DOI 10.1007/s10546-006-9061-9Boundary-Layer Meteorology (2006) © Springer 2006

A LARGE-EDDY SIMULATION AND LAGRANGIAN STOCHASTICSTUDY OF HEAVY PARTICLE DISPERSION IN THE

CONVECTIVE BOUNDARY LAYER

XUHUI CAI∗, RUI ZHANG and YAN LIDepartment of Environmental Sciences, Peking University, Beijing 100871, China

(Received in final form 26 January 2006)

Abstract. Large-eddy simulation and Lagrangian stochastic dispersion models were used tostudy heavy particle dispersion in the convective boundary layer (CBL). The effects of var-ious geostrophic winds, particle diameters, and subgrid-scale (SGS) turbulence were inves-tigated. Results showed an obvious depression in the vertical dispersion of heavy particlesin the CBL and major vertical stratification in the distribution of particle concentrations,relative to the passive dispersion. Stronger geostrophic winds tended to increase the disper-sion of heavy particles in the lower CBL. The SGS turbulence, particularly near the surface,markedly influenced the dispersion of heavy particles in the CBL. For reference, simulationsusing passive particles were also conducted; these simulation results agreed well with resultsfrom previous convective tank experiments and numerical simulations.

Keywords: Convective boundary layer, Dispersion, Heavy particles, Lagrangian stochasticmodel, Large-eddy simulation, Subgrid scale.

1. Introduction

The dispersion of heavy particles in the atmospheric boundary layer (ABL)is essential to many environmental processes such as the diffusion of airpollutant particulates and mineral dust. Typically, particulate air pollutantsor natural dust emitted from or near the Earth’s surface undergo turbulentdispersion in the ABL before being deposited back to the surface or occa-sionally carried into the free atmosphere (e.g., during a sandstorm event).Over the past few decades, experiments (e.g., Willis and Deardorff, 1976;Sanyder and Lawson, 2002) and numerical simulations (e.g., Deardorff,1974; Lamb, 1978) have illustrated the crucial importance of turbulencestructure and particle properties in ABL dispersion (Sawford and Guest,1991; Wang and Stock, 1993).

The dispersion of passive tracers or particles in the turbulent atmo-sphere can be simulated using Lagrangian stochastic (LS) models. This

∗ E-mail: xhcai@pku.edu.cn

XUHUI CAI ET AL.

method has increased in theoretical robustness following Thomson’s (1987)well-mixed criteria and recently has been widely applied (e.g., Stohl et al.,1998; Kurbanmuradov and Sabelfeld, 2000). There are two ways to con-duct a LS simulation: either, one reconstructs the fluctuation velocities ofparticles along their travelling trajectories using Eulerian statistics of tur-bulence, or one calculates particle motion using the output of a large-eddysimulation (LES) model with resolvable velocity fields and subgrid-scale(SGS) turbulence parameters. As early as the 1970s (Lamb, 1978), the lattermethod had been successfully applied to verify the tracer dispersion in theconvective boundary layer (CBL) estimated by convective tank experiments(Willis and Deardorff, 1976).

For heavy particles with non-negligible mass, inertia and gravity forcesmust be considered, and the dispersion process becomes more complex.One important difference in heavy particle dispersion compared with pas-sive particle dispersion is the “crossing-trajectory effect” (Yudine, 1959)caused by the body force. Wilson (2000) demonstrated that, for practi-cal applications, a rather simple LS model that does not consider parti-cle inertia may be sufficient for modelling heavy particle dispersion in theatmospheric surface layer. However, greater conceptual clarity may enhanceour understanding of physical processes throughout the entire depth of theABL and may help in quantitatively determining the relative importance offactors in this process.

LES can resolve large-scale turbulent flows explicitly and parameterisethe smaller and relatively homogenous part of turbulence using a SGSscheme. After the pioneering work of Deardorff (1972), this techniquebecame an effective tool for studying the properties and structure of tur-bulence, particularly in the CBL. An advantage of using LES to study thedispersion process is that only the SGS part of turbulence need be treatedas a stochastic process. Therefore, a more realistic and explicit simulationof the dispersion process is possible.

Turbulence in the CBL is dominated by large eddies or thermals withscales comparable to the CBL depth, and where sensible heating of the sur-face plays a dominant role in maintaining the turbulence structure. It isbelieved that this structure is responsible for the special dispersion patternsin the CBL as shown by laboratory experiments (Willis and Deardorff,1976; Sanyder and Lawson, 2002). The CBL is typical over land surfacesduring daytime and over oceans during cold air breakouts. The occur-rence of sandstorms, a typical type of mineral dust dispersion in the atmo-sphere, favours an unstable atmosphere (Qian et al., 1997). Furthermore,turbulence and the CBL structure have been reliably constructed using LES(Nieuwstadt et al., 1993). Our study also used the LES method to examinethe dispersion of heavy particles in the CBL.

MODELLING HEAVY PARTICLE DISPERSION

Specifically, we use the LES–LS method to investigate how gravity, par-ticle inertia, and the geostrophic wind influence dispersion. For simplicity,we consider only dispersion over a flat, homogeneous surface, and do notconsider the feedback effects of particle motion on air flow. The paper isorganised as follows: Section 2 overviews the two models and details theparameter settings for the simulations. Section 3 provides the numericalsimulation results and a discussion, and a summary is given in Section 4.

