140
Air Pollution and Turbulence: Modeling and Applications
mind, the main idea of the said approach is to obtain an eddy diffusivity scheme for
practical applications in air pollution modeling, which reveals the essential features
of turbulent diffusion, but which as far as possible preserves the simplicity and fl exibility of the K-theory formulation. Degrazia et al. (1997, 2000) propose the vertical
profi les of diffusion coeffi cients obtained by means of spectral techniques.
Several diffi culties arise in the evaluation of the transversal turbulent diffusion. It
is often, and not always correctly, hypothesized that,
h
y
K
K
≅
(5.13)
where K y is the transversal turbulent diffusion (supposing the wind is blowing in the
direction of x-axis).
Another problem in the simulation of horizontal dispersion is the numerical error
associated with the advection of the pollutant due to part of the mean wind. Such
error is linked to the size of the spatial grid used to schematize the diffusion fi eld,
and can turn out to be greater than parameter K h itself.
In the past, the most complex simulation techniques made widespread use of
dynamic grid models, in particular with the application of the numerical method of
fi nite differences and K closure. However, several major limitations of such applications have come to light:
1. The numerical approximation of the advection term often produces a fi ctitious diffusion.
2. K closure is a fundamentally incorrect approximation in strong turbulence
conditions.
3. Since the concentrations are calculated as spatial means within threedimensional cells of the grid, it is diffi cult to compare them with measurements carried out at single points in space.
4. It is diffi cult to link K eddy coeffi cients with experimental measurements in
the atmosphere.
5. A correct application of K closure requires that the grid dimensions be
smaller than those of the pollutant cloud, a condition that is diffi cult to
satisfy the proximity to the source.
5.3.5 BOX MODELS
Of the Eulerian models, box models constitute the most simple mathematical
approach, since they ignore the spatial structure of phenomena. They assume that
the pollutants are uniformly distributed within a parallelepiped. From a theoretical
point of view, this is equivalent to assuming infi nite diffusion coeffi cients, which
provoke an instantaneous propagation of the pollutant introduced into the box under
consideration. The pollutant present in the box originates from internal sources or
from external contributions transported by wind or fl ows through the summit as a
consequence of variations in height of the box itself, which generally coincides with
the ML height.
© 2010 by Taylor and Francis Group, LLC
Air Pollution and Turbulence: Modeling and Applications
mind, the main idea of the said approach is to obtain an eddy diffusivity scheme for
practical applications in air pollution modeling, which reveals the essential features
of turbulent diffusion, but which as far as possible preserves the simplicity and fl exibility of the K-theory formulation. Degrazia et al. (1997, 2000) propose the vertical
profi les of diffusion coeffi cients obtained by means of spectral techniques.
Several diffi culties arise in the evaluation of the transversal turbulent diffusion. It
is often, and not always correctly, hypothesized that,
h
y
K
K
≅
(5.13)
where K y is the transversal turbulent diffusion (supposing the wind is blowing in the
direction of x-axis).
Another problem in the simulation of horizontal dispersion is the numerical error
associated with the advection of the pollutant due to part of the mean wind. Such
error is linked to the size of the spatial grid used to schematize the diffusion fi eld,
and can turn out to be greater than parameter K h itself.
In the past, the most complex simulation techniques made widespread use of
dynamic grid models, in particular with the application of the numerical method of
fi nite differences and K closure. However, several major limitations of such applications have come to light:
1. The numerical approximation of the advection term often produces a fi ctitious diffusion.
2. K closure is a fundamentally incorrect approximation in strong turbulence
conditions.
3. Since the concentrations are calculated as spatial means within threedimensional cells of the grid, it is diffi cult to compare them with measurements carried out at single points in space.
4. It is diffi cult to link K eddy coeffi cients with experimental measurements in
the atmosphere.
5. A correct application of K closure requires that the grid dimensions be
smaller than those of the pollutant cloud, a condition that is diffi cult to
satisfy the proximity to the source.
5.3.5 BOX MODELS
Of the Eulerian models, box models constitute the most simple mathematical
approach, since they ignore the spatial structure of phenomena. They assume that
the pollutants are uniformly distributed within a parallelepiped. From a theoretical
point of view, this is equivalent to assuming infi nite diffusion coeffi cients, which
provoke an instantaneous propagation of the pollutant introduced into the box under
consideration. The pollutant present in the box originates from internal sources or
from external contributions transported by wind or fl ows through the summit as a
consequence of variations in height of the box itself, which generally coincides with
the ML height.
© 2010 by Taylor and Francis Group, LLC