38
Air Pollution and Turbulence: Modeling and Applications
time (or distance) from the source and just a function of the turbulence (e.g., large
eddy length and velocity scales). The eddy diffusivity Equation 3.5 can accurately
represent the near source diffusion in weak winds. For this, eddy diffusivities should
be considered as functions of not only turbulence but also distance from the source
(Arya, 1995).
For short times
L
(
)
i
t
T
<<
, ρ τ ≈
L ( ) 1
i
, and
=
2
2 2
i
i
X
v t
(3.14)
the fl uid particle plume growth is linear with time.
Kampé de Fériet expressed Taylor’s equation in the form of Equation 3.7 which
can be recast into
⎛
⎞
=
− ξρ ξ ξ
⎜
⎟
⎝
⎠
∫
L
2
2
2
L
L
L
0
2
( ) d
i
i
i
i
X UT
i
i
X
X
v T
UT
(3.15)
where
U is the horizontal mean wind velocity
X is the particle mean displacement X = Ut
ξ is the rescaled time τ L i
T .
From Equation 3.15 we obtain the nondimensionalized formula
L
2
L
2
2
L
L
0
( )d
2
i
i
i
i
t T
i
i
X
t
T
T
v
⎛
⎞
=
−ξ ρ ξ ξ
⎜
⎟
⎝
⎠
∫
(3.16)
Equation 3.16 yields the nondimensionalized mean square generalized displacement
2
i
X as a universal function of the normalized time L i
t T (Degrazia et al., 1991)
⎛
⎞
= ⎜
⎟
⎝
⎠
2
2
2
L
L
2
i
i
i
i
i
X
t
g
T
v T
(3.17)
It is worth recalling at this point that Equation 3.17 yields the form of the universal
function for the nondimensionalized dispersion as proposed earlier by Draxler (1976).
Notice also that the left-hand side of Equation 3.17 is normalized in terms of the
velocity fl uctuation variance and its Lagrangian integral timescale. The right-hand
side of Equation 3.17 yields the form of the universal function
L
( / )
i
i
g t T . In this latter
form, Taylor’s model suggests that the relevant parameters for the determination of
2
i
X are the Lagrangian quantities
2
i
v and L i
T .
3.1.1.1 Some Considerations about Taylor’s Model
In the analysis of the diffusion of tracers in the PBL, Equation 3.7 can be explored
in two complementary ways. In a fi rst approach, one aims at predicting values and/
or forms for the mean square generalized displacement
2 ( )
i
X t by assuming (physically) reasonable expressions for the Lagrangian velocity variances σ
2
( )
i
and for
© 2010 by Taylor and Francis Group, LLC
Air Pollution and Turbulence: Modeling and Applications
time (or distance) from the source and just a function of the turbulence (e.g., large
eddy length and velocity scales). The eddy diffusivity Equation 3.5 can accurately
represent the near source diffusion in weak winds. For this, eddy diffusivities should
be considered as functions of not only turbulence but also distance from the source
(Arya, 1995).
For short times
L
(
)
i
t
T
<<
, ρ τ ≈
L ( ) 1
i
, and
=
2
2 2
i
i
X
v t
(3.14)
the fl uid particle plume growth is linear with time.
Kampé de Fériet expressed Taylor’s equation in the form of Equation 3.7 which
can be recast into
⎛
⎞
=
− ξρ ξ ξ
⎜
⎟
⎝
⎠
∫
L
2
2
2
L
L
L
0
2
( ) d
i
i
i
i
X UT
i
i
X
X
v T
UT
(3.15)
where
U is the horizontal mean wind velocity
X is the particle mean displacement X = Ut
ξ is the rescaled time τ L i
T .
From Equation 3.15 we obtain the nondimensionalized formula
L
2
L
2
2
L
L
0
( )d
2
i
i
i
i
t T
i
i
X
t
T
T
v
⎛
⎞
=
−ξ ρ ξ ξ
⎜
⎟
⎝
⎠
∫
(3.16)
Equation 3.16 yields the nondimensionalized mean square generalized displacement
2
i
X as a universal function of the normalized time L i
t T (Degrazia et al., 1991)
⎛
⎞
= ⎜
⎟
⎝
⎠
2
2
2
L
L
2
i
i
i
i
i
X
t
g
T
v T
(3.17)
It is worth recalling at this point that Equation 3.17 yields the form of the universal
function for the nondimensionalized dispersion as proposed earlier by Draxler (1976).
Notice also that the left-hand side of Equation 3.17 is normalized in terms of the
velocity fl uctuation variance and its Lagrangian integral timescale. The right-hand
side of Equation 3.17 yields the form of the universal function
L
( / )
i
i
g t T . In this latter
form, Taylor’s model suggests that the relevant parameters for the determination of
2
i
X are the Lagrangian quantities
2
i
v and L i
T .
3.1.1.1 Some Considerations about Taylor’s Model
In the analysis of the diffusion of tracers in the PBL, Equation 3.7 can be explored
in two complementary ways. In a fi rst approach, one aims at predicting values and/
or forms for the mean square generalized displacement
2 ( )
i
X t by assuming (physically) reasonable expressions for the Lagrangian velocity variances σ
2
( )
i
and for
© 2010 by Taylor and Francis Group, LLC