2. Methods

2.1. The LES model

The Peking University large-eddy simulation model (PKU-LES; Cai andChen, 1995) was used to determine CBL turbulence fields and their tem-poral evolution. This model resolved the partial differential equations ofu, v, w, and θ explicitly, where u, v, w are three components of resolvablevelocity, and θ is resolvable potential temperature. The scheme of SGS tur-bulence closure and the parameterisation of lower boundary conditions inthis model were the same as those by Mason (1989), i.e., assumptions ofeddy viscosity and Monin–Obukhov similarity were applied, respectively.It should be noted that the Monin–Obukhov theory was applied locally,i.e., between each grid of the lowest model level and the lower boundary(surface), the flux–profile relationship (Businger et al., 1971) was appliedto mean winds, mean potential temperature, and surface friction veloc-ity and heat flux. Surface heat flux was prescribed for the simulation.The Coriolis force and the background pressure gradient were consideredin the geostrophic wind term (Meong, 1984) in the governing equation.Moisture effects were omitted for simplicity. Numerically, this model waspseudo-spectral in the horizontal and used second-order, centred, finitedifferencing in the vertical as in the method by Moeng (1984). The Adams–Bashforth scheme was used for time integration. The lateral boundary con-ditions were periodic, and the upper boundary condition was given as zerovertical velocity, stress, and heat flux (Moeng, 1984). This model has beenused for a series of studies of the CBL structure and surface influences aswell as dispersion properties (Cai and Chen, 1997, 2000, 2003; Cai et al.,2002). Further details and evaluation of this model have been provided byCai and Chen (1995).

The model domain was set as 6.4 km ×6.4 km ×1.5 km with uniformgrids 64 × 64 × 30 in the x, y, and z directions; the spatial resolution was100 m ×100 m ×50 m. The model was initialised by an ideal potential tem-perature profile with a temperature of 290 K from the surface to 700 mand an inversion layer with ∂θ/∂z = 0.02 K m−1 above. A small random

XUHUI CAI ET AL.

TABLE IRelevant parameters for the LES runs.

Run Q0 (K m s−1) Ug (m s−1) zi (m) w∗ (m s−1) t∗ (s) 〈u∗〉 (m s−1)

Q2U0 0.2 0 1000 1.89 529 0.17Q2U2 0.2 2 1000 1.89 529 0.2Q2U4 0.2 4 1000 1.89 529 0.28

Here w∗ is the convective velocity defined by Deardorff (1970); t∗ =zi/w∗ is the turnovertime; Q0 is the surface heating rate; Ug is the geostrophic wind; zi is the CBL depth;〈u∗〉 is mean friction velocity.

fluctuation of potential temperature was superimposed on the lowest tengrid levels to begin the simulation run. By specifying a Coriolis parame-ter f =0.9×10−4 rad s−1, the model usually integrated for approximately 3 hbefore approaching a quasi-stationary state. Three simulation runs were per-formed with geostrophic winds of 0, 2, and 4 m s−1 in the x direction andwith a kinematic heat flux of 0.2 K m s−1 at the surface. The surface rough-ness length was set at 0.1 m for all simulation runs. The total simulation timeof each run was 18,000 s with a timestep of 2 s, and simulation results for thelast 6000 s (which were statistically steady) were stored at time intervals of50 s. Thus, there were 120 turbulence fields for each LES run to drive par-ticle dispersion. Relevant parameters for the LES runs are summarised inTable I. Mean friction velocity 〈u∗〉 was determined by calculating a hori-zontal average for the lower boundary of the LES model point by point andthen determining the average for the entire simulation period. Thus, 〈u∗〉 wasnever zero even when the geostrophic wind was zero.

2.2. The LS model for passive particles

The LS model simulates atmospheric dispersion by tracking a large num-ber of particles released continuously or instantaneously from the source.Wilson and Sawford (1996) and Wilson (2000) have extensively reviewedthis method. In the context of a LES, the turbulence velocity ui in the CBLcan be separated into a resolvable part and a SGS part, as follows:

ui = ui +ui, (1)

where ui = (u, v, w) is the resolvable part of the LES, and ui is the SGSfluctuating velocity that must be treated as stochastic in the LS model. Ifwe write a particle’s Lagrangian velocity as Ui , then its trajectory is deter-mined as

dxi = Ui dt, (2)

MODELLING HEAVY PARTICLE DISPERSION

where dxi is the Lagrangian trajectory displacement, and t is time. Fora passive particle or an element of the fluid, the Lagrangian velocity Ui

equals the Eulerian velocity ui at each point of the trajectory, i.e., Ui = ui .We can also partition the Ui into the resolved part Ui and the SGS partUi and have Ui = ui and Ui =ui . So the calculation of the Lagrangian dis-placement can also be separated into two parts, dxi = (dxi)resolved + (dxi)SGS.The resolved part (dxi)resolved is calculated explicitly by the resolved veloc-ity, (dxi)resolved = Uidt .

The stochastic part can be written as (dxi)SGS =Uidt , and with the SGSparticle velocity written as the stochastic differential equation:

dUi =aidt +bijdζj , (3)

where ζj is a random number with zero mean and a Gaussian probabilitydistribution, and its variance is equal to dt . The ai and bij are coefficientsto be determined, and the subscripts i and j take values of 1, 2, and 3, rep-resenting the three components of the Cartesian coordinates, respectively.According to Kolmogorov’s similarity theory for the statistics of velocityincrements over a small time interval dt (Wilson, 2000),

bij = (C0ε)1/2δij , (4)

where ε is the dissipation rate of turbulent kinetic energy, C0 is a constant,and δij is the Kronecker function. The other coefficient ai should satisfythe Fokker–Planck equation with a well-mixed constraint:

∂pE

∂t+ui

∂pE

∂xi

+ ∂aipE

∂ui

− 12C0ε

∂2pE

∂ui∂uj

=0, (5)

where pE is the probability density function (pdf) of the SGS Eulerianvelocity, which can be regarded as given. Though ai is not uniquely definedfrom (5) for a multidimensional condition, for one-dimensional and homo-geneous and isotropic turbulence, ai can be defined uniquely by addition-ally assuming a Gaussian pdf of pE, as follows (Sawford, 1993; Wilson andSawford, 1996):

ai =−C0ε

2σ 2i

Ui, (6)

where σi is the standard deviation of the SGS turbulent velocity. Note, thesame subscript i does not sum up in (6).

