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.     

High-Order Statistics of Temperature Fluctuations in an Unstable Atmospheric Surface Layer over Grassland

Skewness(S) and kurtosis(K) of temperature in the surface layer over a grassland are investigated under unstable thermal stratifications. We find that both skewness and kurtosis generally obey Monin–Obukhov similarity theory and tend to be constant values(1.5 and 5.3, respectively) when the stability parameter z/L -2. Quantitative formulas of the similarity functions are proposed. The temperature probability density function(PDF) is close to Gaussian in near neutral stratification and non-Gaussian in unstable stratification. The influence of coherent motions on the PDF behavior is analyzed using the quadrant analysis technique. It shows that PDF behaviors are controlled by ejections and sweeps. The results also indicate that the PDF type of the ejections always follows a Gaussian distribution, while the PDF of the sweeps changes with stability.     

The Relationship between Skewness and Kurtosis of A Diffusing Scalar

It has been demonstrated that in turbulent dispersion, there exists a quadratic relationship between the skewness (S) and kurtosis (K) statistics obtained from continuous, elevated sources of scalar contaminant released into both convective and stable atmospheric boundary layers. Specifically, one observes that where A and B are empirically fitted constants that depend on the flow. For two reasons, this is potentially useful information in regard to modelling the probability density function (PDF) of a diffusing scalar. First, since many PDFs have a signature relationship between their skewness and kurtosis, candidate models can immediately be either accepted or rejected depending upon whether they conform to the quadratic curve that is observed experimentally. Second, if one intends to model the PDF by inverting a limited number of moments, the task is reduced when there is a functional relationship between the standardized third and fourth moments. The aforementioned relationship has been corroborated by others who have examined data over a wide range of experimental configurations. However, from one flow to another, there appears to be a non-negligible variability in the two fitting constants of the quadratic curve. In this paper we put forth a framework to help explain this phenomenon, and we also attempt to predict how these parameters vary in space and/or time. Our point is illustrated with well-resolved data from a wind-tunnel, grid-turbulence, plume experiment.     

Statistics of atmospheric turbulence within a natural black spruce forest canopy

Turbulence statistics were measured in a natural black-spruce forest canopy in southeastern Manitoba, Canada. Sonic anemometers were used to measure time series of vertical wind velocity (w), and cup anemometers to measure horizontal wind speed (s), above the canopy and at seven different heights within the canopy. Vertical profiles were measured during 25 runs on eight different days when conditions above the canopy were near-neutral.Profiles of s and of the standard deviation ( _w ) of w show relatively little scatter and suggest that, for this canopy and these stability conditions, profiles can be predicted from simple measurements made above the canopy. Within the canopy, a negative skewness and a high kurtosis of the w-frequency distributions indicate asymmetry and the persistence of large, high-velocity eddies. The Eulerian time scale is only a weak function of height within the canopy.Although w-power spectra above the canopy are similar to those in the free atmosphere, we did not observe an extensive inertial subrange in the spectra within the canopy. Also, a second peak is present that is especially prominent near the ground. The lack of the inertial subrange is likely caused by the presence of sources and sinks for turbulent kinetic energy within our canopy. The secondary spectral peak is probably generated by wake turbulence caused by form drag on the wide, horizontal spruce branches.     

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.     

Turbulence structure in a deciduous forest

Three-dimensional wind velocity components were measured at two levels above and at six levels within a fully-leafed deciduous forest. Greatest shear occurs in the upper 20% of the canopy, where over 70% of the foliage is concentrated. The turbulence structure inside the canopy is characterized as non-Gaussian, intermittant and highly turbulent. This feature is supported by large turbulence intensities, skewness and kurtosis values and by the large infrequent sweeps and ejections that dominate tangential momentum transfer. Considerable day/night differences were observed in the vertical profiles of the mean streamwise wind velocity and turbulence intensities since the stability of the nocturnal boundary layer dampens turbulence above and within the canopy.     

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 note on the height dependence of the skewness and the kurtosis of the vertical turbulent velocity in the neutral surface layer

Several experimental results indicate that S _w (the skewness of the vertical turbulent velocity) increases with height, and K _w (the kurtosis) decreases with height in the neutral surface layer. The measured behavior of S _w and K _w with height under neutral conditions is shown to be consistent with available models for the gradient-diffusion and the pressure-velocity correlation. In addition, an estimate of the mode of the probability density distribution turns out to be helpful in interpreting the height dependence of K _w.     

