136
Air Pollution and Turbulence: Modeling and Applications
is that the down-gradient transport hypothesis is inconsistent with observed features of
turbulent diffusion in the upper portion of the mixed layer (ML), where counter-gradient
material fl uxes are known to occur (Deardoff and Willis, 1975).
In addition, unlike molecular diffusion, turbulent diffusion is not a property of
fl uids, but of the turbulence itself or of fl ows, and it may vary greatly from one fl ow
to another and from one region to another of the same fl ow. The above relations are
essentially based only on a qualitative analogy between molecular and turbulent diffusion. For the fi rst-order closure to be realistic, the mean concentration fi eld must
have a much larger timescale than that of turbulent transport.
Despite these well-known limits, the K-closure is widely used in several atmospheric conditions, because it describes the diffusive transport in an Eulerian framework, where almost all measurements are Eulerian in character. It produces results
that agree with experimental data as well as any more complex model, and it is not
as computationally expensive as higher-order closures.
The reliability of the K-approach strongly depends on the way the eddy diffusivity
is determined on the basis of the turbulence structure of the PBL, and on the model’s
ability to reproduce experimental diffusion data. A great variety of formulations
exist (Ulke, 2000). Most of them are based on similarity theory, and give different
results for the same atmospheric stability, as well as discontinuities and jumps at the
transition between different stability regimes of the PBL.
The tensor K (3 × 3) of turbulent diffusion, whose elements can be extrapolated
from experimental measurements, is introduced in Equation 5.3. Then, by also applying the following approximations:
The K tensor is diagonal.
•
The molecular diffusion is negligible.
•
c
•
represents the concentration of a nonreactive pollutant (thus
–
S = S).
Equation 5.3 can be written in the form:
∂ = − ⋅∇ + ∇⋅ ∇ +
∂
c
c
K c S
t
u
(5.5)
Equation 5.5 can be integrated (analytically or numerically) if input data for u, K,
and S are provided, together with the initial and boundary conditions for
– c .
Eulerian models and K models mainly differ in the functions utilized for the K
coeffi cients and the techniques used for the integration of Equation 5.5.
Equation 5.5 can be resolved in two ways:
1. With analytic methods, obtaining exact solutions
2. With numerical methods, obtaining approximate solutions
5.3.2 ANALYTICAL SOLUTIONS
Analytical solutions of equations are of fundamental importance in understanding and
describing physical phenomena. Analytical solutions (as opposed to numerical ones)
© 2010 by Taylor and Francis Group, LLC
Air Pollution and Turbulence: Modeling and Applications
is that the down-gradient transport hypothesis is inconsistent with observed features of
turbulent diffusion in the upper portion of the mixed layer (ML), where counter-gradient
material fl uxes are known to occur (Deardoff and Willis, 1975).
In addition, unlike molecular diffusion, turbulent diffusion is not a property of
fl uids, but of the turbulence itself or of fl ows, and it may vary greatly from one fl ow
to another and from one region to another of the same fl ow. The above relations are
essentially based only on a qualitative analogy between molecular and turbulent diffusion. For the fi rst-order closure to be realistic, the mean concentration fi eld must
have a much larger timescale than that of turbulent transport.
Despite these well-known limits, the K-closure is widely used in several atmospheric conditions, because it describes the diffusive transport in an Eulerian framework, where almost all measurements are Eulerian in character. It produces results
that agree with experimental data as well as any more complex model, and it is not
as computationally expensive as higher-order closures.
The reliability of the K-approach strongly depends on the way the eddy diffusivity
is determined on the basis of the turbulence structure of the PBL, and on the model’s
ability to reproduce experimental diffusion data. A great variety of formulations
exist (Ulke, 2000). Most of them are based on similarity theory, and give different
results for the same atmospheric stability, as well as discontinuities and jumps at the
transition between different stability regimes of the PBL.
The tensor K (3 × 3) of turbulent diffusion, whose elements can be extrapolated
from experimental measurements, is introduced in Equation 5.3. Then, by also applying the following approximations:
The K tensor is diagonal.
•
The molecular diffusion is negligible.
•
c
•
represents the concentration of a nonreactive pollutant (thus
–
S = S).
Equation 5.3 can be written in the form:
∂ = − ⋅∇ + ∇⋅ ∇ +
∂
c
c
K c S
t
u
(5.5)
Equation 5.5 can be integrated (analytically or numerically) if input data for u, K,
and S are provided, together with the initial and boundary conditions for
– c .
Eulerian models and K models mainly differ in the functions utilized for the K
coeffi cients and the techniques used for the integration of Equation 5.5.
Equation 5.5 can be resolved in two ways:
1. With analytic methods, obtaining exact solutions
2. With numerical methods, obtaining approximate solutions
5.3.2 ANALYTICAL SOLUTIONS
Analytical solutions of equations are of fundamental importance in understanding and
describing physical phenomena. Analytical solutions (as opposed to numerical ones)
© 2010 by Taylor and Francis Group, LLC