Turbulence and Dispersion of Contaminants in the Planetary Boundary Layer 45
The relation between Lagrangian and Eulerian spectra can be found by considering
the following expression (Equation 3.26):
∞
=
ρ τ
π τ τ
∫
0
( ) 4
( ) cos2
d
i
i
F n
n
(3.37)
where F i (n) is the Eulerian spectrum normalized by σ
2
i .
The substitution of Equation 3.35 in Equation 3.37 yields
( )
∞
= β ρ β τ
π β τ τ
∫
L
L
0
( ) 4
cos2
d
i
i
i
i
i
F n
n
from Equations 3.35 and 3.37 we obtain
( )
= β
β
L ( )
i
i
i
i
nF n
nF n
(3.38)
By using Equation 3.38, the generalized dispersion parameter and the eddy diffusivity, respectively Equations 3.29 and 3.30, can be related to the scale factor β i by the
following relationships:
∞
π
= σ
β β
π
∫
2
2
2 2
2
0
sin ( )
( )
d
( )
i
i
i
i i
n t
X
t
F n
n
n t
(3.39)
and
∞
σ
π
⎛
⎞ =
β β
⎜
⎟
⎝
⎠
π ∫
2
2
0
d 1
sin(2 )
( )
d
d 2
2
i
i
i i
i
nt
X
F n
n
t
n
(3.40)
Equations 3.39 and 3.40 can be transformed to (Batchelor, 1949; Pasquill and Smith,
1983; Degrazia and Moraes, 1992)
∞
σ β
π β
= π ∫
2
2
2
2
2
0
sin ( / )
( )
d
i i
i
i
i
n t
X
Fn
n
n
(3.41)
and
∞
α
σ β
π β
⎛
⎞
=
=
⎜
⎟
⎝
⎠
π ∫
2
2
0
d 1
sin(2 / )
( )
d
d 2
2
i i
i
i
i
nt
K
X
Fn
n
t
n
(3.42)
Equation 3.42 has an asymptotic behavior when travel time becomes large (lim τ → ∞)
that has the effect of selecting F i (n) at the origin of the frequency space (Degrazia
and Moraes, 1992). As a consequence, the rate of dispersion becomes independent of
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