Lagrangian Stochastic Dispersion Model Research Articles

Lagrangian stochastic dispersion modelling for the simulation of the release of contaminants from tall and low sources

A Lagrangian particle model is used to study the dispersion of pollutants released in two different tracer experiments: Prairie Grass and Copenhagen. The model is based on a generalized form of the Langevin equation. Turbulence inputs are parameterized according to a scheme able to generate continuous values in all stability conditions and in all heights in the planetary boundary layer. Predicted values of ground-level concentration agree quite well with observed ones. The major progress of this paper is as follows: For air quality modeling it is highly necessary to include a parameterization that allows a correct description of the turbulent transport of contaminants released simultaneously from low and tall sources.

Application of a model system for the study of transport and diffusion in complex terrain to the TRACT experiment

The transport and diffusion processes of a tracer gas released near the ground in the Rhine valley region, in Central Europe, during the 1992 TRACT field experiment, are simulated by a computational model system for complex terrain. This system (RMS) is composed of the prognostic mesoscale model RAMS, the Lagrangian stochastic dispersion model SPRAY and the interface code MIRS, which links RAMS to SPRAY. Three flow simulations were performed, with different initialisations and the one showing the best agreement with the measured flow was selected for the simulation of the TRACT tracer experiment. Tracer concentrations measured by an array of samplers at ground level and by an airplane aloft, are used to evaluate the 3-D concentration field simulated by the model system. The analysis of the simulation results generated by RMS shows that our model system very well reproduces the general behaviour of the contaminant plume, the temporal and spatial distribution of the concentration and the location of the concentration maxima.

Shared- and distributed-memory parallelization of a Lagrangian atmospheric dispersion model

This paper describes parallelization of a 3-D Lagrangian stochastic atmospheric dispersion model using both distributed- and shared-memory methods. Shared-memory parallelism is implemented through the use of OpenMP compiler directives. Distributed-memory parallelism relies on the MPI message-passing library. One or both (using MPI for inter-node and OpenMP for intra-node communication) of the parallel modes can be used depending upon the requirements of the problem and the computational platform available. The distributed-memory version achieves a nearly linear decrease in execution time as the number of processors is increased. As the number of particles per processor is lowered, performance is limited by the decrease in work per processor and by the need to produce one set of output files. The shared-memory version achieves a speedup factor of ∼1.4 running on machines with four processors.

Estimation Of The Lagrangian Structure Function Constant C0 From Surface-Layer Wind Data

Eulerian turbulence observations, madein the surface layer under unstable conditions (z/L > 0),by a sonic anemometer were used to estimatethe Lagrangian structure function constant O. Twomethods were considered. The first one makes use of arelationship, widely used in the Lagrangian stochasticdispersion models, relating O to the turbulent kineticenergy dissipation rate e, wind velocity variance andLagrangian decorrelation time. The second one employsa novel equation, connecting O to the constant of thesecond-order Eulerian structure function. Beforeestimating O, the measurements were processed in orderto discard non-stationary cases at least to a firstapproximation and cases in which local isotropy couldnot be assumed. The dissipation e was estimated eitherfrom the best fit of the energy spectrum in theinertial subrange or from the best fit of the third-orderlongitudinal Eulerian structure function. Thefirst method was preferred and applied to the subsequentpart of the analysis. Both methods predict thepartitioning of O in different spatial components as aconsequence of the directional dependence of theEulerian correlation functions due to the isotropy.The O values computed by both methods are presented anddiscussed. In conclusion, both methods providerealistic estimates of O that compare well withprevious estimations reported in the literature, evenif a preference is to be attributed to the second method.

An approximate analytical model for footprint estimation of scalar fluxes in thermally stratified atmospheric flows

An approximate analytical model was developed to estimate scalar flux footprint in thermally stratified atmospheric surface layer flows. The proposed model was based on a combination of Lagrangian stochastic dispersion model results and dimensional analysis. The main advantage of this model is its ability to analytically relate atmospheric stability, measurement height, and surface roughness length to flux and footprint. Flux estimation by the proposed model was in good agreement with those calculated by detailed Eulerian and Lagrangian models. Measured water vapor fluxes collected along a downwind transect of a transition from a desert to an irrigated potato site were also used to assess the proposed model performance in the field. It was found that the model well reproduced the measured flux evolution with downwind distance.

