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.     

Dynamic parameters in models of atmospheric vortex structures

Vortex simulation and the computation of fields of dynamic parameters of vortex structures (velocity, rotor velocity, and helicity) are carried out with the use of exact hydrodynamic equations in a cylindrical coordinate system. Components of centripetal and Coriolis accelerations are taken into account in the initial equations. Internal and external solutions are defined. Internal solutions ignore the disturbances of the pressure field, but they are considered in external solutions. The simulation is carried out so that the effect of accounting for spatial coordinates on the structure of the above fields is pronounced. It is shown that the initial kinetic energy of rotating motion transforms into the kinetic energy of radial and vertical velocity components in models with centripetal acceleration. In models with Coriolis acceleration, the Rossby effect is clearly pronounced. The method of an “inverse problem” is used for finding external solutions, i.e., reconstruction of the pressure field at specified velocity components. Computations have shown that tangential components mainly contribute to the velocity and helicity vortex moduli at the initial stage.     

Projecting uncertainty onto marine megafauna trajectories

In this study, a method is proposed for estimating the uncertainty of a Lagrangian pathway calculated from an undersampled ocean surface velocity field. The primary motivation and application for this method is the differentiation between active and passive movements for sea turtles whose trajectories are observed with satellite telemetry. Synthetic trajectories are launched within a reconstructed surface velocity field and integrated forward in time to produce likely trajectories of an actual turtle or drifter. Uncertainties in both the initial conditions at launch and the velocity field along the trajectory are used to yield an envelope of possible synthetic trajectories for each actual trajectory. The juxtaposition of the actual trajectory with the resulting cloud of synthetic trajectories provides a means to distinguish between active and passive movements of the turtle. The uncertainty estimates provided by this model may lead to improvements in our understanding of where and when turtles are engaged in specific behaviors (i.e. migration vs. foraging)—for which potential management efforts may vary accordingly.     

Directed drifter launch strategies for Lagrangian data assimilation using hyperbolic trajectories

The dependence of the fidelity of a Lagrangian data assimilation scheme on the initial launch locations of the observed drifters is studied in the context of a reduced gravity, primitive equation model of mid-latitude circulations. A directed launch strategy, based on tracking the Lagrangian manifolds emanating from strongly hyperbolic regions in the flow field, is developed. In a series of twin assimilation experiments, the rate of convergence of the data assimilating scheme is shown to be consistently higher in such directed launches compared to those produced from randomly selected initial drifter positions. By directing initial drifter positions along the out-flowing branch of identifiable Lagrangian boundaries, the relative dispersion of the drifters, the overall data coverage and the sampling of high kinetic energy features in the flow are optimized. In general, the performance of the assimilation procedure is shown to depend strongly on the independence of the observed drifter trajectories and the temporal persistence of the corrections provided by the data.     

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.     

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.     

Enhanced estimation of sonobuoy trajectories by velocity reconstruction with near-surface drifters

An investigation to improve trajectory prediction using Lagrangian data is presented. The velocity field of a data assimilating model, EAS-16, is corrected using drifter observations taken during an experiment off Taiwan. The results are tested using another independent Lagrangian data set provided by sonobuoys launched in the same area. The latter have instrument chains that extend well into the water column. Consequently the corrected model velocities were projected into the water column in order to calculate sonobuoy trajectories. The drifter and sonobuoy trajectories both show two distinct regimes in the considered area of approximately 1/2° square. One regime is dominated by shelf dynamics, the other by meandering of the Kuroshio, with a sharp boundary dividing the two. These two regimes are not reproduced by the trajectories of the EAS-16 model. When the drifter data are blended with the model velocities, synthetic sonobuoy trajectories track the observed ones much better, and the two regimes are clearly depicted. Two different methods for the velocity reconstruction are tested. One is based on a variational approach and the other on a normal mode decomposition. Both methods show qualitatively similar improvements in the prediction of sonobuoys trajectories, with a quantitative improvement in the total rms error of approximately 50% and 25%, respectively.     

