A modified Euler–Lagrange transformation for particle orbits in nonlinear progressive waves

This paper presents a modified Euler–Lagrange transformation method to obtain the third-order trajectory solution in a Lagrangian form for the water particles in nonlinear water waves. We impose the assumption that the Lagrangian wave frequency is a function of wave steepness and an arbitrary vertical position for each water particle. Expanding the unknown function in a small perturbation parameter and using a successive expansion in a Taylor series for the water particle path and the period of a particle motion, the third-order asymptotic expressions for the Lagrangian particle trajectories, the mass transport velocity and the period of particle motion can be derived directly in Lagrangian form. The wave frequency and mean level of the particle motion in Lagrangian form differ from those of the Eulerian. Finally, the third-order asymptotic solution obtained is uniformly valid in contrast with early works containing resonant terms presented by Wiegel [1964. Oceanographical Engineering. Prentice-Hall, New Jersey, pp. 37–40] (Eqs. (B.1) and (B.1), (B.2) in Appendix B) or Chen et al.[2006. Theoretical analysis of surface waves shoaling and breaking on a sloping bottom. Part 2 nonlinear waves. Wave motion, 43, 356–369] based on a straightforward expansion for two-dimensional progressive waves.     

A Lagrangian mean theory on coastal sea circulation with inter-tidal transports I. Fundamentals

A brief review is made on the theory of the Lagrangian residual circulation and inter-tidal transports in a convectively weakly nonlinear system. In the review the emphasis is put on the systematical development of the theory and its weakness of convectively weakly nonlinear approximation. The fundamentals of a Lagrangian tidally-averaged theory on circulation with inter-tidal transport processes have been proposed for a general nonlinear coastal/estuarine system. The Lagrangian residual velocity is strictly de- fined, and it has been verified to be able to embody the velocity field of circulation. A new concept of the concentration for inter- tidal transport processes is presented. The concentration describing the inter-tidal transport processes should be a ＂Lagrangian inter-tidal concentration＂ defined and named, but not the Eulcrian tidally-averaged concentration used traditionally. The circulation described here contains a set of infinite temporal-spatial fields of velocity/concentration, each of which corresponds to a specific value of tidal phases varying continuously over one tidal cycle. When the convectively weakly nonlinear condition（ with a smaller order of eddy diffusion and sources） is approximately satisfied, a set of infinite temporal-spatial fields of velocity/concentration can be reduced to a single one.. the mass transport velocity/the Eulerian tidally averaged concentration as exhibited traditionally.     

论狭窄潮汐水道中sigma坐标下的余流

在受波动影响的近岸浅水区域,运用sigma坐标是计算平均水位附近的余流的有效途径。本项研究在理论上分析了在狭窄潮汐水道中sigma坐标下的余流的物理意义,并运用一系列的理想化数值模型对分析结果进行了验证。对于浅水波,sigma层和水体中的波动面相一致,因而斯托克斯速度及其分量可以用sigma坐标上的速度来表达。一个sigma层上的余流（即sigma余流）是位于这一sigma层平均深度上的欧拉余流和斯托克斯速度垂向分量的和,可以被看做是半拉格朗日余流。因为斯托克斯速度的垂向分量比其水平分量小一个量级,sigma余流可看做为欧拉余流的近似。在sigma层上的物质输运余流是sigma余流和斯托克斯速度水平分量的和,在大小和方向上和拉格朗日余流近似。     

Lagrangian measurements and modelling of fluid advection in the inner surf and swash zones

New laboratory and field data are presented on fluid advection into the swash zone. The data illustrate the region of the inner surf zone from which sediment can be directly advected into the swash zone during a single uprush, which is termed the advection length. Experiments were conducted by particle tracking in a Lagrangian reference frame, and were performed for monochromatic breaking waves, solitary bores, non-breaking solitary waves and field conditions. The advection length is normalised by the run-up length to give an advection ratio, A, and different advection ratios are identified on the basis of the experimental data. The data show that fluid enters the swash zone from a region of the inner surf zone that can extend a distance seaward of the bore collapse location that is approximately equal to half of the run-up length. This region is about eight times wider than the region predicted by the classical swash solution of Shen and Meyer [Shen, M.C., Meyer, R.E., 1963. Climb of a bore on a beach. Part 3. Runup. Journal of Fluid Mechanics 16, 113–125], as illustrated by Pritchard and Hogg [Pritchard, D., Hogg, A.J., 2005. On the transport of suspended sediment by a swash event on a plane beach. Coastal Engineering 52, 1–23]. Measured advection ratios for periodic waves show no significant trend with Iribarren number, consistent with self-similarity in typical swash flows. The data are compared to recent characteristic solutions of the non-linear shallow water wave (NLSW) equations and both finite difference and finite volume solutions of the NLSW equations.     

