# constant analytical ```EDDY DIFFUSIVITY PARAMETERIZATIONS AND IMPROVE FEATURES IN AN
ANALYTICAL MULTYLAYER DISPERSION MODEL
T. Tirabassi1, F. Baffo1 and D.M. Moreira2
Abstract. The advection-diffusion equation has been widely applied in operational atmospheric dispersion
models to predict ground-level concentrations due to low and tall stacks emissions. Analytical solutions of
equations are of fundamental importance in understanding and describing physical phenomena, since they
are able to take into account all the parameters of a problem, and investigate their influence. Unfortunately,
no general solution is known for equations describing the transport and dispersion of air pollution. There are
some specific solutions based on particular assumptions concerning the turbulent state of the atmosphere, the
best-known being the so-called Gaussian solution, where wind and dispersion parameters coefficients are
constant with height.
We present an improvement of an analytical model proposed by Vilhena et al. (1998). The model is based on
a discretization of the Planetary Boundary Layer (PBL) in N sub-layers; in each sub-layer the advectiondiffusion equation is solved by the Laplace transform technique, considering an average value for eddy
diffusivity and wind speed.
Resumo. A equa&ccedil;&atilde;o de difus&atilde;o-advec&ccedil;&atilde;o tem sido amplamente aplicada em modelos de dispers&atilde;o
atmosf&eacute;rica para predizer concentra&ccedil;&otilde;es ao n&iacute;vel do solo devido &agrave;s emiss&otilde;es de fontes altas e baixas.
Solu&ccedil;&otilde;es anal&iacute;ticas destas equa&ccedil;&otilde;es s&atilde;o de fundamental import&acirc;ncia no entendimento e descri&ccedil;&atilde;o de
fen&ocirc;menos f&iacute;sicos, desde que elas s&atilde;o capazes de levar em conta todos os par&acirc;metros de um problema e
investigar sua influ&ecirc;ncia. Infelizmente, nenhuma solu&ccedil;&atilde;o geral &eacute; conhecida para as equa&ccedil;&otilde;es que descrevem
o transporte e dispers&atilde;o de poluentes atmosf&eacute;ricos. Existem algumas solu&ccedil;&otilde;es espec&iacute;ficas baseadas em
hip&oacute;teses particulares devido ao estado da atmosfera, bem como a conhecida solu&ccedil;&atilde;o Gaussiana, onde o
vento e os par&acirc;metros de dispers&atilde;o s&atilde;o constantes com a altura.
Apresentamos neste trabalho uma melhoria no modelo anal&iacute;tico proposto por Vilhena et al. (1998). O
um valor m&eacute;dio para o coeficiente de difus&atilde;o e velocidade do vento.
Key words: air pollution modelling, solutions of advection-diffusion equation, eddy coefficients.
______________________________________________________
1 – Institute ISAC of CNR, Via Gobetti 101, 40129 Bologna, Italy
Phone: +39 051 6399601 Fax: +39 051 6399658 E.mail: t.tirabassi@ isac.cnr.it
2 – ULBRA, Av. Miguel Tostes, 101, Canoas, Brazil
Phone: 51 477-9285 Fax: 51 477-1313 Email: davidson@ulbra.tche.br
INTRODUCTION
The Eulerian approach for modelling the statistical properties of concentrations of contaminants
in a turbulent flow like the Planetary Boundary Layer (PBL) is widely used in the field of air
pollution studies.
One of the most commonly used dispersion equation closures is based on the gradient transport
hypothesis which, in analogy to molecular diffusion, assumes that turbulence causes a net
movement of material down the gradient of material concentration, at a rate that is proportional to
the magnitude of the gradient (Pasquill and Smith, 1983).
The simplicity of the approach (the so-called K-theory of turbulent diffusion) has led to the
widespread use of this theory as the mathematical basis for simulating atmospheric dispersion.
However, K-closure has its own limits. In contrast to molecular diffusion, turbulent diffusion is
scale-dependent. This means that the diffusion rate of a cloud of material generally depends on the
cloud dimensions and intensity of turbulence. As the cloud grows, larger eddies are incorporated in
the expansion process, so that a progressively larger fraction of turbulent kinetic energy is available
for the cloud expansion. However, eddies much larger than the cloud itself are relatively
unimportant in its expansion. Thus, the gradient-transfer theory works well when the dimension of
the dispersed material is much larger than the size of the turbulent eddies involved in the diffusion
process, i.e. for ground-level releases and large travel times. Strictly speaking, one should introduce
a diffusion coefficient function not only of atmospheric stability and release height, but also of
travel time or distance from source; however, such time-dependence makes it difficult to treat the
diffusion equation in a fixed-coordinate system, when multiple sources must be dealt with
simultaneously. Otherwise, one should limit the application of the gradient theory to large travel
times (Pasquill and Smith, 1983). A further problem is that the down-gradient transport hypothesis
is inconsistent with the observed features of turbulent diffusion in the upper portion of the mixed
layer, where countergradient material fluxes are known to occur (Deardoff and Willis, 1975).
Despite these well known limits, the K-closure is widely used in several atmospheric conditions,
because it describes the diffusive transport in an Eulerian framework where almost all
measurements are Eulerian in character. It produces results that agree with experimental data as
well as any more complex model, and it is not computationally expensive, as is the case of higher
order closures.
The reliability of the K-approach strongly depends on the way the eddy diffusivity is determined
on the basis of the turbulence structure of the PBL, and on the model’s ability to reproduce
experimental diffusion data. A great variety of formulations exist (Ulke, 2000).
Bearing the K-theory limitations in mind, the main idea of the approach proposed is to obtain an
eddy diffusivity scheme for practical applications in an analytical multilayer dispersion model, as
well as a variable vertical discretization in order to represent better transport and diffusion
phenomena at the ground, at the top of the boundary layer and near the source.
THE MODEL
The concentration turbulent fluxes are often assumed to be proportional to the mean
concentration gradient. This assumption, along with the equation of continuity, leads to the
advection-diffusion equation. For a Cartesian coordinate system, in which the x direction
coincides with that one of the average wind, the steady state advection-diffusion equation is
written as:






