Particle simulations of dispersion using observed

meandering and turbulence

Dean Vickers� and L. Mahrt

College of Oceanic and Atmospheric Sciences, COAS Admin Bldg 104

Oregon State University, Corvallis, OR, U.S.A.

Danijel Belusic

Department of Geophysics, Faculty of Science

University of Zagreb, Zagreb, Croatia

�Corresponding author address: Dean Vickers, College of Oceanic and Atmospheric Sciences, COAS Admin

Bldg 104, Oregon State University, Corvallis, OR, U.S.A.

E-mail: vickers@coas.oregonstate.edu

Abstract

A Lagrangian stochastic particle model driven by observed winds from a net-

work of 13 sonic anemometers is used to simulate the transport of contaminates due

to meandering of the mean wind vector and diffusion by turbulence. The turbulence

and the meandering motions are extracted from the observed velocity variances us-

ing a variable averaging window width. Such partitioning enables determination of

the separate contributions from turbulence and meanderingto the total dispersion.

The turbulence is described by a Markov Chain Monte Carlo process based on the

Langevin equation using the observed turbulence variances.

The meandering motions, not the turbulence, are primarily responsible for the

1-h averaged horizontal dispersion as measured by the travel time dependence of the

particle position variances. As a result, the 1-h averaged horizontal concentration

patterns are often characterized by streaks and multi-modal distributions. Time

series of concentration at a fixed location are highly nonstationary even when the

1-h averaged spatial distribution is close to Gaussian. Theresults show that the

contribution of meandering to the time-averaged horizontal dispersion is dominant

under all atmospheric conditions: weak and strong winds, and unstable and stable

stratification.

Keywords: dispersion, meandering, particle model, weak winds

1

1. Introduction

Despite much attention over the last several decades, understanding of dispersion processes

remains limited, especially in weak wind conditions. It haslong been known that lateral dis-

persion appears to be enhanced in weak wind conditions relative to predictions based only on

the dispersive capabilities of turbulent mixing (e.g., Hanna 1983; Hanna, 1990). These difficult

to model conditions have been termed “light and variable winds,” and more recently, “mean-

dering of the mean wind vector” (e.g., Mylne, 1992; Etling, 1990). Standard Gaussian plume

models are inadequate for such conditions. Numerous authors have suggested a wide variety

of approaches to explicitly account for meandering, although a consensus has not been reached

(e.g., Gifford, 1960; Sagendorf and Dickson, 1974; Kristensen et al. 1981; Hanna, 1983; Oettl

et al., 2001; Anfossi et al., 2006).

Anfossi et al. (2005) and Oettl et al. (2005) proposed that inlight wind conditions, me-

andering motions are always present, regardless of stability. Vickers and Mahrt (2007) found

that on scales larger than turbulence and less than a few hours (mesoscale), variations in the

cross-wind velocity variance at a given site were not related to the local mean flow or the tur-

bulence. Their analysis was based on nine different tower datasets including grassland, brush

rangeland, snow covered plain, the ocean, three different pine forests, an aspen forest and an

urban site. These three studies agree that meandering motions always seem to be present, yet

appear to be unpredictable based on local variables. The meandering motions may originate

from a variety of physical mechanisms and may have been generated elsewhere and propagated

to the measurement site. Except for a few well-defined case studies, the source of meandering

motion is generally not well known or predictable (e.g., Mahrt et al., 2001a).

As a result of inadequacies associated with plume and puff models, Lagrangian particle

2

models became widely used (e.g., Brusasca et al., 1992). These studies are often limited due

to a lack of spatial information and may rely on a single anemometer with the assumption

of horizontal homogeneity in the wind field. More recently, dispersion in the weakly stable

boundary layer has been studied using Lagrangian particle models driven by wind fields from

Large Eddy Simulation (LES) models (e.g., Brown et al., 1994; Kosovic and Curry, 2000; Weil

et al., 2004). However, LES models cannot handle the intermittent nature of strongly stable

boundary layers, and therefore their usefulness is limitedfor such cases.

Dispersion models have been coupled to regional atmospheric models, such as RAMS and

MM5. A problem with this approach is that while observational studies have demonstrated that

meandering motions are almost always present, regional atmospheric models under-represent

mesoscale motions in the stable boundary layer (Zagar et al., 2006). For example, thermally

driven flows in the model are often too strong and too steady, while observations indicate that

the flow is nonstationary (Doran and Horst, 1981), and periodically disturbed by downward

transport of momentum associated with turbulence bursts (Mahrt et al., 2001b; Soler et al.,

2002).

