A Lagrangian Solution To The Relationship Between A Distributed Source And Concentration Profile

A new solution is presented to the problem ofrelating source strength and concentration profiles within a plant canopy. The solution is based on the Lagrangian dispersion theory developed by G. I. Taylor in 1921. A dispersion matrix is derived that relates the source and concentration profiles based on profiles of the turbulent length and velocity scales. The matrix translates the effects of persistence (a temporal effect) into spatialcoordinates and represents the change from near-field to far-field in acontinuous fashion, successfully accounting for both regimes. A test ofthe new model using wind-tunnel data showed excellent quantitative agreement between model and measurements. A comparison was also made withM. R. Raupach's localized near-field theory, which underestimated the near-field effect in the wind-tunnel data and relative to the new model.     

热浮升烟流扩散的拉格朗日粒子模拟

本文建立了一个处理对流边界层热浮升烟流扩散的拉格朗日粒子模式。模式既考虑了对流边界层的特殊气流结构，并作了均匀湍流参数化的简化；同时提出了在拉格朗日模式中合理计入热浮升烟流抬升影响的近似方法。模拟计算结果表明:烟流热浮力的影响使得地面最大浓度值远比被动烟流的低，而且出现位置离源更远。模式计算与外场试验结果合理地一致。模式物理概念明确合理，输入参数少，计算量小，具有简单实用的优点，适合日常环境应用需要。     

The Turbulent Lagrangian Time Scale in Forest Canopies Constrained by Fluxes,Concentrations and Source Distributions

One-dimensional Lagrangian dispersion models, frequently used to relate in-canopy source/sink distributions of energy, water and trace gases to vertical concentration profiles, require estimates of the standard deviation of the vertical wind speed, which can be measured, and the Lagrangian time scale, T _L , which cannot. In this work we use non-linear parameter estimation to determine the vertical profile of the Lagrangian time scale that simultaneously optimises agreement between modelled and measured vertical profiles of temperature, water vapour and carbon dioxide concentrations within a 40-m tall temperate Eucalyptus forest in south-eastern Australia. Modelled temperature and concentration profiles are generated using Lagrangian dispersion theory combined with source/sink distributions of sensible heat, water vapour and CO₂. These distributions are derived from a multilayer Soil-Vegetation-Atmospheric-Transfer model subject to multiple constraints: (1) daytime eddy flux measurements of sensible heat, latent heat, and CO₂ above the canopy, (2) in-canopy lidar measurements of leaf area density distribution, and (3) chamber measurements of CO₂ ground fluxes. The resulting estimate of Lagrangian time scale within the canopy under near-neutral conditions is about 1.7 times higher than previous estimates and decreases towards zero at the ground. It represents an advance over previous estimates of T _L , which are largely unconstrained by measurements.     

Dispersion in sheared Gaussian homogeneous turbulence

By integrating the Fokker-Planck equation corresponding to a Lagrangian stochastic trajectory model, which is consitent with the selection criterion of Thomson (1987), an analytical solution is given for the joint probability density functionp(x_i, u_i, t) for the position (x _i) and velocity (u _i) at timet of a neutral particle released into linearly-sheared, homogeneous turbulence. The solution is compared with dispersion experiments conforming to the restrictions of the model and with a shortrange experiment performed in highly inhomogeneous turbulence within and above a model crop canopy. When the turbulence intensity, wind shear and covariance are strong, the present solution is better than simpler solutions (Taylor, 1921; Durbin, 1983) and as good as any numerical Lagrangian stochastic model yet reported.     

