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 d z to the orientation = 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 d 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 /P_E irrotational are zero-spin models, having d z=0, is proven     

On the Formulation of Lagrangian Stochastic Models of Scalar Dispersion Within Plant Canopies

Lagrangian stochastic models, quadratic in velocity and satisfying the well-mixed condition for two-dimensional Gaussian turbulence, are used to make predictions of scalar dispersion within a model plant canopy. The non-uniqueness associated with satisfaction of the well-mixed condition is shown to be non-trivial (i.e. different models produce different predictions for scalar dispersion). The best agreement between measured and predicted mean concentrations of scalars is shown to be obtained with a small sub-class of optimal models. This sub-class of optimal models includes Thomson's model (J. Fluid Mech. 180, 529–556, 1987), the simplest model that satisfies the well-mixed condition for Gaussian turbulence, but does not include two other models identified recently as being in optimal agreement with the measured spread of tracers in a neutral boundary layer. It is therefore demonstrated that such models are not universal, i.e. applicable to a wide range of flows without readjustment of model parameters. Predictions for scalar dispersion in the model plant canopy are also obtained using the model of Flesch and Wilson (Boundary-Layer Meteorol. 61, 349–374, 1992). It is shown that, when used with a Gaussian velocity distribution or a maximum-missing-information velocity distribution, which accounts for the measured skewness and kurtosis of velocity statistics, the agreement between predictions obtained using the model of Flesch and Wilson and measurements is as good as that obtained using Thomson's model.     

On The Dynamical Content Of Lagrangian Stochastic Models In The Well-Mixed Class

It is shown how the correspondence between Lagrangian stochasticmodels and second-moment closures of the scalar-flux equation can be exploited to distinguishbetween Lagrangian stochastic models in the well-mixed class. It is found that physically realisticclosures of the scalar-flux equation correspond to Lagrangian stochastic models that have non-zero`spin' and so produce spiralling tracer-particle trajectories, whilst `zero-spin'models correspond to the isotropic-production model of scalar-fluxes.Lagrangian stochastic models consistent with rapid distortion theory and Speziale's transformation rule for the Reynolds stressequations in the extreme limit of two-dimensional turbulence are also shown to have non-zero spin.The residual non-uniqueness associated with satisfaction of thewell-mixed condition and the specification of mean spin is shown to be related to the helicity oftracer-particle trajectories. Investigations are also made of the influence upon turbulent dispersion oftime-dependent spin and of mean rotations of the fluctuating Lagrangian acceleration vector(i.e., second-order spin).     

Non-Gaussian Lagrangian Stochastic Model for Wind Field Simulation in the Surface Layer

Wind field simulation in the surface layer is often used to manage natural resources in terms of air quality, gene flow(through pollen drift), and plant disease transmission(spore dispersion). Although Lagrangian stochastic(LS) models describe stochastic wind behaviors, such models assume that wind velocities follow Gaussian distributions. However,measured surface-layer wind velocities show a strong skewness and kurtosis. This paper presents an improved model, a non-Gaussian LS model, which inco...     

A Lagrangian Stochastic Model for Sea-Spray Evaporation in the Atmospheric Marine Boundary Layer

The dispersion of heavy particles subjected to a turbulent forcing is often simulated with Lagrangian stochastic models. Although these models have been employed successfully over land, the implementation of traditional LS models in the marine boundary layer is significantly more challenging. We present an adaptation of traditional Lagrangian stochastic models to the atmospheric marine boundary layer with a particular focus on the representation of the scalar turbulence for temperature and humidity. In this new model, the atmosphere can be stratified and the bottom boundary is represented by a realistic wavy surface that moves and deforms. Hence, the correlation function for the turbulent flow following a particle is extended to the inhomogenous, anisotropic case. The results reproduce behaviour for scalar Lagrangian turbulence in a stratified airflow that departs only slightly from the expected behaviour in isotropic turbulence. When solving for the surface temperature and the radius of evaporating heavy water droplets in the airflow, the modelled turbulent forcing on the particle also behaves remarkably well. We anticipate that this model will prove especially useful in the context of sea-spray dispersion and its associated sensible heat, latent heat, and gas fluxes between spray droplets and the atmosphere.     

