Acta Geophysica
vol. 56, no. 1, pp. 234-256
DOI: 10.2478/s11600-007-0041-3
Particle simulations of dispersion
using observed meandering and turbulence
Dean VICKERS1, Larry MAHRT1, and Danijel BELUŠIĆ2
1
College of Oceanic and Atmospheric Sciences, Oregon State University, Corvallis, OR, USA
e-mail: [email protected]
2
Department of Geophysics, University of Zagreb, Zagreb, Croatia
Abstract
A Lagrangian stochastic particle model driven by observed winds from a network 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 using a variable averaging window width. Such partitioning enables determination of
the separate contributions from turbulence and meandering to 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. The results show that meandering dominates the travel-time dependence of the horizontal dispersion under all
atmospheric conditions: weak and strong winds, and unstable and stable stratification.
Key words: dispersion, meandering, particle model, weak winds.
1. INTRODUCTION
Despite much attention over the last several decades, understanding of dispersion
processes remains limited, especially in weak wind conditions. It has long been known
that lateral dispersion appears to be enhanced in weak wind conditions relative to pre© 2008 Institute of Geophysics, Polish Academy of Sciences
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dictions based only on the dispersive capabilities of turbulent mixing (e.g., Hanna
1983, 1990). These difficult to model conditions have been termed “light and variable
winds,” and more recently, “meandering 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 in light wind conditions, meandering 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 turbulence. 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 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, Kosović 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 limited for 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
(Žagar 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 the distance 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
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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.
In this study we use fast response wind observations from a network of 3-D sonic
anemometers to drive a simple particle model where the meandering and turbulence
vary in space (horizontal and vertical) and time. The meandering motions are explicitly resolved and the turbulence 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 turbulence and meandering components.
Section 4 describes the particle model. The results are presented in Section 5 and the
conclusions are summarized in Section 6.
2. DATASET
The dataset used to drive the simulations is from the Cooperative Atmosphere-Surface
Exchange 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 (Fig. 1). The six satellite stations have sonic anemometer measurements at 5-m
above ground only, while the central tower, located at the origin in Fig. 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 ve-
Fig. 1. The seven tower locations (squares) in CASES-99 and the xy-domain used in the simulations.
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locity 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. Despite these problems, CASES99 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-m level 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 weak meandering motions compared to other locations. Vickers and Mahrt (2007) compared the mesoscale
cross-wind velocity variance for nine different tower datasets and found that CASES99 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, except in 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, 2007, van den
Kroonenberg and Bange 2007). Motions on timescales smaller than the gap scale have
properties associated with turbulence while motions larger than the gap scale do not.
Motions at averaging time scales greater than the gap scale are primarily 2-dimensional, unlike turbulence. The fluxes of heat and momentum 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 relationship 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 (Fig. 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 2006).
The velocity component variances are partitioned as
2
2
2
σ uR
= σ uT
+ σ uM
,
(1)
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Fig. 2. Representation of the average stability dependence of the gap timescale separating turbulence and larger scale motions at 5 m above ground (Vickers and Mahrt 2003, their eqs. 1214). The bulk Richardson number is proportional to the potential temperature gradient and inversely proportional to the mean wind speed squared. Positive values indicate stable stratification.
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 turbulence strength is related to the vertical wind shear
and the vertical temperature stratification through Monin–Obukhov similarity theory,
while the mesoscale motions are not (Smedman 1988).
The partitioning of the flow is imperfect because sometimes turbulence and
mesoscale motions 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 averaging width classifies large boundary layer scale eddies
in convective conditions as part of the mean flow when they do not directly contribute
to the vertical heat flux near the surface. Such classification may tend to make our turbulent mixing in unstable conditions weaker than that reported in previous studies
which may have included boundary-layer scale eddies as turbulence. When such largescale eddies do contribute to the flux they are included as turbulence.
The velocity component variances can be computed for any scale by summing
the orthogonal multiresolution modes to obtain the variance, similar to integrating the
spectra in Fourier analysis. The method can also be posed in terms of simple Rey-
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nold’s decomposition. For example, the turbulent component of the velocity variance
can be computed as
u′ = u − u ,
(2)
σ uT2 = [u ′u ′] ,
(3)
where the overbar denotes a local averaging time equal to the gap 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 recordscale velocity variance is given by
u ′ = u − [u ] ,
(4)
2
= [ u′u ′] .