In this study, we used the turbulence field output from the LES modelto drive the Lagrangian particle dispersion model. Because most turbulentenergy could be directly resolved by the LES model, only the stochasticpart of SGS turbulence had to be treated by the LS dispersion model. One

XUHUI CAI ET AL.

advantage of this method is that the smaller SGS turbulence could be rea-sonably assumed as homogeneous in the interior of the CBL, and we coulddetermine the coefficients ai from (6). However, in the regime near the sur-face, the SGS turbulence is very important to particle dispersion; this tur-bulence cannot be assumed to be homogeneous, and the method may failif homogeneity is assumed. To consider near-surface effects, we added theMonin–Obukhov similarity formulation between the lowest LES velocitylevel (e.g., 0.5�z, or 25 m as used in this study) and the surface for the LSdispersion. That is, we applied the surface-layer relationships for the meanwinds, turbulent velocity variance, and dissipation rate in this layer. Theapplication was done point by point to the lowest model grids, correspond-ing to the lower boundary treatment in the LES model. Accordingly, Equa-tion (6) could be rewritten in this layer as (Thomson, 1987; Wilson, 2000;Stohl and Seibert, 2002)

ai =−C0ε

2σ 2i

Ui + 12

∂σ 2i

∂z

(U 2

i

σ 2i

+1)

δi3. (7)

Note again that the same subscript i also does not sum up in Equation (7),and the terms for horizontal non-homogeneity of the SGS turbulence areneglected, since they are likely to be less significant than the vertical term.

Thus, with the SGS kinetic energy and dissipation rate provided bythe LES model for the upper levels, the flux–profile relationship (Busingeret al., 1971), and the surface-layer parameterisation for σi and ε adoptedfrom Pasquill and Smith (1983; their equations 2.79, 2.89a, and 2.90a), thetrajectory of a passive particle could be deduced within the entire CBLby Equations (2), (3), (4), (6), and (7) and the resolved turbulence fields.Matching of the surface-layer treatment to the first layer of the LES modelis necessary, both for the mean flow and for the turbulence. The surfacelayer mean winds were matched from the surface to those at each grid ofthe lowest LES model level. Matching of the surface-layer turbulence to thefirst layer of the LES model is shown in Appendix A.

2.3. The treatment for heavy particles

For heavy particles, inertia and gravity effects must also be considered.First, we define the aerodynamic response time for Stokes particles as

τa = D2ρ

18µ, (8)

where D is the diameter of the particle, ρ is its density, and µ is the viscos-ity coefficient of the fluid. Then, by introducing inertia and gravity force,the governing equation of heavy particle motion can be written as (e.g.,Wang and Stock, 1993)

MODELLING HEAVY PARTICLE DISPERSION

dUi

dt= (ui − Ui)

τa

F (Rep)+gδi3, (9)

where ui is the Eulerian velocity of the particle’s trajectory, Ui is particlevelocity, and g is the gravity acceleration; F is a correction coefficient forpractical particles as a function of the particle Reynolds number Rep. Wefurther assumed that Rep was small: F =1 (Wang and Stock, 1993; Wilson,2000).

Equations (9) and (2) can be used to determine the trajectory of aheavy particle if the Eulerian velocity ui , which drives the particle motion,is given. In the case of the LES–LS method, a major part of the driv-ing velocity for particles was given explicitly by the LES model as theresolvable field of turbulence. The SGS part of the turbulent fluctua-tion could also be reconstructed by the LS model described above fora passive particle or fluid element. The difficulty at this point, however,was that the trajectories of heavy and passive particles did not coincidewith each other. The following methods were taken to circumvent thisdifficulty.

Because the trajectory departure of a heavy particle from a passive par-ticle is small in SGS turbulence, we assumed that the SGS driving veloc-ity for a heavy particle could reasonably approximate that for a passiveparticle. Thus Equations (9) and (2) could be applied directly throughEquations (1) and (3). The discrete formula for the particle velocity on itstrajectory for successive timesteps can be written as

Ui = U 0i + (ui − U 0

i )

τa

dt +gdtδi3 = U 0i + (ui +ui − U 0

i )

τa

dt +gdtδi3, (10)

where U 0i is the particle velocity at the previous timestep.

As Wilson (2000) noted, an alternative and more simplified method isto directly simulate the stochastic velocity of the heavy particle. Here, wepartitioned the particle velocity into three parts:

Ui = Ui +Ui +wgδi3, (11)

where wg =gτa is the equilibrium settling velocity, and Ui and Ui represent,respectively, the contribution of the resolved and SGS fluid velocities to theparticle. Strictly speaking, both Ui and Ui are unknown. However, in thisstudy, Ui could be approximated by the resolved velocity of the LES, u, ifthe inertia effect on the particle was negligible. This point will be addressedlater. Ui may also be treated by the LS model in the form of Equation (3),but the coefficients ai and bij should be modified for heavy particle disper-sion (Sawford and Guest, 1991; Wilson, 2000). Considering that SGS tur-bulence is usually much weaker relative to the resolvable, energetic large

XUHUI CAI ET AL.

eddies in the CBL (except near the surface), refinement of the LS modelto treat heavy particles may have negligible effects. Thus, we adopted thesame LS model for passive particles to represent the motion of heavy par-ticles. That is, we assumed that heavy particles “passively” respond to SGSfluctuations. Thus, Equation (3) was used to determine Ui , and Equations(11) and (2) were used to calculate the particle trajectory.

2.4. Simulation cases for particle dispersion

Heavy particles of mineral sand with densities of ρ =2600 kg m−3 were sim-ulated in this study. The particles were assumed rigid and orbicular withthree diameters (53, 80, and 104 µm) and were used in dispersion runs withthree geostrophic winds (0, 2, and 4 m s−1). First, however, simulations ofpassive particles were conducted to evaluate the LES–LS model. Additionaltrials were included to examine the influence of SGS turbulence, particu-larly near-surface SGS turbulence, on the simulation results. Table II pro-vides details of the LS simulation cases and their relevant parameters.

