Page 371 - Fundamentals of Air Pollution 3E

326 20. Air Pollution Modeling and Prediction

level, such as automobile pollutants in urban areas, and to continuous or
instantaneous ground-level area sources. It is not appropriate for elevated
point source diffusion until the plume has grown larger than the space
scale. Numerical rather than analytical solutions of Eq. (20-18) are used.
Errors in advection may completely overshadow diffusion. The amplifi-
cation of random errors with each succeeding step causes numerical instabil-
ity (or distortion). Higher-order differencing techniques are used to avoid
this instability, but they may result in sharp gradients, which may cause
negative concentrations to appear in the computations. Many of the numeri-
cal instability (distortion) problems can be overcome with a second-moment
scheme (9) which advects the moments of the distributions instead of the
pollutants alone. Six numerical techniques were investigated (10), including
the second-moment scheme; three were found that limited numerical distor-
tion: the second-moment, the cubic spline, and the chapeau function.
In the application of gradient transfer methods, horizontal diffusion is
frequently ignored, but the variation in vertical diffusivity must be approxi-
mated (11-14).

D. Trajectory Models
In its most common form, a trajectory model moves a vertical column,
with a square cross section intersecting the ground, at the mean wind
speed, with pollutants added to the bottom of the column as they are
generated by each location over which the column passes. Treatment of
vertical dispersion varies among models, from those which assume immedi-
ate vertical mixing throughout the column to those which assume vertical
dispersion using a vertical coefficient K z with a suitable profile (15).
Modeling a single parcel of air as it is being moved along allows the
chemical reactions in the parcel to be modeled. A further advantage of
trajectory models is that only one trajectory is required to estimate the
concentration at a given endpoint. This minimizes calculation because con-
centrations at only a limited number of points are required, such as at
stations where air quality is routinely monitored. Since wind speed and
direction at the top and the bottom of the column are different, the column
is skewed from the vertical. However, for computational purposes, the
column is usually assumed to remain vertical and to be moved at the wind
speed and direction near the surface. This is acceptable for urban application
in the daytime, when winds are relatively uniform throughout the lower
atmosphere.
Trajectory models of a different sort are used for long-range transport,
because it is necessary to simulate transport throughout a diurnal cycle in
which the considerable wind shear at night transports pollutants in different
directions. Expanding Gaussian puffs can be used, with the expanded puff
breaking, at the time of maximum vertical mixing, into a series of puffs