Comments on the “shear effect” and diffusion in the Lagrangian framework

We indentify three different types of Lagrangian coordinate systems that are used in oceanography. These are: true Lagrangian coordinates (TLC), Lagrangian coordinates (LC), and averaged Lagrangian coordinates (ALC). The diffusion process is studied in each of these coordinate systems. At large scales the eddydiffusivity is proven to be independent of molecular diffusivity, providing the spectrum of turbulent kinetic energy varies as scale raised to a power less than 4 1/3. The shear effect is examined using solutions to the averaged Lagrangian diffusion equation obtained by Okuboet al. (1983). In Eulerian coordinates both advection and diffusion are necessary for the occurrence of the shear effect, while in ALC timedependent dispersion coefficients are necessary for the process. In TLC we use the method of Taylor (1921) to study the dispersion of material by a velocity field, that from the Eulerian perspective, consists of turbulent motion across a uniform shear. The transformation of the above Eulerian velocity field into TLC results in a uniform deformation field and turbulent motion both along and across the shear. This work shows how dispersion of material is related to the turbulent Eulerian velocity and uniform velocity gradients. The instantaneous rate of change of variance of a spreading patch of material is completely specified by the instantaneous divergence obtained over the area occupied by the patch (Kawai, 1976). This relationship is shown to depend upon the fact that at any particular instant it is possible to define TLC that are equivalent to the Eulerian coordinates. In order to describe patch spreading from divergence measured over longer periods it is also necessary to consider other dispersive processes.Contribution number of the Newfoundland Institute of Cold Ocean Science.     

Modified PIC Method for Sea Ice Dynamics

The sea ice cover displays various dynamical characteristics such as breakup, rafting, and ridging under external forces. To model the ice dynamic process accurately, the effective numerical modeling method should be established. In this paper, a modified particle-in-cell （PIC） method for sea ice dynamics is developed coupling the finite difference （FD） method and smoothed particle hydrodynamics （SPH）. In this method, the ice cover is first discretized into a series of lagrangian ice particles which have their own sizes, thicknesses, concentrations and velocities. The ice thickness and concentration at Eulerian grid positions are obtained by interpolation with the Gaussian function from their surrounding ice particles. The momentum of ice cover is solved with FD approach to obtain the Eulerian cell velocity, which is used to estimate the ice particle velocity with the Gaussian function also. The thickness and concentration of ice particles are adjnsted with particle mass density and smooth length, which are adjusted with the redistribution of ice particles. With the above modified PIC method, numerical simulations for ice motion in an idealized rectangular basin and the ice dynamics in the Bohai Sea are carried out. These simulations show that this modified PIC method is applicable to sea ice dynamics simulation.     

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。     

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.     

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 hybrid Lagrangian-Eulerian numerical model for sea-ice dynamics

A hybrid Lagrangian-Eulerian(HLE) method is developed for sea ice dynamics,which combines the high computational efficiency of finite difference method(FDM) with the high numerical accuracy of smoothed particle hydrodynamics(SPH).In this HLE model,the sea ice cover is represented by a group of Lagrangian ice particles with their own thicknesses and concentrations.These ice variables are interpolated to the Eularian gird nodes using the Gaussian interpolation function.The FDM is used to determine the ice velocities at Eulerian grid nodes,and the velocities of Lagrangian ice particles are interpolated from these grid velocities with the Gaussian function also.The thicknesses and concentrations of ice particles are determined based on their new locations.With the HLE numerical model,the ice ridging process in a rectangular basin is simulated,and the simulated results are validated with the analytical solution.This method is also applied to the simulation of sea ice dynamics in a vortex wind field.At last,this HLE model is applied to the Bohai Sea,and the simulated concentration,thickness and velocity match the satellite images and the field observed data well.     

Numerical simulation of impact loads using a particle method

The violent free-surface motions interacting with structures are investigated using the moving particle semi-implicit (MPS) method, which was originally proposed by Koshizuka and Oka (1996) for incompressible flow simulation. In the present numerical method, a more efficient algorithm for Lagrangian moving particles is used for solving various highly nonlinear free-surface problems without using the Eulerian approach or the grid system. Therefore, the convection terms and time derivatives in the Navier–Stokes equation can be calculated more directly without any numerical diffusion, instabilities, or topological failure. In particular, the MPS method is applied to the simulation of liquid-entry and slamming problems, such as wet-drop (liquid–liquid collision) tests in an LNG tank and slamming loads (solid–liquid collision) on rigid plates with various incident angles. The numerical results are in good agreement with available experimental data.     

