Pollutant Dispersion Simulation in the ABL by the GILTT Method
183
∂
∂
∂
∂
∂
∂
∂
⎛
⎞
⎛
⎞
+
+
=
+
+
⎜
⎟
⎜
⎟
⎝
⎠
⎝
⎠
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
⎛
⎞
⎛
⎞
⎛
⎞
−
β
−
β
−
β
+
⎜
⎟
⎜
⎟
⎜
⎟
⎝
⎠
⎝
⎠
⎝
⎠
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
⎛
⎞
−τ
−
τ
−
τ
⎜
⎟
⎝
⎠
∂
∂
∂
∂
2
2
( , , )
( , , )
( , , )
( , , )
( , , )
( , , )
( , , )
( , , )
( , , )
( , , )
(
x
z
c x z t
c x z t
c x z t
c x z t
c x z t
u
w
K
K
t
x
z
x
x
z
z
c x z t
c x z t
c x z t
u
w
z
t
z
x
z
z
c x z t
c x z t
c
u
w
t
t
x
t
⎛
⎞ +
⎜
⎟
⎝
⎠
∂
⎛
⎞
⎛
⎞
∂
∂
∂
∂
∂
∂
⎛
⎞
⎛
⎞
+
β
+
τ
⎜
⎟
⎜
⎟
⎜
⎟
⎜
⎟
⎝
⎠
⎝
⎠
∂
∂
∂
∂
∂
∂
⎝
⎠
⎝
⎠
, , )
( , , )
( , , )
x
x
x z t
z
c x z t
c x z t
K
K
z
x
x
t
x
x
(7.6)
where β = 0.5S k σ w T l , for 0 < z < h, x > 0 and t > 0. Equation 7.6 is subjected to the
boundary conditions:
∂
=
=
∂
( , , ) 0 at
0,
z
c x z t
K
z
h
z
(7.6a)
to the initial condition:
( , ,0) 0 at = 0
c x z =
t
(7.6b)
to the source condition:
δ −
(0, , )
(
) at = 0
s
uc z t = Q z H
x
(7.6c)
and for far away from the source we have
∗
∗
∂
=
∂
( , , ) 0 at =
c L z t
x L
x
(7.6d)
where
c now represents the crosswind integrated concentration (g/m 2 )
h is the boundary layer height (m)
H s is the height of the source (m)
L * is far away from the source (m)
Q is the emission rate (g/s)
K x and K z are the longitudinal and vertical eddy diffusivities (m 2 /s), respectively
δ is the Dirac delta function
Here u
– , w
– are functions of height z and K x and K z are also functions of source
distance x.
In order to solve problem (Equation 7.6), taking advantage of the well-known
solution of the stationary problem with advection in the x-direction by the GILTT
method (Moreira et al., 2005b), we apply the Laplace transform technique in the t
variable. This procedure leads to the stationary problem:
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