基于拉格朗日分析法的中尺度涡精细三维结构研究及其误差估计

在前人的工作中,拉格朗日分析法被用来演示大尺度环流,同时拉格朗日拟序结构可以较好的演示中尺度涡两维结构的发展过程。然而,很少研究关注怎么利用拉格朗日分析法针对中尺度涡三维结构进行演示。与以往利用欧拉方法研究中尺度涡三维结构的工作不同,我们利用拉格朗日分析法,从另一个视角来研究涡旋结构。我们在海山上方模拟出一个理想的气旋涡,涡旋内的下沉流和涡旋旁的上升流形成一个闭合的环流。这种结构很难从欧拉角度来演示。然而,粒子的运动轨迹很好地展示了整个循环:流体在涡旋中旋转下沉,汇聚到底层的上升流区,并通过上升流返回到海表面。我们也将拉格朗日分析法应用于真实的模拟结果中。作为中国南海的一个重要现象,靠近越南中部的海域中的偶极子（反气旋涡/气旋涡）,关于其结构的研究已经比较成熟了,但这些研究主要关注的是海面过程。通过拉格朗日分析,我们很好的演示了偶极子的三维结构:流体在反气旋涡(气旋涡)内部旋转上升(下沉)。更重要的是,粒子的轨迹表明,这两个涡旋之间不存在水团交换,因为强边界急流将它们彼此分开。以上结论均得到了计算误差估计的可信度支持。尽管在强辐散流和强垂直扩散流中,计算误差逐渐增大,但是在一定的时间步长和积分周期内,计算误差始终保持在一个较小的值。     

拉格朗日描述下三层流体系统中界面内波的三阶渐近解

在流体力学中,描述流体运动有Lagrange方法和Euler方法.Euler方法是通过观测通过空间各固定位置点处流体质点的运动行为来描述流体运动规律,而Lagrange方法是跟踪各个流体质点,通过观测它们在时空运动中所走过的路径来描述流体的运动规律.在数学处理上,Euler方法较Lagrange方法简单,但Lagrange方法可以完全描述流体运动的整个流场的所有特性,而Euler方法却无法描述每个流体质点的运动轨迹.本文,我们研究具有刚性边界的三层流体系统中的界面内波,其中上层流体的密度比下层流体的密度大.通过在界面处引入朗格朗日匹配条件并使用微扰法得到了拉格朗日描述下的界面内波的一阶解、二阶解及三阶解,给出了质量输运速度、波频率、平均水平和质点运动轨迹的解.结果表明对于质量输运速度、波频率、平均水平和质点运动轨迹在界面处会有不连续性,但是我们发现在满足一定的三层流体水深比和密度比条件时这种不连续性将会消失.     

Assimilation of simulated float data in Lagrangian coordinates

We implement an approach for the accurate assimilation of Lagrangian data into regional general ocean circulation models. The forward model is expressed in Lagrangian coordinates and simulated float data are incorporated into the model via four-dimensional variational data assimilation. We show that forward solutions computed in Lagrangian coordinates are reliable for time periods of up to 100 days with phase speeds of 1 m/s and deformation radius of 35 km. The position and depth of simulated floats are assimilated into the viscous, Lagrangian shallow water equations. The weights for the errors in the model and data are varied and the assimilation results react appropriately. We show the effect of different spatial and temporal samplings of float data on all Lagrangian trajectories in the computational domain. At the end of the assimilation period, results from the Lagrangian shallow water equations could be interpolated and used as initial and boundary conditions in an Eulerian general ocean circulation model.     

三维潮致拉格朗日余流的数值计算及其在渤海中的应用

根据冯士筰教授于1987年导出的三维质量输运速度的控制方程组出发,在垂直方向上用三次样条函数为基函数作谱展开,水平方向上应用有限差分格式来求解方程组,然后将所提出的计算方法应用于夏季渤海环流,取湍粘性系数随深度变化模式,通过计算可知,潮致余流改变了一般认为夏季渤海环流为一逆时针方向大环流的简单结论,并得到了观测的初步支持。     

A note on the hydrodynamics of a tail tube buoy

This note presents some analytical results for a tail–tube buoy configuration frequently used in wave energy conversion. The overall approach is based on Falnes and McIver's (Falnes, J., McIver, P., 1985. Surface wave interactions with systems of oscillating bodies and pressure distributions. Applied Ocean Research 7 (4), 225–234) extension to floating oscillating water columns of Evans' (Evans, D.V., 1982. Wave power absorbtion by systems of oscillating surface pressure distributions. Journal of Fluid Mechanics 114, 481–499) theory of oscillating pressure distributions. The diffraction air-flow flux through the tube and the diffraction wave force on the flotation collar are obtained using the formulation of Garrett (1970, 1971) (Garrett, C.J.R., 1970. Bottomless harbours. Journal of Fluid Mechanics 43 (3), 433–449. Garrett, C.J.R., 1971. Wave forces on a circular dock. Journal of Fluid Mechanics 46 (1), 129–139). Results can be used in sizing the tube and collar for efficient energy conversion.     

