182
Air Pollution and Turbulence: Modeling and Applications
are often called nonlocal fl uxes. Local K-theory is a method for parameterizing the
effects of turbulent mixing based on how small eddies will mix quantities along a
local gradient of the transported quantity.
Some decades ago, it was noted that in the upper part of convectively driven
boundary layers, the fl ux of scalars are counter to the gradient of the mean scalar profi le (Deardoff, 1966). The mean potential temperature gradient and the fl ux
change sign at different levels introduce a certain region in the convective boundary
layer (CBL) where they have the same sign. This was in contrast with the common
view in fi rst-order turbulent closure that turbulent diffusion is down gradient. In
order to describe diffusion also in these regions, Ertel (1942) and Deardoff (1966,
1972) proposed to modify the usual applied fl ux–gradient relationship in K-theory
approach according to
∂
⎛
⎞
= −
− γ
′ ′
⎜
⎟
⎝
⎠
∂
z
c
w c
K
z
(7.4)
where γ represents the counter-gradient term.
Many schemes and parameterizations for counter-gradient term have been
developed (e.g., Wyngaard and Brost, 1984; Fiedler and Moeng, 1985; Holtslag
and Moeng, 1991; Wyngaard and Weil, 1991; Holtslag and Boville, 1993; Hamba,
1993; Robson and Mayocchi, 1994; Zilitinkevich et al., 1999). In this chapter,
without losing generality, we use the parameterization proposed by van Dop and
Verver (2001), which is based on the work of Wyngaard and Weil (1991):
⎡
⎤
σ
∂
∂
∂
⎛
⎞
+
+ τ
= −
′ ′
⎢
⎥
⎜
⎟
⎝
⎠∂
∂
∂
⎣
⎦
L
1
2
w
k
w
z
S
c
T
w c
K
z
t
z
(7.5)
where
S k is the skewness of the vertical turbulent velocity (w′), that is, = ′
′
3
2 3 / 2
/( )
k
S w w
σ w is the vertical turbulent velocity standard deviation (m/s)
T Lw is the Lagrangian timescale (s)
τ is the relaxation time (s)
The second term in the operator (in the brackets) represents the nonlocal countergradient term.
Using Equations 7.4 and 7.5, the turbulence closure problem is solved without
obeying Fick’s law, being called non-Fickian closure (also known as nonlocal closure). The non-Fickian closure allows the investigation of more energetic eddies in
different heights and the effect of the asymmetric transport in the computation of
the pollutant concentration considering in a more complete way the structure of the
turbulent dispersion.
Applying Equations 7.2 and 7.5 in Equation 7.1, the crosswind integrated transient
advection–diffusion equation, in the Eulerian framework, for a Cartesian coordinate
system in which the x-direction coincide with that of the average wind, is written as
(Buske et al., 2009)
© 2010 by Taylor and Francis Group, LLC
Air Pollution and Turbulence: Modeling and Applications
are often called nonlocal fl uxes. Local K-theory is a method for parameterizing the
effects of turbulent mixing based on how small eddies will mix quantities along a
local gradient of the transported quantity.
Some decades ago, it was noted that in the upper part of convectively driven
boundary layers, the fl ux of scalars are counter to the gradient of the mean scalar profi le (Deardoff, 1966). The mean potential temperature gradient and the fl ux
change sign at different levels introduce a certain region in the convective boundary
layer (CBL) where they have the same sign. This was in contrast with the common
view in fi rst-order turbulent closure that turbulent diffusion is down gradient. In
order to describe diffusion also in these regions, Ertel (1942) and Deardoff (1966,
1972) proposed to modify the usual applied fl ux–gradient relationship in K-theory
approach according to
∂
⎛
⎞
= −
− γ
′ ′
⎜
⎟
⎝
⎠
∂
z
c
w c
K
z
(7.4)
where γ represents the counter-gradient term.
Many schemes and parameterizations for counter-gradient term have been
developed (e.g., Wyngaard and Brost, 1984; Fiedler and Moeng, 1985; Holtslag
and Moeng, 1991; Wyngaard and Weil, 1991; Holtslag and Boville, 1993; Hamba,
1993; Robson and Mayocchi, 1994; Zilitinkevich et al., 1999). In this chapter,
without losing generality, we use the parameterization proposed by van Dop and
Verver (2001), which is based on the work of Wyngaard and Weil (1991):
⎡
⎤
σ
∂
∂
∂
⎛
⎞
+
+ τ
= −
′ ′
⎢
⎥
⎜
⎟
⎝
⎠∂
∂
∂
⎣
⎦
L
1
2
w
k
w
z
S
c
T
w c
K
z
t
z
(7.5)
where
S k is the skewness of the vertical turbulent velocity (w′), that is, = ′
′
3
2 3 / 2
/( )
k
S w w
σ w is the vertical turbulent velocity standard deviation (m/s)
T Lw is the Lagrangian timescale (s)
τ is the relaxation time (s)
The second term in the operator (in the brackets) represents the nonlocal countergradient term.
Using Equations 7.4 and 7.5, the turbulence closure problem is solved without
obeying Fick’s law, being called non-Fickian closure (also known as nonlocal closure). The non-Fickian closure allows the investigation of more energetic eddies in
different heights and the effect of the asymmetric transport in the computation of
the pollutant concentration considering in a more complete way the structure of the
turbulent dispersion.
Applying Equations 7.2 and 7.5 in Equation 7.1, the crosswind integrated transient
advection–diffusion equation, in the Eulerian framework, for a Cartesian coordinate
system in which the x-direction coincide with that of the average wind, is written as
(Buske et al., 2009)
© 2010 by Taylor and Francis Group, LLC