海冰动力学的混合拉格朗日-欧拉数值方法

综合考虑欧拉坐标下有限差分法(FDM)在海冰动力学计算中的效率,以及拉格朗日坐标下光滑质点流体动力学方法(SPH)对海冰流变行为的精确模拟,本文发展了一种海冰动力学的混合拉格朗日-欧拉(HLE)数值方法。该方法首先在拉格朗日坐标下将海冰离散为若干个具有厚度、密集度的海冰质点,并由这些海冰质点通过Gauss函数对欧拉网格上的海冰参量进行积分插值;然后,在欧拉坐标下对海冰动量方程进行差分计算以确定各网格节点的海冰速度,并由此采用Gauss函数积分插值出拉格朗日坐标下各海冰质点的速度分布;最后,通过对海冰质点运动和分布的计算,确定出各海冰质点的位置、厚度和密集度等参量。采用该HLE方法对规则区域内的海冰堆积过程和涡动风场作用下的海冰动力演化趋势进行了数值试验;最后,采用该HLE方法对渤海海冰的动力过程进行了72h数值模拟,其计算结果与卫星遥感图像和现场观测资料吻合较好。以上计算结果均表明该HLE方法在海冰动力学数值模拟中具有较高的计算效率和模拟精度,可用于海冰动力过程的数值模拟。     

Numerical modelling of Lagrangian residual current in the Bohai Strait

The Bohai Strait is the only passage-way of the semienclosed Bohai Sea into the outer ocean. The various dynamical processes have great effects on both sides of the strait, especially on the Bohai Sea. Tidal current dominates in the strait, and among the tida! components the M, component tide is prevailing.The seawater in the strait is homogeneous vertically almost the whole year because of strong tidal mixing. The M, component tide was simulated numerically using two dimensional barotropical model, and the Eulerian tide-induced residual currents were also deduced. The results showed that the tidal currents were dominated by the coastal line and topography. There exist many eddies in the tide-induced residual current fields, particularly near the northern coast.On the basis of the simulation of the tidal currents, the movements of the water particles in the strait were tracked by means of Lagrangian techniques. Consequently the Lagrangian residual currents were obtained. Somewhat similar to the Eulerian     

Particle trajectories of nonlinear gravity waves in deep water

The paper presents a numerical method for calculating the particle trajectories of nonlinear gravity waves in deep water. Particle trajectories, mass-transport velocity and Lagrangian wave period can be accurately determined by the proposed method. The high success rate of the proposed method is examined by comparing the present results with those of (a) Longuet-Higgins, M.S., 1986, 1987. Eulerian and Lagrangian aspects of surface waves. Journal of Fluid Mechanics 173, 683-707 and (b) Lagrangian moments and mass transport in Stokes waves. Journal of Fluid Mechanics 179, 547-555. It is shown that the dimensionless mass-transport velocity can exceed 10% for large waves, and the Lagrangian wave period is much larger than the Eulerian wave period for large 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.     

Resolution dependent relative dispersion statistics in a hierarchy of ocean models

In order to determine the effect of Eulerian spatial resolution on the two particle statistics of synthetic drifter trajectories, we examine a hierarchy of ocean models, starting from 2D turbulence simulations, progressing to idealized simulations of a buoyant coastal jet with ROMS, and finally to realistic HYCOM simulations of the Gulf Stream. In each case, particle dispersion at large time and space scales is found to be controlled by energetic meso-scale features of the flow that are relatively insensitive to the resolution of finer scale motions. In all cases, time-distance graphs given in terms of computed Finite Scale Lyapunov Exponents show an expected increase in the extent of exponential scaling with increasing spatial smoothing of the velocity field. The limiting value of the FSLE at small separation distances is found to scale remarkably well with the resolution of Eulerian velocity gradients as given by the average of positive Okubo–Weiss parameter values.     