河口垂向拉格朗日余环流研究

应用弱非线性理论导出了狭窄型河口内拉格朗日余流的控制方程和溶解质长期输运方程,并应用流速分解法和B—样条配置技术建立了潮汐及拉格朗日余流的数值模型,应用欧拉—拉格朗日方法和B—样条配置技术建立了长期物质输运方程的数值模型,对狭窄型河口内潮致—斜压拉格朗日余环流进行了研究。结果表明,河口垂向拉格朗日余环流,不同于欧拉余环流,呈现一种较复杂的流型,在盐水入侵界的上游可能存在一涡旋。作者提出,将河口按余环流生成机制和水动力特征分为斜压,潮控及河流三个区。     

Interaction of regular waves with a group of dual porous circular cylinders

Diffraction of linear waves around a group of dual porous cylinders consisting of a thin and porous outer cylinder with an impermeable inner cylinder is investigated analytically based on the eigenfunction expansion method proposed by Spring and Monkmeyer [Spring BH, Monkmeyer PL. Interaction of plane waves with vertical cylinders. In: Proceedings 14th international coastal engineering conference. 1974. p. 1828–47] and further modified by Linton and Evans [Linton CM, Evans DV. The interaction of waves with arrays of vertical circular cylinders. Journal of Fluid Mechanics 1990;215:549–69]. The present formulation is an extension of the work of Wang and Ren [Wang KH, Ren X. Wave interaction with a concentric porous cylinder system. Ocean Engineering 1994;21(4):343–60], wherein; the interaction of linear waves with a single concentric porous cylinder system was studied. This paper aims at investigating the influence of multiple interactions between the cylinders in the group on the hydrodynamic wave forces, wave run-up and free-surface elevation in their vicinity. Further, the study focuses on the variation of the forces and run-up on the individual cylinders within the group compared to that on isolated cylinders.     

Numerical simulation of nonlinear wave and force generated by a wedge-shape wave maker

Nonlinear waves and forces induced by a wedge-shape wave maker were simulated in a potential-theory-based fully nonlinear 2D Numerical Wave Tank (NWT). The NWT is developed in a time domain by using Boundary Element Method (BEM) including Mixed Eulerian–Lagrangian method (MEL) and Runge–Kutta 4th-order (RK4) integration as a time marching process. For ensuring accurate nonlinear free surface both material-node and semi-Lagrangian approach are independently developed for crosschecking. The acceleration-potential scheme is used for obtaining accurate time derivative of velocity potential. The developed NWT is utilized to calculate water particle velocity and a series of higher-harmonic force components on the wave maker. The added-mass and radiation-damping coefficients of the wave maker are also obtained from the least-square method. The simulation results are compared with the experimental and numerical results of other researchers. To compare the relative importance of free-surface and body-surface nonlinearities, a body nonlinear formulation is independently developed. Force by body nonlinear method is in good agreement with fully nonlinear result in case of low body-stroke frequency.     

Statistics from Lagrangian observations

We review statistical analyses of Lagrangian data from the ocean. These can be grouped into studies involving single particles and those with pairs or groups of particles. Single particle studies are the most common. The prevalent analysis involves binning velocities geographically to estimate the Eulerian means and lateral diffusivities. However single particle statistics have also been used to study Rossby wave propagation, the influence of bottom topography and eddy heat fluxes. Other studies have used stochastic models to simulate dispersion, calculated Lagrangian frequency spectra and examined the relation between Lagrangian and Eulerian integral scales. Studies involving pairs of particles are fewer, and the results are not well-established yet. There are indications that pair separations grow exponentially in time below the deformation radius, as is also the case in the stratosphere. The behavior at larger scales is less clear, indicating either a turbulent cascade or dispersion by the sheared large-scale circulation. In addition, three or more particles can be used to measure relative vorticity and divergence.     

