Pollutant Dispersion Simulation in the ABL by the GILTT Method
185
To determine the unknown coeffi cient c
–
n (x,r), we replace Equation 7.9 with
Equation 7.8 and taking moments, that is, applying the operator
Ψ
∫ 0 ( ) ( )
h
m z dz, we
come out with the result:
=
=
=
=
=
=
Ψ
Ψ
+
Ψ
Ψ
=
′
′
Ψ
Ψ
+
Ψ
Ψ
+
′′
′
′
−
λ
Ψ
Ψ
+
Ψ
Ψ
+
′ ′
−
∑
∑
∫
∫
∑
∑
∫
∫
∑
∑
∫
∫
0
0
0
0
0
0
0
0
2
0
0
0
0
( , )
( ) ( )d
( , )
( ) ( )d
( , )
( ) ( )d
( , )
( ) ( )d
( , )
( ) ( )d
( , )
( ) ( )d
h
h
N
N
n
n
m
n
n
m
n
n
h
h
N
N
n
x n
m
n
x n
m
n
n
h
h
N
N
n
n
z n
m
n
z n
m
n
n
n
c x r u
z
z z
c x r w
z
z z
c x r K
z
z z
c x r K
z
z z
c x r
K
z
z z
c x r K
z
z z
c
=
=
=
=
=
=
βΨ
Ψ
−
β Ψ
Ψ
+
′
′
−
β Ψ
Ψ
−
β Ψ
Ψ
+
′
′
′
′
+
λ β Ψ
Ψ
−
β Ψ
Ψ
′ ′
∑
∑
∫
∫
∑
∑
∫
∫
∑
∑
∫
∫
0
0
0
0
0
0
0
0
2
0
0
0
0
( , )
( ) ( )d
( , )
( ) ( )d
( , )
( ) ( )
( , ) ( )
( ) ( )d
( , )
( ) ( )d
( , ) ( ) ( ) ( )d
h
h
N
N
n
m
n
n
m
n
n
h
h
N
N
n
n
m
n
n
m
n
n
h
h
N
N
n
n
n
m
n
n
m
n
n
x r r
z
z z
c x r r
z
z z
c x r
u
z
z dz
c x r
u
z
z z
c x r
w
z
z z
c x r
w
z
z z
=
=
=
=
=
+
−
τ
Ψ
Ψ
−
τ
Ψ
Ψ
+
′
−
τ
Ψ
Ψ
+
β Ψ
Ψ
+
′
′
+
β
Ψ Ψ
+
β Ψ
Ψ
′′
′
′
′ ′
∑
∑
∫
∫
∑
∑
∫
∫
∑ ∫
∫
2
0
0
0
0
0
0
0
0
0
0
0
( , )
( ) ( )d
( , )
( ) ( )d
( , )
( ) ( )d
( , )
( ) ( )d
( , ) (
) ( ) ( )d
( , )
( ) ( )d
h
h
N
N
n
n
m
n
n
m
n
n
h
h
N
N
n
n
m
n
xn
m
n
n
h
h
N
n
x
n
m
n
x n
m
n
c x r r
z
z z
c x r r u
z
z z
c x r r w
z
z z
c x r
K
z
z z
c x r
K
z
z z
c x r
K
z
z
=
=
=
=
=
+
+
β Ψ
Ψ
+
τ
Ψ
Ψ
+
′
′
′
+
τ
Ψ
Ψ
−
Ψ
Ψ
′
′
∑
∑
∑
∫
∫
∑
∑
∫
∫
0
0
0
0
0
0
0
0
0
( , ) (
) ( ) ( )d
( , )
( ) ( )d
( , )
( ) ( )d
( , )
( ) ( )d
N
n
h
h
N
N
n
x
n
m
n
x n
m
n
n
h
h
N
N
n
x n
m
n
n
m
n
n
z
c x r
K
z
z z
c x r r K
z
z z
c x r r K
z
z z
c x r r
z
z z
(7.10)
© 2010 by Taylor and Francis Group, LLC