U c    K x c     K y c     K z c 
x x  x  y  y  z  z 
(1)
where c denotes the average concentration, U the mean wind speed in x direction and Kx , Ky and
Kz are the eddy diffusivities. The cross-wind integration of the equation (1) (neglecting the
U
c
y

x
 
z 

Kz
y
 c 
(2)
z 

subject to the boundary conditions of zero flux at the ground and at PBL top, and a source with
emission rate Q at height H s :
y
U c ( 0 ,z)=Q(z-H s )
Kz
c
in x = 0
(3)
y
z
0
in z = 0, zi
(4)
where c y now represents the average cross-wind integrated concentration, and z i is the height of
the PBL.
Bearing in mind the dependence of the Kz coefficient and wind speed profile U on variable z,
following Vilhena et al. (1998), the height z i of a PBL is discretized in N sub-intervals, in such a
way that, within each interval, Kz(z) and U(z) assume the average value:
zn
 K z (z)dz
zn 1
(5)
zn
 U z (z)dz
z n  z n 1 zn 1
(6)
1
Kn 
Un 
z n  z n 1
1
Therefore the solution of problem (2) is reduced to the solution of N problems of the type:

Un

y
cn  K z
x
2
z
2
y
cn
zn-1  z  zn
with
(7)
for n = 1: N, where cny denotes the concentration at the nth sub-interval. To determine the 2N
integration constants, additional (2N-2) conditions, namely continuity of concentration and flux at
interface, are considered:
y
y
c n  c n 1
n = 1,2,...(N-1)
y
y
c n
Kn
z
(8)
 K n 1
 c n 1
n = 1,2,...(N-1)
z
(9)
Applying the Laplace transform in equation (7) results:

2
z
2
UnS y
U
y
y
c n (s,z) 
c n (s,z)= 
c n ( 0 ,z)
Kn
Kn

(10)

in which c ny ( s, z )  L c ny ( x, z ); x  s , which has the well-know solution:
c n (s,z) = Ane  Rn z  Bne Rn z 
y
Q
 R (z H s )
R (z H s )
(e n
e n
)
2 Ra
where
Rn = 
UnS
Kn
and
Ra = 
U n SKn
(11)
Finally, by applying the interface and boundary conditions a linear system for the integration
constants is obtained. Henceforth, the concentration is obtained by inverting numerically the
transformed concentration c y using Gaussian quadrature scheme:
M
c ny ( x, z )   A j .
j 1
M
c ny ( x, z )   A j .
j 1

1
2

Pj 
. An exp  

x 



 Pj U n  
.z   Bn exp 
.z  
 xK n  
xK n 



Pj U n
(12)

 Pj U n 
Pj 
PjU n 
. An exp  
.z   B n exp 
.z 

 xK n 
x 
xK n 





 
Pj U n
Q
. exp   ( z  H s )
xK n
P j K nU n  
x


Pj U n
  exp  ( z  H
s)


xK n


 
 