Tracer experiments are limited because they reveal only thedistance dependence of the

plume spread by measuring concentrations at a fixed set of receptors, while numerous studies

have shown that plume spread is best described by travel time, not distance from the source. The

influence of meandering implied from plume spread in tracer experiments is strongly dependent

on the mean advective wind speed. In stronger winds, the plume has less time to disperse

due to turbulence or change direction due to meandering before arriving at the fixed receptors.

This forced dependence on the mean wind speed has contributed to the notion that meandering

motions are only important for dispersion in very weak winds.

3

In this study we use fast response wind observations from a network of 3-D sonic anemome-

ters to drive a simple particle model where the meandering and turbulence vary in space (hor-

izontal and vertical) and time. The meandering motions are explicitly resolved and the turbu-

lence is modeled. We are unaware of any previous study using the observed flow field from

a network of towers to drive 3-D particle simulations. The dataset is described in Section 2.

Section 3 documents the method used for partitioning the observed velocity variances into tur-

bulence and meandering components. Section 4 describes theparticle model. The results are

presented in Section 5 and the conclusions are summarized inSection 6.

2. Dataset

The dataset used to drive the simulations is from the Cooperative Atmosphere-Surface Ex-

change Study - 1999 (CASES-99) grassland site in rural Kansas, USA, during October (Poulos

et al., 2002; Fritz et al., 2003; Sun et al., 2002). The measurements resolve the turbulence and

mean flow at seven towers located within a 600-m diameter circle (Figure 1). The six satel-

lite stations have sonic anemometer measurements at 5-m above ground only, while the central

tower, located at the origin in Figure 1, has seven vertical levels of sonic anemometers at 1, 5,

10, 20, 30, 40 and 50 m.

The temporal resolution of the dataset (20 hz sampling) far exceeds the spatial resolution.

There is an unavoidable mis-match in the scales of temporal and spatial velocity variability that

can be measured practically in the field. Spatial variability on scales less than the tower spacing

is not resolved. Because the towers are unevenly spaced and only the central tower contains

vertical information, the resolved scales of spatial variability vary with position in the network.

4

Despite these problems, CASES-99 is the best network we are aware of for applying the particle

model.

The dataset consists of 199 1-h data records where the wind measurements passed quality

control procedures for all 13 sonic anemometers. 135 of these records (68%) are during stable

conditions as determined by the sign of the heat flux at the 5-mlevel on the central tower. The

1-h average wind speed at 5 m on the central tower for all records is 3.2 m s�1, and ranges from

0.4 to 9.1 m s�1. There are 61 weak wind records where the network-average (seven towers)

5-m wind speed is less than 2 m s�1. The wind speeds are computed as vector averages, and as

such are smaller than the 1-h average of the instantaneous wind speed.

The CASES-99 dataset could be characterized as one with weakmeandering motions com-

pared to other locations. Vickers and Mahrt (2007) comparedthe mesoscale cross-wind velocity

variance for nine different tower datasets and found that CASES-99 ranked eighth out of nine.

The only site with weaker meandering motions than CASES-99 was a snow covered, treeless

flat coastal plain near Barrow, Alaska. They found that in general, larger mesoscale velocity

variance was associated with complex terrain sites, exceptin thermally driven flow situations

(e.g., nocturnal drainage flows). Sites in flat terrain tended to have weaker meandering.

3. Velocity decomposition

Temporal variations in the velocity components are partitioned into mean motions (mesoscale

or meandering motions) and turbulent motions using a variable averaging width based on the

gap region in the multiresolution decomposition (cospectra) of the heat flux (Vickers and Mahrt,

2003; 2006; Acevedo et al., 2006; Acevedo et al., 2007; van den Kroonenberg and Bange,

5

2007). Motions on timescales smaller than the gap scale haveproperties associated with turbu-

lence while motions larger than the gap scale do not. Motionsat averaging time scales greater

than the gap scale are primarily 2-dimensional, unlike turbulence. The fluxes of heat and mo-

mentum at time scales greater than the gap scale are often erratic, a strong function of averaging

time and can be of either sign, and unlike the fluxes on turbulence scales, have no clear relation-

ship to the local shear or stratification (Vickers and Mahrt,2003). The timescale associated with

the gap region has previously been shown to be a strong function of bulk stability (Figure 2).

Here, we calculate the gap scale seperately for each 1-h record using an automated algorithm

that examines the heat flux cospectra (Vickers and Mahrt, 2005).