Lagrangian Stochastic Modeling of Dispersion in the Stable Boundary Layer

Lagrangian stochastic models are well-suited for modeling dispersion in the stable boundary layer, especially in complex terrain. This note briefly describes the formulations and application of a Lagrangian stochastic model to predict dispersion of tracers released within nocturnal drainage flows.     

A Stochastic Single-Particle Lagrangian Model for the Concentration Fluctuations in a Plume Dispersing Inside an Urban Canopy

A new approach for estimating concentration fluctuations intensity in dense built-up environments using a Lagrangian stochastic (LS) particle model is described. Following past success in modelling the dynamics of concentration variance as a diffusion-advection process, the ensemble-averaged concentration variance is represented by particles that advect and diffuse throughout the computational domain. The calculation of the concentration variance is addressed by assuming an appropriate distribution of effective variance sources for a given mean concentration field. Dissipation is treated by allowing the variance carried by every particle to decay exponentially with a locally-estimated decay time. The approach has the benefit of easily handling complex boundary conditions. It can also be easily and naturally implemented as an extension to an existing LS model, which is used for mean concentration estimations. The method differs from existing two-particle methods that demand knowledge of the structure function of the flow. It is also more computationally efficient than micro-mixing approaches that involve maintaining high population levels of particles in every grid volume. The model is compared with high frequency concentration measurements, taken as part of the JU2003 (Joint Urban 2003) experiment that was carried out in Oklahoma City. Good agreement is observed.     

A Lagrangian Stochastic Model for Heavy Particle Dispersion in the Atmospheric Marine Boundary Layer

The dispersion of heavy particles and pollutants is often simulated with Lagrangian stochastic (LS) models. Although these models have been employed successfully over land, the free surface at the air-sea interface complicates the implementation of traditional LS models. We present an adaptation of traditional LS models to the atmospheric marine boundary layer (MBL), where the bottom boundary is represented by a realistic wavy surface that moves and deforms. In addition, the correlation function for the turbulent flow following a particle is extended to the anisotropic, unsteady case. Our new model reproduces behaviour for Lagrangian turbulence in a stratified air flow that departs only slightly from the expected behaviour in isotropic turbulence. When solving for the trajectory of a heavy particle in the air flow, the modelled turbulent forcing on the particle also behaves remarkably well. For example, the spectrum of the turbulence at the particle location follows that of a massless particle for time scales approximately larger than the Stokes’ particle response time. We anticipate that this model will prove especially useful in the context of sea-spray dispersion and its associated momentum, sensible and latent heat, and gas fluxes between spray droplets and the atmosphere.     

Turbulence Scale Dependence of the Richardson Constant in Lagrangian Stochastic Models

We investigate the relative dispersion properties of the well-mixed class of Lagrangian stochastic models. Dimensional analysis shows that, given a model in the class, its properties depend solely on a non-dimensional parameter, which measures the relative weight of Lagrangian-to-Eulerian scales. This parameter is formulated in terms of Kolmogorov constants, and model properties are then studied by modifying its value in a range that contains the experimental variability. Large variations are found for the quantity, g* = 2gC₀^{− 1}, where g is the Richardson constant.     

Modelling Small-Scale Drifting Snow with a Lagrangian Stochastic Model Based on Large-Eddy Simulations

Observations of drifting snow on small scales have shown that, in spite of nearly steady winds, the snow mass flux can strongly fluctuate in time and space. Most drifting snow models, however, are not able to describe drifting snow accurately over short time periods or on small spatial scales as they rely on mean flow fields and assume equilibrium saltation. In an attempt to gain understanding of the temporal and spatial variability of drifting snow on small scales, we propose to use a model combination of flow fields from large-eddy simulations (LES) and a Lagrangian stochastic model to calculate snow particle trajectories and so infer snow mass fluxes. Model results show that, if particle aerodynamic entrainment is driven by the shear stress retrieved from the LES, we can obtain a snow mass flux varying in space and time. The obtained fluctuating snow mass flux is qualitatively compared to field and wind-tunnel measurements. The comparison shows that the model results capture the intermittent behaviour of observed drifting snow mass flux yet differences between modelled turbulent structures and those likely to be found in the field complicate quantitative comparisons. Results of a model experiment show that the surface shear-stress distribution and its influence on aerodynamic entrainment appear to be key factors in explaining the intermittency of drifting snow.     

