A Velocity–Dissipation Lagrangian Stochastic Model for Turbulent Dispersion in Atmospheric Boundary-Layer and Canopy Flows

An extended Lagrangian stochastic dispersion model that includes time variations of the turbulent kinetic energy dissipation rate is proposed. The instantaneous dissipation rate is described by a log-normal distribution to account for rare and intense bursts of dissipation occurring over short durations. This behaviour of the instantaneous dissipation rate is consistent with field measurements inside a pine forest and with published dissipation rate measurements in the atmospheric surface layer. The extended model is also shown to satisfy the well-mixed condition even for the highly inhomogeneous case of canopy flow. Application of this model to atmospheric boundary-layer and canopy flows reveals two types of motion that cannot be predicted by conventional dispersion models: a strong sweeping motion of particles towards the ground, and strong intermittent ejections of particles from the surface or canopy layer, which allows these particles to escape low-velocity regions to a high-velocity zone in the free air above. This ejective phenomenon increases the probability of marked fluid particles to reach far regions, creating a heavy tail in the mean concentration far from the scalar source.     

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.     

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.     

A closure to derive a three-dimensional well-mixed trajectory-model for non-Gaussian, inhomogeneous turbulence

A three-dimensional Lagrangian stochastic (LS) model to evaluate pollutant dispersion in the atmospheric boundary layer has been developed. The model satisfies the well-mixed criterion of Thomson and allows for inhomogeneous, skew turbulence. Making use of the spherical reference frame, one of the possible solutions has been obtained. A skewed joint probability density function (PDF), which reproduces the given velocity moments (means, variances, skewness and covariances), has been built-up by a linear combination of eight Gaussian PDFs. In order to verify consistency with the well-mixed criterion, the long term results have been compared with the theoretical behaviour. A comparison between our model and Thomson's published algorithms was also carried out. By comparing wind-tunnel data and numerical predictions, a further validation of our LS model has been obtained. From an analysis of the numerical results, we can state that our model is able to evaluate dispersion in the case of complex flows where the application of previous models is unsuccessful.     

A Homogeneous Langevin Equation Model, Part i: Simulation of Particle Trajectories in Turbulence with a Skewed Velocity Distribution

A Lagrangian stochastic model for the time evolution of the velocity of a fluid particle is presented. This model is based on a one-dimensional generalized Langevin equation, and assumes the velocity probability distribution of the turbulent fluid is skewed and spatially homogeneous. This has been shown to be an effective approach to simulating vertical dispersion in the convective boundary layer. We use a form of the Langevin equation that has a linear (in velocity) deterministic acceleration and a random acceleration that is a non-Gaussian, skewed process. For the case of homogeneous fluid velocity statistics, this 'linear-skewed' Langevin equation can be integrated explicitly, resulting in an efficient numerical simulation method. Model simulations were tested using cases for which exact, analytic statistical properties of particle velocity are known. Results of these tests show that, for homogeneous turbulence, a linear-skewed Langevin equation model can overcome the difficulties encountered in applying a Langevin equation with a skewed random acceleration. The linear-skewed Langevin equation model results are compared to results of a 'nonlinear-Gaussian' Langevin equation model, and show that the linear-skewed model is significantly more efficient.     

A Homogeneous Langevin Equation Model, Part ii: Simulation of Dispersion in the Convective Boundary Layer

We present a Lagrangian stochastic model of vertical dispersion in the convective boundary layer (CBL). This model is based on a generalized Langevin equation that uses the simplifying assumption that the skewed vertical velocity probability distribution is spatially homogeneous. This approach has been shown to account for two key properties of CBL turbulence associated with large-scale coherent turbulent structures: skewed vertical velocity distributions and long velocity correlation time. A 'linear-skewed' form of the generalized Langevin equation is used, which has a linear (in velocity) deterministic acceleration and a skewed random acceleration. 'Reflection' boundary conditions for selecting a new velocity for a particle that encounters a boundary were investigated, including alternatives to the standard assumption that the magnitudes of the particle incident and reflected velocities are positively correlated. Model simulations were tested using cases for which exact, analytic statistical properties of particle velocity and position are known, i.e., well-mixed spatial and velocity distributions. Simulations of laboratory experiments of CBL dispersion show that (1) the homogeneous linear-skewed Langevin equation model (as well as an alternative 'nonlinear-Gaussian' Langevin equation model) can simulate the important aspects of dispersion in the CBL, and (2) a negatively-correlated-speed reflection boundary condition simulates the observed dispersion of material near the surface in the CBL significantly better than alternative reflection boundary conditions. The homogeneous linear-skewed Langevin equation model has the advantage that it is computationally more efficient than the homogeneous nonlinear-Gaussian Langevin equation model, and considerably more efficient than inhomogeneous Langevin equation models.     

