Impact of horizontal eddy diffusivity on Lagrangian statistics for coastal pollution from a major marine fairway

Lagrangian trajectory methods are often applied as deterministic transport models, where transport is due strictly to advection without taking into account stochastic elements of particle dispersion, which raises questions about validity of the model results. The present work investigates the impact of horizontal eddy diffusivity for a case study of coastal pollution in the Gulf of Finland, where the pollutants are assumed to originate from a major fairway and are transported to the coast by surface currents. Lagrangian trajectories are calculated using the TRACMASS model from velocity fields calculated by the Rossby Centre circulation model for 1982 to 2001. Three cases are investigated: (1) trajectory calculation without eddy diffusivity, (2) stochastic modelling of eddy diffusivity with a constant diffusion coefficient and (3) stochastic modelling of eddy diffusivity with a time- and space-variable diffusion coefficient. It is found that the eddy diffusivity effect increases the spreading rate of initially closely packed trajectories and the number of trajectories that eventually reach the coast. The pattern of most frequently hit coastal sections, the probability of hit to each such section and the time the pollution spends offshore are virtually invariant with respect to inclusion of eddy diffusivity.     

Identification of the environmentally safe fairway in the South-Western Baltic Sea and Kattegat

Application of the preventive techniques for the optimisation of fairways in the south-western Baltic Sea and the Kattegat in terms of protection of the coastal regions against current-driven surface transport of adverse impacts released from vessels is considered. The techniques rely on the quantification of the offshore domains (the points of release of adverse impacts) in terms of their ability to serve as a source of remote, current-driven danger to the nearshore. An approximate solution to this inverse problem of current-driven transport is obtained using statistical analysis of a large pool of Lagrangian trajectories of water particles calculated based on velocity fields from the Denmark’s Meteorological Institute (DMI)/BSH cmod circulation model forced by the DMI-HIRHAM wind fields for 1990–1994. The optimum fairways are identified from the spatial distributions of the probability of hitting the coast and for the time (particle age) it takes for the pollution to reach the coast. In general, the northern side of the Darss Sill area and the western domains of the Kattegat are safer to travel. The largest variations in the patterns of safe areas and the properties of pollution beaching occur owing to the interplay of water inflow and outflow. The gain from the use of the optimum fairways is in the range of 10–30?% in terms of the decrease in the probability of coastal hit within 10?days after pollution release or an increase by about 1–2?days of the time it takes for the hit to occur.     

Modelling environmentally friendly fairways using Lagrangian trajectories: a case study for the Gulf of Finland,the Baltic Sea

We address possibilities of minimising environmental risks using statistical features of current-driven propagation of adverse impacts to the coast. The recently introduced method for finding the optimum locations of potentially dangerous activities (Soomere et al. in Proc Estonian Acad Sci 59:156–165, 2010) is expanded towards accounting for the spatial distributions of probabilities and times for reaching the coast for passively advecting particles released in different sea areas. These distributions are calculated using large sets of Lagrangian trajectories found from Eulerian velocity fields provided by the Rossby Centre Ocean Model with a horizontal resolution of 2 nautical miles for 1987–1991. The test area is the Gulf of Finland in the northeastern Baltic Sea. The potential gain using the optimum fairways from the Baltic Proper to the eastern part of the gulf is an up to 44% decrease in the probability of coastal pollution and a similar increase in the average time for reaching the coast. The optimum fairways are mostly located to the north of the gulf axis (by 2–8 km on average) and meander substantially in some sections. The robustness of this approach is quantified as the typical root mean square deviation (6–16 km) between the optimum fairways specified from different criteria. Drastic variations in the width of the ‘corridors’ for almost optimal fairways (2–30 km for the average width of 15 km) signifies that the sensitivity of the results with respect to small changes in the environmental criteria largely varies in different parts of the gulf.     