The velocity component variances are partitioned as�2uR = �2uT + �2uM (1)

where subscript u denotes the u component of the wind for example, R denotes the total variance

for the record (1 hour), T denotes the part due to turbulence and M the part due to mesoscale

motions (meandering). Motions on scales greater than the record length are excluded from

consideration here. It is important to partition the velocity variances because turbulence and

meandering motions are generated by different physics and they have different influences on the

plume. Turbulence is dispersive and dilutes a plume in a Lagrangian sense, while meandering

primarily advects the plume when the scale of such motion exceeds the plume width. The

meandering motions are “dispersive” in a Eulerian sense in that they reduce the time-averaged

concentration of a tracer at a point in space. The turbulencestrength is related to the vertical

wind shear and the vertical temperature stratification through Monin-Obukov similarity theory,

while the mesoscale motions are not (Smedman, 1988).

6

The partitioning of the flow is imperfect because sometimes turbulence and mesoscale mo-

tions overlap in scale, in which case identifying a gap region becomes problematic and the

automated algorithm fails. Such cases are rare for the present data. Use of the variable aver-

aging width classifies large boundary layer scale eddies in convective conditions as part of the

mean flow when they do not directly contribute to the verticalheat flux near the surface. Such

classification may tend to make our turbulent mixing in unstable conditions weaker than that re-

ported in previous studies which may have included boundary-layer scale eddies as turbulence.

When such large-scale eddies do contribute to the flux they are included as turbulence.

The velocity component variances can be computed for any scale by summing the orthog-

onal multiresolution modes to obtain the variance, similarto integrating the spectra in Fourier

analysis. The method can also be posed in terms of simple Reynold’s decomposition. For

example, the turbulent component of the velocity variance can be computed as

u0 = u� u (2)�2uT = [u0u0℄ (3)

where the overbar denotes a local averaging time equal to thegap scale (e.g.,�60 s for stable

conditions or�600 s for unstable conditions) and the square brackets denote averaging over

the record length. The timescale associated with the overbar defines the upper limit of scales

of motion included in the variance while the averaging associated with the brackets reduces

random sampling error. Using the same notation, the record-scale velocity variance is given by

u0 = u� [u℄ (4)�2uR = [u0u0℄: (5)

7

The mesoscale component of the velocity variance can then becomputed as a residual using Eq.

(1) as long as the window width associated with the local averaging time (overbar in Eq. 2) is

an integer factor of the record length. The mesoscale component could also be computed using

a bandpass filter where scales between the gap scale and 1 hourare retained. The strength of

the meandering motions will be characterized in terms of thetwo horizontal velocity variances

as �uvM = (�2uM + �2vM )1=2: (6)

4. Particle model

a. Spatial interpolation of the mean flow

The time-dependent mean wind components are spatially interpolated onto a 3-D rectangu-

lar grid of 10-m resolution in x and y and 1-m resolution in z every time step. All particles

within a grid box are advected using the single set of mean wind components for that grid box.

The advantage of the grid approach is that a larger number of particles can be simulated for the

same computer time. The approach assumes that spatial variation of the mean wind within a

grid box is negligible. The chosen grid size is much less thanthe tower spacing, and therefore

the results using the grid approach are not noticeably different than results found by spatially

interpolating the mean wind to each individual particle location.

The mean wind at a grid point is calculated at every model timestep as�(x; y; z) = Pwk�kPwk (7)

8

where� represents u, v or w and the sum is over all 13 measurement locations and 12 virtual

sites (see below) and the overbars here denote a centered, running-mean time average using a

window width equal to the gap scale (Section 3). The weights (wk) are assigned to be inversely

proportional to the distance between the measurement location (xk; yk; zk) and the grid point

(x; y; z) as

wk = 3� �3 + � (8)� = (xk � x)2X2 + (yk � y)2Y 2 + (zk � z)2Z2 : (9)

For the CASES-99 network,X andY are specified to increase with the radial distance (R)

from the central tower to account for the greater horizontalresolution of the measurements at

smallR (Figure 1), andZ is chosen to increase with height to capture the stronger vertical

gradients in the mean wind near the surface. We selectX = Y = max [110; R+ 25℄ where all

quantities are in meters, andZ = 2.5 for z< 6 andZ = 6 for z> 6. For those observations

wherej xk � x j> X or j yk � y j> Y or j zk � z j> Z, the weight is set to zero.X, Y andZ define a maximum radius of influence around each measurement.Our conclusions are not

sensitive to the precise specification ofX, Y andZ.