A Semi-analytical Model for Short-Range Near-Ground Continuous Dispersion

We present a semi-analytical model for the dispersion of passive scalars from continuous ground sources up to distances of a few hundred metres. We attempt to cope with problems typical of analytical models, such as the correct representation of the near-ground concentration and lateral dispersion, while avoiding the use of any empirical parameters. A previous analysis of Prairie Grass Project (PGP) data has shown that the near-ground, cross-wind integrated concentration decreases as some power of the distance from the source that is, itself, distance dependent. As the conventional power-law model is incapable of reproducing this behaviour, we propose a model in which the vertical diffusivity depends on both the height as a power law, and on the distance from the source. For this equation, we construct an infinite-series formal solution, with the first term used as an approximation. A set of equations based on this approximation and on Monin–Obukhov similarity theory is proposed for the the vertical diffusivity, from which the cross-wind integrated concentration is derived analytically. We further construct a simple empirical model for the distance-dependent vertical diffusivity. For the plume lateral width, a Langevin stochastic model depending on the plume height is proposed, whose formal analytical solution is used to derive a set of equations for the cloud width, which are easily solved numerically, with the results validated against PGP data. We apply four statistical measures to evaluate the performance of the model, including the computation of the 95% confidence intervals, for which we find very good agreement. Implementation of this model is extremely simple and computationally efficient.     

Large-Eddy Simulation of air Pollution Dispersion in the Nocturnal Cloud-Topped Atmospheric Boundary Layer

Effects of stratocumulus clouds on the dispersion of contaminants are studied in the nocturnal atmospheric boundary layer. The study is based on a large-eddy simulation (LES) model with a bulk parametrization of clouds. Computations include Lagrangian calculations of atmospheric dispersion of a passive tracer released from point sources at various heights above the ground. The results obtained show that the vertical diffusion is non-Gaussian and depends on the location of a source in the boundary layer.     

Forward-in-time and Backward-in-time Dispersion in the Convective Boundary Layer: the Concentration Footprint

The turbulence field obtained using a large-eddy simulation model is used to simulate particle dispersion in the convective boundary layer with both forward-in-time and backward-in-time modes. A Lagrangian stochastic model is used to treat subgrid-scale turbulence. Results of forward dispersion match both laboratory experiments and previous numerical studies for different release heights in the convective boundary layer. Results obtained from backward dispersion show obvious asymmetry when directly compared to results from forward dispersion. However, a direct comparison of forward and backward dispersion has no apparent physical meaning and might be misleading. Results of backward dispersion can be interpreted as three-dimensional or generalized concentration footprints, which indicate that sources in the entire boundary layer, not only sources at the surface, may influence a concentration measurement at a point. Footprints at four source heights in the convective boundary layer corresponding to four receptors are derived using forward and backward dispersion methods. The agreement among footprints derived with forward and backward methods illustrates the equivalence between both approaches. The paper shows explicitly that Lagrangian simulations can yield identical footprints using forward and backward methods in horizontally homogeneous turbulence.     

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.     

Estimating a Lagrangian Length Scale Using Measurements of CO2 in a Plant Canopy

Analytical Lagrangian equations capable of predicting concentration profiles from known source distributions offer the opportunity to calculate source/sink distributions through inverted forms of these equations. Inverse analytical Lagrangian equations provide a practical means of estimating source profiles using concentration and turbulence measurements. Uncertainty concerning estimates of the essentially immeasurable Lagrangian length scale ( ${\mathcal{L}}$ ), a key input, impedes the operational practicality of this method. The present study evaluates ${\mathcal{L}}$ within a corn canopy by using field measurements to constrain an analytical Lagrangian equation. Measurements of net CO₂ flux, soil-to-atmosphere CO₂ flux, and in-canopy profiles of CO₂ concentration provided the information required to solve for ${\mathcal{L}}$ in a global optimization algorithm for 30-min time intervals. For days when the canopy was a strong CO₂ sink, the optimization frequently located ${\mathcal{L}}$ profiles that follow a convex shape. A constrained optimization then fit the profile shape to a smooth sigmoidal equation. Inputting the optimized ${\mathcal{L}}$ profiles in the forward and inverse Lagrangian equations leads to strong correlations between measured and calculated concentrations and fluxes. Coefficients of the sigmoidal equation were specific to each 30-min period and did not scale with any measured variable. Plausible looking ${\mathcal{L}}$ profiles were associated with negative bulk Richardson number values. Once the canopy senesced, a simple eddy diffusivity profile sufficed to relate concentrations and sources in the analytical Lagrangian equations.     