σ uR
(5)
The mesoscale component of the velocity variance can then be computed 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 hour are retained. The strength of the meandering motions will be characterized in terms of the two horizontal velocity variances as
2
2
σ uvM = (σ uM
+ σ vM
)
1/ 2
4.
.
(6)
PARTICLE MODEL
4.1 Spatial interpolation of the mean flow
The time-dependent mean wind components are spatially interpolated onto a 3-D rectangular 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 than the tower spacing, and therefore the results using the grid approach are
not noticeably different from 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 time step as
ϕ ( x, y, z ) =
Σwkϕ k
,
Σwk
(7)
where φ 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
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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
3−ς
,
3+ς
(8)
( xk − x ) 2 ( y k − y ) 2 ( zk − z ) 2
+
+
.
X2
Y2
Z2
(9)
wk =
ς=
For the CASES-99 network, X and Y are specified to increase with the radial distance R from the central tower to account for the greater horizontal resolution of the
measurements at small R (Fig. 1), and Z is chosen to increase with height to capture
the stronger vertical gradients in the mean wind near the surface. We select
X = Y = max [110, R+25] where all quantities are in meters, and Z = 2.5 for z < 6
and Z = 6 for z > 6. For those observations where xk − x > X or yk − y > Y or
zk − z > Z , the weight is set to zero. X, Y and Z define a maximum radius of influence
around each measurement. Our conclusions are not sensitive to the precise specification of X, Y and Z.
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 / z0 ) −ψ m ) ,
(10)
v ( z ) = av ( ln( z / z0 ) −ψ m ) ,
(11)
following Monin–Obukhov similarity theory for the layer between the surface and the
5-m measurement level. The coefficients au and av at each satellite station are computed based on the 5-m wind measurement, a fixed roughness length of 3 cm and the
stability function ψm . 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 Obukhov length scale calculated using the turbulence fluxes
at the 5 m level. The virtual w( z ) values are defined by specifying a linear profile of
mean vertical motion between 5 m and the surface, where w 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 horizontal variability of the mean flow near the
surface.
4.2 Turbulence
The turbulence is described by a Markov Chain Monte Carlo process with one step
memory as
ϕ ′ ( t + δt ) = βϕ ϕ ′ ( t ) + σ ϕ ( t ) (1 − βϕ2 ) γ ( t ) ,
1/ 2
(12)
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where φ represents u, v or w and primes denote fluctuations due to turbulence. βφ is the
Lagrangian autocorrelation parameter for the velocity components and γ is from a random sequence 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, σφ(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 σφ in eq. (12). The time-dependent standard deviations of the velocity components at the particle location are determined for 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 β,
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 be calculated from the observations. Following Venkatram et al. (1984) and others, we formulate the Lagrangian integral timescales for the three velocity components as
TLϕ =
lm
σϕ
=
κz
,
Φm [σ ϕ ]
(13)
where lm is the turbulence length scale from classical mixing length theory (e.g., Tennekes and Lumley 1972), z is particle height above ground, Φm is the nondimensional
wind shear of Monin–Obukhov similarity theory and κ = 0.4 is von Karman’s constant. For simplicity, we use one value of TLφ for the entire record such that Φm is
computed using the 1-h average stability and [σφ] is the 1-h average of the velocity
standard deviation at the central tower.
The nondimensional wind shear is formulated as a function of the stability parameter z/L as
Φm ( z / L) = [1 − 16( z / L)]
Φm ( z / L) = 1 +
−1/ 4
,
z/L<0,
z
−d ⎛
z⎡
z ⎞⎤
L
⎢a + b e ⎜1 + c − d ⎟⎥ ,
L⎣
L ⎠⎦
⎝
(14)
z/L>0,
(15)
where L is the Obukhov length scale, a = 1, b = 0.667, c = 5, and d = 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 constant equal to the flux
measurement 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 Lagrangian timescale formulation (eq. 13) are between 2 and 5 s for the
three wind components for neutral and stable conditions, with larger values (5 to 20 s)
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D. VICKERS et al.
Fig. 3. Contour plot of the formulation for the Lagrangian integral timescale [s] in z/L-σφ space
for fixed z = 5 m. Dots show all the CASES-99 data records. The three panels are for the
φ = u, v and w wind components. Contour lines are drawn for 0.5, 1, 2, 5, 10, 20 and 50 seconds. Values of z/L > 3.9 have been set to 3.9 for display purposes.
for unstable conditions (Fig. 3). The longer timescale for the most unstable conditions
is consistent with larger turbulent eddies associated with surface heating. The formulated TLφ is found to be relatively constant for a wide range of stabilities because Φm
and σφ are inversely correlated (Fig. 3).