A set of “virtual” mean wind measurements at z = 1 and 3 m are constructed at the six

satellite stations where the horizontal wind components assume a log-linear profile given by

u(z) = au (ln(z=zo)� m) (10)v(z) = av (ln(z=zo)� m) (11)

following Monin-Obukov similarity theory for the layer between the surface and the 5-m mea-

surement level. The coefficientsau andav at each satellite station are computed based on the

9

5-m wind measurement, a fixed roughness length of 3 cm and the stability function m. The

stability function is specified to be a function of z=L following Paulson (1970) for unstable

conditions and Beljaars and Holtslag (1991) for stable conditions where L is the Obukov length

scale calculated using the turbulence fluxes at the 5 m level.The virtualw(z) values are defined

by specifying a linear profile of mean vertical motion between 5 m and the surface, wherew is

assumed to vanish. One could use Eqs. (7-9) instead of the above method, however, use of the

virtual sites is thought to better represent the horizontalvariability of the mean flow near the

surface.

b. Turbulence

The turbulence is described by a Markov Chain Monte Carlo process with one step memory

as �0(t+ Æt) = ���0(t) + ��(t)(1� �2�)1=2 (t) (12)

where� represents u, v or w and primes denote fluctuations due to turbulence.�� is the La-

grangian autocorrelation parameter for the velocity components and is from a random se-

quence with zero mean and unit variance (Wang and Stock, 1992; Avila and Raza, 2005). The

stochastic approach is based on the Langevin equation. In the model, there is a separate Eq.

(12) for each individual particle.

The observed standard deviations of the velocity components (��(t) in Eq. 12) are computed

using the same centered, running windows described above for the mean flow. Motions on

scales larger than the gap scale are classified as mean flow, and thus do not contribute to�� in

Eq. (12). The time-dependent standard deviations of the velocity components at the particle

10

location are determined every time step using the spatial interpolation method described above

(Eqs. 7-9). The stochastic part of the turbulence (2nd term on the right hand side of Eq. 12) is

allowed to vary in both space and time.

The problem is not closed without specifying the autocorrelation parameter�, which is

a function of the Lagrangian integral timescale and the model time step (e.g., Zannetti, 1990).

The Lagrangian integral timescale is not known and can not becalculated from the observations.

Following Venkatram et al. (1984) and others, we formulate the Lagrangian integral timescales

for the three velocity components asTL� = lm�� = �z�m[��℄ (13)

where lm is the turbulence length scale from classical mixing lengththeory (e.g., Tennekes

and Lumley, 1972), z is particle height above ground,�m is the nondimensional wind shear of

Monin-Obukov similarity theory and� = 0.4 is von Karman’s constant. For simplicity, we use

one value of TL� for the entire record such that�m is computed using the 1-h average stability

and[��℄ is the 1-h average of the velocity standard deviation at the central tower.

The nondimensional wind shear is formulated as a function ofthe stability parameter z=Las

�m(z=L) = (1� 16(z=L))�1=4 ; z=L < 0 (14)�m(z=L) = 1 + zL [a+ be�d zL (1 + � d zL)℄ ; z=L > 0 (15)

where L is the Obukov length scale,a = 1, b = 0:667, = 5, andd = 0:35 following

the standard Businger-Dyer form for unstable conditions (z=L < 0) and that of Beljaars and

Holtslag (1991) for stable conditions. Here, z is a constantequal to the flux measurement

11

height (5 m).

In the current context, the Lagrangian integral timescale can be thought of as the time over

which the velocity of a particle is self-correlated. For CASES-99, typical values of the La-

grangian timescale formulation (Eq. 13) are between 2 and 5 sfor the three wind components

for neutral and stable conditions, with larger values (5 to 20 s) for unstable conditions (Figure

3). The longer timescale for the most unstable conditions isconsistent with larger turbulent

eddies associated with surface heating. The formulated TL� is found to be relatively constant

for a wide range of stabilities because�m and�� are inversely correlated (Figure 3).

A condition for Lagrangian particle modeling is that the time stepÆt be less than the La-

grangian integral timescale TL�. The approach taken here is to specify a fixedÆt for all data

records that satisfies the conditionÆt <TL for the most restrictive case with large z=L and large��. In this approach,Æt is fixed and a variable� is calculated as�� = exp(�Æt=TL�) (16)

where again� represents u, v or w,� is the Lagrangian autocorrelation parameter and TL is the

Lagrangian integral timescale (e.g., Zannetti, 1990).

For our choice ofÆt= 0.5 s and the above formulation of TL�, a typical value of� for all

three wind components is about 0.8, except for unstable conditions where� can exceed 0.9.

Sensitivity tests with the particle model show that the 1-h average concentration fields are not

sensitive to factor of two variations in the estimate of the Lagrangian integral timescale. For

example, the percent change in� corresponding to an increase in TL from 4 to 8 s is only

about 6%, while changes in concentration patterns from the particle model are not significant

for changes in� of less than 10%.