 
 
(13)
Solution (12) is valid for layers that do not contain the contaminant source. At the same time,
solution (13) can be used to evaluate the concentration field in the layer that contains the
continuous source.
Here, Aj and Pj are the weights and roots of the Gaussian quadrature scheme. In the present
study, M=8 was considered, because this value provides the required accuracy with small
computational effort. Obviously, the greater the number of layers (N), the more accurate the
concentration pattern calculated, although the relative code running time is consequently greater.
Moreover, the strata in which the PBL is divided are not constant in thickness. A more detailed
description is required of wind and diffusion coefficients in proximity to the ground, where their
gradients are high and more strongly influence pollutant dispersion. Therefore layers close to the
terrestrial surface are assigned a smaller thickness than those located higher up.
PARAMETERIZATION OF THE VERTICAL TURBULENT DIFFUSION COEFFICIENT
The literature reports many, greatly varied formulae for the calculation of the vertical turbulent
diffusion coefficient (Ulke, 2000).
Some of them are presented here and will be used in the following section to assess the relative
model performances:
Formulae of Degrazia: employed throughout the PBL
Unsatable condition
L  0 (Degrazia et al. , 1997):
13


 z 
z 
z
K z  0.22 w* hz h 1 3 1   1  exp  4   0.0003 exp 8 
zi 
 zi  

 zi 
(14)
Stable condition L  0 (Degrazia et al., 2000; Degrazia et al. , 2001):
Kz 
0.31  z h u z
1  3.7 z 
(15)
where z is height; zi the thickness of the stable layer;   L1  z z i 5 4 , where L is the MoninObukhov length.
Similarity formulation: employed only within the Surfer Layer (Panofsky and Dutton, 1988).
Kz 
ku* z
 h z L 
(16)
The function  h is calculated with the formulae of Dyer:
z

Unstable conditions: z / L  0  h  1  16 
L

Neutral conditions: z / L  0
h  1
Stable conditions: z / L  0
h  1 5
1 2
z
L
Formulae of Lamb and Durran: employed throughout the PBL in unstable conditions
(Seinfeld and Pandis, 1997).
43
z
 z 
k zz  w z i 2.5 k  1  15 
L
 zi  
14
0
z
 0.05
zi

z
z
k zz  w z i 0.021  0.408   1.351
 zi 

 zi

 z
k zz  w z i 0.2 exp 6  10
 zi

2

z
  4.096

 zi



3

z
  2.560

 zi



4



0.05 
z
 0.6
zi
0 .6 
z
 1 .1
zi
(17)
z
 1 .1
zi
k zz  w z i 0.0013
Formulae of Myrup and Ranzieri: employed throughout the PBL in neutral conditions
(Seinfeld and Pandis, 1997).
k zz  ku z
z
 0 .1
zi

z
k zz  ku z1.1  
zi 

0.1 
k zz  0
z
 1 .1
zi
z
 1.1
zi
(18)
Formula of Shir: employed throughout the PBL in neutral conditions (Seinfeld and Pandis,
1997).
 8 fz 

k zz  ku z exp  
u
 

(19)
Formulae of Lamb et al. : employed throughout the PBL in neutral conditions (Seinfeld and
Pandis, 1997).
2
3
4
2
 zf 
 zf 
 zf  
u 
4
 2  zf 
7.396 *10  6.082 *10    2.532   12.72   15.17  
k zz 
f 
 u 
 u 
 u 
 u  

k zz  0
for
 zf 
0     0.45
 u 
for
 zf 
   0.45
 u 
(20)
where f represents the Coriolis coefficient: f  1.46 *10 4
Formula of Businger and Arya: employed throughout the PBL in stable conditions (Seinfeld
and Pandis, 1997).
Kz 
 8 fz 
ku* z

exp  
0.74  4.7z L 
 u* 
(21)
Formulae of Troen and Mahrt: employed throughout the PBL (Pleim e Chang, 1992).
Unstable conditions
 zi