A condition for Lagrangian particle modeling is that the time step δt be less than
the Lagrangian integral timescale TLφ . The approach taken here is to specify a fixed δt
for all data records that satisfies the condition δt < TL for the most restrictive case
with large z/L and large σφ . In this approach, δt is fixed and a variable β is calculated
as
βϕ = exp ( −δt / TLϕ ) ,
(16)
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where again φ represents u, v or w, β is the Lagrangian autocorrelation parameter and
TL is the Lagrangian integral timescale (e.g., Zannetti 1990).
For our choice of δt = 0.5 s and the above formulation of TLφ , a typical value of
β for all three wind components is about 0.8, except for unstable conditions where β
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 β 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 β of less than 10%.
4.3 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 (δt = 0.5 s) as
x p (t + δt ) = x p (t ) + (u + u′)δt ,
(17)
with analogous equations for the y and z directions. The overbars denote the mean
wind components, which include meandering, and the primes denote the turbulent
components. The particles 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 spatially and 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 sufficient. In the case where the record-mean
flow is removed from the 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).
4.4 Measures of dispersion
The time-averaged travel time dependence of particle dispersion is quantified in terms
of the statistics
(
) ⎦⎥
2
σ x2 (t ) = ⎡ x p (t ) − ⎡⎣ x p (t ) ⎤⎦ ⎤ ,
⎣⎢
(18)
with analogous equations for the y and z directions, where xp(t) denotes the particle xposition after travel time t and the brackets here denote an average over all particles
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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 longer
travel times. When a significant fraction of the particles for a given travel time have
left the model domain, σx can artificially decrease with increasing travel time because
the remaining particles 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 all records.
The time-averaged horizontal dispersion will be represented in terms of
σ xy = (σ x2 + σ y2 )
1/ 2
,
(19)
because the meaning of the along-and cross-wind directions becomes blurred when the
record-mean wind is removed (Section 4.5). Note that the particles are dispersed in 3
spatial dimensions such that σxy 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.
4.5 Removing the record-mean flow
A practical difficulty with the CASES-99 network is that for any significant recordmean wind, the particles leave the domain after only very short travel times. To obtain
dispersion statistics for longer travel times, the record-mean wind is removed. 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) (Fig. 4).
Theoretically, removing the record-mean wind should have no influence on the
particle dispersion, however, in practice the standard deviation of the particle horizontal positions tends to be slightly larger after removing the mean wind because the particles see more spatial variability 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 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
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Fig. 4. Standard deviation of particle horizontal position for simulations 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.
each individual 5-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.
4.6 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 concentration Cn (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 concentration (CWIC) and vertically integrated concentration (VTIC) are computed as
CWIC ( x, z ) =
+∞
∑C
n
∆y ,
(20)
y =−∞
+∞
VTIC ( x, y ) = ∑ Cn ∆z .
(21)
z =0
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5.
RESULTS
5.1 Turbulent diffusion
Plume spread from the parameterized turbulence in the particle model is compared to
the classical Pasquill–Gifford diffusion curves in Fig. 5 (e.g., Stern et al. 1984). The
diffusion parameters, σy and σz , from our particle model were obtained by running the
model with all meandering motions turned off. The main difference is found for 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 model for the unstable
case is more similar to the widely used interpolation formula for small and large diffusion times based on Taylor (1921),
Fig. 5. Vertical (σz) and cross-wind (σy) 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.
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σy =
σz =
σv t
(1 + 0.5 t / TLy )1/ 2
σw t
(1 + 0.5 t / TLz )1/ 2
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,
(22)
,
(23)
(e.g., Venkatram et al. 1984) than to the Pasquill–Gifford curves. 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 σy and σz 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.
5.2 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 recordmean wind can be retained such that the spatial distribution of particles can be exam-
Fig. 6. Examples of the 1-h average vertically integrated concentration (VTIC × 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. See color version of this figure in electronic edition.
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Table 1
1-h average mean wind U (m s ), 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 Figs. 6-9. Subscripts u, v and w refer to the along-wind,
cross-wind and vertical directions.
–1
σu
σv
σw
–1.2
–1.4
0.55
0.70
0.50
0.80
0.20
0.30
1.8
1.8
0.19
0.13
0.19
0.16
0.08
0.04
Record
U
z/L
29116
29814
1.5
0.9
29202
29204
1.2
0.4
ined. In stronger mean flow, the particles leave the model domain too quickly. For
contrast, we selected two weak-wind stable (nocturnal) cases and two weak-wind unstable (daytime) cases.