12

c. Particles

Given the mean wind components (Section 4.1) and the turbulent wind components (Section

4.2) at the particle location, the particle position (xp; yp; zp) is updated every time step (Æt = 0.5

s) as xp(t+ Æt) = xp(t) + (u+ u0) Æt (17)

with analogous equations for the y and z directions. The overbars denote the mean wind com-

ponents, which include meandering, and the primes denote the turbulent components. The par-

ticles are assumed to behave as fluid elements and to travel with the local fluid velocity where

molecular diffusion is ignored. Prior to the simulation, the horizontal wind components may be

rotated into a coordinate system aligned with the spatiallyand temporally (1-h) averaged flow

such that x becomes the downwind direction and y the cross-wind direction.

One particle is released every time step throughout the simulation and followed until it exits

the domain. The release is a continuous point source with no initial diffusion prescribed. In

sensitivity studies, the number of particles released was increased until the diagnostics became

insensitive to a further increase. A release rate of two particles per second was found to be suf-

ficient. In the case where the record-mean flow is removed fromthe wind components (Section

4.5), particles are released at a height of 5 m above ground from the central tower site because

that level and location have the best spatial data coverage.For simulations with the record-mean

flow retained, the particle release point is adjusted in (x,y) to maximize particle lifetimes inside

the domain. For simplicity, particles “bounce” off the lower boundary at z equal zero (perfect

reflection).

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d. Measures of dispersion

The time-averaged travel time dependence of particle dispersion is quantified in terms of the

statistics �2x(t) = [ (xp(t)� [xp(t)℄)2 ℄ (18)

with analogous equations for the y and z directions, where xp(t) denotes the particle x-position

after travel time t and the brackets here denote an average over all particles released during the

record. Initially there are zero particles in the domain. The number of particles increases with

time until at some point it reaches a quasi-steady state because the number of particles leaving

the domain is approximately offset by the number of new particles released. If no particle ever

left the domain, there would be 7200 particles (particle release rate of 2 per second) in the

domain at the end of the simulation.

The brackets in Eq. (18) denote an average over all particles, and therefore an average

over several thousands of realizations of particle position for fixed travel time. The actual

number of samples of particle position (for a fixed travel time) decreases with increasing travel

time because particles can leave the model domain for longertravel times. When a significant

fraction of the particles for a given travel time have left the model domain,�x can artificially

decrease with increasing travel time because the remainingparticles become clustered near the

edge of the domain. As a consequence, travel times where the fraction of particles remaining

in the domain is less than 0.85 are excluded from analyses. The maximum potential travel time

retained for computing the statistics is fixed at 480 s for allrecords.

The time-averaged horizontal dispersion will be represented in terms of

14

�xy = (�2x + �2y)1=2 (19)

because the meaning of the along- and cross-wind directionsbecomes blurred when the record-

mean wind is removed (Section 4.5). Note that the particles are dispersed in 3 spatial dimensions

such that�xy by itself is an incomplete description of the total dispersion, however, our main

focus here is to contrast the contributions to the horizontal dispersion from turbulence and

meandering.

e. Removing the record-mean flow

A practical difficulty with the CASES-99 network is that for any significant record-mean

wind, the particles leave the domain after only very short travel times. To obtain dispersion

statistics for longer travel times, the record-mean wind isremoved. Removing the record-mean

wind clearly has a large impact on the spatial distribution of particles, however, it has little

impact on the travel-time dependence of particle dispersion as calculated with Eqs. (18-19)

(Figure 4).

Theoretically, removing the record-mean wind should have no influence on the particle dis-

persion, however, in practice the standard deviation of theparticle horizontal positions tends to

be slightly larger after removing the mean wind because the particles see more spatial variabil-

ity near the center of the domain compared to near the edge of the domain. The differences are

apparently an artifact of the tower geometry. Such differences are small enough that they do not

influence our main conclusions.

The record-mean flow is removed as follows. Prior to the simulation, the network-average,

record-average 5-m u and v wind components are computed and subtracted from the time series

15

at each of the seven 5-m measurement sites. With this approach, the remaining record-average

5-m horizontal wind is small but nonzero for each individual5-m station. For the other levels

on the central tower and the virtual sites, the local record-average u and v wind components are

removed such that the record-mean wind is identically zero.The record-mean vertical motion

is always removed for each site individually.

f. Concentration patterns

Statistics are computed by overlaying a counting grid over the domain. The resolution of

the counting grid is chosen to be the same as the one used for the spatial interpolation described

above. The time-averaged concentrationCn (particles m�3) in a grid box is computed as the

average number of particles in the grid box over the entire simulation divided by the grid box

volume. For the simulations with the record-mean flow retained, the cross-wind integrated