Comparison between Eulerian diagnostics and finite-size Lyapunov exponents computed from altimetry in the Algerian basin

Transport and mixing properties of surface currents can be detected from altimetric data by both Eulerian and Lagrangian diagnostics. In contrast with Eulerian diagnostics, Lagrangian tools like the local Lyapunov exponents have the advantage of exploiting both spatial and temporal variability of the velocity field and are in principle able to unveil subgrid filaments generated by chaotic stirring. However, one may wonder whether this theoretical advantage is of practical interest in real-data, mesoscale and sub-mesoscale analysis, because of the uncertainties and resolution of altimetric products, and the non-passive nature of biogeochemical tracers. Here we compare the ability of standard Eulerian diagnostics and the finite-size Lyapunov exponent in detecting instantaneous and climatological transport and mixing properties in the south-western Mediterranean. By comparing with sea-surface temperature patterns, we find that the two approaches provide similar results for slowly evolving eddies like the first Alboran gyre. However, the Lyapunov exponent is also able to predict the (sub-)mesoscale filamentary processes occurring along the Algerian current and above the Balearic Abyssal Plain. Such filaments are also observed, with some mismatch, in sea-surface temperature patterns. Climatologies of Lyapunov exponents do not show any compact relation with other Eulerian diagnostics, unveiling a different structure even at the basin scale. We conclude that filamentation dynamics can be detected by reprocessing available altimetric data with Lagrangian tools, giving insight into (sub-)mesoscale stirring processes relevant to tracer observations and complementing traditional Eulerian diagnostics.     

Dispersion of surface drifters and model-simulated trajectories

From a data set encompassing the years 1990–2008 pairs of surface drifters with maximum initial separations of 5, 10 and 25 km have been identified. Model trajectories have been calculated using the same initial positions and times as the selected pairs of surface drifters. The model trajectories are based on the TRACMASS trajectory code and driven by the ocean general circulation model NEMO. The trajectories are calculated off-line, i.e. with the stored velocity fields from the circulation model. The sensitivity of the trajectory simulations to the frequency of the stored velocity fields was tested for periods of 3 and 6 h as well as 5 days. The relative dispersion of the surface-drifter and model trajectories has been compared, where the latter was found to be too low compared to the relative dispersion of the drifters.Two low-order trajectory sub-grid parameterisations were tested and successfully tuned so that the total amplitude of the relative dispersion of the model trajectories is similar to that associated with the drifter trajectories. These parameterisations are, however, too simple for a correct simulation of Lagrangian properties such as the correlation time scales and the variance of the eddy kinetic energy.The importance of model-grid resolution is quantified by comparing the relative dispersion from an eddy-permitting and a coarse-resolution model, respectively. The dispersion rate is halved with the coarse grid. The consequences of the two-dimensionality of the trajectories is evaluated by comparing the results obtained with the 2D and the Lagrangian 3D trajectories. This shows that the relative dispersion is 15% stronger when the trajectories are freely advected with the 3D velocity field.     

Modelling coastal connectivity in a Western Boundary Current: Seasonal and inter-annual variability

Understanding the transport and distribution of marine larvae by ocean currents is one of the key goals of population ecology. Here we investigate circulation in the East Australian Current (EAC) and its impact on the transport of larvae and coastal connectivity. A series of Lagrangian particle trajectory experiments are conducted in summer and winter from 1992-2006 which enables us to investigate seasonal and inter-annual variability. We also estimate a mean connectivity state from the average of each of the individual realisations. Connectivity patterns are related to the movement of five individual larval species (two tropical, two temperate and one invasive species) and are found to be in qualitative agreement with historical distribution patterns found along the coast of SE Australia.We use a configuration of the Princeton Ocean Model to investigate physical processes in the ocean along the coast of SE Australia where the circulation is dominated by the EAC, a vigorous western boundary current. We assimilate hydrographic fields from a ∼10?\km global analysis into a ∼3?\km resolution continental shelf model to create a high-resolution hindcast of ocean state for each summer and winter from 1992-2006. Particles are released along the coast of SE Australia, and at various isobaths across the shelf (25-1000 m) over timescales ranging from 10-90 days. Upstream of the EAC separation point across-shelf release location dominates the particle trajectory length scales, whereas seasonality dominates in the southern half of the domain, downstream of the separation point.Lagrangian probability density functions show dispersion pathways vary with release latitude, distance offshore and the timescale of dispersion. Northern (southern) release sites are typified by maximum (minimum) dispersal pathways. Offshore release distance also plays a role having the greatest impact at the mid-latitude release sites. Maximum alongshore dispersion occurs at the mid-latitude release sites such as Sydney. Seasonal variability is also greatest at mid-latitudes, associated with variations in the separation point of the EAC. Climatic variations such as El Niño and La Niña are also shown to play a role in dictating the connectivity patterns. La Niña periods have a tendency to increase summer time connectivity (particularly with offshore release sites) while El Niño periods are shown to increase winter connectivity.The EAC acts as a barrier to the onshore movement of particles offshore, which impacts on the connectivity of offshore release sites. Consequentially particles released inshore of the EAC jet exhibit a greater coastal connectivity than those released offshore of the EAC front. The separation point of the EAC also dictates connectivity with more sites being connected (with lower concentration) downstream of the separation point of the EAC. These results can provide a useful guide to the potential connectivity of marine populations, or the spread of invasive pests (via ballast water or release of propagules from established populations).     