基于拉格朗日拟序结构的黑潮区域海流运输结构提取和分析

海流的拉格朗日运动对于研究物质输送有着重要意义,拉格朗日拟序结构(LCSs)作为研究海流结构的新型方法,相比于传统欧拉方法更为客观。本文提出了一种新的计算LCSs束的方法,基于25年的平均速度场,利用变分方法计算得到黑潮区域的气候态LCSs,并通过简化合并的方法得到了气候态LCSs束,该LCSs束能够突出地显示出海流特性和运输模式,其代表的平均拉格朗日环流有很强的约束作用,且具有鲁棒性。最终我们获得了气候态下12个月份的流场结构图,揭示了月周期性拉格朗日环流规律。本文还利用虚拟粒子输运、多年浮标轨迹以及气候态温盐异常3种方法进行了验证,与拉格朗日运输模式相吻合,证明了海流拉格朗日拟序结构的准确性和可靠性。     

Application of VOF in Calculating Wave Energy Loss due to Spilling Breaker

- A large amount of experimental analysis and systematical theoretical calculation has been done by the authors to solve the problem of wave transformation and breaking, considering the effect of both current and topography, but only the wave energy loss due to spilling breaker in the surf zone has been discussed in this paper. Based on test result analysis and calculation with the Stream Function Wave Theory, the wave velocity field at breaking points has been obtained, and it is used to calculate the wave heights after breaking by the VOF (Volume of Fluid) method, in which the governing equations are continuity equation and Navier-Stokes Equation for imcompressible fluids. In the present paper, the improved VOF technique is used to calculate the wave heights of stable regular waves after breaking. Results fit the test data well, which shows that the VOF method is suitable to numerical simulation of regular waves after breaking. Besides, the breaker coefficient B of regular waves in the bore model is a     

Interactions of nonlinear gravity waves and uniform current in Lagrangian system

The particle trajectory on a weakly nonlinear progressive surface wave obliquely interacting with a uniform current is studied by using an EulerLagrange transformation. The thirdorder asymptotic solution is a periodic bounded function of Lagrangian labels and time, which imply that the entire solution is uniformlyvalid. The explicit parametric solution highlights the trajectory of a water particle and mass transport associated with a particle displacement can now be obtained directly in Lagrangian form. The angular frequency and Lagrangian mean level of the particle motion in Lagrangian form differ from those of the Eulerian. The variations in the water particle orbits resulting from the oblique interaction with a steady uniform current of different magnitudes are also investigated.     

Numerical models for evolution of extreme wave groups

The paper considers the application of two numerical models to simulate the evolution of steep breaking waves. The first one is a Lagrangian wave model based on equations of motion of an inviscid fluid in Lagrangian coordinates. A method for treating spilling breaking is introduced and includes dissipative suppression of the breaker and correction of crest shape to improve the post breaking behaviour. The model is used to create a Lagrangian numerical wave tank, to reproduce experimental results of wave group evolution. The same set of experiments is modelled using a novel VoF numerical wave tank created using OpenFOAM. Lagrangian numerical results are validated against experiments and VoF computations and good agreement is demonstrated. Differences are observed only for a small region around the breaking crest.     

Assessment of water exchange between a discharge region and the open sea – A comparison of different methodological concepts

Two different methods of estimating the water exchange through the Baltic coastal region of Laxemar have been used, consisting of particle trajectories and passive tracers. Water is traced from and to a small discharge region near the coast. The discharge material in this region is treated as zero-dimensional particles or tracers with neutral buoyancy. The real discharge material could be a leakage of radio-nuclides through the sea floor from an underground repository of nuclear waste.Water exchange rates between the discharge region and the model domain are estimated using both forward and backward trajectories as well as passive tracers. The Lagrangian trajectories can account for the time evolution of the water exchange while the tracers give one average age per model grid box. Water exchange times such as residence time, age and transient times have been calculated with trajectories but only the average age (AvA) for tracers. The trajectory calculations provide a more detailed time evolution than the tracers.On the other hand the tracers are integrated “on-line” simultaneously in the sea circulation model with the same time step while the Lagrangian trajectories are integrated “off-line” from the stored model velocities with its inherent temporal resolution, presently 1 h. The sub-grid turbulence is parameterised as the Laplacian diffusion for the passive tracers and with an extra stochastic velocity for trajectories. The importance of the parameterised sub-grid turbulence for the trajectories is estimated to give an extra diffusion of the same order as the Laplacian diffusion by comparing the Lagrangian dispersions with and without parameterisation. The results of the different methods are similar but depend on the chosen diffusivity coefficient with a slightly higher correlation between trajectories and tracers when integrated with a lower diffusivity coefficient.