Some Aspects of Atmospheric Dispersion in the Stratified Atmospheric Boundary Layer over Homogeneous Terrain

The ability to simulate atmospheric dispersion with models developed for applied use under stable atmospheric stability conditions is discussed. The paper is based on model simulations of three experimental data sets reported in the literature. The Hanford data set covered weakly stable conditions, the Prairie Grass experiments covered both weakly stable and very stable atmospheric conditions, and the Lillestrøm experiment was carried out during very stable conditions. Simulations of these experiments reported in the literature for eight different models are discussed. Applied models based on the Gaussian plume model concept with the spread parameters described in terms of the Pasquill stability classification or Monin–Obukhov similarity relationships are used. Other model types are Lagrangian particle models which also are parameterized in terms of Monin–Obukhov similarity relationships. The applied models describe adequately the dispersion process in a weakly stable atmosphere, but fail during very stable atmospheric conditions. This suggests that Monin–Obukhov similarity theory is an adequate tool for the parameterization of the input parameters to atmospheric dispersion models during weakly stable conditions, but that more detailed parameterisations including other physical processes than those covered by the Monin–Obukhov theory should be developed for the very stable atmosphere.     

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.     

Lagrangian statistical simulation of the turbulent motion of heavy particles

A Lagrangian stochastic model for the motion of heavy particles has been developed by coupling a stochastic model for the motion of fluid elements to the Stokes equations of motion of a particle in a turbulent flow. The effects of crossing trajectories and continuity are incorporated by generalising Csanady's (1963) ideas developed for stationary homogeneous turbulence; effects of turbulence inhomogeneity and nonstationarity are embodied in the stochastic model for the fluid motion.The model has been used particularly to examine the effects of turbulence nonstationarity through simulations of the dispersion of heavy particles in the decaying homogeneous turbulence which is obtained by Taylor-transforming grid turbulence. Significant differences from the stationary case occur, mainly as a result of the growth of the turbulent time scale with time.The importance of the source location in influencing both passive scalar and particle dispersion in grid turbulence is highlighted by the model and can be simply accounted for by nondimensionalisation using the r.m.s. turbulence velocity at the source and the mean travel time from the grid to the source as velocity and time scales, respectively. Reconciliation of the three different experiments of Snyder and Lumley (1971), Wells and Stock (1983) and Ferguson (1986) reporting heavy particle flow and dispersion statistics in wind tunnel grid turbulence has been attempted using this nondimensionalisation. A good correspondence between the various data sets was not obtained because the source in the Wells and Stock, and Ferguson experiments was located at the grid where the self-similar development of the turbulence which underlies the scaling is not appropriate.The model matches the data for the heaviest particles used by Snyder and Lumley reasonably well. For very light particles, it correctly reverts to the passive scalar limit, while the experimental data in general do not properly approach this limit.     

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 incorporates controllable non-Gaussian random variables to simulate the targeted non-Gaussian velocity distribution with more accurate skewness and kurtosis.Wind velocity statistics generated by the non-Gaussian model are evaluated by using the field data from the Cooperative Atmospheric Surface Exchange Study,October 1999 experimental dataset and comparing the data with statistics from the original Gaussian model.Results show that the non-Gaussian model improves the wind trajectory simulation by stably producing precise skewness and kurtosis in simulated wind velocities without sacrificing other features of the traditional Gaussian LS model,such as the accuracy in the mean and variance of simulated velocities.This improvement also leads to better accuracy in friction velocity(i.e.,a coupling of three-dimensional velocities).The model can also accommodate various non-Gaussian wind fields and a wide range of skewness–kurtosis combinations.Moreover,improved skewness and kurtosis in the simulated velocity will result in a significantly different dispersion for wind/particle simulations.Thus,the non-Gaussian model is worth applying to wind field simulation in the surface layer.     