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.     

ATTILA: atmospheric tracer transport in a Lagrangian model

The model ATTILA has been developed to treat the global-scale transport of passive trace species in the atmosphere within the framework of a general circulation model (GCM). ATTILA runs online within the GCM ECHAM4 and advects the centroids of 80.000 to 190.000 constant mass air parcels. Each trace constituent is thereby represented by a mass mixing ratio in each parcel. ATTILA contains state-of-the-art parameterizations of convection, turbulent boundary layer mixing and inter-parcel transport, and provides an algorithm to map the tracer concentrations from the trajectories to the ECHAM model grid. The transport characteristics of ATTILA are evaluated against observations and the standard semi-Lagrangian transport scheme of ECHAM by two experiments. (1) We simulate the distribution of the short-lived tracer radon (²²²Rn) in order to examine fast vertical transport over continents, and long-range transport from the continents to remote areas. (2) We simulate the distribution of radiocarbon (¹⁴C) from nuclear weapon tests in order to examine upper tropospheric and stratospheric transport characteristics. Contrary to the semi-Lagrangian scheme, ATTILA shows a greatly reduced meridional transport in the upper troposphere and lower stratosphere, and a reduced downward flux from the stratosphere to the troposphere, especially in mid-latitudes. Since ATTILA is a numerically non-diffusive scheme, it is able to maintain steep gradients, which compare better to the observations than the rather smooth gradients produced by the semi-Lagrangian scheme.     

Instability in Lagrangian stochastic trajectory models,and a method for its cure

A number of authors have reported the problem of unrealistic velocities (“rogue trajectories”) when computing the paths of particles in a turbulent flow using modern Lagrangian stochastic (LS) models, and have resorted to ad hoc interventions. We suggest that this problem stems from two causes: (1) unstable modes that are intrinsic to the dynamical system constituted by the generalized Langevin equations, and whose actual triggering (expression) is conditional on the fields of the mean velocity and Reynolds stress tensor and is liable to occur in complex, disturbed flows (which, if computational, will also be imperfect and discontinuous); and, (2) the “stiffness” of the generalized Langevin equations, which implies that the simple stochastic generalization of the Euler scheme usually used to integrate these equations is not sufficient to keep round-off errors under control. These two causes are connected, with the first cause (dynamical instability) exacerbating the second (numerical instability); removing the first cause does not necessarily correct the second, and vice versa. To overcome this problem, we introduce a fractional-step integration scheme that splits the velocity increment into contributions that are linear (U _i ) and nonlinear (U _i U _j ) in the Lagrangian velocity fluctuation vector U, the nonlinear contribution being further split into its diagonal and off-diagonal parts. The linear contribution and the diagonal part of the nonlinear contribution to the solution are computed exactly (analytically) over a finite timestep Δt, allowing any dynamical instabilities in the system to be diagnosed and removed, and circumventing the numerical instability that can potentially result in integrating stiff equations using the commonly applied explicit Euler scheme. We contrast results using this and the primitive Euler integration scheme for computed trajectories in a drastically inhomogeneous urban canopy flow.     

Flux Footprints in the Convective Boundary Layer: Large-Eddy Simulation and Lagrangian Stochastic Modelling

We investigated the flux footprints of receptors at different heights in the convective boundary layer (CBL). The footprints were derived using a forward Lagrangian stochastic (LS) method coupled with the turbulent fields from a large-eddy simulation model. Crosswind-integrated flux footprints shown as a function of upstream distances and sensor heights in the CBL were derived and compared using two LS particle simulation methods: an instantaneous area release and a crosswind linear continuous release. We found that for almost all sensor heights in the CBL, a major positive flux footprint zone was located close to the sensor upstream, while a weak negative footprint zone was located further upstream, with the transition band in non-dimensional upwind distances −X between approximately 1.5 and 2.0. Two-dimensional (2D) flux footprints for a point sensor were also simulated. For a sensor height of 0.158 z _i, where z _i is the CBL depth, we found that a major positive flux footprint zone followed a weak negative zone in the upstream direction. Two even weaker positive zones were also present on either side of the footprint axis, where the latter was rotated slightly from the geostrophic wind direction. Using CBL scaling, the 2D footprint result was normalized to show the source areas and was applied to real parameters obtained using aircraft-based measurements. With a mean wind speed in the CBL of U = 5.1 m s⁻¹, convective velocity of w _* = 1.37 m s⁻¹, CBL depth of z _i = 1,000 m, and flight track height of 159 m above the surface, the total flux footprint contribution zone was estimated to range from about 0.1 to 4.5 km upstream, in the case where the wind was perpendicular to the flight track. When the wind was parallel to the flight track, the total footprint contribution zone covered approximately 0.5 km on one side and 0.8 km on the other side of the flight track.     