Using the LES results for the quasi-stationary CBL with a depth ofapproximately 1000 m (Table I), 100,000 particles were instantaneously

TABLE II

LS simulation cases and parameters.

Case Release (m) Settling velocity Particle diameter SGS turbulence Monin–Obukhovheight hs wg (m s−1) D (µm) similarity

P 60 – – Yes NoPH 200 – – Yes NoGHI 200 – 80 yes NoPHI 200 – – Yes NoPS 60 – – Yes YesG2S 60 0.2 53 Yes YesG4S 60 0.4 80 Yes YesG6S 60 0.6 104 Yes YesPN 60 – – No NoG2N 60 0.2 53 No NoG4N 60 0.4 80 No NoG6N 60 0.6 104 No NoG2 60 0.2 53 Yes NoG4 60 0.4 80 Yes NoG6 60 0.6 104 Yes No

Case labels: P, passive; G, gravity; I, inertia effect; H, higher release; N, no SGS turbu-lence; S, near-surface Monin–Obukhov similarity adopted.

MODELLING HEAVY PARTICLE DISPERSION

released in the LS model. The particles were evenly distributed on a hor-izontal level of the model domain at a specified source height. Initially,random velocities with Gaussian pdf distributions and SGS velocity vari-ances σ 2

u = σ 2v = σ 2

w = 23e2 of the LES field were specified for each particle

(where e2 is the SGS turbulence kinetic energy). All particles were trackedfor 100 min (6,000 s) after release. To make the simulation results quantita-tively comparable to those of previous work, the statistical and normalisa-tion method followed that by Lamb (1982). Concentration was normalisedby the source strength, and the diffusion characteristics were normalised byCBL scales, i.e., the CBL height zi , convective velocity w∗, and turnovertime t∗ = zi/w∗.

Marking the strength of the instantaneous release (total release numberof particles) as S and the number concentration at time t and space point(x, y, z) as c(x, y, z, t), the normalised concentration of the mean verticaldispersion could then be written as (Lamb, 1982)

C(Z,T )= zi

S

∫∫

c(x, y, z, t)dxdy, (12)

where Z = z/zi is the non-dimensional height, T = t/t∗ is the non-dimensional dispersion time, and is the horizontal domain. Note thatC(Z,T ) is equivalent to the crosswind integrated concentration of a con-tinuous point source if streamwise dispersion is neglected.

It is possible to realistically simulate dispersion in the CBL for a con-tinuous point source, particularly in the case of a non-zero mean wind.However, a much greater particle release and longer simulation time arerequired for a statistically steady result and may essentially change the CBLheight and create problems in the resulting statistics. If we focus on verticaldispersion in the CBL and neglect the influence of horizontal dispersion,it is preferable to simulate the temporal–vertical evolution described by(12) rather than to realistically simulate the downwind–vertical dispersion.Lamb (1982) examined these two methods and found only minor differ-ences in the respective results if a mean wind speed criterion was satisfied.The former is more computationally efficient and more easily obtains a sta-tistically steady result.

The surface boundary in the LS model was assumed to have perfectreflection. This assumption may be reasonable for the position treatmentof heavy particle simulation if we consider a dust source area that isapproaching certain equilibrium between sand particles touching the sur-face and particles emitting from the surface. This boundary treatmentis apparently inappropriate for the velocity simulation of heavy particlesbecause the gravity effect occurs continuously along the trails of heavy par-ticles. However, at this stage, we did not tackle the gravity effect in detail.

XUHUI CAI ET AL.

By neglecting streamwise dispersion and considering the fact that the lat-eral boundaries are periodic in LES fields, we treated the particle disper-sion at lateral boundaries in the LS model as follows. If a particle movedout from one side of the horizontal domain, it would re-enter symmet-rically from the opposite side; if the particle went out from the down-wind boundary, it would re-enter from an upwind boundary. By using thismethod, the total number of simulated particles was kept constant duringone simulation run.

3. Results and Discussion

3.1. Major characteristics of the simulated CBL

To show the general performance of the PKU–LES model, Figure 1displays portions of the simulation results. Figure 1a is an example ofthe instantaneous w field at an x − z cross-section for the LES simula-tion case Q2U0 (see Table I). The figure clearly shows well-defined andrandomly located convective thermals extending vertically throughout theentire depth of the CBL. Figure 1b shows the typical structure of weakerand wider downdrafts and stronger and narrower updrafts at a level in theCBL interior (z= 300 m). Profiles of the horizontal statistics of turbulenceand mean properties are also shown in Figure 1c–g, i.e., potential temper-ature 〈θ〉, the horizontal wind component 〈u〉, turbulent velocity variances〈u2 + u2〉, 〈v2 + v2〉 and 〈w2 + w2〉, vertical velocity skewness 〈w3〉/〈w2〉3/2,and heat flux 〈wθ + wθ〉; the angle brackets 〈〉 indicate horizontal aver-ages over the model domain, and labels with and without overbars indicateresolvable and SGS quantities. These results show typical characteristics ofthe CBL and are quantitatively comparable to results from previous LESstudies (e.g., Moeng, 1984; Mason, 1989; Schmit and Schumann, 1989;Nieuwstadt et al., 1993).