A skewed meandering plume model for concentration statistics in the convective boundary layer

The meandering plume technique, which assumes that the total plume dispersion can be split into independent meander and relative dispersion components, is especially suited for modelling concentration (fluctuation) statistics in the convective boundary layer (CBL) with its large-scale turbulent motions. We develop a simple and practical meandering plume model for CBL applications that accounts for the skewed and inhomogeneous turbulence characteristics of the convective flow. The meander component is derived from a one-particle Lagrangian stochastic dispersion model by requiring that the meander and relative dispersion components correctly balance the first two total dispersion moments. Balancing of the third total moment implies a skewed relative dispersion, for which a bi-Gaussian distribution is used. The relative dispersion variance is parameterised with an extended asymptotic formulation for travel times much smaller than the Lagrangian integral time scale. For large travel times, the relative dispersion variance approaches the total dispersion variance. The in-plume fluctuations in the relative coordinate system are accounted for via the gamma probability density function. Laboratory data and large-eddy simulation results on total, relative and meander spreads are used to examine the model parameterisations and results. A requirement of the meandering plume model, that it should give the same mean concentration distribution as that obtained by the one-particle Lagrangian approach, is virtually fulfilled. Comparison of the model predictions of the concentration fluctuation intensity with existing laboratory data highlights the important contribution of in-plume fluctuations, which are normally neglected in meandering plume models. The paper also describes limitations of the new model and indicates the scope for further refinements.

Gas diffusion in a convection layer near a coastal region

Abstract Thermal internal boundary layer (TIBL) in onshore wind has a significant influence on dispersion of air pollutants and causes fumigation which brings high ground level concentration (GLC) of pollutants. A wind tunnel experiment was conducted in the thermally stratified wind tunnel of Nagasaki Research & Development Center, MHI, Japan, for the purpose of investigating the effect of TIBL on GLC and turbulent statistical properties. Numerical simulations using a Lagrangian stochastic dispersion model were also conducted and its results were compared with wind tunnel results and filed observation. Concerning GLC, they showed good agreement and this confirms the usefulness of a Lagrangian stochastic model in practical application to predict GLC in coastal areas.

A novel approach to atmospheric dispersion modelling: The Puff-Particle Model

The Puff–Particle Model (PPM) uses a new approach to problems of three-dimensional atmospheric dispersion from micro- to mesoscale. The pollutant particles are grouped in clusters treated as Gaussian puffs, which are dispersed making use of the concept of relative diffusion. The centre of mass of each puff is moved along a stochastic trajectory. This trajectory is derived from particle trajectories given by a Lagrangian stochastic-dispersion model. In this way, the PPM retains the advantages of traditional puff models and those of particle models, and is able to take into account the correct probability density function of the stochastic velocity components. The effect of meandering (caused by turbulent eddies larger than the puff but not resolved by the flow field) is simulated by the puff centre trajectories, yielding a complete description of dispersion. The PPM is validated using measurements from three tracer experiments in Copenhagen, Lillestrøm and Kincaid.

On Trajectory Curvature as a Selection Criterion for Valid Lagrangian Stochastic Dispersion Models

Recently Wilson and Flesch (Boundary-Layer Meteorology, 84, 411-426, 1997) suggested that the average increment 〈dθ z〉 to the orientation θ = arctan(w‘/u‘) of the Lagrangian velocity-fluctuation vector can be used to distinguish the better Lagrangian stochastic models within the well-mixed class. Here it is demonstrated that the specification of 〈dθ z〉 constitutes neither a sufficient or universally applicable criterion to distinguish the better Lagrangian stochastic models within the well-mixed class. The hypothesis made by Wilson and Flesch that Lagrangian stochastic models with φ/PE irrotational are ‘zero-spin’ models, having 〈dθ z〉=0, is proven

A Two-Dimensional Lagrangian Stochastic Dispersion Model for Convective Boundary Layers with Wind Shear

Rotach, Gryning and Tassone constructed a two-dimensional Lagrangian stochastic model to describe the dispersion of passive tracers in turbulent boundary layers with stabilities ranging from ideally-neutral (w* = 0) to fully-convective (u* = 0). They found that the value of the Kolmogorov constant, C0, as determined by optimizing model agreement with the measured spread of passive tracers, was dependent upon stability. Here, it is shown that the non-uniqueness, associated with satisfaction of the well-mixed condition, can be exploited to construct an alternative version of the model of Rotach et al. for which C0 = 3 is universally applicable over the entire range of stabilities under consideration. This alternative model is shown to be in very close agreement with predictions, obtained in large-eddy simulations, for the dispersion of passive tracers in turbulent boundary layers with stabilities ranging from ideally-neutral to fully-convective.