Stochastic Lagrangian trajectory model for drifting objects in the ocean

The prediction of drifting object trajectories in the ocean is a complex problem plagued with uncertainties. This problem is usually solved simulating the possible trajectories based on wind and advective numerical and/or instrumental data in real time, which are incorporated into Lagrangian trajectory models. However, both data and Lagrangian models are approximations of reality and when comparing trajectory data collected from drifter exercises with respect to Lagrangian models results, they differ considerably. This paper introduces a stochastic Lagrangian trajectory model that allows quantifying the uncertainties related to: (i) the wind and currents numerical and/or instrumental data, and (ii) the Lagrangian trajectory model. These uncertainties are accounted for within the model through random model parameters. The quantification of these uncertainties consists in an estimation problem, where the parameters of the probability distribution functions of the random variables are estimated based on drifter exercise data. Particularly, it is assumed that estimated parameters maximize the likelihood of our model to reproduce the trajectories from the exercise. Once the probability distribution parameters are estimated, they can be used to simulate different trajectories, obtaining location probability density functions at different times. The advantage of this method is that it allows: (i) site specific calibration, and (ii) comparing uncertainties related to different wind and currents predictive tools. The proposed method is applied to data collected during the DRIFTER Project (eranet AMPERA, VI Programa Marco), showing very good predictive skills.     

The Airshed for Ozone and Fine Particulate Pollution in the Eastern United States

— To examine the spatial scales associated with atmospheric pollutants such as ozone (O₃) and fine particulate matter (PM_2.5), we employ the following five techniques: (1) Analysis of the persistence of high O₃ concentrations aloft; (2) spatial and lag correlations between the short-term components (i.e., weather-induced variations) in the time series of O₃ and PM_2.5 throughout the eastern United States; (3) analysis of mixed-layer forward trajectories compiled at different locations on a climatological basis to identify the potential region covered in 1-day of atmospheric transport; (4) analysis of three-dimensional Lagrangian trajectories of tracer particles for three high-O₃ episode events in the summer of 1995; and (5) analysis of the spatial extent over which emissions have an impact through photochemical model simulations. Regardless of the method chosen, the results demonstrate that pollutants such as O₃ and PM_2.5 have the potential to affect regions having spatial scales of several hundred kilometers. This finding has implications to regulatory policies for addressing the pollution problem, and for optimally designing monitoring networks for such pollutants.     

Calculating the macrodispersion coefficient of the ensemble averaged solute transport equation in the discrete domain

Recognizing the spatial heterogeneity of hydraulic parameters, many researchers have studied the solute transport by both groundwater and channel flow in a stochastic framework. One of the methodologies used to up‐scale the stochastic solute transport equation, from a point‐location scale to a grid scale, is the cumulant expansion method combined with the calculus for the time‐ordered exponential and the calculus for the Lie operator. When the point‐location scale transport equation is scaled up to the grid scale, using the cumulant expansion method, a new dispersion coefficient emerges in the dispersive term of the solute transport equation in addition to the molecular dispersion coefficient. This velocity driven dispersion is called ‘macrodispersion’. The macrodispersion coefficient is the integral function of the time‐ordered covariance of the random velocity field. The integral is calculated over a Lagrangian trajectory of the flow. The Lagrangian trajectory depends on the following: (i) the spatial origin of the particle; (ii) the time when the macrodispersion is calculated; and (iii) the mean velocity field along the trajectory itself. The Lagrangian trajectory is a recursive function of time because the location of the particle along the trajectory at a particular time depends on the location of the particle at the previous time. This recursive functional form of the Lagrangian trajectory makes the calculation of the macrodispersion coefficient difficult. Especially for the unsteady, spatially non‐stationary, non‐uniform flow field, the macrodispersion coefficient is a highly complex expression and, so far, calculated using numerical methods in the discrete domains. Here, an analytical method was introduced to calculate the macrodispersion coefficient in the discrete domain for the unsteady and steady, spatially non‐stationary flow cases accurately and efficiently. This study can fill the gap between the theory of the ensemble averaged solute transport model and its numerical implementations. Copyright © 2011 John Wiley & Sons, Ltd.     

A connection between the Lagrangian stochastic–convective and cumulant expansion approaches for describing solute transport in heterogeneous porous media

The equation describing the ensemble-average solute concentration in a heterogeneous porous media can be developed from the Lagrangian (stochastic–convective) approach and from a method that uses a renormalized cumulant expansion. These two approaches are compared for the case of steady flow, and it is shown that they are related. The cumulant expansion approach can be interpreted as a series expansion of the convolution path integral that defines the ensemble-average concentration in the Lagrangian approach. The two methods can be used independently to develop the classical form for the convection–dispersion equation, and are shown to lead to identical transport equations under certain simplifying assumptions. In the development of such transport equations, the cumulant expansion does not require a priori the assumption of any particular distribution for the Lagrangian displacements or velocity field, and does not require one to approximate trajectories with their ensemble-average. In order to obtain a second-order equation, the cumulant expansion method does require truncation of a series, but this truncation is done rationally by the development of a constraint in terms of parameters of the transport field. This constraint is less demanding than requiring that the distribution for the Lagrangian displacements be strictly Gaussian, and it indicates under what velocity field conditions a second-order transport equation is a reasonable approximation.     

