Turbulence and Dispersion of Contaminants in the Planetary Boundary Layer 53
physical state characterized by a diversifi ed phenomenology and hence values of ψ ε
and
*
( )
c
m i
f
must be necessarily obtained from heuristic arguments (observations).
In the case of horizontal homogeneity, the evolution of CBL is controlled by the vertical transport of heat. As a consequence, the present analysis will focus on the vertical
eddy diffusivity. This eddy diffusivity can be derived from Equation 3.74 by assuming
−
⎡
⎤
⎛ ⎞
⎛
⎞
⎛
⎞
=
=
−
−
−
⎢
⎥
⎜ ⎟
⎜
⎟
⎜
⎟
⎝ ⎠
⎝
⎠
⎝
⎠
λ
⎣
⎦
1
*
w
w
( )
0.55
1 exp 4
0.0003exp 8
( )
c
m
c
m
i
i
i
z
z
z
z
f
z
z
z
(3.76)
where λ
=
−
−
−
w
( )
1.8 [1 exp( 4( / )) 0.0003exp(8( / ))]
c
m
i
i
i
z
zz
zz
is the value of the vertical wavelength at the spectral peak, which was obtained from empirical data by
Caughey and Palmer (1979). To proceed, the vertical eddy diffusivity described in
terms of energy-containing eddies and a function of the downwind distance X and the
height z can be obtained from Equations 3.74 and 3.76 using c w = 0.36, as follows:
(
)
( )
(
)
( )
{
}
4/3
1/3
2/3
1/3
5/3
0
0.12
1 exp 4 /
0.0003exp 8 /
*
d
sin 3.17 1 exp 4 /
0.0003exp 8 /
(1
)
z
i
i
i
i
i
K
z z
z z
w z
n
z z
z z
Xn
n
n
ε
−
∞
ε
⎡
⎤
=
ψ
−
−
−
⎣
⎦
′
⎡
⎤
−
−
−
ψ
′
⎣
⎦
′
×
+ ′
∫
(3.77)
The formula (3.77) allows to describe the turbulent dispersion in the CBL in terms of
the energy-containing eddies and of memory effect of the turbulent fi eld represented
by the source distance X. The dissipation function ψ ε can be evaluated from the following expression, which was obtained from empirical fi tting by HØ ´ jstrup (1982)
−
ε
⎡
⎤
⎛
⎞ ⎛
⎞
ψ =
−
−
+
⎢
⎥
⎜
⎟
⎜
⎟ ⎝
⎠
⎝
⎠
⎢
⎥
⎣
⎦
1 2
2
2 3
1 3
1
0.75
i
z
z
z
L
(3.78)
where L is the Monin–Obukhov length in the convective surface layer. The behavior
of the vertical eddy diffusivity for two different heights, as given by Equation 3.77, is
presented in Figure 3.1. From Figure 3.1, we see that the eddy diffusivity is initially
zero, increases with time at fi rst linearly and then more slowly and fi nally tends to
a constant value, which can be obtained from Equations 3.75 and 3.76. This turbulent parameterization describes the asymptotic vertical eddy diffusivity far from the
source and can be written as (Degrazia et al., 2001)
(
)
( )
ε ⎡
⎤
=
ψ
−
−
−
⎣
⎦
4 3
1 3
*
0.19
1 exp 4
0.0003 exp 8
z
i
i
i
K
z z
z z
w z
(3.79)
Figure 3.2 exhibits the behavior of vertical profi le of
*
z
i
K w z as given by Equation
3.79. This profi le represents a well-behaved eddy diffusivity with a maximum in the
© 2010 by Taylor and Francis Group, LLC
physical state characterized by a diversifi ed phenomenology and hence values of ψ ε
and
*
( )
c
m i
f
must be necessarily obtained from heuristic arguments (observations).
In the case of horizontal homogeneity, the evolution of CBL is controlled by the vertical transport of heat. As a consequence, the present analysis will focus on the vertical
eddy diffusivity. This eddy diffusivity can be derived from Equation 3.74 by assuming
−
⎡
⎤
⎛ ⎞
⎛
⎞
⎛
⎞
=
=
−
−
−
⎢
⎥
⎜ ⎟
⎜
⎟
⎜
⎟
⎝ ⎠
⎝
⎠
⎝
⎠
λ
⎣
⎦
1
*
w
w
( )
0.55
1 exp 4
0.0003exp 8
( )
c
m
c
m
i
i
i
z
z
z
z
f
z
z
z
(3.76)
where λ
=
−
−
−
w
( )
1.8 [1 exp( 4( / )) 0.0003exp(8( / ))]
c
m
i
i
i
z
zz
zz
is the value of the vertical wavelength at the spectral peak, which was obtained from empirical data by
Caughey and Palmer (1979). To proceed, the vertical eddy diffusivity described in
terms of energy-containing eddies and a function of the downwind distance X and the
height z can be obtained from Equations 3.74 and 3.76 using c w = 0.36, as follows:
(
)
( )
(
)
( )
{
}
4/3
1/3
2/3
1/3
5/3
0
0.12
1 exp 4 /
0.0003exp 8 /
*
d
sin 3.17 1 exp 4 /
0.0003exp 8 /
(1
)
z
i
i
i
i
i
K
z z
z z
w z
n
z z
z z
Xn
n
n
ε
−
∞
ε
⎡
⎤
=
ψ
−
−
−
⎣
⎦
′
⎡
⎤
−
−
−
ψ
′
⎣
⎦
′
×
+ ′
∫
(3.77)
The formula (3.77) allows to describe the turbulent dispersion in the CBL in terms of
the energy-containing eddies and of memory effect of the turbulent fi eld represented
by the source distance X. The dissipation function ψ ε can be evaluated from the following expression, which was obtained from empirical fi tting by HØ ´ jstrup (1982)
−
ε
⎡
⎤
⎛
⎞ ⎛
⎞
ψ =
−
−
+
⎢
⎥
⎜
⎟
⎜
⎟ ⎝
⎠
⎝
⎠
⎢
⎥
⎣
⎦
1 2
2
2 3
1 3
1
0.75
i
z
z
z
L
(3.78)
where L is the Monin–Obukhov length in the convective surface layer. The behavior
of the vertical eddy diffusivity for two different heights, as given by Equation 3.77, is
presented in Figure 3.1. From Figure 3.1, we see that the eddy diffusivity is initially
zero, increases with time at fi rst linearly and then more slowly and fi nally tends to
a constant value, which can be obtained from Equations 3.75 and 3.76. This turbulent parameterization describes the asymptotic vertical eddy diffusivity far from the
source and can be written as (Degrazia et al., 2001)
(
)
( )
ε ⎡
⎤
=
ψ
−
−
−
⎣
⎦
4 3
1 3
*
0.19
1 exp 4
0.0003 exp 8
z
i
i
i
K
z z
z z
w z
(3.79)
Figure 3.2 exhibits the behavior of vertical profi le of
*
z
i
K w z as given by Equation
3.79. This profi le represents a well-behaved eddy diffusivity with a maximum in the
© 2010 by Taylor and Francis Group, LLC