190
Air Pollution and Turbulence: Modeling and Applications
7.4 TURBULENT PARAMETERIZATIONS AND
DISPERSION DIFFUSION EXPERIMENTS
In the sequel, we report the parameterizations adopted in the simulations reported in
this chapter. In fact, we need to recall that the choice of the turbulent parameterization represents a fundamental aspect for pollutant dispersion modeling (Moreira et
al., 2005c). Indeed, to evaluate the 3D concentration in the ground-level centerline
(Equation 7.20), we need to know the lateral dispersion parameter σ y . For a CBL, we
used the lateral dispersion parameter σ y derived by Degrazia et al. (1998a), which
has the form:
∞
σ
′
=
ψ
′
π
+ ′
′
∫
2
2
1 / 3
2
5 / 3 2
0
0.21
d
sin (2.26
)
(1
)
y
n
Xn
h
n
n
(7.21)
Here X is an adimensional distance (X = xw * /uh), w * is the convective velocity scale,
n′ is adimensional frequency, and h is the top of the CBL. Equation 7.21 contains
the unknown function ψ; the molecular dissipation of turbulent velocity is a leading
destruction term in equations for the budget of second-order moments, and according to Hφjstrup (1982), reads like
1/2
2
2 / 3
1/3
1
0.75
z
z
h
L
−
⎡
⎤
⎛
⎞ ⎛
⎞
ψ =
−
+
⎢
⎥
⎜
⎟ ⎜
⎟
−
⎝
⎠ ⎝
⎠
⎢
⎥
⎣
⎦
(7.22)
where L is the length of Monin–Obukhov defi ned in the surface boundary layer.
In terms of the convective scaling parameters, the vertical eddy diffusivity can be
formulated as (Degrazia et al., 1997)
1/3
1/3
*
4
8
0.22
1
1 exp
0.0003exp
z
K
z
z
z
z
w h
h
h
h
h
⎡
⎤
⎛ ⎞ ⎛
⎞
⎛
⎞
⎛ ⎞
=
−
−
−
−
⎜ ⎟ ⎜
⎟
⎜
⎟
⎜ ⎟
⎢
⎥
⎝ ⎠ ⎝
⎠
⎝
⎠
⎝ ⎠
⎣
⎦
(7.23)
To represent the near-source diffusion under weak wind conditions, the eddy diffusivities should be considered as functions not only of turbulence (e.g., large eddy
length and velocity scales), but also of distance from the source (Arya, 1995).
Following this idea, Degrazia et al. (2002) proposed for the unstable boundary layer
the ensuing algebraic formulation for the eddy diffusivities:
( )
( )
( )
( ) ( )
( )
α
⎡
⎤
ψ
+
ψ
⎢
⎥
⎣
⎦
=
⎡
⎤
+
ψ
⎢
⎥
⎣
⎦
2/3
4/3
2/3
2/3
*
1/2 1/3
*
*
*
2
1/3
2/3
*
1 / 21 / 3
*
*
0.583
/
0.55 /
1.03
0.55 /
2.06
i
i
m i
m
i
m
i
i
w hc
z h
X
z h
c
f
X
K
z h
f
c
f X
(7.24)
where c v,w = 0.36, c u = 0.3, (f m
* ) i is the normalized frequency of the spectral peak,
namely,
© 2010 by Taylor and Francis Group, LLC
Air Pollution and Turbulence: Modeling and Applications
7.4 TURBULENT PARAMETERIZATIONS AND
DISPERSION DIFFUSION EXPERIMENTS
In the sequel, we report the parameterizations adopted in the simulations reported in
this chapter. In fact, we need to recall that the choice of the turbulent parameterization represents a fundamental aspect for pollutant dispersion modeling (Moreira et
al., 2005c). Indeed, to evaluate the 3D concentration in the ground-level centerline
(Equation 7.20), we need to know the lateral dispersion parameter σ y . For a CBL, we
used the lateral dispersion parameter σ y derived by Degrazia et al. (1998a), which
has the form:
∞
σ
′
=
ψ
′
π
+ ′
′
∫
2
2
1 / 3
2
5 / 3 2
0
0.21
d
sin (2.26
)
(1
)
y
n
Xn
h
n
n
(7.21)
Here X is an adimensional distance (X = xw * /uh), w * is the convective velocity scale,
n′ is adimensional frequency, and h is the top of the CBL. Equation 7.21 contains
the unknown function ψ; the molecular dissipation of turbulent velocity is a leading
destruction term in equations for the budget of second-order moments, and according to Hφjstrup (1982), reads like
1/2
2
2 / 3
1/3
1
0.75
z
z
h
L
−
⎡
⎤
⎛
⎞ ⎛
⎞
ψ =
−
+
⎢
⎥
⎜
⎟ ⎜
⎟
−
⎝
⎠ ⎝
⎠
⎢
⎥
⎣
⎦
(7.22)
where L is the length of Monin–Obukhov defi ned in the surface boundary layer.
In terms of the convective scaling parameters, the vertical eddy diffusivity can be
formulated as (Degrazia et al., 1997)
1/3
1/3
*
4
8
0.22
1
1 exp
0.0003exp
z
K
z
z
z
z
w h
h
h
h
h
⎡
⎤
⎛ ⎞ ⎛
⎞
⎛
⎞
⎛ ⎞
=
−
−
−
−
⎜ ⎟ ⎜
⎟
⎜
⎟
⎜ ⎟
⎢
⎥
⎝ ⎠ ⎝
⎠
⎝
⎠
⎝ ⎠
⎣
⎦
(7.23)
To represent the near-source diffusion under weak wind conditions, the eddy diffusivities should be considered as functions not only of turbulence (e.g., large eddy
length and velocity scales), but also of distance from the source (Arya, 1995).
Following this idea, Degrazia et al. (2002) proposed for the unstable boundary layer
the ensuing algebraic formulation for the eddy diffusivities:
( )
( )
( )
( ) ( )
( )
α
⎡
⎤
ψ
+
ψ
⎢
⎥
⎣
⎦
=
⎡
⎤
+
ψ
⎢
⎥
⎣
⎦
2/3
4/3
2/3
2/3
*
1/2 1/3
*
*
*
2
1/3
2/3
*
1 / 21 / 3
*
*
0.583
/
0.55 /
1.03
0.55 /
2.06
i
i
m i
m
i
m
i
i
w hc
z h
X
z h
c
f
X
K
z h
f
c
f X
(7.24)
where c v,w = 0.36, c u = 0.3, (f m
* ) i is the normalized frequency of the spectral peak,
namely,
© 2010 by Taylor and Francis Group, LLC