A note on the derivation of potential energy for two-dimensional water waves

The potential energy available in a two-dimensional progressive water wave can be calculated in numerous ways. One derivation of this energy based on the first law of thermodynamics and on the linearized velocity potential for waves, is presented in this paper. The energy densities and total energy expressions are given for deep and finite depth water waves. It is also shown that the travelling component of the energy for deep water waves is the potential energy component.     

A note on wave-induced set-up by obliquely incident waves

Water level variations due to obliquely incident, shoaling and breaking waves on a plane sloping beach were discussed recently by Hsu et al. (Coastal Engineering, 53, 865–877, 2006). An inconsistency in this work with respect to the set-down, and its implications to circulation offshore of the breakpoint, was pointed out by Shi and Kirby (Coastal Engineering, 55, 1246 – 1249, 2008). Here we extend that discussion to include the surfzone momentum balance and wave-induced set-up. We discuss some remaining inconsistencies in the approximation of the surfzone momentum balance, derive and present a consistent approximation, and validate the new approximation through numerical comparison to a more exact model.     

Shallow water gravity waves: A note on the particle orbits

When surface gravity waves of small amplitude progress in shallow water of constant mean depth, the fluid particle orbits are observed to be oval, where the longer axis of the oval is parallel to the flat bottom, and at the bottom the orbits are straight lines. Potential flow, upon which the standard wave theory is based, predicts that the oval orbits are ellipses, but by a rather lengthy mathematical procedure that is founded on the questionable assumption of irrotationality. Using a more elementary and physical method, that does not employ the irrotational assumption, the elliptical orbits can be understood much more easily. The elementary method features a balance of two oppositely directed forces on each fluid particle: the outward centrifugal force and the inward pressure force.     

Remarks on potential vorticity mixing over topography and momentum conservation

Recent works have shown that parameterizing eddy fluxes as a downgradient diffusion of potential vorticity (PV) generates circulation over topography. This result is examined in the context of a simple barotropic quasi-geostrophic model. It is shown that, unless a constraint on the eddy diffusivity is maintained, the PV mixing parameterization creates a fictitious torque which generates angular momentum. These considerations indicate that momentum conservation is not maintained in recent models that purport to show generation of net alongslope flow associated with PV mixing over topography.     

A note on the direct injection of turbulence by breaking waves

We investigate the turbulence induced by wave-breaking at the ocean surface. Two recent models use a mechanism of direct depth injection of turbulent kinetic energy (TKE) by breaking waves. Those models aim to reproduce the near-surface mean and turbulent properties, in particular the TKE dissipation rates. Of critical importance are the injection depth of each breaking wave and the size distribution of those breaking waves. The models by Sullivan et al. (2007) and by Kudryavtsev et al. (2008) have very different parameterizations, and those differences are reviewed here and compared to available observations. Using realistic parameterizations in these models leads to TKE injections too shallow to compare to observations, in particular for developed seas. The near-surface turbulence is thus still not well understood to the zeroth order. For instance, whether developed seas produce deeper or shallower mixing than young seas is neither well understood nor well modelled. Additional dedicated measurements as well as investigations of breaking non-breaking wave interactions are needed.     

Run-up of solitary waves on slopes with different profiles

The problem of sea-wave run-up on a beach is discussed within the framework of exact solutions of a nonlinear theory of shallow water. Previously, the run-up of solitary waves with different forms (Gaussian and Lorentzian pulses, a soliton, special-form pulses) has already been considered in the literature within the framework of the same theory. Depending on the form of the incident wave, different formulas were obtained for the height of wave run-up on a beach. A new point of this study is the proof of the universality of the formula for the maximum height of run-up of a solitary wave on a beach for the corresponding physical choice of the determining parameters of the incident wave, so that the effect of difference in form is eliminated. As a result, an analytical formula suitable for applications, in particular, in problems related to tsunamis, has been proposed for the height of run-up of a solitary wave on a beach.     

A note on the horizontal momentum exchange in combined waves and current

The horizontal exchange of momentum due to the organized motion in combined waves and current has been analyzed. The combination of the vertical orbital wave motion and the mean current gives a periodic variation in the horizontal velocity in addition to the wave orbital motion. This periodic variation, combined with the wave orbital motion, gives a significant contribution to the momentum exchange. Two examples are considered, the interaction of a pure wave motion and a current normal to the direction of wave propagation, and a wave driven longshore current with an undertow velocity profile. It is demonstrated that the new contribution changes the resulting momentum exchange considerably.     

Rossby number dependence on meander dynamics of a potential vorticity front

Semi-geostrophic dynamics of jets are studied using a potential vorticity front in an equivalent barotropic model. Meandering processes of the front are examined in the thin-jet limit on a -plane by a curvilinear coordinate system. For calculated along-front velocity fields, asymmetrical profiles are caused by meandering. This asymmetry of the velocity profile is enhanced as the Rossby number becomes large. Using the along-front velocity fields, the normal velocity of front is expressed so that the Rossby number is explicitly included. This expression can be rewritten in the form of the mKdV equation.     