(CWIC) and vertically integrated (VTIC) concentrations are computed as

CWIC(x; z) = +1Xy=�1Cn�y (20)V TIC(x; y) = +1Xz=0Cn�z: (21)

5. Results

a. Turbulent diffusion

Plume spread from the parameterized turbulence in the particle model is compared to the

classical Pasquill-Gifford diffusion curves in Figure 5 (e.g., Stern et al., 1984). The diffusion

16

parameters�y and�z from our particle model were obtained by running the model with all

meandering motions turned off. The main difference is foundfor the unstable case, where the

rate of increase in plume spread with downwind distance slows down more quickly in the model

than in the Pasquill-Gifford curves. The agreement is closer for the stable case. Comparisons

between the formulation of turbulent diffusion in the particle model and the Pasquill-Gifford

curves are confounded due to the widely different approaches.

The decrease in the rate of plume spread in the particle modelfor the unstable case is more

similar to the widely used interpolation formula for small and large diffsion times based on

Taylor (1921),

�y = �v t(1 + 0:5 t=TLy)1=2 (22)�z = �w t(1 + 0:5 t=TLy)1=2 (23)

(e.g., Venkatram et al., 1984) than to the Pasquill-Giffordcurves. Plugging in the formulation

of TL (Eq. 13) with the observed stability and velocity variance into Eqs. (22-23) predicts a

time-dependent plume spread for�y and�z that is similar to that found using the particle model

turbulence formulation. We conclude that the formulation of turbulent diffusion in the particle

model agrees reasonably well with standard formulations.

b. Spatial and temporal distributions

In this section, results are discussed for four weak-wind case studies (Table 1). These

records were selected because they have weak mean flow, and therefore, the record-mean wind

can be retained such that the spatial distribution of particles can be examined. In stronger mean

flow, the particles leave the model domain too quickly. For contrast, we selected two weak-wind

17

stable (nocturnal) cases and two weak-wind unstable (daytime) cases.

The 1-h averaged vertically integrated spatial concentration patterns (Figures 6-7) do not

agree with the traditional concepts of a plume due to streaksof high concentration generated

by meandering. These spatial streaks survive 1-h time averaging despite attempts by the tur-

bulence to smooth the concentration pattern. It is not uncommon to find an area of minimum

concentration in the record-mean downwind direction from the source, on what would be the

plume centerline, and areas of maximum time-averaged concentration located on the edges of

the plume. This double-maximum structure is associated with a bimodal probability distribution

function of the wind direction. A trimodal structure is apparent for record 29116 in Figure 7.

The wind direction often jumps between preferred modes rather than oscillates back and forth.

This behavior occurred in a number of datasets examined by Mahrt (2007).

Even when the distribution ofCn(x,y) appears close to Gaussian, as for record 29814 in

Figure 7, time series of the short-term averaged concentration are often highly non-stationary

(Figure 8), in contrast to conceptual plume models. High short-lived concentrations are often

followed or preceded by longer periods of very low concentration, and the 1-h average concen-

tration at a point is typically dominated by a few short-termevents. The meandering motions

determine when and where the events of high concentration occur. Strong nonstationarity of the

concentration on very short timescales (seconds) was foundin the simulations of Farrell et al.

(2002).

After averaging the patterns in Figure 8 over all 61 weak windrecords, the maximum con-

centrations are found for an azimuthal angle of zero and the angular distribution is approxi-

mately Gaussian. This is because there is no preferred direction or time-history of the meander-

ing motions, as suggested by Mahrt (2007) for a variety of datasets.

18

c. Horizontal dispersion

The main physical process contributing to particle horizontal dispersion is meandering (Fig-

ure 9). The horizontal dispersion is not strongly related tothe turbulence strength (Figure 9a)

because meandering motions dominate horizontal dispersion and the meandering motions are

not correlated with the turbulence strength. When meandering is turned off in the model, the

horizontal dispersion is strongly related to the turbulence mixing strength. This result appears

to apply for all wind speeds and stabilities.

The importance of the meandering is demonstrated by the small scatter and strong rela-

tionship between the “true” particle horizontal dispersion, as determined by the Lagrangian

particle simulations (�xy), and the meandering velocity scale (�uvM ) (Figure 9b). In contrast,

there is no clear relationship between particle dispersionand the turbulence-scale fluctuations

(Figure 9a). Apparently, the turbulence plays a minor role for horizontal dispersion compared

to the meandering. This result suggests that efforts to improve dispersion models should be

directed towards a better understanding of the source of meandering motions. The simulations

in Figure 9 were performed with the record-mean flow removed,and as such all the records,

even the strong wind speed cases, are represented. The estimates of dispersion in Figure 9 are

vertically integrated quantities. The approximately linear relationship (Figure 9b) between the

meandering velocity scale and the horizontal dispersion issummarized in Table 2. Although

the meandering velocity scale predicts the horizontal dispersion from the Lagrangian particle

model quite well, it is not available in models.