The 1-h averaged vertically integrated spatial concentration patterns (Figs. 6-7)
do not agree with the traditional concepts of a plume due to streaks of high concentra-
Fig. 7. Angular distribution of the 1-h average vertically integrated concentrations (VTIC × 103)
for the same records as Fig. 6, for arcs at distances of 100 m (solid), 200 m (dashed) and 500 m
(dotted).
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tion generated by meandering. These spatial streaks survive 1-h time averaging despite
attempts by the turbulence 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 timeaveraged 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 Fig. 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 of Cn ( x, y ) appears close to Gaussian, as for record
29814 in Fig. 7, time series of the short-term averaged concentration are often highly
non-stationary (Fig. 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 concentration at a point is typically dominated by a few
short-term events. The meandering motions determine when and where the events of
high concentration occur. Strong nonstationarity of the concentration on very short
timescales (seconds) was found in the simulations of Farrell et al. (2002).
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 Fig. 6.
The x-axis shows the time-dependence during the 1-h record, not travel time. See color version
of this figure in electronic edition.
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D. VICKERS et al.
After averaging the patterns in Fig. 8 over all 61 weak wind records, the maximum concentrations are found for an azimuthal angle of zero and the angular distribution is approximately Gaussian. This is because there is no preferred direction or timehistory of the meandering motions, as suggested by Mahrt (2007) for a variety of datasets.
5.3 Horizontal dispersion
The main physical process contributing to particle horizontal dispersion is meandering
(Fig. 9). The horizontal dispersion is not strongly related to the turbulence strength
(Fig. 9a) because meandering motions dominate horizontal dispersion and the mean-
·
·
·
Fig. 9. 1-h averaged vertically integrated horizontal dispersion (σxy) as a function of σuv 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 s for all 199 records.
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dering 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 relationship between the “true” particle horizontal dispersion, as determined by
the Lagrangian particle simulations (σxy), and the meandering velocity scale (σuvM)
(Fig. 9b). In contrast, there is no clear relationship between particle dispersion and the
turbulence-scale fluctuations (Fig. 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 Fig. 9 were performed with the record-mean flow removed, and as such all the records, even the strong wind speed
cases, are represented. The approximately linear relationship (Fig. 9b) between the
meandering velocity scale and the horizontal dispersion is summarized 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.
T ab le 2
Relationship between particle horizontal dispersion and the meandering velocity scale
in terms of two linear regression forms: (i) σxy = a + b σuvM t, and (ii) σxy = c σuvM t,
for the data in Fig. 9b. 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
The interpretation that meandering dominates the total horizontal dispersion depends on the decomposition of the velocity fluctuations into turbulence and largerscale motions (Section 3). We suspect that many previous studies included meandering velocity fluctuations in their estimate of turbulence. Because the gap scale separating higher frequency turbulence from lower frequency meandering typically ranges
from tens of seconds to a few minutes in stable conditions, 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 meandering, 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 σuvM, although the strong relationship may be specific to this
dataset, where spatial variability in the wind field is not large partly because of the
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D. VICKERS et al.
limited spatial domain of the tower network. Changes in wind direction at one of the
towers are often accompanied by 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).
5.4 Relative importance of meandering
The fraction of the total horizontal dispersion attributable to meandering can be estimated as
σ
FM = xyM ,
(24)
σ xyR
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
2
2
2
σ xyR
≠ σ xyT
+ σ xyM
.
(25)
In practice, it is possible to find unphysical values of FM greater than unity because in
the simulations 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 turbulence, used to evaluate eq. (24) and Fig. 10, are performed in 2-D mode where w′ and w are set to zero.
The main point demonstrated by FM (Fig. 10) is that the meandering motions
primarily determine the horizontal dispersion, not the turbulence. The surprisingly
large values of FM for unstable conditions may be partly due to an underestimation of
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.
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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 Fig. 10), where FM is reduced by about 10%
for 5-m wind speeds above 2 m s–1 compared 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 timescale associated with turbulence, determined individually for each data record based
on the heat flux cospectra, 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 dependence 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 monotonic 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 modes. Case studies are required to determine if
the physics of such motions can be uncovered. For practical modeling, the impact of
such mesoscale modes need to be described statistically, 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.
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D. VICKERS et al.
A c k n o w l e d g m e n t s . 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).
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Received 26 June 2007
Accepted 26 September 2007
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