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11.2 General Gaussian Dispersion Model 321

temperatures, and humidity. Less obvious, but equally important, are vertical
motions that influence air parcel motion. Atmospheric stability affects the vertical
motion of air parcels. The temperature difference between the air parcel and
atmosphere causes vertical motion, at least near the emission source, but the con-
vective circulation thus established is affected directly by the stability of the
atmosphere. Winds tend to be turbulent and gusty when the atmosphere is unstable,
and this type of weather causes air pollutants to disperse erratically. Subsidence
occurs in larger scale vertical circulation as air from high-pressure areas replaces
that carried aloft in adjacent low-pressure systems. This often brings very dry air
from a high altitude to a low level.

11.2.4 Adiabatic Lapse Rate of Temperature

In the analysis above, we assumed constant air temperature and molar weight,
however, both temperature T and molar weight M change with elevation, and the
change in M is not as important as that of temperature. For an adiabatic air parcel in
the atmosphere, it may produce work to the surroundings. The rate of temperature
change over elevation can be derived from the first law of thermodynamics [21]as

dT g
¼ ð11:10Þ
dz c p;a þ h dw=dTð Þ
fg
where c p;a is the air heating capacity, w is the water vapor mass fraction in the air
(kg vapor/air). Since most air contains less than a few percent of water vapor,
except for those in cloud or fogs, the effect of water vapor can be ignored and we
can use a simpler equation that is meant to be for dry air

dT g
¼ ð11:11Þ
dz c p;a
This is also called the adiabatic lapse rate of dry air temperature. Temperature
decreases with elevation and it is a straight line in a plot of elevation versus
temperature (Fig. 11.4). With a typical air heating capacity of c p;a ¼ 1;006 J/kg.K
2
and g ¼ 9.81 m/s , we can estimate the adiabatic lapse rate of dry air temperature
dT ¼ 0.975 or 9.75 K/km. This is an estimation for guidance only. The actual value
dz
changes with location and time. As dry air moves vertically, its temperature
changes at about 1 °C per 100 m [21]. For air with moisture, the temperature change
could be 5–10 °C per km, depending on its water content, and air up in the sky
reaches saturation easily to form clouds.
The change of temperature over elevation is the main reason behind atmospheric
stability, which is an important factor in air dispersion.