Accurate representation of geostrophic and hydrostatic balance in unstructured mesh finite element ocean modelling

Accurate representation of geostrophic and hydrostatic balance is an essential requirement for numerical modelling of geophysical flows. Potentially, unstructured mesh numerical methods offer significant benefits over conventional structured meshes, including the ability to conform to arbitrary bounding topography in a natural manner and the ability to apply dynamic mesh adaptivity. However, there is a need to develop robust schemes with accurate representation of physical balance on arbitrary unstructured meshes. We discuss the origin of physical balance errors in a finite element discretisation of the Navier–Stokes equations using the fractional timestep pressure projection method. By considering the Helmholtz decomposition of forcing terms in the momentum equation, it is shown that the components of the buoyancy and Coriolis accelerations that project onto the non-divergent velocity tendency are the small residuals between two terms of comparable magnitude. Hence there is a potential for significant injection of imbalance by a numerical method that does not compute these residuals accurately. This observation is used to motivate a balanced pressure decomposition method whereby an additional “balanced pressure” field, associated with buoyancy and Coriolis accelerations, is solved for at increased accuracy and used to precondition the solution for the dynamical pressure. The utility of this approach is quantified in a fully non-linear system in exact geostrophic balance. The approach is further tested via quantitative comparison of unstructured mesh simulations of the thermally driven rotating annulus against laboratory data. Using a piecewise linear discretisation for velocity and pressure (a stabilised P₁P₁ discretisation), it is demonstrated that the balanced pressure decomposition method is required for a physically realistic representation of the system.     

基于拉格朗日方法的南海自动剖面浮标轨迹模拟系统*

南海自动剖面浮标轨迹模拟系统包括高分辨率模式流场、拉格朗日追踪模型和垂向浮标运动参数化方案等三个核心部分。该系统可在南海范围内模拟两类自动剖面浮标: 传统自动剖面浮标(停滞深度为1000m, 最大下潜深度为2000m)和新型深海自动剖面浮标(停滞深度为距海底500m)。通过对南海现有的6个传统浮标的模拟, 该系统可以预测其100d内的漂流轨迹。通过与真实浮标轨迹数据的对比, 验证了该模拟系统的准确性。此外, 根据该系统, 我们初步探讨了深海自动剖面浮标阵列(时空分辨率为2°×2°×30d)在南海内区布放方案的可行性。该模拟系统的建立和完善将有助于对现有传统剖面浮标布放策略进行优化, 并对未来深海剖面浮标在南海的推广应用提供初步的理论依据。     

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.     

Improved boundary equations for elliptic water wave models

Approximate equations for the elliptic mild slope equation are derived based on Padé approximation and used as absorbing boundaries. The new boundary equations can absorb the incident waves for high range of wave incidence angle. In addition, the new boundary equations can be used iteratively to refine the solution by eliminating reflections from the boundaries. An iterative conjugate gradient scheme has been used to solve the elliptic water wave equation and the new boundary equations. The model developed can accommodate for wave diffraction, refraction and reflections from structures with wide range of wave angles at the boundaries. The new model has been tested for several cases. The model compares very well with other models.     

An analytical calculation of two-dimensional dispersion

The methods of Okuboet al. (1976a) are used to calculate the Lagrangian deformations and diffusivities of a cluster of drifters. A solution of the two-dimensional first-order advection-diffusion equation (Okuboet al., 1983a) is then used to calculate the dimensions and orientation of the cluster from these Lagrangian deformations and diffusivities. The solution is shown to be internally consistent (to give cluster areas that are consistent with the observed cluster areas) to within a 0.5% error. As time progresses a larger portion of the dispersion is caused by the diffusivities rather than the deformations. In the experiments analyzed the Lagrangian deformations and diffusivities are generally observed to increase at a constant rate over time intervals of about one hour. Dimensional arguments suggest that Lagrangian diffusivities increase proportional tot ² and the deformations proportional tot ^1,5 for time intervals large compared to the period required to spread from a point source to the initial cluster dimensions. Small quadratic velocity gradients cause the solution of the first order advection-diffusion equation to overestimate cluster spreading. Most of the displacement (once motion due to the mean velocity and linear deformations is extracted) is caused by scales of motion much smaller than the cluster. This explains the relatively small magnitude of the errors caused by parameterizing quadratic and other statistically significant nonlinear shears as a component of the eddy-diffusivity.     

