Mathematical Air Pollution Models: Eulerian Models
135
The new u and c are introduced into Equation 5.1; after several calculations and
hypothesizing a wind with divergence nil, the following is obtained:
∂ = − ⋅∇ − ∇⋅
+ ∇ +
′ ′
∂
2
c
c
c
D c S
t
u
u
(5.3)
This equation includes some new variables (those with an apex) whose values are
unknown. The appearance of new terms in equations for mean quantities leads to
a number of unknowns greater than the number of equations. Thus, the system of
equations is not closed and is therefore irresolvable. To close it, in fact, new equations of variance and covariance (second-order moments) would be required, but this
would only shift the problem to a higher order since it would yield further unknown
quantities that are third-order moments. Now, if it were decided to fi nd equations for
the third-order moments, this would yield unknowns of a higher order, that is, fourthorder moments, requiring the introduction of new equations. Iterating the procedure,
the conclusion would be reached that the number of unknowns is always greater than
the number of equations. A solution to this problem consists of utilizing only a fi nite
number of equations, relative to a certain number of unknowns, parameterizing the
remaining ones in terms of known quantities.
The most classic and widely used approach to obviate this problem is the parameterization of second-order moments, assuming a hypothetical analogy between
molecular diffusion and the turbulent transfers. Such approach is referred to as the
K-theory or fl ux-gradient theory, as it assumes that the fl ow of a given fi eld is proportional to the gradient of an appropriate mean variable. This is a fi rst-order closure
of the set of equations under examination, since it conserves the equations relative to
the fi rst moments and parameterizes the second moments:
= − ∇
′ ′
c
K c
u
(5.4)
where K is the eddy diffusivity coeffi cient.
The simplicity of the K-theory of turbulent diffusion has led to its widespread use
as the mathematical basis for simulating urban, photochemical pollution. However,
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. However, eddies
much larger than the cloud itself are relatively unimportant in its expansion. Thus, the
gradient-transfer theory works well when the dimension of dispersed material is much
larger than the size of turbulent eddies involved in the diffusion process, that is, for
ground-level emissions and for large travel times. Strictly speaking, one should introduce
a diffusion coeffi cient function not only of atmospheric stability and emission height but
also of the travel time or distance from source. However, such time-dependence makes
it diffi cult to treat the diffusion equation in a fi xed-coordinate system where multiple
sources have to be treated simultaneously. Otherwise, one should limit the application
of the gradient theory to large travel times (Pasquill and Smith, 1983). A further problem
© 2010 by Taylor and Francis Group, LLC
135
The new u and c are introduced into Equation 5.1; after several calculations and
hypothesizing a wind with divergence nil, the following is obtained:
∂ = − ⋅∇ − ∇⋅
+ ∇ +
′ ′
∂
2
c
c
c
D c S
t
u
u
(5.3)
This equation includes some new variables (those with an apex) whose values are
unknown. The appearance of new terms in equations for mean quantities leads to
a number of unknowns greater than the number of equations. Thus, the system of
equations is not closed and is therefore irresolvable. To close it, in fact, new equations of variance and covariance (second-order moments) would be required, but this
would only shift the problem to a higher order since it would yield further unknown
quantities that are third-order moments. Now, if it were decided to fi nd equations for
the third-order moments, this would yield unknowns of a higher order, that is, fourthorder moments, requiring the introduction of new equations. Iterating the procedure,
the conclusion would be reached that the number of unknowns is always greater than
the number of equations. A solution to this problem consists of utilizing only a fi nite
number of equations, relative to a certain number of unknowns, parameterizing the
remaining ones in terms of known quantities.
The most classic and widely used approach to obviate this problem is the parameterization of second-order moments, assuming a hypothetical analogy between
molecular diffusion and the turbulent transfers. Such approach is referred to as the
K-theory or fl ux-gradient theory, as it assumes that the fl ow of a given fi eld is proportional to the gradient of an appropriate mean variable. This is a fi rst-order closure
of the set of equations under examination, since it conserves the equations relative to
the fi rst moments and parameterizes the second moments:
= − ∇
′ ′
c
K c
u
(5.4)
where K is the eddy diffusivity coeffi cient.
The simplicity of the K-theory of turbulent diffusion has led to its widespread use
as the mathematical basis for simulating urban, photochemical pollution. However,
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. However, eddies
much larger than the cloud itself are relatively unimportant in its expansion. Thus, the
gradient-transfer theory works well when the dimension of dispersed material is much
larger than the size of turbulent eddies involved in the diffusion process, that is, for
ground-level emissions and for large travel times. Strictly speaking, one should introduce
a diffusion coeffi cient function not only of atmospheric stability and emission height but
also of the travel time or distance from source. However, such time-dependence makes
it diffi cult to treat the diffusion equation in a fi xed-coordinate system where multiple
sources have to be treated simultaneously. Otherwise, one should limit the application
of the gradient theory to large travel times (Pasquill and Smith, 1983). A further problem
© 2010 by Taylor and Francis Group, LLC