3.2. Reference runs of passive particle dispersion

Passive particle dispersion in the CBL was examined first to verify theLES–LS method used in this study; these results served as a reference forthe following simulations of heavy particles. Two reference runs with differ-ent particle release heights in the CBL condition were carried out (runs Pand PH, see Table II). The timestep in the calculation of particle trajecto-ries was set at 0.1TL, with TL the minimum time scale of SGS turbulenceamong three velocity components, given as

TL =min[2σ 2u /(C0ε),2σ 2

v /(C0ε),2σ 2w/(C0ε)], (13)

MODELLING HEAVY PARTICLE DISPERSION

0 2 4x (km)

0

0.5

1

z/z

i

.

a

0 2 4 5 6x (km)

0

1

2

3

4

5

6

y (k

m)

b

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

290 292 294 296 298 300 302 304 306

c

<q> (K)

z / z

i

-1 0 1 2 3 4

d

<u>(ms-1

)0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5

e

SGS

<u2> / w2

z / z

i

z / z

i

Q2U0 Q2U2 Q2U4

Q2U0 Q2U2 Q2U4

Q2U0 Q2U2 Q2U4

Q2U0 Q2U2 Q2U4

Q2U0 Q2U2 Q2U4

Q2U0 Q2U2 Q2U4

Q2U0 Q2U2 Q2U4

Total

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

z / z

i

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

z / z

i

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4 f

SGS

z / z

i

0.0 0.1 0.2 0.3 0.4 0.5

<w2> / w2*

-1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 -1.0 0.0 0.2 0.4 1.6 1.8 1.0

hg

w-Skewness

i

SGS

<wq >/Q0

z /z

i

Total

Total

SGS

Total

1 3 5 6 1 3

* <v2> / w2*

Figure 1. (a) Instantaneous vertical velocity at an x–z section, (b) vertical velocity at anx −y plane 300 m above the surface, (c) potential temperature, (d) mean speed of the u com-ponent, (e) turbulent velocity variance of the u component, (f) turbulent velocity variance ofthe v component, (g) turbulent velocity variance of the w component, (h) vertical velocityskewness, and (i) heat flux. These profiles are based on results averaged over the last 6000 sof the simulation.

where the constant C0 =3.0, and σu, σv and σw are the standard deviationsof SGS turbulent velocity components.

Figure 2 shows the contours of normalised concentration C (definedby Equation (12)) as a function of non-dimensional dispersion time t/t∗and dimensionless height Z = z/zi . Because non-dimensional travel timecan also be regarded as a dimensionless downwind distance X =

(xzi

)w∗Um

,with Um the mean wind speed in the CBL, we may also view the resultsshown in Figure 2 as dispersion plumes in an X − Z section (Lamb,1982). Using similar heights to those in the convective tank experiment byWillis and Deardorff (1974), our simulation results (runs P and PH drivenby LES run Q2U0) showed good agreement with the laboratory exper-iment results and previous numerical results (Lamb, 1982). For a near-surface release (h/zi ≈ 0.06; Figure 2a), the maximum of the simulated

XUHUI CAI ET AL.

plume initially moved horizontally near the surface and then “lifted off”quickly, ascending towards the top of the CBL. For an elevated releasesource in the lower boundary layer (h/zi ≈ 0.2; Figure 2b), the plumedescended in the initial stage until eventually contacting the ground fromwhich it was then swept up by updrafts and carried to higher levels of theCBL. Quantitative comparisons also showed good agreement between oursimulation and laboratory experiment results. The contour of concentrationC = 2.5 intersected the surface at approximately a similar downwind dis-tance (or dispersion time) from the source (0.6t/t∗), and after a dispersiontime of approximately 4t/t∗, the plume mixed well with an average plumeheight of approximately 0.5zi . However, for the elevated release, the sim-ulated “rebound” of the plume tongue for C = 1.2 was not as strong asthat found by the water tank experiment; this discrepancy can be attributedto the SGS scheme of the LES and the tracer buoyancy in the laboratorystudy (Lamb, 1982; Gopalakrishnan and Avissar, 2000). Note that, for clar-ity, contours with values larger than 5 are not shown in the figure or thefigures hereafter.

3.3. Inertia effect

Accurate prediction of heavy particle dispersion in a turbulent flow requires,at least, consideration of inertia and body forces. The existence of inertiaslows the response of particles to the fluctuating velocity of their circum-stance; inertia also causes the particle trajectory to depart from that of thefluid element originally containing it (Sawford and Guest, 1991). Previousstudies (Yudine, 1959; Csanady, 1963; Hajji et al., 1996) have suggested thatparticle inertia may be negligible in typical atmospheric problems. Usingresults of a field experiment of released glass beads, Wilson (2000) com-pared a simulated series of heavy particle dispersion with different model

1

00.5

1

0 2 4 5 6 7 8 9 10 1

t / t1

00.5

1

z /

zi

a

b

*

3

10 2 4 5 6 7 8 9 10 113

Figure 2. Normalised concentration and time evolution for passive particles released (a) 60and (b) 200 m above the surface. Geostrophic wind Ug = 0 m s−1. Contours were 0.1, 0.5,0.75, 1, 1.25, 1.5, 2, 3, 4, and 5. Contours with values larger than 5 are not shown.

MODELLING HEAVY PARTICLE DISPERSION

refinements. He concluded that it is sufficient to adapt a simple methodto simulate heavy particle dispersion in the atmospheric surface layer forbeads 50–100 µm in size and when the ratio of the particle inertia timescaleto the turbulence timescale is small. That is, by simply superposing a grav-itational settling velocity on the heavy particle and shortening the turbu-lent velocity timescale along the particle trajectory, particle dispersion canbe determined by a LS model.

To explicitly and quantitatively evaluate the inertia effect on heavy par-ticle dispersion throughout the entire depth of the CBL, and to examinethe possibility of ignoring the inertia effect hereafter in our LES–LS model,we conducted a pair of specially designed simulation runs. In one run (runGHI listed in Table II), the dispersion of heavy particles with diametersof 80µm was simulated in the CBL with a 2 m s−1 geostrophic wind. Thetimestep in this case was set to 0.05τa, i.e., each particle would “walk”20 steps in one inertia–response time scale. To isolate the inertia effect ondispersion, we intentionally deactivated the gravity effect. Correspondingly,a companion run was carried out with purely passive particles and thesame timestep (i.e., 0.05τa); the passive particles were treated as heavy par-ticles with diameters of 80µm, but the inertia effect was deactivated (casePHI in Table II). Simulation results of these two runs agreed very well.Figure 3 compares ground concentrations and average plume heights. Aftera long dispersion time, slight departures of ground concentrations betweeninertia and passive runs did appear (Figure 3a). The maximum relativeerror between the two cases reached 2%, with a mean error of only 0.43%.For average plume heights, the results of the two runs almost coincided(Figure 3b). These results demonstrated that the inertia effect on heavy par-ticle dispersion in the CBL is trivial for the diameters we considered; thus,we ignored the inertia effect hereafter in this study.