海上溢油粒子追踪预测模型中的两种数值方法比较

在海上溢油粒子追踪预测模型中,关键的是对拉格朗日微分方程的求解。本文首先通过数值实验比较了欧拉法和龙格-库塔法求解拉格朗日溢油轨迹微分方程的优劣,然后将其应用到2005年4月3日发生在大连附近的“ARTEAGA”油轮溢油事故的油膜粒子追踪模型中。数值实验和应用结果表明,在近岸不均匀流场下,用龙格-库塔方法解拉格朗日油粒子微分方程比用欧拉法求解精度高,用龙格-库塔方法模拟“ARTEAGA”油轮轨迹及其扩散范围与实际观测更为接近,而用欧拉法模拟溢油扩散的面积偏大。     

Tide-induced Lagrangian residual velocity and dynamic analysis based on field observations in the inner Xiangshan Bay,China

In the Xiangshan Bay at the east coast of China, coastal marine pollution is conspicuous and severe in recent years. As transport of the pollutants is closely related to the coastal circulation, there is a great practical significance to investigate the circulation in this area. In this work, the surface pattern and vertical profiles of Lagrangian residual velocity (LRV) were studied based on field observation data from the inner Xiangshan Bay. By tracking GPS-GPRS drifters’ trajectories, the surface LRV pattern is going out in the central deep trough and flowing inwards near the shoreside. Combined with data from two mooring stations, vertical profiles of LRV is flowing out at surface and flowing in at the bottom, consistent with the gravitational circulation induced by baroclinic effects at the estuary. However, according to the diagnostic analysis, the main mechanism driving the residual current is barotropic rather than baroclinic. The LRV equation is controlled by the tidally-averaged barotropic pressure gradient force, tidal body force and tidally-averaged turbulent stress, while the tidally-averaged baroclinic pressure gradient force is one order of magnitude less than other forces. Additionally, the tidally mean eddy viscosity coefficient which is used in the expression of tidally-averaged turbulent stress might be not adequate and requires further studies.     

Centrifugal forces estimated from trajectories of drifting buoys in winding paths of the Kuroshio

Radii and angular velocities in the motions of drifting buoys deployed in the Kuroshio are estimated by fitting circles to the trajectories of two drifting buoys, one with a drogue at 300 m depth and the other at 800 m depth. The buoys were deployed in the Kuroshio where it was flowing counter-clockwise around the large cold water mass south of Honshu. The same technique was applied to two drifting buoys with drogues at 300 m depth placed in the Kuroshio where it flowed clockwise around Oshima Island in Sagami Bay. The centrifugal forces were 7% and 6% as large as the Coriolis forces in the Kuroshio around the cold water mass, and they were –56% and –42% as large as the Coriolis forces in the current around the Oshima Island. The temperature gradient observed in the Oshima-West Channel suggested that the pressure gradient there was smaller due to the centrifugal force acting against the Coriolis force than the pressure gradient to be balanced with the Coriolis force.     

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.     

On two small-scale circulation mechanisms of fine-aerosol transport in the atmospheric surface layer

Small-scale processes are taken to mean the disturbances of the atmospheric basic background which are caused by the thermal inhomogeneity of the underlying surface and under which one can neglect the effects of both centripetal and Coriolis accelerations. Slight disturbances suggest the use of linearized hydrothermodynamic equations of a weakly compressible atmosphere. Two models are considered. In one of the models, circulation over a weakly sloping barchan is analyzed using a refined model of mountain—valley circulation (the well-known Prandtl model). The other model, which is a model of a thermal spot in a geostrophic flow, can conditionally be called “anticonvective.” This problem is solved using the method of universal functions for parabolic equations with variable coefficients.     

Numerical Simulation of Local Scour Around A Large Circular Cylinder Under Wave Action

A horizontal two- dimensional numerical model is developed for estimation of sediment transport and sea bed change around a large circular cylinder under wave action. The wave model is based on an elliptic mild slope equation. The wave-induced current by the gradient of radiation stress is considered and a depth integrated shallow water equation is applied to the calculation of the current. The mass transport velocity and the bed shear stress due to streaming are considered, which are important factors affecting the sediment transport around a structure due to waves, especially in reflective areas. Wave-current interaction is taken into account in the model for computing the bed shear stress. The model is implemented by a finite element method. The results of this model are compared with those from other methods and agree well with experimental data.