The interpretation that meandering dominates the total horizontal dispersion depends on the

decomposition of the velocity fluctuations into turbulenceand larger-scale motions (Section 3).

We suspect that many previous studies included meandering velocity fluctuations in their esti-

19

mate of turbulence. Because the gap scale separating higherfrequency turbulence from lower

frequency meandering typically ranges from tens of secondsto a few minutes in stable condi-

tions, use of a conventional averaging time of say 10 to 30 minutes to compute the turbulent

standard deviations of the wind components would inadvertently include significant meander-

ing, and therefore underestimate the influence due to meandering. The distinction is important

because the turbulence is dispersive in a Lagrangian sense while the meandering motions are

not.

Figure 9 and Table 2 suggest that horizontal dispersion is predictable in terms of Eulerian

measurements of�uvM , although the strong relationship may be specific to this dataset, where

spatial variability in the wind field is not large partly because of the limited spatial domain of

the tower network. Changes in wind direction at one of the towers are often accompanied by

a similar changes at other towers. The above approach is not closed because the meandering

velocity scale is not available in atmospheric models, and it does not appear to be predictable in

terms of other model variables (Vickers and Mahrt, 2007). For the CASES-99 data we find no

clear relationship between the meandering velocity scale and the mean wind speed or stability

(not shown).

d. Relative importance of meandering

The fraction of the total horizontal dispersion attributable to meandering can be estimated

as FM = �xyM�xyR (24)

20

where the numerator is computed using the particle model with the turbulence turned off, and

the denominator is computed using the full model with turbulence. Interpretation of FM is not

straightforward because �2xyR 6= �2xyT + �2xyM : (25)

In practice, it is possible to find unphysical values of FM greater than unity because in the sim-

ulations without turbulence the particles tend to stay at the release height of 5 m, where the

potential horizontal variability of the meandering motions is highest because of the locations of

the anemometers. To eliminate this influence on FM , the simulations with and without turbu-

lence, used to evaluate Eq. (24) and Figure 10, are performedin 2-D mode where w’ andw are

set to zero.

The main point demonstrated by FM (Figure 10) is that the meandering motions primarily

determine the horizontal dispersion, not the turbulence. The surprisingly large values of FMfor unstable conditions may be partly due to an underestimation of the dispersive capability of

the turbulence in the particle model, at least compared to the Pasquill-Gifford curves (Section

5.1). For very short travel times in near-neutral and stronger wind speed conditions, FM is a

minimum of about 50%, while FM is a maximum for stable conditions with weaker turbulence.

For near-neutral and unstable conditions, FM increases with increasing travel time for travel

times less than a few minutes, indicating that the relative contribution to the dispersion from

turbulent mixing is a maximum near the source, where the particles are closest together. The

meandering becomes relatively more effective at dispersion for longer travel times. A weak

wind speed dependence of FM is found for the stable cases (123 stable records with z=L > 0.1

in Figure 10), where FM is reduced by about 10% for 5-m wind speeds above 2 m s�1 compared

21

to wind speeds below 2 m s�1 (not shown).

6. Conclusions and future directions

A simple particle trajectory model was developed to study meandering motions using the

observed temporal and spatial variability of the wind field in CASES-99. Meandering was

defined to include all motions on timescales between the maximum turbulent timescale, deter-

mined individually for each data record based on the heat fluxcospectra, and a fixed 1-h record

length.

The meandering motions, not the turbulence, are primarily responsible for the horizontal

dispersion. As a consequence, spatial streaks and bimodal (or trimodal) 1-h averaged spatial

distributions were commonly observed. Double maximum patterns with higher time-averaged

concentrations on the edges of the plume were not uncommon. The meandering determines

the locations of the streaks. Time series of the 30-s averaged concentration were highly non-

stationary, even when the 1-h averaged spatial distribution was close to Gaussian. In contrast

to traditional thinking of a plume, the time-averaged concentrations were dominated by a few

short-lived very high concentration events followed or preceded by much longer periods of zero

or very low concentration. Meandering type motions appear to dominate the travel-time depen-

dence of the horizontal dispersion in all atmospheric conditions, including weak winds, strong

winds, and unstable and stable conditions.