On the mechanisms of the saltating motion of bedload

Saltation of sediment particles is an important pattern of bedload transport.Based on force analysis for sediment particles,a Lagrangian model was proposed for the saltating motion of bedload in river flows,which was then solved with numerical method.Simulation results on the saltating trajectories neglecting particle rotation and turbulence effects compare fairly well with experimental observations.The mean values of the saltation parameters (saltation height,length and velocity) also agree well with the previous experimental data.Based on the numerical results,regression equations for the dimensionless saltation height,length and velocity were presented.Using the numerically achieved characteristics of the sediment saltation,we also obtained mathematical expression for the sediment transport rate.The studies in this paper are significant for its contribution to mechanism of the bedload motion and the computation of sediment transport rate.     

Eddy diffusivity derived from drifter data for dispersion model applications

Ocean transport and dispersion processes are at the present time simulated using Lagrangian stochastic models coupled with Eulerian circulation models that are supplying analyses and forecasts of the ocean currents at unprecedented time and space resolution. Using the Lagrangian approach, each particle displacement is described by an average motion and a fluctuating part. The first one represents the advection associated with the Eulerian current field of the circulation models while the second one describes the sub-grid scale diffusion. The focus of this study is to quantify the sub-grid scale diffusion of the Lagrangian models written in terms of a horizontal eddy diffusivity. Using a large database of drifters released in different regions of the Mediterranean Sea, the Lagrangian sub-grid scale diffusion has been computed, by considering different regimes when averaging statistical quantities. In addition, the real drifters have been simulated using a trajectory model forced by OGCM currents, focusing on how the Lagrangian properties are reproduced by the simulated trajectories.     

Numerical reconstruction of trajectory of small-size surface drifter in the Mediterranean sea

In this study, we addressed the effects of wind-induced drift on Lagrangian trajectories of surface sea objects using high-resolution ocean forecast and atmospheric data. Application of stochastic Leeway model for prediction of trajectories drift was considered for the numerical reconstruction of the Elba accident that occurred during the period 21.06.2009–22.06.2009: a person on an inflatable raft was lost in the vicinity of the Elba Island coast; from the initial position, the person on a raft drifted southwards in the open sea and later he was found on a partially deflated raft during rescue operation. For geophysical forcing, we used high-resolution currents from the Mediterranean Forecasting System and atmospheric wind from the European Centre for Medium-Range Weather Forecasts. To investigate the effect of wind on trajectory behavior, numerical simulations were performed using different categories of drifter-like particles similar to a person on an inflatable raft. An algorithm of spatial clustering was used to differentiate the most probable search areas with a high density of particles. Our results showed that the simulation scenarios using particles with characteristics of draft-limited sea drifters provided better prediction of an observed trajectory in terms of the probability density of particles.     

Designing a monitoring network for detecting groundwater pollution with stochastic simulation and a cost model

A method is presented to design monitoring networks for detecting groundwater pollution at industrial sites. The goal is to detect the pollution at some distance from the site’s boundary so that it can be cleaned up or hydrologically contained before contaminating groundwater outside the site. It is assumed that pollution may occur anywhere on the site, that transport is by advection only and that no retardation and chemical reactions take place. However, the approach can be easily extended to include designated (and uncertain) source areas, dispersion and reactive transport. The method starts from the premise that it is impossible to detect 100% of all the contaminant plumes with reasonable costs and therefore seeks a balance between the risk of pollution and network density. The design approach takes account of uncertainty in the flow field by simulating realisations of conductivity, groundwater head and associated flow fields, using geostatistical simulation and a groundwater flow model. The realisations are conditioned to conductivity and head observations that may already be present on the site. The result is an ensemble of flow fields that is further analysed using a particle track program. From this the probability of missing a contaminant plume originating anywhere on the terrain can be estimated for a given network. From this probability follows the risk, i.e. the expected costs of an undetected pollution. The total costs of the monitoring strategy are calculated by adding the risk of pollution to the costs of installing and maintaining the monitoring wells and the routinely performed chemical analyses. By repeating this procedure for networks of varying well numbers, the best network is chosen as the one that minimises total cost. The method is illustrated with a simulated example showing the added worth of exploratory wells for characterising hydraulic conductivity of a site.     