论边界层中的大气扩散PDF模式

基于大气扩散Ｋ理论，用作为风速脉动均方差和拉氏时间尺度函数的湍流交换系数，得到了直接利用风速脉动几率密度而不用扩散参数的大气扩散ＰＤＦ模式。分别研究了对流边界层上升气流区与下降区垂直速度的统计特征，求得双正态ＰＤＦ模式。在给定ＣＢＬ自身参数如对流特征速度ｗ＊，顶高ｈｉ和源高度上的平均风速时，该模式计算出的无量纲浓度分布与室内外测试结果一致。     

Development and Validation of a Lagrangian Probability Density Function Model of Horizontally-Homogeneous Turbulence Within and Above Plant Canopies

Second- and third-order turbulence closure models have met with mixed success when applied to the prediction of turbulent flows within and above plant canopies and model predictions are typically no better than those obtained using simpler two-equation (k-e){(k-\varepsilon)} models. This is because the local gradient diffusion approximation (a crucial model requirement) cannot represent accurately turbulent transport that is dominated by the presence of ejections and sweeps whose length scales are comparable with the canopy height. To make progress, turbulent transport must be treated without approximation, as in Lagrangian probability density function (PDF) models. This study is the first to develop and to validate a PDF model of horizontally-homogeneous canopy flow. The model relies upon a prescribed length-scale that has been used elsewhere in the modelling of turbulent flows. Model predictions compare favourably with measurements of neutrally stratified turbulent flows within and above canopies of mature corn and forested eucalypt trees.     

Probability distribution function of a forced passive tracer in the lower stratosphere

The probability distribution function （PDF） of a passive tracer, forced by a ＂mean gradient＂, is studied. First, we take two theoretical approaches, the Lagrangian and the conditional closure formalisms, to study the PDFs of such an externally forced passive tracer. Then, we carry out numerical simulations for an idealized random flow on a sphere and for European Center for Medium-Range Weather Forecasts （ECMWF） stratospheric winds to test whether the mean-gradient model can be applied to studying stratospheric tracer mixing in midlatitude surf zones, in which a weak and poleward zonal-mean gradient is maintained by tracer leakage through polar and tropical mixing barriers, and whether the PDFs of tracer fluctuations in midlatitudes are consistent with the theoretical predictions. The numerical simulations show that when diffusive dissipation is balanced by the mean-gradient forcing, the PDF in the random flow and the Southern-Hemisphere PDFs in ECMWF winds show time-invariant exponential tails, consistent with theoretical predictions. In the Northern Hemisphere, the PDFs exhibit non-Gaussian tails. However, the PDF tails are not consistent with theoretical expectations. The long-term behavior of the PDF tails of the forced tracer is compared to that of a decaying tracer. It is found that the PDF tails of the decaying tracer are time-dependent, and evolve toward flatter than exponential.     

Probability distributions of concentration fluctuations of a weakly diffusive passive plume in a turbulent boundary layer

Results are presented from an experimental investigation of turbulent dispersion of a saline plume of large Schmidt number (Sc=830) in a turbulent boundary-layer shear flow simulated in a laboratory water channel. The dispersion measurements are obtained in a neutrally buoyant plume from an elevated point source over a range of downstream distances, where both plume meandering and fine-structure variations in the instantaneous plume are important. High-resolution measurements of the scalar fluctuations in the plume are made with a rake of conductivity probes from which probability distributions of concentration at various points throught the plume are extracted from the time series.Seven candidate probability distributions were tested, namely, the exponential, lognormal, clipped normal, gamma, Weibull, conjugate beta, andK-distributions. Using the measured values of the conditional mean concentration, , and the conditional fluctuation intensity,i _p , the Weibull distribution provided the best match to the skewness and kurtosis over all downstream fetches. The skewness and kurtosis were always overpredicted by the lognormal probability density function (pdf), and underpredicted by the gamma pdf. The conjugate beta distribution for which the model parameters are determined using a method of moments based on the fluctuation intensity,i _p , and skewness,S _p , was capable of modeling the distribution of scalar concentration over a wide range of positions in the plume.     