The origin and dynamics of the meandering motions is still unknown. Well defined mono-

tonic gravity waves and solitons have been successfully examined from observations, however,

the majority of the records in our datasets represent a more complex superposition of different

22

modes. Case studies are required to determine if the physicsof such motions can be uncov-

ered. For practical modeling, the impact of such mesoscale modes need to be described statisti-

cally, perhaps in terms of probability distributions. While the characteristics and strength of the

mesoscale modes do not seem to be well related to local variables, it may be possible to frame

the probability distributions in terms of gross characteristics of the setting, such as the presence

of complex terrain, the role of drainage flows or the bulk vertical structure of the flow.

Acknowledgments.

DV and LM would like to acknowledge Contract W9FN05C0067 from the Army Research

Office and Grant 0607842-ATM of the Physical and Dynamical Meteorology Section of the

National Science Foundation. The work of DB was partially supported by the Croatian Ministry

of Science, Education and Sports (project 119-1193086-1323).

23

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List of Figures

28

FIG. 1. The seven tower locations (squares) in CASES-99 and the xy-domain used in the

simulations.

29

FIG. 2. Representation of the average stability dependence of the gap timescale separating tur-

bulence and larger scale motions at 5 m above ground (Vickersand Mahrt, 2003, their Eqs.

12-14). The bulk Richardson number is proportional to the potential temperature gradient and

inversely proportional to the mean wind speed squared. Positive values indicate stable stratifi-

cation.

30

FIG. 3. Contour plot of the formulation for the Lagrangian integral timescale (s) in z=L-��space for fixed z = 5 m. Dots show all the CASES-99 data records.The three panels are for

the� = u, v and w wind components. Contour lines are drawn for 0.5, 1,2, 5, 10, 20 and 50

seconds. Value of z=L > 3.9 have been set to 3.9 for display purposes.

31

FIG. 4. Standard deviation of particle horizontal position forsimulations with and without the

mean flow for travel times of 60 and 120 s. Each dot represents one of the relatively weak wind

speed 1-h records in CASES-99.

32

FIG. 5. Vertical (�z) and cross-wind (�y) plume dispersion parameters due to turbulence only

(solid curves) as a function of distance downwind for an unstable example (upper curve) and

a stable example (lower curve) compared to the Pasquill-Gifford curves (dashed) for Pasquill

stability categories A-F, where category A is the most unstable and F the most stable.

33

FIG. 6. Examples of the 1-h average vertically integrated concentration (VTIC x 103) for 4 weak

wind records (Table 1) in the xy-plane. The winds have been rotated such that the network-

average record-average mean wind at the 5-m level is from left to right. The particle release

point is (x,y,z) = (-330,0,5). Upper 2 panels are unstable, and bottom 2 panels are stable.

34

FIG. 7. Angular distribution of the 1-h average vertically integrated concentrations (VTIC x

103) for the same records as Figure 6 for arcs at distances of 100 m(solid), 200 m (dashed) and

500 m (dotted).

35

FIG. 8. Contour plot of 30-s average vertically integrated particle counts on an arc 300 m from

the source as a function of simulation time and azimuthal angle for the same records as Figure

6. The x-axis shows the time-dependence during the 1-h record, not travel time.

36

FIG. 9. 1-h averaged vertically integrated horizontal dispersion (�xy) as a function of�uv times

travel time for a) turbulence, b) meandering and c) all motions on scales less than 1 hour. Each

dot represents a discrete travel time of 60, 120, etc... 480 sfor stable (blue) and unstable (red)

periods for all 199 records.

37

FIG. 10. Fraction of horizontal dispersion associated with meandering (FM ) as a function of

travel time for stable (solid curve, 123 records), near-neutral (dashed, 34 records) and unstable

(dots, 42 records) stability classes. The near-neutral class is defined as -0.1� z=L � 0.1.

38

List of Tables

39

TABLE 1. 1-h average mean wind U (m s�1), stability parameter z=L and the turbulence velocity

standard deviations (m s�1) for the 5-m level on the central tower for the 4 records in Figures

6-9. Subscripts u, v and w refer to the along-wind, cross-wind and vertical directions.

Record U z=L �u �v �w29116 1.5 -1.2 0.55 0.50 0.20

29814 0.9 -1.4 0.70 0.80 0.30

29202 1.2 1.8 0.19 0.19 0.08

29204 0.4 1.8 0.13 0.16 0.04

40

TABLE 2. Relationship between particle horizontal dispersion and the meandering velocity

scale in terms of two linear regression forms: i)�xy = a + b�uvM t, and ii)�xy = c �uvM t, for

the data in Figure 10b. N is the number of samples and R2 is the fraction of variance explained

by the regression. Coefficient a has units of m.

stability a b c N R2 i) R2 ii)

unstable 16.6 0.75 0.89 336 0.88 0.88

stable 10.8 0.79 0.87 1189 0.90 0.90

41