Simulation of the Lagrangian tide-induced residual velocity in a tide-dominated coastal system: a case study of Jiaozhou Bay,China

The Eulerian residual transport velocity and the first-order Lagrangian residual velocity for weakly nonlinear systems have been used extensively in the past to depict inter-tidal mass transport. However, these could not explain the observed net surface sediment transport pattern in Jiaozhou Bay (JZB), located on the western Yellow Sea. JZB is characterized by strong tidal motion, complex topography and an irregular coastline, which are features of typical nonlinear systems. The Lagrangian residual velocity, which is applicable to general nonlinear systems, was simulated with the water parcel tracking method. The results indicate that the composition of the Lagrangian residual velocity at different tidal phases coincides well with the observed net surface sediment transport pattern. The strong dependence of water flushing time on the initial tidal phase can also be explained by the significant intra-tidal variation of the Lagrangian residual velocity. To investigate the hydrodynamic mechanism governing the nonlinearity of the M ₂ tidal system, a set of nonlinearity indexes were defined and analysed. In the surface layer, horizontal advection is the main contributor to the strong nonlinearity near the bay mouth, while in the bottom layer, the strong nonlinearity near the bay mouth may result from the vertical viscosity and horizontal advection.

     

A quantitative characterization of the seasonal Lagrangian circulation of the Gulf of California from a three-dimensional numerical model

A set of four indices that quantify Lagrangian properties of the Gulf of California seasonal circulation were implemented from outputs of a three-dimensional numerical model. From trajectories of particles seeded over the entire Gulf, we calculated for 12 one-month periods the following indices: net and total distance traveled by the particles, the number of particles that are found within an area centered on the release positions after one month, and time taken by particles to escape from a 50-km-radius circle. These indices can be used for studies on transport of inert properties and passive planktonic organisms such as eggs and early-stage larvae; their use is illustrated for typical summer and winter conditions in the Gulf of California. These indices show the potential for connecting areas separated by a few hundreds of km along the eastern side of the Gulf, due to the strong seasonal up-gulf and down-gulf current. In the Northern Gulf, large displacements occur at the borders of the basin-wide seasonally reversing eddy that dominates the large-scale circulation (cyclonic in summer, anticyclonic in winter). On the other hand, the potential for self-recruitment areas is found as particles can be trapped for longer than one month within these eddies, as well as in smaller ones in the Northern Gulf, and near the coast of the peninsular side of the Southern Gulf, where current speeds are slow and many small capes and islands are present.     

Simulation of the Lagrangian tide-induced residual velocity in a tide-dominated coastal system: a case study of Jiaozhou Bay, China

The Eulerian residual transport velocity and the first-order Lagrangian residual velocity for weakly nonlinear systems have been used extensively in the past to depict inter-tidal mass transport. However, these could not explain the observed net surface sediment transport pattern in Jiaozhou Bay (JZB), located on the western Yellow Sea. JZB is characterized by strong tidal motion, complex topography and an irregular coastline, which are features of typical nonlinear systems. The Lagrangian residual velocity, which is applicable to general nonlinear systems, was simulated with the water parcel tracking method. The results indicate that the composition of the Lagrangian residual velocity at different tidal phases coincides well with the observed net surface sediment transport pattern. The strong dependence of water flushing time on the initial tidal phase can also be explained by the significant intra-tidal variation of the Lagrangian residual velocity. To investigate the hydrodynamic mechanism governing the nonlinearity of the M ₂ tidal system, a set of nonlinearity indexes were defined and analysed. In the surface layer, horizontal advection is the main contributor to the strong nonlinearity near the bay mouth, while in the bottom layer, the strong nonlinearity near the bay mouth may result from the vertical viscosity and horizontal advection.     

Drifter observations of the Gulf of Maine Coastal Current

Two-hundred and twenty seven satellite-tracked drifters were deployed in the Gulf of Maine (GoM) from 1988 to 2007, primarily during spring and summer. The archive of tracks includes over 100,000 km logged thus far. Statistics such as transit times, mean velocities, response to wind events, and preferred pathways are compiled for various areas of the coastal GoM. We compare Lagrangian flow with Eulerian estimates from nearby moorings and evaluate drifter trajectories using Ekman theory and 3-D ocean circulation models.     

