Turbulence and Dispersion of Contaminants in the Planetary Boundary Layer 59
with
(
)
⎛
⎞
⎛ ⎞
= +
= +
=
=
=
= −
⎜
⎟
⎜ ⎟
⎝ ⎠
⎝
⎠
1 3
5 6
1 3
1
2
1
1
1
,
1
,
,
2 ,
1 ,
i
i
i
i
i
u
v
w
i
i
W
W
B s
A a
m
m m
b
B s
5 6
2
0
1
2
3
55
70
725
935
,
,
,
,
,
27
9
72
216
i
i
i
i
i
C
C
C
C
A ab
B b
−
−
= −
=
= −
=
=
=
(3.98)
According to Degrazia and Anfossi (1998) the initial 1-D component of spectrum
can be written as
5/3
( ,0)
,
, ,
(1
)
i
i
i
a
F k
i u v w
b k
=
=
+
(3.99)
where
−
ε
=
π
ψ
5/3
2/3 2
*
5/3
*
(0.98 2 ) ( )
[( ) ]
/
c
i
i
i
i
m i
a
czz
z
w f
=
π
*
(1.5/2 )( / ) (1 ( ) )
/
c
i
ii
m i
b
zz z
f
, with c i = α i (0.5 ± 0.05)(2πκ) −2/3 , α i = 1, 4/3, 4/3 for
u, v, and w, respectively (Champagne et al., 1977)
=
− κ
1/3
*
* 0
( ) ( ( / L))
i
w
u
z
,
=
( ) ( / )
c
m i
i i
f
zGz , G u = 1.5, G v = 1.5, and
=
−
−
−
×
1.8[1 exp (4 / ) 0.0003
w
i
G
z z
exp (8 / )]
i
z z
The CBL is nonisotropic only in the vertical direction, due to the heat fl ux in the surface. To calculate the w component of the spectrum, we consider that for a particular
time instant t there is a relationship between the 1-D spectrum and the 3-D average
spectrum given by the following expression:
= α
∫
∫
0
0
1
( , )d
( , )
( )
( , )
1
( , )d
t
w
w
t
F k t t
T
F k t
k
E k t
E k t t
T
(3.100)
where the ratio between the two integrals is a weight function that indicates that
the w component takes part in the construction of the 3-D spectrum and α(k) is the
proportionality constant.
The solution of Equation 3.100 provides the vertical component as a function of
the 3-D spectrum:
0
( , )
( ,0)exp
( , )d
t
w
w
F k t F k
Q k s s
⎡
⎤
′
=
⎢
⎥
⎣
⎦
∫
(3.101)
In this case F w (k,0) is given by Equation 3.99 where
∂
= α
+
′
∂
1
( , )
( , )
( ) ( , )
( , )
Q k s
Q k s
k Q k s
Q k s
s
(3.101a)
© 2010 by Taylor and Francis Group, LLC
with
(
)
⎛
⎞
⎛ ⎞
= +
= +
=
=
=
= −
⎜
⎟
⎜ ⎟
⎝ ⎠
⎝
⎠
1 3
5 6
1 3
1
2
1
1
1
,
1
,
,
2 ,
1 ,
i
i
i
i
i
u
v
w
i
i
W
W
B s
A a
m
m m
b
B s
5 6
2
0
1
2
3
55
70
725
935
,
,
,
,
,
27
9
72
216
i
i
i
i
i
C
C
C
C
A ab
B b
−
−
= −
=
= −
=
=
=
(3.98)
According to Degrazia and Anfossi (1998) the initial 1-D component of spectrum
can be written as
5/3
( ,0)
,
, ,
(1
)
i
i
i
a
F k
i u v w
b k
=
=
+
(3.99)
where
−
ε
=
π
ψ
5/3
2/3 2
*
5/3
*
(0.98 2 ) ( )
[( ) ]
/
c
i
i
i
i
m i
a
czz
z
w f
=
π
*
(1.5/2 )( / ) (1 ( ) )
/
c
i
ii
m i
b
zz z
f
, with c i = α i (0.5 ± 0.05)(2πκ) −2/3 , α i = 1, 4/3, 4/3 for
u, v, and w, respectively (Champagne et al., 1977)
=
− κ
1/3
*
* 0
( ) ( ( / L))
i
w
u
z
,
=
( ) ( / )
c
m i
i i
f
zGz , G u = 1.5, G v = 1.5, and
=
−
−
−
×
1.8[1 exp (4 / ) 0.0003
w
i
G
z z
exp (8 / )]
i
z z
The CBL is nonisotropic only in the vertical direction, due to the heat fl ux in the surface. To calculate the w component of the spectrum, we consider that for a particular
time instant t there is a relationship between the 1-D spectrum and the 3-D average
spectrum given by the following expression:
= α
∫
∫
0
0
1
( , )d
( , )
( )
( , )
1
( , )d
t
w
w
t
F k t t
T
F k t
k
E k t
E k t t
T
(3.100)
where the ratio between the two integrals is a weight function that indicates that
the w component takes part in the construction of the 3-D spectrum and α(k) is the
proportionality constant.
The solution of Equation 3.100 provides the vertical component as a function of
the 3-D spectrum:
0
( , )
( ,0)exp
( , )d
t
w
w
F k t F k
Q k s s
⎡
⎤
′
=
⎢
⎥
⎣
⎦
∫
(3.101)
In this case F w (k,0) is given by Equation 3.99 where
∂
= α
+
′
∂
1
( , )
( , )
( ) ( , )
( , )
Q k s
Q k s
k Q k s
Q k s
s
(3.101a)
© 2010 by Taylor and Francis Group, LLC