Statistical Variability of Dispersion in the Convective Boundary Layer: Ensembles of Simulations and Observations

A Lagrangian particle dispersion model (LPDM) driven by velocity fields from large-eddy simulations (LESs) is used to determine the mean and variability of plume dispersion in a highly convective planetary boundary layer (PBL). The total velocity of a “particle” is divided into resolved and unresolved or random (subfilter scale, SFS) velocities with the resolved component obtained from the LES and the SFS velocity from a Lagrangian stochastic model. This LPDM-LES model is used to obtain an ensemble of dispersion realizations for calculating the mean, root-mean-square (r.m.s.) deviation, and fluctuating fields of dispersion quantities. An ensemble of 30 realizations is generated for each of three source heights: surface, near-surface, and elevated. We compare the LPDM calculations with convection tank experiments and field observations to assess the realism of the results. The overall conclusion is that the LPDM-LES model produces a realistic range of dispersion realizations and statistical variability (i.e., r.m.s. deviations) that match observations in this highly convective PBL, while also matching the ensemble-mean properties. This is true for the plume height or trajectory, vertical dispersion, and the surface values of the crosswind-integrated concentration (CWIC), and their dependence on downstream distance. One exception is the crosswind dispersion for an elevated source, which is underestimated by the model. Other analyses that highlight important LPDM results include: (1) the plume meander and CWIC fluctuation intensity at the surface, (2) the applicability of a similarity theory for plume height from a surface source to only the very strong updraft plumes—not the mean height, and (3) the appropriate variation with distance of the mean surface CWIC and the lower bound of the CWIC realizations for a surface source.     

ON THE PDF MODELS FOR ATMOSPHERIC DIFFUSION IN BOUNDARY LAYER

In this paper, taking its turbulent exchange coefficient as a function of the Lagrangian timescale and standard variance of the turbulence in atmosphere, the atmospheric dispersion PDFmodels are obtained on the basis of atmospheric diffusion K-theory. In the model the statistics ofwind speed are directly used as its parameters instead of classic dispersion parameters. The bi-Gaussian PDF is derived in convective boundary layer (CBL), from the statistics of verticalvelocity in both of the downdraft and updraft regions that are investigated theoretically in the otherpart of this paper. Giving the driven parameters of the CBL (including the convective velocity scalew* and the mixing depth h_i) and the time-averaged wind speed at release level, the PDF model isable to simulate the distribution of concentration released at any levels in the CBL. The PDF'ssimulations are fairly consistent with the measurements in CONDORS experiment or the resultsbrought out by some numerical simulations.     

An Advanced Puff Model Based On A Mixed Eulerian/Lagrangian Approach For Turbulent Dispersion In The Convective Boundary Layer

An advanced model aimed at describing the problem of dispersion in theconvective boundary layer is proposed. The pollutant particles are groupedin clusters and modelled as Gaussian puffs. The expansion of each puff ismodelled according to the concept of relative dispersion and expressed interms of the spectral properties of the energy containing eddies of the turbulent field. The centre of mass of each puff is moved along a stochastic trajectory, obtained using a Lagrangian stochastic model and filtering the velocity with a recursive Kalman filter. At any instant, a filtering procedure, depending both on travel time and on puff size, acts to select spectral components involved in the expansion and in the meandering of the puff. Such an approach requires only a moderate number of puff releases, so that the proposed model is faster to run than a standard Lagrangian model. On the other hand, unlike the traditional puff model, it allows us to simulate both expansion and meandering of the puff. Therefore, it is well suited to simulate dispersion when the turbulent structures are larger thanthe plume dimensions, as for example in convective conditions. Being based onspectral formulations in both Eulerian and Lagrangian parts, the model is consistent in all the turbulent parameterizations utilised. Comparisons with a standard Lagrangian particle model as well as with a classical convective experimental dataset show good performance of the proposed model.     

