Mathematical Air Pollution Models: Eulerian Models
133
Finally, models can be classifi ed on the basis of the time resolution of the concentrations produced:
Episodic models (temporal resolution of less than an hour)
•
Short-time models (temporal resolutions greater than or equal to an hour,
•
and less than or equal to 24 h)
Climatological models (with resolution greater than 24 h, generally sea•
sonal or annual)
5.3 THEORETICAL CHARACTERISTICS OF
MATHEMATICAL MODELS
The theoretical approach to the problem essentially assumes four basic forms. In the
K approach, diffusion is considered, at a fi xed point in space, proportional to the local
gradient of the concentration of the diffused material. Consequently, it is fundamentally Eulerian since it considers the motion of fl uid within a spatially fi xed system of
reference. Such models are most suited to confronting complex problems, for example, the dispersion of pollutants over complex terrain or the diffusion of noninert pollutants. They are based on the numerical resolution, on a fi xed spatial-temporal grid,
of the equation of the mass conservation of the pollutant chemical species.
Among the Eulerian models, box models constitute the most simple mathematical
approach since they neglect the spatial structure of phenomena. They assume that the
pollutants are uniformly distributed within a parallelepiped (box). From the theoretical
viewpoint, this is equivalent to assuming infi nite diffusion coeffi cients that provoke
an instantaneous propagation of the pollutant within the considered box. The pollutant present in the box originates from internal sources or from external contributions
transported by the wind or fl ows across the summit due to variations in the height of
the box itself, which generally coincides with the height of the mixing layer.
Lagrangian models differ from Eulerian ones in adopting a system of reference
that follows atmospheric motions. Initially, the term Lagrangian was used only
to refer to the box or moving box models that followed the mean wind trajectory.
Currently, this class includes all models that decompose the pollutant cloud into discrete “elements,” such as segments, puffs, or computer particles. In particle models,
pollutant dispersion is simulated through the motion of computer particles whose
trajectories allow the calculation of the concentration fi eld of the emitted substance.
The underlying hypothesis is that the combination of the trajectories of such particles
to simulate the paths of the air particles situated, at the initial moment, in the same
position. The motion of the particles can be reproduced both in a deterministic and
in a stochastic way. Gaussian models are theoretically based upon an exact, but not
realistic, solution of the equation of transport and diffusion in the atmosphere, in
cases where both wind and turbulent diffusion coeffi cients are constant with height.
The solution is forced to represent real situations by means of empirical parameters, referred to as “sigmas.” They can be either stationary (the time-independent
plume models) or time-dependent (puff models). The name given to these models
is derived from the fact that the pollutant distribution, both vertical and transverse
to wind direction, is described by the famous curve discovered by the physicistmathematician Gauss. The various versions of Gaussian models essentially differ in
© 2010 by Taylor and Francis Group, LLC
133
Finally, models can be classifi ed on the basis of the time resolution of the concentrations produced:
Episodic models (temporal resolution of less than an hour)
•
Short-time models (temporal resolutions greater than or equal to an hour,
•
and less than or equal to 24 h)
Climatological models (with resolution greater than 24 h, generally sea•
sonal or annual)
5.3 THEORETICAL CHARACTERISTICS OF
MATHEMATICAL MODELS
The theoretical approach to the problem essentially assumes four basic forms. In the
K approach, diffusion is considered, at a fi xed point in space, proportional to the local
gradient of the concentration of the diffused material. Consequently, it is fundamentally Eulerian since it considers the motion of fl uid within a spatially fi xed system of
reference. Such models are most suited to confronting complex problems, for example, the dispersion of pollutants over complex terrain or the diffusion of noninert pollutants. They are based on the numerical resolution, on a fi xed spatial-temporal grid,
of the equation of the mass conservation of the pollutant chemical species.
Among the Eulerian models, box models constitute the most simple mathematical
approach since they neglect the spatial structure of phenomena. They assume that the
pollutants are uniformly distributed within a parallelepiped (box). From the theoretical
viewpoint, this is equivalent to assuming infi nite diffusion coeffi cients that provoke
an instantaneous propagation of the pollutant within the considered box. The pollutant present in the box originates from internal sources or from external contributions
transported by the wind or fl ows across the summit due to variations in the height of
the box itself, which generally coincides with the height of the mixing layer.
Lagrangian models differ from Eulerian ones in adopting a system of reference
that follows atmospheric motions. Initially, the term Lagrangian was used only
to refer to the box or moving box models that followed the mean wind trajectory.
Currently, this class includes all models that decompose the pollutant cloud into discrete “elements,” such as segments, puffs, or computer particles. In particle models,
pollutant dispersion is simulated through the motion of computer particles whose
trajectories allow the calculation of the concentration fi eld of the emitted substance.
The underlying hypothesis is that the combination of the trajectories of such particles
to simulate the paths of the air particles situated, at the initial moment, in the same
position. The motion of the particles can be reproduced both in a deterministic and
in a stochastic way. Gaussian models are theoretically based upon an exact, but not
realistic, solution of the equation of transport and diffusion in the atmosphere, in
cases where both wind and turbulent diffusion coeffi cients are constant with height.
The solution is forced to represent real situations by means of empirical parameters, referred to as “sigmas.” They can be either stationary (the time-independent
plume models) or time-dependent (puff models). The name given to these models
is derived from the fact that the pollutant distribution, both vertical and transverse
to wind direction, is described by the famous curve discovered by the physicistmathematician Gauss. The various versions of Gaussian models essentially differ in
© 2010 by Taylor and Francis Group, LLC