A Comparison of Higher-Order Vertical Velocity Moments in the Convective Boundary Layer from Lidar with In Situ Measurements and Large-Eddy Simulation

We have analyzed measurements of vertical velocity w statistics with the NOAA high resolution Doppler lidar (HRDL) from about 390 m above the surface to the top of the convective boundary layer (CBL) over a relatively flat and uniform agricultural surface during the Lidars-in-Flat-Terrain (LIFT) experiment in 1996. The temporal resolution of the zenith-pointing lidar was about 1 s, and the range-gate resolution about 30 m. Vertical cross-sections of w were used to calculate second- to fourth-moment statistics of w as a function of height throughout most of the CBL. We compare the results with large-eddy simulations (LES) of the CBL and with in situ aircraft measurements. A major cause of the observed case-to-case variability in the vertical profiles of the higher moments is differences in stability. For example, for the most convective cases, the skewness from both LES and observations changes more with height than for cases with more shear, with the observations changing more with stability than the LES. We also found a decrease in skewness, particularly in the upper part of the CBL, with an increase in LES grid resolution.     

Turbulence in an almond orchard: Vertical variations in turbulent statistics

Three-dimensional wind velocity components were measured above and within a uniform almond orchard. Turbulent statistics associated with the turbulent flow inside the canopy are examined in detail. Turbulence in an almond orchard is characterized by relatively high turbulent intensities and large skewness and kurtosis values. These results indicate that the frequency distribution of wind velocity components is non-Gaussian. Conditional sampling of the turbulent measurements show that large, infrequent sweeps provide the predominant mechanism for tangential momentum stress in the canopy crown. Deep inside the canopy, a secondary wind maximum and small, but positive, tangential momentum stresses are observed.     

Turbulent transport of scalar quantities within and above a paddy field

An experiment was conducted to study turbulent transport processes of scalar quantities within and above a rice plant canopy. A sonic anemometer-thermometer and a Lyman- humidiometer were used to measure the turbulent fluxes of sensible and latent heat and related turbulence statistics within a paddy field. The sensible and latent heat fluxes measured at two heights within and above the plant canopy showed that the upper layer of this plant canopy was an active source region and that the source strength of sensible and latent heat depended on the solar radiation and physiology of rice plants. Analysis of joint probability distributions of w and T and of w and q within this plant canopy showed that downdrafts were remarkably efficient for upward transport of sensible and latent heat in the daytime. The vertical fluxes of temperature and humidity variance were also divergent from the upper layer of plant canopies. The power spectra of temperature and humidity within the plant canopy decreased rapidly in the high frequency range, compared with the - 2/3 law relationship of nS(n) vs log n observed above plant canopies.     

On Reynolds Averaging of Turbulence Time Series

We show that validity of Reynolds averaging for estimating the (ensemble) mean of a turbulence time series requires that the series values be both stationary and uncorrelated. In strict statistical terminology, these two conditions are jointly designated as independent identically distributed (i.i.d.). Moreover, we show that when the series values are correlated, knowledge of the correlation between the values is needed to obtain a reliable estimate of the mean. Last, we contend that a viable averaging algorithm must be Reynolds number (Re) dependent, requiring one version for low Re (Gaussian) turbulence and another for high Re (non-Gaussian) turbulence. Alternatively the median (as opposed to the mean) is recommended as a measure of the central tendency of the turbulence probability density function.     

On the downward transport of turbulent kinetic energy in the surface layer over plant canopies

Downward fluxes of turbulent kinetic energy have been frequently observed in the air layer just above plant canopies. In order to investigate the mechanism for such downward transport, analysis of observational data is attempted. Height-dependency of turbulent kinetic energy flux and turbulence statistics including higher order moments is represented as a function of a non-dimensional height z/H, where z is an observational height and H an average height of plant canopies. Downward fluxes and non-Gaussianity of wind velocity fluctuations are predominant just above plant canopies and decrease with increasing height. The downward flux is closely related to the high intensity of turbulence and the non-Gaussianity of wind velocity fluctuations, especially with a positive skewness in the longitudinal wind and a negative skewness in the vertical wind. The analysis method of conditional sampling and averaging is applied to the present observations. The results show that the predominance of the intermittent inrush phase over the intermittent ejection phase leads to the above-mentioned non-Gaussianity. Finally, a simple explanation is given in order to interpret the turbulent flow structure in the air layer near the plant canopies, which is associated with the downward energy transport process.     