0 2 4 70

1

2

3

4a

Nor

mal

ized

con

cent

ratio

n

t / t* t / t*

PHI GHI

0

200

400

600

800b

Mea

n pl

ume

heig

ht (

m)

PHI GHI

1 3 5 6 0 2 4 71 3 5 6

Figure 3. Comparison of the inertia particle run GHI and passive particle run PHI: (a) nor-malised surface concentration and (b) mean plume height.

XUHUI CAI ET AL.

Interestingly, the results of simulation run PHI with a small timestep(0.05τa) agreed very well with those of run PH with a larger timestep of0.1TL (figures not shown). This agreement suggests that a timestep as largeas 0.1TL is acceptable for LES–LS simulations of particle trajectories in theCBL. The above findings are meaningful for computational efficiency. Byusing a powerful PC (e.g., a Pentium 4 with 2.4 GHz CPU, 1024 MB EMSmemory), a simulation with small timesteps (0.05τa) and no inertia effect(passive case) requires a computational time of approximately 12 h. In con-trast, if gravity and the inertia effect were also considered, the CPU timecould increase to 35 h, which is unacceptable if multiple cases are beingexamined. By ignoring the inertia effect and using a timestep of 0.1TL, theCPU time becomes much more acceptable.

3.4. Dispersion of heavy particles

By ignoring the inertia effect, the following simulations included only a set-tling velocity to reflect the gravity effect of heavy particles. Combining theeffect of the resolvable turbulence velocity from the LES and the LS modelrepresenting SGS fluctuation, Equation (11) was used to calculate heavyparticle trajectories.

Figure 4 shows the dispersion of heavy particles with different diame-ters under the condition of zero geostrophic wind (simulation cases G2S,G4S, and G6S, driven by LES run Q2U0; see Tables I and II). Contoursshown in the figure are for a non-dimensional, crosswind-integrated con-centration, C. As illustrated in the figure, the gravity force reshaped thedispersion pattern of heavy particles in the CBL. Dispersion of particleswith a 53-µm diameter differed markedly from that of particles with a104-µm diameter in the CBL. Although the former case also showed thegravity influence (Figure 4a), it was still comparable to that of passive par-ticle dispersion (Figure 2a). For the heaviest case (104 µm), the majorityof the particles could not be carried to the upper part of the CBL froma near-surface release; the concentration above the surface layer could beas low as 10% of the passive counterpart when approaching a quasi-steadystate after a longer dispersion time (e.g., t/t∗ > 4 in Figure 4c). Note thata 100-µm diameter is usually used to discriminate between “sand” and“dust” amongst natural particles (Bagnold, 1941). For CBL scaling, weexamined the ratio of the particle settling velocity to the convective veloc-ity, i.e., wg/w∗, and obtained 0.11, 0.21, and 0.32, respectively, for parti-cle diameters of 53, 80, and 104 µm. This result indicates that the settlingvelocity of particles with 104-µm diameter approached more than 30% ofthe convective velocity. Thus, it is unsurprising that the dispersion patternchanged in this case because it would be difficult for typical updrafts in theCBL to lift such heavy particles.

MODELLING HEAVY PARTICLE DISPERSION

00.5

1

0 2 3 4 7 8 9 10

t / t 11

00.5

1

z / z

a

00.5

1b

c

5 61

0 2 3 4 7 8 9 10 115 61

0 2 3 4 7 8 9 10 115 61

*

i

Figure 4. Normalised concentration and temporal evolution for particles with diameters of(a) 53, (b) 80, and (c) 104 µm. The release height was 60 m above the surface. Geostrophicwind Ug =0 m s−1. Contours are the same as those in Figure 2.

Figure 5 shows the influence of gravity on particle dispersion. Resultsof three heavy particle runs and a passive particle run under conditions ofzero geostrophic wind (simulation runs G2S, G4S, G6S, and PS, driven byLES run Q2U0; see Tables I and II) are shown. Comparing the tempo-ral variation of C at the ground surface (Figure 5a) reveals that the sur-face concentration for passive and small–medium particles (D =53,80µm)increased quickly, reached a maximum after release, decreased to a min-imum by the dispersion time t/t∗ ≈ 2 (because of the “lift-off” processin the CBL), and then gradually approached a quasi-steady value. Forthe heaviest particles (D = 104 µm), the surface concentration reached ahigher value at the initial stage, increased continuously until t/t∗ ≈ 4, andthen gradually approached a much higher value. The surface concentra-tion clearly increased with increasing particle diameters after the disper-sion reached the quasi-steady state (t/t∗ ≈ 4). This concentration value for53-µm particles is nearly twice that for passive particles; for particles witha 104-µm diameter, the value was more than tenfold greater. Profiles ofparticle concentrations for these cases show strong relationships with sur-face concentrations (Figure 5b; mean profiles averaged from t/t∗ = 4 − 10).The passive case has a well-mixed profile throughout almost the entireCBL (approximately 0.1–0.9zi). However, non-zero gradients were presentfor profiles from the heavy particle cases, and concentrations in the inte-rior of CBL decreased substantially corresponding to increases in surfaceconcentrations.

XUHUI CAI ET AL.

0 4 8 100

2

4

6

8

10

12

14

16

18

20

12

a

Nor

mal

ized

con

cent

ratio

n

t / t *

PS G2S G4S G6S

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.01 0.1 1 10

b

Normalized concentration

z / z

i

PS G2S G4S G6S

62

Figure 5. (a) Normalised surface concentration and temporal evolution and (b) normalisedconcentration profiles. Symbols indicate passive and heavy particle runs. Geostrophic windUg =0 m s−1.