Lagrangian simulations of unstable gravity-driven flow of fluids with variable density in randomly heterogeneous porous media

A new Lagrangian particle model based on smoothed particle hydrodynamics (SPH) is developed and used to simulate Darcy scale flow and transport in porous media. The method has excellent conservation properties and treats advection exactly. The Lagrangian method is used in stochastic analysis of miscible density-driven fluid flows. Results show that heterogeneity significantly increases dispersion and slows development of Rayleigh–Taylor instability. The presented numerical examples illustrate the advantages of Lagrangian methods for stochastic transport simulations.     

Difference between the Lagrangian trajectories and Eulerian residual velocity fields in the southwestern Yellow Sea

The responses to tidal and/or wind forces of Lagrangian trajectories and Eulerian residual velocity in the southwestern Yellow Sea are investigated using a high-resolution circulation model. The simulated tidal harmonic constants agree well with observations and existing studies. The numerical experiment reproduces the long-range southeastward Eulerian residual current over the sloping bottom around the Yangtze Bank also shown in previous studies. However, the modeled drifters deployed at the northeastern flank of the Yangtze Bank in the simulation move northeastward, crossing over this strong southeastward Eulerian residual current rather than following it. Additional sensitivity experiments reveal that the influence of the Eulerian tidal residual currents on Lagrangian trajectories is relatively weaker than that of the wind driven currents. This result is consistent with the northeastward movement of ARGOS surface drifters actually released in the southwestern Yellow Sea. Further experiments suggest that the quadratic nature of the bottom friction is the crucial factor, in the southwestern Yellow Sea, for the weaker influence of the Eulerian tidal residual currents on the Lagrangian trajectories. This study demonstrates that the Lagrangian trajectories do not follow the Eulerian residual velocity fields in the shallow coastal regions of the southwestern Yellow Sea.     

A Lagrangian approach to modelling stable isotopes in precipitation over mountainous terrain

A Lagrangian (Rayleigh) distillation model is used to track the evolution of stable isotopes in precipitation over mountainous terrain from the Pacific Coast of Canada to two alpine field sites in the Canadian Rocky Mountains. Precipitation δ¹⁸O at Vancouver constrains the model and air–mass back trajectories provide the water vapour pathway for 10 winter storm events. Isotopic values along storm pathways are modelled with a classical Rayleigh model that prescribes a linear decrease in temperature and pressure from initial to final conditions, and two models that account directly for orographic precipitation processes by: (i) applying an orographic rainfall model and (ii) using North American Regional Reanalysis data to calculate the change in vapour content along storm pathways. All models are significant predictors of snowpack δ¹⁸O, but the orographic model provides the best fit to precipitation‐weighted δ¹⁸O for each storm. The improvement in modelled δ¹⁸O by accounting for terrain along storm trajectories illustrates the need to account for orographically driven moisture loss when modelling vapour transport to ice core sites with mountainous upwind terrain. This finding is also applicable to isotopic studies of paleoaltimetry and source areas of groundwater recharge. Copyright © 2011 John Wiley & Sons, Ltd.     

First-principles derivation of reactive transport modeling parameters for particle tracking and PDE approaches

Both Eulerian and Lagrangian reactive transport simulations in natural media require selection of a parameter that controls the “promiscuity” of the reacting particles. In Eulerian models, measurement of this parameter may be difficult because its value will generally differ between natural (diffusion-limited) systems and batch experiments, even though both are modeled by reaction terms of the same form. And in Lagrangian models, there previously has been no a priori way to compute this parameter. In both cases, then, selection is typically done by calibration, or ad hoc. This paper addresses the parameter selection problem for Fickian transport by deriving, from first principles and D (the diffusion constant) the reaction-rate-controlling parameters for particle tracking (PT) codes and for the diffusion–reaction equation (DRE). Using continuous time random walk analysis, exact reaction probabilities are derived for pairs of potentially reactive particles based on D and their probability of reaction provided that they collocate. Simultaneously, a second PT scheme directly employing collocation probabilities is derived. One-to-one correspondence between each of D, the reaction radius specified for a PT scheme, and the DRE decay constant are then developed. These results serve to ground reactive transport simulations in their underlying thermodynamics, and are confirmed by simulations.