Turbulence and Dispersion of Contaminants in the Planetary Boundary Layer 51
order of magnitude of ε determined only by those quantities that characterize the
large energy-containing eddies.
The analytical integration of Equation 3.67 over the whole frequency domain
leads to the following Eulerian turbulent velocity variance:
( )
( )
−
∞
ε
∗
∗
∗
⎛
⎞
⎡
⎤
ψ
⎜
⎟
⎢
⎥
⎝
⎠
σ =
+
⎢
⎥
⎡
⎤
⎡
⎤
⎢
⎥
⎢
⎥
⎣
⎦
⎢
⎥
⎣
⎦
⎣
⎦
∫
2 3
5 3
2
2
5 2
0
1.06
1 1.5
d
i
i
ic
c
c
m
m
i
i
z
c z
w
z
nz
n
U f
U f
(3.68)
and
( )
ε
∗
∗
⎛
⎞
ψ
⎜
⎟
⎝
⎠
σ =
⎡
⎤
⎢
⎥
⎣
⎦
2 3
2
2
2 3
1.06 i
i
ic
c
m i
z
c
w
z
f
(3.69)
which is used to normalize the spectrum so that the normalized Eulerian spectrum
can be written as follows:
( )
( )
−
∗
∗
⎧
⎫
⎪
⎪
=
=
+
⎨
⎬
⎡
⎤
⎡
⎤
σ
⎪
⎪
⎢
⎥
⎢
⎥
⎣
⎦
⎣
⎦
⎩
⎭
5 3
2
( )
( / )
( )
1 1.5
E
ic
E
ic
c
c
ic
m
m
i
i
S n
z
nz U
F n
U f
f
(3.70)
Substituting Equation 3.69 and β =
σ
0.55
ic
ic
U
(Degrazia and Anfossi, 1998) in
Equation 3.42 yields
( )
∗
ε
∗
⎛
⎞
ψ
⎜
⎟
⎝
⎠
σ β =
π
⎡
⎤
⎢
⎥
⎣
⎦
1 3
1 2
2
1 3
0.09
2
i
i
ic ic
c
m i
z
Uc w
z
f
(3.71)
and
( )
ε
∗
⎛
⎞
ψ
⎜
⎟
⎝
⎠
π ≡ =
β
⎡
⎤
⎢
⎥
⎣
⎦
1 3
1 2
1 3
11.76
2
i
i
i
c
ic
m i
z
c
z
t
z
a
X
U
f
(3.72)
where a time-to-space transposition is applied to the time dependency in Equation
3.42 to yield a spatially dependent K α , with
=
*
(
/ )
i
X
xw Uz , a nondimensional
distance defi ned by the ratio of travel time x/U and the convective timescale z i /w * .
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