3.5. Impact of background winds

Figures 6 and 7 show contours of nondimensional concentration C withdifferent particle diameters under geostrophic winds of 2 m s−1 (runs G2S,G4S, and G6S driven by LES run Q2U2) and 4 m s−1 (runs G2S, G4S, andG6S driven by LES run Q2U4), respectively. Dispersion patterns for par-ticles of three diameters were similar to those of the corresponding onesunder zero background wind (Figure 4). However, stronger backgroundwinds clearly favoured particle suspension from the surface and into thelower CBL (e.g., z/zi <0.5), with a large concentration gradient there. Forexample, the 0.5 contour maintains a height of approximately 0.1–0.2zi inFigure 7c, while the corresponding contour remains near the surface inFigure 4c for the dispersion time t/t∗ > 1. These results agree with theintuitive assumption that stronger wind conditions lead to greater dustemissions from the surface. The dispersion found by this study can be sep-arated into two parts, one near the surface and another in the interiorof the CBL. Heavy particles tend to fall to the surface where they accu-mulate and form high concentrations. Background wind adds extra meanstress and therefore additional mechanical turbulent energy in the near-surface layer. This mechanism could be very important in the process ofheavy particles accumulated near the surface being ejected into the interiorof the CBL and then carried upwards by energetic thermals or large eddies.Thus, SGS turbulence near the surface should be treated properly in simu-lations of heavy particle dispersion, as detailed in Section 3.7 below.

3.6. Impact of SGS turbulence

Although large eddies in the CBL were directly resolved by the LES modelin this study and used by the LS model to drive heavy particle dispersion,

MODELLING HEAVY PARTICLE DISPERSION

0 1 2 4 6 7 8 9 10 110

0.51

0 1 2 4 6 7 8 10 1

t / t 1

00.5

1

z / z

i

a

0 1 2 4 6 7 8 9 10 110

0.51

b

c

*

3 5

3 5

953

Figure 6. Normalised concentration and temporal evolution for particles with diameters of(a) 53, (b) 80, and (c) 104 µm. Geostrophic wind Ug =2 m s−1. Contours are the same as inFigure 2.

00.5

1

0 2 5 6 7 8 9 10 1

t / t

10

0.51

z / z

i

a

1

00.5

1b

c

*

3 4

0 2 5 6 7 8 9 10 111 3 4

0 2 5 6 7 8 9 10 111 3 4

Figure 7. The same as in Figure 6 but with geostrophic wind Ug =4 m s−1 for (a) 53, (b) 80,and (c) 104 µm.

unresolved SGS turbulence also contributes to dispersion, especially nearthe surface where small eddies dominate and the Lagrangian integral timescale is small (Wyngaard et al., 1997). For a passive tracer, the SGS partof the turbulence may be unimportant; Gopalakrishnan and Avissar (2000)obtained reasonable LS simulation results by deactivating the SGS effectwith a LES model. However, the gravity effect on the heavy particles com-plicates the dispersion process and results in a non-negligible influenceon the dispersion pattern, as shown above. For a better understanding ofheavy particle dispersion and the reliability of the LES–LS simulation used

XUHUI CAI ET AL.

0 4 8 100.0

0.2

0.4

0.6

0.8

12

a

z / z

i

t / t t / t

0

5

10

15

20 b

No

rmal

ized

co

nce

ntr

atio

n

PS G2S G4S G6S PN G2N G4N G6N

2 6 0 4 8 10 122

Figure 8. Dispersion results with and without SGS turbulence: (a) mean plume height and(b) normalised surface concentration.

in this study, further investigation into the impact of SGS turbulence wasneeded. Thus, we intentionally deactivated the SGS and reran the simula-tion cases for passive and heavy particles; results of these simulations werecompared with the original results.

Figure 8 shows the time evolution of mean plume heights and surfaceconcentrations. Results for four pairs of runs with and without SGS tur-bulence are shown (runs PS, G2S, G4S, and G6S vs. PN, G2N, G4N, andG6N; all runs were driven by LES run Q2U0). For all cases, a short disper-sion time led to a small SGS effect. The difference was trivial for passiveparticles either including or not including the SGS turbulence. However, forlonger dispersion times, e.g., when t/t∗ was larger than 2–3, the SGS effectbecame substantial for heavy particles. These results suggest that the maineffect of SGS turbulence is to increase the possibility of heavy particlesbeing suspended in the atmosphere, e.g., an increase in mean plume height(Figure 8a) and decrease in surface concentrations (Figure 8b). WithoutSGS turbulence, heavy particles tend to fall to the surface and accumu-late. However, interestingly, the influence of the SGS effect was most seri-ous for small to medium particles (53 and 80µm). The influence was eitherrelatively small for the heaviest particles (104 µm) or negligible for passiveparticles. In both cases, SGS turbulence may be less important, becauseeither the role of gravity or the energetic large eddies in the CBL may bedominant.

3.7. Influence of near-surface turbulence

It is well-known that turbulence near the surface cannot be resolved wellby the LES method and that SGS turbulence dominates at the surface(Mason, 1994; Kljun et al., 2002). This issue may be greater in heavy par-ticle dispersion because heavy particles tend to accumulate near the sur-face. Further, SGS parameterisation near the surface in a LES may be

MODELLING HEAVY PARTICLE DISPERSION

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8a

z / z

i

0 4 8 10 120.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8b

z / z

i

t / t*

PS G2S G4S G6S P G2 G4 G6

2 60 4 8 10 12

t / t*

2 6

Figure 9. Comparison of mean plume heights for different treatments of surface SGS tur-bulence. Solid symbols, Monin–Obukhov similarity adopted; open symbols, parameteriseddirectly from the LES results. (a) geostrophic wind Ug =0 m s−1 and (b) Ug =4 m s−1.