Turbulent dispersion in coastal atmospheric boundary layers: An application of a Lagrangian model

A Lagrangian model is applied to simulate the dispersion of passive tracers (in particular, water vapour) in coastal atmospheric boundary layers under onshore wind conditions. When applied to convective boundary layers over uniform surfaces, the model gives results in agreement with those of similar studies. Numerical simulation of turbulent dispersion in coastal areas also reproduces the basic features known from experimental studies. Under onshore wind conditions, the humidity field is plume-shaped with the maximum vertical transport being over land downstream of the coast line. The model shows that the surface sensible heat flux over land, the static stability of the onshore air flow and the onshore wind speed are the most important factors determining the basic features of turbulent dispersion in coastal areas.     

Scaling of the Vertical Spreading of a Plume of a Passive Tracer in a Rough-Wall Neutral Boundary Layer

The influence of surface roughness on the dispersion of a passive scalar in a rough wall turbulent boundary layer has been studied using wind-tunnel experiments. The surface roughness was varied using different sizes of roughness elements, and different spacings between the elements. Vertical profiles of average concentration were measured at different distances downwind of the source, and the vertical spread of the plume was computed by fitting a double Gaussian profile to the data. An estimate of the integral length scale is derived from the turbulence characteristics of the boundary layer and is then used to scale the measured values of plume spread. This scaling reduces the variability in the data, confirming the validity of the model for the Lagrangian integral time scale, but does not remove it entirely. The scaled plume spreading shows significant differences from predictions of theoretical models both in the near and in the far field. In the region immediately downwind of the source this is due to the influence of the wake of the injector for which we have developed a simple model. In the far field we explain that the differences are mainly due to the absence of large-scale motions. Finally, further downwind of the source the scaled values of plume spread fall into two distinct groups. It is suggested that the difference between the two groups may be related to the lack of dynamical similarity between the boundary-layer flows for varying surface roughness or to biased estimates of the plume spread.     

An Analytical Footprint Model For Non-Neutral Stratification

We propose an analytical model for the so-called footprint of scalar fluxes in the atmospheric boundary layer. It is the generalization of formulations already given in the literature, which allows to account for thermal stability. Our model is only marginally more complicated than these, and it is therefore simple enough to be applicable for a routine footprint analysis within long-term measurements. The mathematical framework of our model is a stationary gradient diffusion formulation with height-independent crosswind dispersion. It uses the solution of the resulting two-dimensional advection – diffusion equation for power law profiles of the mean wind velocity and the eddy diffusivity. To find the adjoint Monin–Obukhov similarity profile, we propose two different approaches, a purely analytical one and a simplenumerical error minimalization.     

Restriction on the timestep to be used in stochastic Lagrangian models of turbulent dispersion

By considering two analytical solutions of G. I. Taylor (1921) for dispersion in homogeneous turbulence, we derive a quantitative upper limit for the timestep dt to be used in the stochastic Lagrangian model; a more severe upper limit will probably exist in inhomogeneous turbulence. For practical purposes, there is no lower limit to the timestep.     

Dispersion of passive tracer in the atmospheric convective boundary layer with wind shears: a review of laboratory and numerical model studies

Summary Paper reviews recent laboratory and numerical model studies of passive gaseous tracer dispersion in the atmospheric convective boundary layer (CBL) with surface and elevated wind shears. Atmospheric measurement data used for validation of these two model techniques are briefly discussed as well. A historical overview is given of laboratory studies of dispersion in the atmospheric CBL. Model studies of tracer dispersion in two CBL types, the (i) non-steady, horizontally homogeneous CBL and (ii) quasi-stationary, horizontally heterogeneous CBL, are reviewed. The discussion is focused on the dispersion of non-buoyant plume emitted from a point source located at different elevations within the CBL. Approaches towards CBL modeling employed in different laboratory facilities (water tanks and wind tunnels) are described. The reviewed numerical techniques include Large Eddy Simulation (LES) and Lagrangian modeling. Numerical data on dispersion in the sheared CBL is analyzed in conjunction with experimental results from wind-tunnel CBLs.     