Measurements and modelling of the advection of suspended sediment in the swash zone by solitary waves

Novel laboratory experiments and numerical modelling have been performed to study the advection scales of suspended sediment in the swash zone. An experiment was designed specifically to measure only the sediment picked up seaward of the swash zone and during bore collapse. The advection scales and settling of this sediment were measured during the uprush along a rigid sediment-free beach face by a sediment trap located at varying cross-shore positions. Measurements were made using a number of repeated solitary broken waves or bores. Approximately 25% of the pre-suspended sediment picked up by the bores reaches the mid-swash zone (50% of the horizontal run-up distance), indicating the importance of the sediment advection in the lower swash zone. The pre-suspended sediment is sourced from a region seaward of the shoreline (still water line) which has a width of about 20% of the run-up distance. An Eulerian–Lagrangian numerical model is used to model the advection scales of the suspended sediment. The model resolves the hydrodynamics by solving the non-linear shallow water equations in an Eulerian framework and then solves the advection–diffusion equation for turbulence and suspended sediment in a Lagrangian framework. The model provides good estimates of the measured mass and distribution of sediment advected up the beach face. The results suggest that the correct modelling of turbulence generation prior to and during bore collapse and the advection of the turbulent kinetic energy into the lower swash is important in resolving the contribution of pre-suspended sediment to the net sediment transport in the swash zone.     

Simulated eddy induced vertical velocities in a Gulf of Alaska model

Mesoscale eddies can distribute nutrients, heat and fresh water into the Gulf of Alaska (GOA) from the coastal margins. While many studies have investigated the physical characteristics of GOA eddies, their effects on passive-dispersive particles have not been previously simulated to investigate eddy induced upwelling. A climatologically forced Parallel Ocean Program simulation of the north Pacific Ocean with an online particle tracking scheme was used to simulate passive-dispersive particles in the Gulf of Alaska. In-eddy vertical Lagrangian velocities of the particles were calculated both inside and outside the eddies and showed upwelling rates are generally greater inside the eddies where the vertical velocities of the particles ranged from 0.2 to 0.7 m/day.     

Dynamics of “critical” trajectories

Two diagnostic dynamic models for flow in hyperbolic and elliptic regions of a geophysical fluid are developed and compared. As the main interest here is in local dynamical processes, these models are used to study trajectories near stagnation points in the flow field. The simplest model presumes a balance between the Coriolis and geopotential accelerations. This model is equivalent to the classic approach that characterizes these regimes by the quadratic equation for the eigenvalues of the velocity gradient. However, since that model imposes geostrophic dynamics, the eigenvalues of the velocity gradient can be replaced by the local curvature or Hessian of the geopotential scaled by Coriolis. The general model adds both local and inertial accelerations to the dynamical balance. In contrast to the classic result the consequent frequency equation is a quartic that involves both the Hessian of the geopotential field, the components of the velocity gradient, and Coriolis. Roots of this equation give two distinct time scales, which are interpreted as Lagrangian time scales. Motion of the geopotential field produces a third Eulerian time scale. Critical trajectories are those whose initial positions and velocities are such that they are independent of the Lagrangian time scales. These simple models establish that within hyperbolic and elliptic regions of the geopotential field there may be trajectories whose time scales differ radically from even their nearest neighbors.A characteristic of critical trajectories in the ocean is that they often are found near stagnation points. These may be hard to identify even in model simulations, but a similar quantity, the null in the geopotential gradient, might be easier to obtain. To analyze the relation between the critical trajectories, stagnation points, and gradient null, evolution models for the later two objects are proposed. For a steady geopotential all three coincide. However with a time varying geopotential, they are distinct even though all have the same time scale. The analysis provides a metric for the separation of all three objects.     

现场海域水平扩散系数的探讨

根据Taylor的扩散理论,按Hay和Pasquill的拉格朗日流与欧拉流自相关函数相似的假定,用实测的长序列欧拉流资料,计算了不同时间尺度的水平扩散系数。