Numerical Considerations for Lagrangian Stochastic Dispersion Models: Eliminating Rogue Trajectories,and the Importance of Numerical Accuracy

When Lagrangian stochastic models for turbulent dispersion are applied to complex atmospheric flows, some type of ad hoc intervention is almost always necessary to eliminate unphysical behaviour in the numerical solution. Here we discuss numerical strategies for solving the non-linear Langevin-based particle velocity evolution equation that eliminate such unphysical behaviour in both Reynolds-averaged and large-eddy simulation applications. Extremely large or ‘rogue’ particle velocities are caused when the numerical integration scheme becomes unstable. Such instabilities can be eliminated by using a sufficiently small integration timestep, or in cases where the required timestep is unrealistically small, an unconditionally stable implicit integration scheme can be used. When the generalized anisotropic turbulence model is used, it is critical that the input velocity covariance tensor be realizable, otherwise unphysical behaviour can become problematic regardless of the integration scheme or size of the timestep. A method is presented to ensure realizability, and thus eliminate such behaviour. It was also found that the numerical accuracy of the integration scheme determined the degree to which the second law of thermodynamics or ‘well-mixed condition’ was satisfied. Perhaps more importantly, it also determined the degree to which modelled Eulerian particle velocity statistics matched the specified Eulerian distributions (which is the ultimate goal of the numerical solution). It is recommended that future models be verified by not only checking the well-mixed condition, but perhaps more importantly by checking that computed Eulerian statistics match the Eulerian statistics specified as inputs.     

Investigation of droplet dynamics in a convective cloud using a Lagrangian cloud model

A precipitating convective cloud is simulated successfully using the Lagrangian cloud model, in which the flow field is simulated by large eddy simulation and the droplets are treated as Lagrangian particles, and the results are analyzed to investigate precipitation initiation and to examine the parameterization of cloud microphysics. It is found that raindrops appear initially near the cloud top, in which strong turbulence and broadened droplet spectrum are induced by the entrainment of dry air, but high liquid–water mixing ratio is maintained within cloud parts because of insufficient mixing. Statistical analysis of the downward vertical velocity of a droplet W reveals that the transition from cloud droplets to raindrops occurs in the range 20 μm < r < 100 μm, while the variation of W depends on turbulence as well as the droplet radius r. The general pattern of the raindrop size distribution is found to be consistent with the Marshall–Palmer distribution. The precipitation flux can be underestimated substantially, if the terminal velocity $w_{\text{s}}$ is used instead of W, but it is not sensitive to the choice of the critical droplet radius dividing cloud drops and raindrops. It is also found that precipitation starts earlier and becomes stronger if the effect of turbulence is included in the collection kernel.     

Modeling of pollutant dispersion in sea breeze conditions using a Lagrangian model

Summary This work presents a numerical study of non-reactive pollutant dispersion in sea breeze conditions. Sea breeze circulation is investigated using a 3-D mesoscale meteorological model. Simulation was conducted for the area of Tarragona (Spain) which has an important petrochemical industry in the coastal region and complex terrain. Results from the meteorological model were used as input to a Lagrangian particle model in order to analyze the pollutant dispersion of an elevated plume emitting near the shoreline. The simulation was performed for 24 h and an analysis of the meteorological and concentration fields was untertaken for this time period. The results are compared with measured surface data. Good correlation exists between observed and simulated conditions indicating that the coupling of the meteorological and particle models provides a good tool for analyzing air pollution in complex situations.With 13 Figures