  10  :
L

z

Stable or almost neutral conditions  i  10  :
L

where:
z

 h  1  16 
L

h  1 5
z
L

z
k zz  kw z1  
 zi 

z
k zz  ku z 1  
 zi 
2
(22)
 h z L 
(23)
1 2
L0
L0
WIND PARAMETERIZATION
The equations used by the model to calculate mean wind are those of similarity (Panofsky and
Dutton, 1988):
u
u
ka
 z
 z 
ln   m  
 L 
 z0
(24)
where, u* is the scale velocity relative to mechanical turbulence, k a the von Karman constant, and
m the stability function expressed in Businger relations:
z
L
 m    4.7
z
L
for
1 x2 

z
1 x 
  ln 
 m    ln 
  2 arctan x 
2
L
 2 
 2 
1 L0
2
z

with x  1  15 
L

for
1 L0
14
The similarity expression is utilised of within the surfer layer. Alternatively, the wind speed
profile can be described by a power law expressed as follows (Panofsky and Dutton, 1988):
uz  z 
 
u1  z1 
n
(25)
where u z and u1 are the mean wind speeds horizontal to heights z and z1 and n is an exponent that
is related to the intensity of turbolence (Irwin, 1979)
MODEL EVALUATION AGAINST EXPERIMENTAL DATASETS
The new parameterisations of the model have been evaluated using two experimental datasets
with different emission and meteorological scenarios. The Copenhagen field campaign (Gryning
and Lyck, 1984) took place in the suburbs of Copenhagen in 1978. A SF6 tracer was released
without buoyancy from a tower at a height of 115m and collected at ground-level on arcs located
2000, 4000, and 6000 meters from the release point. The site was mainly residential with a
roughness length, z 0 , of 0.6m. The meteorological conditions during the dispersion experiments
ranged from moderately unstable to convective.
The Prairie Grass dataset (Barad, 1958) is composed of dispersion data from a field experiment
conducted in open country ( z 0 was 0.008m) during the summer of 1956 in O’Neill, Nebraska.
Sulphur dioxide was released from a continuous point source at a height of 0.46m and collected at 5
arcs, 50, 100, 200, 400, 800 meters from the source. Here, we use a part of the values of the
crosswind-integrated concentrations as calculated and reported by Van Ulden (1978). The two
experiments cover a vast range of atmospheric turbulence. Figures 1-4 show the eddy coefficient
vertical profiles for the various turbulence regimes found for the experimental datasets.
One of the parameters taken into consideration in the evaluation of the model performance is
mean wind. A comparison was made of the results obtained using three versions of the model. In
the first, the wind profile was calculated with the similarity formulae within the surface layer, and
was assumed constant above it. In the second, a power law was used to represent the wind profile
throughout the entire PBL. In the third, the wind was calculated using the similarity within the
surface layer and the power law above it.
Table 1 presents some performances measures obtained by using the statistical evaluation
procedure described by Hanna (1989) and defined in the following way,
NMSE (normalized mean square) = (Co  C p ) 2 / Co C p
COR (correlation)= (Co  Co )(C p  C p ) /  o p
FA2 = fraction of Co values within a factor two of corresponding Cp values