inadequate due to the strong inhomogeneity of the turbulence. For thesereasons, we included treatment of surface-layer Monin–Obukhov similar-ity in the LS model, even though directly adopting SGS parameters fromthe LES result would have been more straightforward. We then examinedthe influence of including or not including Monin–Obukhov similarity inthe LS simulation. Figure 9 shows the mean plume heights for differentruns with and without a Monin–Obukhov similarity modification underconditions of 0 and 4 m s−1 geostrophic wind. The dispersion results forheavier particles (80 and 104 µm in diameter) were markedly influencedby the treatment of surface SGS turbulence. But the influence was quitedifferent for zero and non-zero background winds. Unreasonable resultsare shown in Figure 9 for the runs using surface SGS turbulence parame-ters obtained directly from the LES outputs. Namely, higher mean plumeheights corresponded to zero background wind, while lower mean plumeheights corresponded to stronger background wind. By adopting Monin–Obukhov similarity, the dispersion results were reasonable, i.e., lower meanplume heights corresponded to a zero geostrophic wind and higher onescorresponded to a stronger wind. For light particles, the dispersion resultwas relatively insensitive to the parameterisation of surface SGS turbulence.For passive particles, the SGS turbulence, including that near the surfacelayer, had little influence on dispersion results, as shown in Figure 8.

4. Summary

Turbulent dispersion of heavy particles in the CBL was investigated by amethod that combined LES and LS simulation. Large-scale motions orturbulent eddies in the CBL were explicitly resolved by the LES modeland used as input fields to drive particle motion in the LS model. The

XUHUI CAI ET AL.

unresolved SGS turbulence was treated as stochastic and its influence wassuperposed on the particle motion. Various factors influencing the disper-sion results were investigated, including inertia and gravity effects on heavyparticles, background wind, and SGS turbulence.

For reference, the dispersion of passive particles in the CBL was firstsimulated by the LES–LS method; these simulation results agreed well withthose of previous laboratory and numerical works (Willis and Deardorff,1974; Lamb, 1982; Gopalakrishnan and Avissar, 2000). The inertia effecton heavy particles was investigated explicitly using a simplified case andrevealed that inertia had only a trivial influence on dispersion results. Thus,only the gravity effect was considered in the remaining simulation runs bysuperposing a settling velocity onto heavy particles.

Heavy particles released near the surface experienced a lift-off process inthe initial stage, though the process was much weaker for very heavy par-ticles (104-µm diameter). Afterwards, equilibrium or a quasi-steady stateof dispersion was achieved. In contrast to a well-mixed vertical distribu-tion of passive particle concentrations in the CBL, heavy particle dispersionshowed obvious gravitational stratification. Heavier particles had larger ver-tical gradients of concentration in the lower part of the CBL, and verticaldispersion was more seriously depressed. The most notable phenomenonmay have been the high accumulation and concentration of heavy particlesnear the surface, corresponding to fewer particles suspended at higher CBLlevels. As the ratio of particle settling velocity to convective scaling velocity(wg/w∗) approached ≈0.3, about 10–15% of particles remained suspendedin the interior of the CBL in contrast to results for passive particles.

Background winds in the CBL may increase the possibility of suspendedheavy particles moving from the near-surface regime to higher levels in theCBL. This agrees with the intuitive assumption that strong winds releasemore particles from the surface. However, SGS turbulence also played animportant role in dispersing heavy particles, although its influence was triv-ial for passive particles. Numerical experiments that “turned off” the effectof SGS turbulence provided entirely different dispersion patterns for heavyparticles. Near-surface SGS turbulence may have contributed more in thisrespect, and the inadequate treatment of near-surface SGS turbulence maylead to unreasonable dispersion results. Coupling Monin–Obukhov simi-larity to the near-surface SGS turbulence estimation seemed to provideimproved and reasonable LS dispersion results for heavy particles.

Acknowledgements

We are highly grateful to the anonymous referee whose critical remarkswere very helpful in improving the initial manuscript. This research was

MODELLING HEAVY PARTICLE DISPERSION

supported by the Chinese National Science Foundation (nos. 40275005 and40233030) and by the National Basic Research and Development Program(2002CB410802).

Appendix A. Matching Surface-layer Turbulence to Upper LayerTurbulence

This LES–LS study used surface-layer Monin–Obukhov similarity for theturbulence dissipation rate and turbulence velocity variances in the near-surface regime. Based on the work of Pasquill and Smith (1983), the fol-lowing relations were used for unstable cases in this study:

ε = u3∗

κz

(φm − z

L

), (A1)

σu =σv =u∗

(12+0.5

zi

|L|)1/3

, (A2)

σw =1.2u∗

[1+ z

|L| +0.8(

z

|L|)2

]1/6

, (A3)

where φm is the nondimensional wind gradient, u∗ is the friction velocity,L is the Obukhov length, and zi is the CBL height. The LES model pro-vided inputs for these variables. Then, σu, σv, σw, and ε were applied toEquation (7) to drive the motion of stochastic particles near the surface.

Above the first grid layer, z1, the SGS kinetic energy e2 and turbu-lence dissipation rate were given by the LES model, with z1 =25 m in thisstudy and the assumption that σ 2

u = σ 2v = σ 2

w = 23e2. Thus, it was necessary

to match the upper and lower layers to prevent a possible discontinuity inthe turbulence level across this height. The following method was used tomatch the results below height z1 from Equations (A1) to (A3) to thosefrom the LES model,

ε = (1−α)εMO +αεLES, (A4)

σ 2i = (1−α2)σ 2

iMO+α2σ 2

iLES, (A5)

where α = z/z1, and σi represents σu, σv and σw. Subscript MO indi-cates the variable evaluated by the Monin–Obukhov relationship Equations(A1)–(A3); subscript LES indicates the variable determined by the LESmodel at height z = z1. The matching scheme was applied point by pointto each grid at the first LES model level.

XUHUI CAI ET AL.

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