COMPARISON AMONG EMPIRICAL PROBABILITY DENSITY FUNCTIONS OF THE VERTICAL VELOCITY IN THE SURFACE LAYER BASED ON HIGHER ORDER CORRELATIONS

Vertical velocity fluctuations were measured in theatmospheric surface layer by means of an ultrasonic anemometer andhigher order correlations were calculated on two time series, recordedin unstable and neutral conditions, and selected for the wholemeasurement period on the basis of the inversion test (stationaritytest). Comparisons have been made between observed and predictedcorrelations by considering Gaussian joint-PDF and Gram-Charlierseries expansions truncated to the fourth and sixth order as doneearlier by Frenkiel and Klebanoff. A bi-Gaussian PDF, given by amixture of two Gaussian PDFs, has also been considered. This lasthas been constructed assuming that either the first three or the firstfour moments are given, and the relationships between correlationfunctions of different order are derived. The departure from Gaussianbehaviour in both stability conditions is derived. Though Gram-Charlier series expansions show a good correspondence toexperimental reality, their use as non-Gaussian probabilitydistributions cannot be suggested in theoretical approaches andshould be considered with care in practical applications, due topossible occurrences of small negative probabilities. The resultsshown in this paper support the applicability of the bi-Gaussian PDFcreated using up to the fourth moment.     

Influence of non-flat terrain and wind direction shear on canopy turbulence

We have analyzed eddy covariance data collected within open canopy to investigate the influence of non-flat terrain and wind direction shear on the canopy turbulence. The study site is located on non-flat terrain with slopes in both south-north and east-west directions. The surface elevation change is smaller than the height of roughness element such as building and tree at this site. A variety of turbulent statistics were examined as a function of wind direction in near-neutral conditions. Heterogeneous surface characteristics results in significant differences in measured turbulent statistics. Upwind trees on the flat and up-sloping terrains yield typical features of canopy turbulence while upwind elevated surface with trees yields significant wind direction shear, reduced u and w skewness, and negligible correlation between u and w. The directional dependence of turbulence statistics is due that strong wind blows more horizontally rather than following terrain, and hence combination of slope related momentum flux and canopy eddy motion decreases the magnitude of Sk _w and r _uw for the downslope flow while it enhances them for the upslope flow. Significant v skewness to the west indicates intermittent downdraft of northerly wind, possibly due to lateral shear of wind in the presence of significant wind direction shear. The effects of wind direction shear on turbulent statistics were also examined. The results showed that correlation coefficient between lateral velocities and vertical velocity show significant dependence on wind direction shear through change of lateral wind shear. Quadrant analysis shows increased outward interaction and reduced role of sweep motion for longitudinal momentum flux for the downslope flow. Multi-resolution analysis indicates that uw correlation shows peak at larger averaging time for the upslope flow than for the downslope flow, indicating that large eddy plays an active role in momentum transfer for the upslope flow. On the other hand, downslope flow shows larger velocity variances than other flows despite similar wind speed. These results suggest that non-flatness of terrain significantly influences on canopy-atmosphere exchange.     

An approximate analytical solution for the deposition of heavy particles released from an elevated line source

This work re-examines and further develops an analytical solution for the deposition swath of heavy particles released in the atmosphere from an elevated source over uniform terrain, correcting the particle diffusivity for the crossing trajectory effect. The revised (approximate) analytical solution proves to be accurate within 20% over a wide range of micrometeorological conditions and particle size, despite its neglect of the turbulence component of the deposition flux. It compares very satisfactorily with experimental data and with the simulations of a Lagrangian stochastic model, provided the variable U(H)/w _g ≤7 (ratio of the mean horizontal wind speed at source height to the particle settling velocity). In this domain of validity, simple formulae relating the statistics of the deposition swath to U(H)/w _g are derived.