Trajectory Curvature As A Selection Criterion For valid Lagrangian Stochastic Dispersion Models

Among well-mixed multi-dimensional Lagrangian stochastic (LS) dispersion models, we observe that those in poorest agreement with observations produce ‘spiralling trajectories,’ with an associated reduction in dispersion. We therefore investigate statistics of increments dθ ' to the orientation θ'= arctan(W'/U') of the Lagrangian velocity-fluctuation vector – as a possible means to distinguish the better LS models within the well-mixed class. ‘Zero-spin’ models, having 〈 dθ'〉 = 0, are found to provide best agreement with observations. It is not clear however, whether imposition of the zero-spin property selects (in conjunction with the well-mixed condition) a unique model.

Experimental and numerical study of a water spray in the wake of an axisymmetric bluff body

An experimental and numerical study was made of a water spray issuing into a recirculating flow behind a bluff body disk mounted in a nozzle. A two-component laser-Doppler/phase-Doppler anomometry system was used to characterize the mean and turbulent dispersed phase. The numerical prediction of the hollow-cone spray was based on an Eulerian-Lagrangian stochastic hybrid model. The continuous gas flow field was predicted using a differential Reynolds stress transport model whereas the particulate droplet flow field was predicted using an improved Lagrangian stochastic dispersion model. Comparison of numerical predictions and experimental measurements was carried out for droplet mean and fluctuating velocities, number mean diameters, and mass fluxes. Results indicate that the droplet dispersion characteristics are strongly influenced by the presence of flow recirculation due to different particle Stokes numbers, yielding recirculation or penetration of particles through the separated flow region. Predictions of droplet axial and radial rms velocities obtained with the Lagrangian stochastic model are less satisfactory than those of other simple gas spray flows.

A two-dimensional Lagrangian stochastic dispersion model for daytime conditions

A two-dimensional (x, z) Lagrangian stochastic dispersion model is presented that is correct (i.e. fulfils the well-mixed condition) for neutral to convective conditions. The probability density function (pdf) of the particle velocities is constructed as a weighted sum of a neutral pdf (u and w jointly Gaussian) and a convective pdf (w skewed, u and w uncorrelated). The transition function ℱ varies continuously with stability and therefore ensures that the model results are not confined to a finite number of stability classes. The model is described in full detail and some sensitivity tests are presented. In particular, the role of Co, the universal constant in the Lagrangian structure function for the inertial subrange, in determining average plume characteristics is discussed. Furthermore, the evolution of average plume-height and -width is investigated for different boundary-layer stabilities ranging from ideally neutral to fully convective. Finally, the model is applied to the situation of a tracer experiment in Copenhagen, and it is shown that the measured surface-concentrations can be well simulated.

A two‐dimensional Lagrangian stochastic dispersion model for daytime conditions

AbstractA two‐dimensional (x, z) Lagrangian stochastic dispersion model is presented that is correct (i.e. fulfils the well‐mixed condition) for neutral to convective conditions. The probability density function (pdf) of the particle velocities is constructed as a weighted sum of a neutral pdf (u and w jointly Gaussian) and a convective pdf (w skewed, u and w uncorrelated). The transition function ℱ varies continuously with stability and therefore ensures that the model results are not confined to a finite number of stability classes.The model is described in full detail and some sensitivity tests are presented. In particular, the role of Co, the universal constant in the Lagrangian structure function for the inertial subrange, in determining average plume characteristics is discussed. Furthermore, the evolution of average plume‐height and ‐width is investigated for different boundary‐layer stabilities ranging from ideally neutral to fully convective. Finally, the model is applied to the situation of a tracer experiment in Copenhagen, and it is shown that the measured surface‐concentrations can be well simulated.

Comparison of closure schemes used to specify the velocity PDF in Lagrangian stochastic dispersion models for convective conditions

