∼ 680 s) than the conventional ferry event. Regarding the effects on beach sediment dynamics, the fast ferry waves were estimated to be very efficient in mobilising the nearshore sediments in contrast to those of the conventional ferry. The fast ferry service appears to generate daily prolonged nearshore events, which contain waves with higher energy than those expected from the normal summer wind wave regime of the area; these events also include some high and very steep waves, which can be particularly erosive. Therefore, fast ferry wakes may have considerable impacts on the seasonal beach sediment dynamics/morphodynamics and the nearshore benthic ecology, as well as they may pose significant risks to bathers, affecting the recreational use of the beaches exposed to fast ferry traffic. Finally, the study has shown that satisfactory field observations of the nearshore characteristics of ship-generated (and wind) waves can be obtained using inshore deployments of calibrated poles, digital video cameras and appropriate image processing algorithms.     

海啸波传播的线性和非线性特征及近海陆架效应影响的数值研究

浅水方程被广泛应用于海啸预警报业务及研究,而针对线性浅水方程与非线性浅水方程在不同海区水深地形条件下的适用范围、计算效率问题是海啸研究人员急需了解的。本文应用基于浅水方程的海啸数值预报模型就海啸波在南海、东海传播的线性、非线性特征以及陆架对其传播之影响进行了数值分析研究。海啸波在深水的传播表征为强线性特征,此时线性系统对海啸波幅的模拟计算具有较高的精度和效率,而弱的非线性特征及弱的色散特征对海啸波幅的预报影响甚微,可以忽略不计。海啸波传播至浅水大陆架后受海底坡度变化、海底粗糙度等因素影响,波动的非线性效应迅速传播、积累,与线性浅水方程计算的海啸波相比表现出较大差异,主要表现为:在南海区,水深小于100m时,海啸波首波以后的系列波动非线性特征比较明显,两者波幅差别较大,但首波波幅的区别不大,因此对于该区域在不考虑海啸爬高的情况下,应用线性系统计算得到的海啸波幅也可满足海啸预警报的要求;在东海区由于陆架影响,海啸波非线性特征明显增强,水深小于100m区域,首波及其后系列波波幅均差异较大,故在该区域必须考虑海啸波非线性作用。本文就底摩擦项对海啸波首波波幅的影响进行了数值对比分析,结果表明:底摩擦作用对海啸波首波波幅影响仅作用于小于100m水深。最后,该文通过敏感性试验,初步分析了陆架宽度及陆架边缘深度对海啸波波幅的影响,得出海啸波经陆架传播共振、变形后,海啸波幅的放大或减小与陆架的宽度及陆架边缘水深有关。     

波流作用下海底边界层沉积物再悬浮与影响因素研究

海底沉积物再悬浮及其分布取决于海洋水动力、沉积物类型与床面形态之间复杂的相互作用,准确地理解和确定沉积物再悬浮过程对于沉积物输运的研究具有重要的意义。本文在祥云湾海洋牧场典型海域开展现场原位观测,获取研究区波浪、海流及悬浮沉积物浓度数据;分析了波、流作用下海底边界层悬浮沉积物垂向分布特征,并探究了海洋水动力和床面形态对悬浮沉积物垂向分布的影响。结果表明,研究区波流之间的相互作用不显著,沉积物再悬浮受控于风暴浪作用,风暴浪作用下底床切应力可以达到沉积物临界切应力的10～15倍,沉积物的再悬浮滞后于风暴浪作用2～3 h。在波浪荷载微小的情况下,悬浮沉积物垂向分布呈现"I"型,波浪荷载下,悬浮沉积物垂向分布呈现幂指函数分布,表现为"L"型;床面形态随波、流作用而演化,影响沉积物的再悬浮过程,u_?w/u_?c=1.00可作为波浪和海流起主导控制作用的床面形态的判别依据,纯波浪荷载作用下的u_?w/u_?c显著高于波浪主控作用下,但二者之间的界线随着波浪荷载的增加而升高。     

Observation and measurement of the bottom boundary layer flow in the prebreaking zone of shoaling waves

In this paper, the characteristics of the bottom boundary layer flow induced by nonlinear, asymmetric shoaling waves, propagating over a smooth bed of 1/15 uniform slope, is experimentally investigated. Flow visualization technique with thin-layered fluorescent dye was first used to observe the variation of the flow structure, and a laser Doppler velocimeter was then employed to measure the horizontal velocity, U.The bottom boundary layer flow is found to be laminar except within a small region near the breaking point. The vertical distribution of the phase-averaged velocity U at each phase is non-uniform, which is directly affected by the mean velocity, . The magnitude of increases from zero at the bottom to a local positive maximum at about z/δ2.02.5 (where z is the height above the sloping bottom and δ is the Stokes layer thickness), then decreases gradually to zero at z/δ6.07.0 approximately, and finally becomes negative as z/δ increases further. Moreover, as waves propagate towards shallower water, the rate of increase in the maximum onshore oscillating velocity component is greater than that of the offshore counterpart except near the breaking point. The free stream velocities in the profiles of the maximum onshore and offshore oscillating velocity components, and are found to appear at z/δ≥6.0. This implies that, if the Stokes layer thickness is used as a length scale, the non-dimensionalized boundary layer thickness remains constant in the pre-breaking zone. Although is greater than and the asymmetry of the maximum free stream velocities (i.e. ) increases with decrease of water depth, a universal similar profile can be established by plotting z/δ versus ( ) or ( ). The final non-dimensional profile is symmetric and unique for the distributions of the maximum onshore and offshore oscillating velocity components within the bottom boundary layer, which are induced by nonlinear, asymmetric shoaling waves crossing the pre-breaking zone.     

弱非线性斯托克斯波在非平整海底上传播的新型抛物型方程

由于缓坡方程计算量大和其本身的缓坡假定而在实际应用中受到了限制,故对斯托克斯波在非平整海底(适用于缓坡和陡坡地形)上传播的Liu和Dingemans的三阶演化方程进行抛物逼近,得到一个新的非线性抛物型方程,它能够包含同类方程未曾考虑的二阶长波效应.通过数值计算结果与Berkhoff等人的经典实验数据的比较,证明所提出的抛物型模型理论具有较高的精度.     

Drag force on a vegetation field due to long-crested and short-crested nonlinear random waves

This paper provides a practical method for estimating the drag force on a vegetation field exposed to long-crested (2D) and short-crested (3D) nonlinear random waves. This is achieved by using a simple drag formula together with an empirical drag coefficient given by Mendez et al. (1999), in conjunction with a stochastic approach. Here the waves are assumed to be a stationary narrow-band random process. Effects of nonlinear waves are included by adopting the Forristall (2000) wave crest height distribution representing both 2D and 3D random waves.