 
FB (fractional bias)= Co  C p / 0.5 Co  C p

where the subscripts o and p refer respectively to observed and predicted quantities, and an
overbar indicates an average.
Table 1. Statistical indices for the tests of different expressions of mean wind.
Wind profile
NMSE
COR
FA2
FB
Similarity
0.06
0.92
1.00
0.07
Power
0.13
0.91
1.00
0.22
Misto
0.12
0.91
1.00
0.20
Analysing the statistical indices in Table 1, it can be seen that there are no significant
differences in the model performances using the different formulae for calculating mean wind,
although the best result was obtained with the Similarity formulae (similarity profile in the surface
layer and constant wind above)
PROFILI CONVETTIVI
Degrazia
Troen e Mahrt
Lamb e Durran
1
0.9
0.8
0.7
z / zi
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.05
0.1
0.15
Kz / w* zi
Figure 1. Convective conditions.
0.2
0.25
PROFILI INSTABILI
Degrazia
Troen e Mahrt
Lumb e Durran
1
0.9
0.8
0.7
z / zi
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.05
Figure 2. Unstable condictions.
0.1
0.15
Kz / w* zi
0.2
0.25
0.3
PROFILI NEUTRI
Degrazia
Similarit&agrave;
Troen e Mahrt
Myrup e Ranzieri
Shir
1
0.9
0.8
0.7
z / zi
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.05
0.1
Figure 3. Neutral condictions.
0.15
0.2
Kz / u * zi
0.25
0.3
0.35
0.4
PROFILI STABILI
Degrazia
Similarit&agrave;
Troen e Mahrt
Businger ed Arya
1
0.9
0.8
0.7
z / zi
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.01
0.02
Kz / u * zi
Figure 4. Stable condictions.
0.03
The values of the statistical indices calculated for the diverse parameterizations of the eddy
diffusion coefficients with the data of Copenhagen are reported in Table 2.
Tabel 2. Statistical indices for different K z profiles with Copenhagen data set.
Kz profile
NMSE
COR
FA2
FB
Degrazia et al.
0.06
0.91
1.00
0.07
Similarity
0.08
0.87
1.00
0.07
Troen and Mart
0.09
0.91
1.00
0.16
Myrup and Ranzieri
0.23
0.69
0.78
0.21
Shir
0.23
0.70
0.78
0.22
Lamb et al.
0.33
0.51
0.78
0.23
12
10
Cp Q-1 10-4 (sm-2 )
8
6
4
Degrazia
Similarit&agrave;
Troen e Mahrt
Myrup e Ranzieri
Shir
Lamb et. al
2
0
0
2
4
6
Co Q-1 10-4 (sm-2 )
8
10
12
Figure 5. Copenhagen dataset. Scatter plot of observed (Co) versus predicted (Cp) crosswindintegrated concentrations normalized with the emission source rate. Points between dashed lines are
in a factor of two
Figure 5 shows a comparison between the data calculated by the model and experimental data.
The points between the dashed lines are in a factor of two.
The values of the statistical indices calculated for the different parameterizations of the eddy
diffusion coefficient with the Prairie Grass data are reported in Table 3, while Figure 6 shows the
comparison between the data calculated by the model and experimental data
200
180
160
Cp Q-1 10-3 (sm-2 )
140
120
100
80
60
40
Degrazia
Similarit&agrave;
Troen e Mahrt
Myrup e Ranzieri
Shir
Lamb et. al
20
0
0
20
40
60
100 120
80
Co Q-1 10-3 (s m-2)
140
160
180
200
Figure 6. Prairie Grass dataset. Scatter plot of observed (Co) versus predicted (Cp) crosswindintegrated concentrations normalized with the emission source rate. Points between dashed lines are
in a factor of two
Table 3. Statistical indices for different K z profiles with Prairie Grass data set
Kz profile
NMSE
COR
FA2
FB
Degrazia et al.
0.07
0.94
0.97
-0.13
Similarity
0.06
0.95
0.97
-0.06
Troen and Mart
0.10
0.93
0.86
-0.15
Myrup and Ranzieri
0.06
0.95
0.97
-0.04
Shir
0.06
0.95
0.97
-0.04
Lamb et al.
0.07
0.95
0.97
0.02
CONCLUSIONS
The parameters considered for the evaluation of the model performances are: mean wind and
the vertical turbulent diffusion coefficient.
Different expressions of the vertical turbulent diffusion coefficient were introduced, together
with different expressions for the calculation of mean wind, available in the literature. A
comparison was made of the concentrations measured and calculated by the model. This was done
through widely used indices for the evaluation of model performances.
On the basis of the values of the indices, it is possible to affirm that, in the case of high
emissions in moderately unstable to convective meteorological conditions, i.e. similar to those of
the Copenhagen campaign, the best performances are obtained using the formulae of Degrazia,
Similarity and Troen and Mahrt, for the calculation of the vertical turbulent diffusion coefficient.
Conversely, in the case of ground-level emissions in meteorological conditions from convective
to stable, i.e. similar to those of the Prairie Grass campaign, the best model performances are
obtained using the formulae of Lamb and Durran for unstable cases, those of Businger and Arya for
stable cases, and those of Myrup and Ranzieri, Shir and Lamb et al. for neutral cases. However, the
performances obtained with the formulae of Degrazia and of Similarity are also good.
As far as the wind profile is concerned, there are non significant differences in the model
performances using the diverse formulae for the calculation of wind, although slightly better
performances are obtained using the Similarity formulae within the surface layer and considering
wind above the said layer.
In all cases, the values of the statistical indices, both in the sensitivity analysis and assessment
of model performances varying the characteristic parameters, in diverse turbulence regimes, turn
out to be good, when compared with those of other models available in the literature (Olesen,
1995).
Such remarks lead to the conclusion that the proposed approach could be used in an operative
model of pollutant dispersion into the atmosphere, as a tool for the evaluation and management of
air quality.
Ackowledgements
The authors thank CNPq (Conselho Nacional de Desenvolvimento Cient&iacute;fico e Tecnol&oacute;gico) and
FAPERGS (Funda&ccedil;&atilde;o de Amparo &agrave; Pesquisa do Estado do Rio Grande do Sul) for the partial
financial support of this work.
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