Pollutant Dispersion Simulation in the ABL by the GILTT Method
181
where
c
– denotes the average concentration of a passive contaminant (g/m 3 )
u
– , v
– , w
– are the mean wind (m/s) components along the x-, y-, and z-axes,
respectively
S is the source term
The terms u c
′ ′ , v c
′ ′, and w c
′ ′ represent, respectively, the turbulent fl uxes of contaminants (g/s m 2 ) in the longitudinal, crosswind, and vertical directions.
Observe that Equation 7.1 has four unknown variables (the concentration c
– and
turbulent fl uxes) that lead us to the known turbulence closure problem. One of the
most widely used closures for Equation 7.1 is based on the gradient transport hypothesis (or K-theory), which, in analogy with Fick’s law of molecular diffusion, assumes
that turbulence causes a net movement of material down the gradient of material
concentration at a rate that is proportional to the magnitude of the gradient (Seinfeld
and Pandis, 1998). So
∂
∂
∂
= −
= −
= −
′ ′
′ ′
′ ′
∂
∂
∂
;
;
x
y
z
c
c
c
u c
K
v c
K
w c
K
x
y
z
(7.2)
where K x , K y , and K z are the Cartesian components of eddy diffusivity (m 2 /s) in the
x-, y-, and z-directions, respectively. In the fi rst-order closure, all the information on
the turbulence complexity is contained in the eddy diffusivities.
Equation 7.2, combined with the continuity equation of mass, leads to the
advection–diffusion equation. For a Cartesian coordinate system, we rewrite the
advection–diffusion equation like (Blackadar, 1997):
⎛
⎞
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
⎛
⎞
⎛
⎞
+
+
+
=
+
+
+
⎜
⎟
⎜
⎟
⎜
⎟
⎝
⎠
⎝
⎠
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
⎝
⎠
x
y
z
c
c
c
c
c
c
c
u
v
w
K
K
K
S
t
x
y
z
x
x
y
y
z
z
(7.3)
The simplicity of the K-theory of turbulent diffusion has led to the widespread use of
this theory as mathematical basis for simulating pollutant dispersion (open country,
urban, photochemical pollution, etc.). But K-closure has its own limits. In contrast to
molecular diffusion, turbulent diffusion is scale-dependent. This means that the rate
of diffusion of a cloud of material generally depends on the cloud dimensions and
the intensity of turbulence. As the cloud grows, larger eddies are incorporated in the
expansion process, so that a progressively larger fraction of turbulent kinetic energy
is available for the cloud expansion.
Another problem is that the down-gradient transport hypothesis is inconsistent
with observed features of turbulent diffusion in the upper portion of the mixed
layer, at convective cases where counter-gradient material fl uxes are known to occur
(Deardoff and Willis, 1975). Because counter-gradient fl uxes are thought to be indicative of boundary layer scale eddies, as opposed to small-scale ones, such fl uxes
© 2010 by Taylor and Francis Group, LLC
181
where
c
– denotes the average concentration of a passive contaminant (g/m 3 )
u
– , v
– , w
– are the mean wind (m/s) components along the x-, y-, and z-axes,
respectively
S is the source term
The terms u c
′ ′ , v c
′ ′, and w c
′ ′ represent, respectively, the turbulent fl uxes of contaminants (g/s m 2 ) in the longitudinal, crosswind, and vertical directions.
Observe that Equation 7.1 has four unknown variables (the concentration c
– and
turbulent fl uxes) that lead us to the known turbulence closure problem. One of the
most widely used closures for Equation 7.1 is based on the gradient transport hypothesis (or K-theory), which, in analogy with Fick’s law of molecular diffusion, assumes
that turbulence causes a net movement of material down the gradient of material
concentration at a rate that is proportional to the magnitude of the gradient (Seinfeld
and Pandis, 1998). So
∂
∂
∂
= −
= −
= −
′ ′
′ ′
′ ′
∂
∂
∂
;
;
x
y
z
c
c
c
u c
K
v c
K
w c
K
x
y
z
(7.2)
where K x , K y , and K z are the Cartesian components of eddy diffusivity (m 2 /s) in the
x-, y-, and z-directions, respectively. In the fi rst-order closure, all the information on
the turbulence complexity is contained in the eddy diffusivities.
Equation 7.2, combined with the continuity equation of mass, leads to the
advection–diffusion equation. For a Cartesian coordinate system, we rewrite the
advection–diffusion equation like (Blackadar, 1997):
⎛
⎞
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
⎛
⎞
⎛
⎞
+
+
+
=
+
+
+
⎜
⎟
⎜
⎟
⎜
⎟
⎝
⎠
⎝
⎠
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
⎝
⎠
x
y
z
c
c
c
c
c
c
c
u
v
w
K
K
K
S
t
x
y
z
x
x
y
y
z
z
(7.3)
The simplicity of the K-theory of turbulent diffusion has led to the widespread use of
this theory as mathematical basis for simulating pollutant dispersion (open country,
urban, photochemical pollution, etc.). But K-closure has its own limits. In contrast to
molecular diffusion, turbulent diffusion is scale-dependent. This means that the rate
of diffusion of a cloud of material generally depends on the cloud dimensions and
the intensity of turbulence. As the cloud grows, larger eddies are incorporated in the
expansion process, so that a progressively larger fraction of turbulent kinetic energy
is available for the cloud expansion.
Another problem is that the down-gradient transport hypothesis is inconsistent
with observed features of turbulent diffusion in the upper portion of the mixed
layer, at convective cases where counter-gradient material fl uxes are known to occur
(Deardoff and Willis, 1975). Because counter-gradient fl uxes are thought to be indicative of boundary layer scale eddies, as opposed to small-scale ones, such fl uxes
© 2010 by Taylor and Francis Group, LLC