针对一次对流云降水反演大气垂直运动速度及雨滴谱

反演大气垂直速度和雨滴谱分布是研究云降水机制和云微物理信息的重要内容,对人工预报天气、干预天气都有重要意义。针对2021年8月29日安徽省内毫米波雷达探测到的一次对流云降水过程,处理毫米波雷达的功率谱数据并进行大气垂直速度和雨滴谱反演。在小粒子示踪法的基础上引入改进小粒子示踪法：选取有效云信号段中最小功率对应的谱点作为反演大气垂直速度的示踪物。首先,根据改进前后的小粒子示踪法分别从功率谱数据中反演大气垂直速度,并跟基数据反演大气速度的结果展开对比分析。进一步得到粒子在静止空气中的下落速度,根据现有粒子下落速度-粒子直径之间的经验公式计算反演粒子直径。研究表明：(1) 采用改进后的小粒子示踪法反演大气垂直速度得到的结果比小粒子示踪法得到的结果更精确,在云层内部两者误差较大;(2) 进一步得到粒子下落速度,结合探测时段的天气状况,得到的粒子速度与大气速度可很好地契合,跟对流云天气情况信息大致吻合;(3) 粒子浓度是反演雨滴谱分布时需要注意的主要参数,云在快速发展过程中,内部粒子持续朝外部扩张,云内部的粒子浓度较小,云边界的粒子浓度反而较大。     

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.     

Dispersion of Heavy Particles Emitted from Area Sources in the Unstable Atmospheric Boundary Layer

Reliable predictions of the daytime dispersal of heavy particles in the unstable atmospheric boundary layer are important in a variety of disciplines. For many applications, particles disperse from area sources near the ground, and corresponding theoretical solutions are desired to reveal insight into the physical processes. Here, theoretical solutions recently developed for neutral conditions are modified to include the effects of atmospheric instability. The Obukhov length L _O and convection velocity w _? are introduced to characterize the patterns of particle dispersion, in additional to friction velocity u _? and settling velocity w _s used in the neutral case. The major effects of atmospheric instability are accounted for by modifying the vertical velocity variance profile and considering the ratio of velocity scales w _?/u _?. Theoretical predictions including the mean concentration profile, plume height, and horizontal transport above the source, and ground deposition flux downwind from the source agree well with large-eddy simulation results while the particle plume is within the atmospheric surface layer. The deposition curve is characterized by a power-law decay whose exponent depends on u _?, w _s, and w _?. A second steeper power-law develops once the plume extends into the mixed layer. This effect is enhanced with increasing atmospheric instability, implying that particles disperse farther from the source.     

On Heavy Particle Dispersion in Turbulent Shear Flows: 3-D Analysis of the Effects of Crossing Trajectories

In the approaches used to predict the dispersion of discrete particles moving in a turbulent flow, the effects of crossing trajectories due to gravity (or any other external force field) are generally accounted for by modifying the integral time scales according to the well-known analysis of Csanady (J Atmos Sci 20:201–208, 1963). Here, an alternative theoretical analysis of the time correlation of the fluid velocity fluctuations along a particle trajectory is presented and applied in a turbulent shear flow. The study is carried out in the frame of three-dimensional Langevin-type stochastic models, where the main unknowns are the drift tensor components rather than the conventional integral time scales of the fluid seen by the particles. Starting from a model for the space-time velocity covariance tensor of the turbulence under the assumption of homogeneous shear flow, the various components of the time correlation tensor of the fluid seen are expressed in the asymptotic case of large mean relative velocity (between the particles and the flow) compared to the particle velocity fluctuations. In order to provide comparison with the generally used expressions arising from isotropic turbulence assumption, we examine also the conventional integral time scales of the fluid seen in the directions parallel and perpendicular to the mean relative velocity. The most prominent deviations from isotropic turbulence are observed when the external force field is in the direction of the mean velocity gradient: in this case the loss of correlation in the mean flow direction is significantly lower than expected in a uniform flow, an observation that is in qualitative agreement with the few available data.     

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.