The Volumetric Particle Approach for Concentration Fluctuations and Chemical Reactions in Lagrangian Particle and Particle-grid Models

A new approach is proposed to predict concentration fluctuations in the framework of one-particle Lagrangian stochastic models. The approach is innovative since it allows the computation of concentration fluctuations in dispersing plumes using a Lagrangian one-particle model with micromixing but with no need for the simulating of background particles. The extension of the model for the treatment of chemically reactive plumes is also accomplished and allows the computation of plume-related chemical reactions in a Lagrangian one-particle framework separately from the background chemical reactions, accounting for the effect of concentration fluctuations on chemical reactions in a general, albeit approximate, manner. These characteristics should make the proposed approach an ideal tool for plume-in-grid calculations in chemistry transport models. The results are compared to the wind-tunnel experiments of Fackrell and Robins (J Fluid Mech, 117:1–26, 1982) for plume dispersion in a neutral boundary layer and to the measurements of Legg et al. (Boundary-Layer Meteorol, 35:277–302, 1986) for line source dispersion in and above a model canopy. Preliminary reacting plume simulations are also shown comparing the model with the experimental results of Brown and Bilger (J Fluid Mech, 312:373–407, 1996; Atmos Environ, 32:611–628, 1998) to demonstrate the feasibility of computing chemical reactions in the proposed framework.     

Dispersion of a Passive Scalar Fluctuating Plume in a Turbulent Boundary Layer. Part III: Stochastic Modelling

We analyze the reliability of the Lagrangian stochastic micromixing method in predicting higher-order statistics of the passive scalar concentration induced by an elevated source (of varying diameter) placed in a turbulent boundary layer. To that purpose we analyze two different modelling approaches by testing their results against the wind-tunnel measurements discussed in Part I (Nironi et al., Boundary-Layer Meteorology, 2015, Vol. 156, 415–446). The first is a probability density function (PDF) micromixing model that simulates the effects of the molecular diffusivity on the concentration fluctuations by taking into account the background particles. The second is a new model, named VP\(\varGamma \), conceived in order to minimize the computational costs. This is based on the volumetric particle approach providing estimates of the first two concentration moments with no need for the simulation of the background particles. In this second approach, higher-order moments are computed based on the estimates of these two moments and under the assumption that the concentration PDF is a Gamma distribution. The comparisons concern the spatial distribution of the first four moments of the concentration and the evolution of the PDF along the plume centreline. The novelty of this work is twofold: (i) we perform a systematic comparison of the results of micro-mixing Lagrangian models against experiments providing profiles of the first four moments of the concentration within an inhomogeneous and anisotropic turbulent flow, and (ii) we show the reliability of the VP\(\varGamma \) model as an operational tool for the prediction of the PDF of the concentration.     

Comparison of the Langrangian Footprint

We compare flux and concentration footprint estimates of athree-dimensional Lagrangian stochastic dispersion modelapplying backward trajectories with the results of ananalytical footprint model by Kormann and Meixner.The comparison is performed for varying stability regimesof the surface layer as well as for different measurementheights. In general, excellent correspondence is found.     

Comparing Two Implementations of a Micromixing Model. Part II: Canopy Flow

The sequential particle micromixing model (SPMMM) is used to estimate concentration fluctuations in plumes dispersing into a canopy flow. SPMMM uses the familiar single-particle Lagrangian stochastic (LS) trajectory framework to pre-calculate the required conditional mean concentrations, which are then used by an interaction by exchange with the conditional mean (IECM) micromixing model to predict the higher-order fluctuations of the scalar concentration field. The predictions are compared with experimental wind-tunnel dispersion data for a neutrally stratified canopy flow, and with a previously reported implementation using simultaneous particle trajectories. The two implementations of the LS–IECM model are shown to be largely consistent with one another and are able to simulate dispersion in a canopy flow with fair to good accuracy.