Lagrangian stochastic dispersion models make use of the probability density function (PDF) of the Eulerian vertical turbulent velocities. For convective conditions, the PDF is often assumed to have a bi-Gaussian form. Using new laboratory measurements of velocity PDFs in the convective boundary layer (CBL), we propose a new closure for constructing this bi-Gaussian PDF and compare results with three other closure schemes in current use. Of the three existing closures, two utilize the second and third moments of the vertical velocity as inputs, while the third one also incorporates the fourth moment. The new closure is defined with the desirable property that it collapses to a simple Gaussian in the limit of zero skewness. The value of an adjustable parameter in this closure scheme is selected using laboratory data for the third and fourth velocity moments. We determine the parameters in the PDF expression obtained using the four closures, and compare them with those derived by fitting velocity PDF data from the convection tank experiments. Significant differences are found between the values of the PDF parameters from the various closures and the water tank data. The performance of the closure schemes is compared by using a Lagrangian stochastic model to compute ground-level crosswind-integrated concentrations from particles released at four source heights. It is shown that the differences between the concentration estimates obtained using various closures increase as the source height increases. Using, as the benchmark, the dispersion results calculated from the Lagrangian stochastic model incorporating the laboratory velocity data without any closure, we recommend our new closure scheme. The results highlight the importance of turbulence observations in the CBL for accurate dispersion modelling.

Lagrangian stochastic dispersion model simulations of tracer data in nocturnal flows over complex terrain

A three-dimensional Lagrangian stochastic dispersion model (LSDM), which has been formulated to predict dispersion in three-dimensional nocturnal flows over complex terrain, is described. The model uses a Langevin equation in the lateral direction and a pure random-walk equation in the vertical direction. The performance of the model has been evaluated using the U.S. Department of Energy's ASCOT tracer databases collected during the 1984 field study in the Brush Creek valley in western Colorado, and the 1991 field study near the Rocky Flats Plant northwest of Denver. Both field experiments were performed in complex terrain regions where three-dimensional drainage flows were observed at night. The input wind fields and turbulence parameters required by the Lagrangian stochastic dispersion model are derived from meteorological data collected during the tracer experiments. The model-predicted concentrations are extensively compared with the corresponding observed concentrations. Several statistical measures calculated to quantitatively assess the performance of the model are presented. The agreement between the predicted and observed concentrations is satisfactory.

Flow Boundaries in Random-Flight Dispersion Models: Enforcing the Well-Mixed Condition

Lagrangian stochastic (LS) dispersion models often use trajectory reflection to limit the domain accessible to a particle. It is shown how the well-mixed condition (Thomson) can he expressed in the Chapman-Kolmogorov equation for a discrete-time LS model to provide a test for the validity of a reflection algorithm. By that means it is shown that the usual algorithm (perfect reflection) is exactly consistent with the wmc when used to bound Gaussian homogeneous turbulence, but that no reflection scheme can satisfy the wmc when applied at a location where the probability distribution for the normal velocity is asymmetric, or locally inhomogeneous. Thus, there is no well-mixed reflection scheme for inhomogeneous or skew turbulence.

A simple way of computing buoyant plume rise in Lagrangian stochastic dispersion models

Abstract A simple and easy to use method to include Eulerian plume rise in Lagrangian particle models is presented. This approach takes into account the vertical variation of wind and stability. Its ability to realistically simulate plume rise and spread, both through numerical experiments under typical atmospheric conditions and by comparison with actual plumes detected by a Differential Lidar in a case study, is shown. Despite its simplicity, our method proved to describe the main plume rise characteristics in a satisfactory way and to yield a fair agreement among observed and predicted plume centreline heights and standard deviations.

Disagreements between gradient-diffusion and Lagrangian stochastic dispersion models, even for sources near the ground

It is well known that if turbulent mass convection is modelled as diffusion, errors result unless trajectories from the source (at h) to the point of observation (Zp) comprise many statistically- independent segments (Taylor, 1921). We show that this is not guaranteed merely by the Lagrangian timescale (r) at the source being small (e.g., source at ground), but that a better criterion is t >> max(r(h), ~Zp)), where t is a typical travel time to zp.

Footprint estimation for scalar flux measurements in the atmospheric surface layer

The flux footprint is the contribution, per unit emission, of each element of a surface area source to the vertical scalar flux measured at height z m ; it is equal to the vertical flux from a unit surface point source. The dependence of the flux footprint on crosswind location is shown to be identical to the crosswind concentration distribution for a unit surface point source; an analytic dispersion model is used to estimate the crosswind-integrated flux footprint. Based on the analytic dispersion model, a normalized crosswind-integrated footprint is proposed that principally depends on the single variable z/z m , where z is a measure of vertical dispersion from a surface source. The explicit dependence of the crosswind-integrated flux footprint on downwind distance, thermal stability and surface roughness is contained in the dependence of z on these variables. By also calculating the flux footprint with a Lagrangian stochastic dispersion model, it is shown that the normalized flux footprint is insensitive to the analytic model assumption of a self-similar vertical concentration profile.