孤立波作用下沿海路基冲刷实验研究

基于波浪水槽实验,对孤立波作用下沿海公路局部冲刷开展实验研究。实验采用1/10与1/20组合坡模拟海岸地形,对岸滩进行逐个多次波浪作用,用地形仪对作用后地形进行测量。通过改变路基在冲刷槽的绝对位置和相对位置,探讨路基位置因素对局部冲刷影响。实验结果表明:孤立波作用下沿海路基向海侧出现明显冲坑,冲刷深度随波浪作用个数增加,最后达到平衡。路基位置是路基局部冲刷主要因素,随着路基位置远离海岸线,路基向海侧冲刷深度呈减小趋势。     

Internal solitary waves in the East China Sea

A European Space Agency＇ s ENVISAT advanced synthetic aperture radar （ASAR） image covering Zhejiang coastal water in the East China Sea （ECS） was acquired on 1 August 2007. This image shows that there are about 20 coherent internal solitary wave （ISW） packets propagating southwestward toward Zhejiang coast. These ISW packets are separated by about 10 kin, suggesting that these ISWs are tide-generated waves. Each ISW packet contains 5-15 wave crests. The wavelengths of the wave crests within the ISW packets are about 300 m. The lengths of the leading wave crests are about 50 km. The ISW amplitude is estimated from solving KdV equation in an ideal two-layer ocean model. It is found that the ISW amplitudes is about 8 m. Further analysis of the ASAR image and ocean stratification profiles show that the observed ISWs are depression waves. Analyzing the tidal current finds that these waves are locally generated. The wavelength and amplitude of the ECS ISW are much smaller than their counter- parts in the South China Sea （SCS）. The propagation speed of the ECS ISW is also an order of magnitude smaller than that of the SCS ISW. The observed ISWs in the ECS happened during a spring tide period.     

Runup of solitary waves on a straight and a composite beach

Particle Image Velocimetry (PIV) and wave gauges have been used to investigate the runup of solitary waves at two different beaches. The first beach is straight with an inclination of 10°, whereas the second is a composite beach with a change in the 10° inclination to 4° at a vertex point above the equilibrium water level. Comparison with numerical simulations using a Navier–Stokes solver with zero viscosity has been performed for the composite beach. Four different amplitudes of incoming solitary waves are investigated.Measurements of the runup show that the composite beach gives a lower runup compared to the straight beach. Furthermore, the composite beach experiences a longer duration of the rundown compared to the straight beach. This is at least partially assumed to be a result of scaling effects, since the fluid above the vertex creates a relatively thinner runup tongue compared to the straight beach scenario.The appearance of a stagnation point at the beach boundary is clearly visible in both the PIV results and the numerical simulation. This stagnation point is originating at the lowermost part of the beach, and is moving upwards with time. It is found that the stagnation point moves faster upwards for the straight beach than for the composite beach. Further, the stagnation point is moving even faster in the numerical simulation, suggesting that the velocity with which the stagnation point moves is influenced by viscous scaling effects.Finally, the numerical simulation seems to capture the physics of the flow well, despite differences in the phase compared to the PIV results. This applies to both the flow field and the surface elevations.     

跃层厚度对内孤立波能量耗散的影响

内孤立波破碎混合是陆架地形上海洋混合过程重要的能量汇,为了探究内孤立波在连续跃层密度分层中能量变化及能量耗散规律,本文使用OpenFOAM建立数值水槽,基于双曲正切曲线设置连续跃层密度分层开展了一系列工况的模拟,针对跃层厚度对内孤立波能量传递及其耗散进行了详细分析。结果表明跃层厚度与跃层处流速剪切存在着负相关关系,随着跃层厚度的增加能量耗散先减小后增大。跃层厚度较小时,跃层处流速剪切强,理查德森数小,易产生开尔文-亥姆霍兹(Kelvin–Helmholtz)不稳定现象;随着跃层厚度的增大,流速剪切减小,理查德森数增大,开尔文-亥姆霍兹不稳定消失;跃层厚度达到临界值后,层结稳定性减小,理查德森数减小,流场的翻转混合过程加强,能量耗散也明显增强。     

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.     

A proposed adiabatic formulation of 3‐dimensional global atmospheric models based on potential vorticity

A 2‐time‐level finite difference atmospheric general circulation model based on the semi‐Lagrangian advection of pseudo potential vorticity (which becomes potential vorticity in that part of the domain where the hybrid vertical coordinate becomes isentropic) has been formulated. At low levels, the hybrid vertical coordinate is terrain following. The problem of isentropic potential vorticity possibly becoming ill‐defined in the regions of planetary boundary layer is thus circumvented. The divergence equation is a companion to the (pseudo) potential vorticity equation and the model is thus called a PV‐D model. Many features of a previously developed shallow water PV‐D model are carried over: a modification of the PV equation needed to give computational stability of long Rossby waves; a semi‐Lagrangian semi‐implicit treatment of both the linear and the nonlinear terms; the use of an unstaggered grid in the horizontal; the use of a nonlinear multigrid technique to solve the nonlinear implicit equations. A linear numerical stability analysis of the model's gravity–inertia waves indicates that the potential temperature needs to be separated into horizontal mean and perturbation parts. This allows an implicit treatment of the vertical advection associated with the mean in the thermodynamic equation. Numerical experiments with developing baroclinic waves have been carried out and give realistic results.     

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.     

江苏沿海波浪多重嵌套模拟研究

根据24年CCMP风场资料和江苏沿海4个方向(N、NE、E和SE)百年一遇风速,构建西北太平洋、东中国海和江苏沿海上述4个方向的百年一遇风场。首次建立一个基于第三代海浪模型SWAN的自西北太平洋、东中国海至江苏沿海的三重嵌套数值模型,以AVISO卫星观测数据和江苏沿海定点实测数据进行验证。以三个计算域4个方向百年一遇风场为驱动风场,驱动该多重嵌套模型,高精度数值模拟江苏沿海4个方向百年一遇有效波高分布并进行分析。结果表明,江苏沿海辐射沙洲地形对有效波高分布影响显著;E向百年一遇风场作用下海域有效波高最大,NE向次之,N向和SE较小。