AIR POLLUTION METEOROLOGY 61
is no wind at ground level, and there usually is some wind
above the ground, mechanical turbulence just above the
ground is common. This type of turbulence increases with
increasing wind speed (at a given height) and is greater over
rough terrain than over smooth terrain. The terrain rough-
ness is usually characterized by a “roughness length” z 0
which varies from about 0.1 cm over smooth sand to a few
meters over cities. This quantity does not measure the actual
height of the roughness elements; rather it is proportional
to the size of the eddies that can exist among the roughness
elements. Thus, if the roughness elements are close together,
z 0 is relatively small.
The relative importance of heat convection and mechan-
ical turbulence is often characterized by the Richardson
number, Ri. Actually, – Ri is a measure of the relative rate
of production of convective and mechanical energy. For
example, negative Richardson numbers of large magnitude
indicate that convection predominates; in this situation, the
winds are weak, and there is strong vertical motion. Smoke
leaving a source spreads rapidly, both vertically and later-
ally (Figure 2). As the mechanical turbulence increases, the
Richardson number approaches zero, and the angular disper-
sion decreases. Finally, as the Richardson number becomes
positive, the stratification becomes stable and damps the
mechanical turbulence. For Richardson numbers above 0.25
(strong inversions, weak winds), vertical mixing effectively
disappears, and only weak horizontal eddies remain.
Because the Richardson number plays such an important
role in the theory of atmospheric turbulence and dispersion,
Table 1 gives a qualitative summary of the implication of
Richardson numbers of various magnitudes.It has been possible to describe the effect of roughness
length, wind speed, and Richardson number on many of the
statistical characteristics of the eddies near the ground quan-
titatively. In particular, the standard deviation of the vertical
wind direction is given by an equation of the form:s
u c
fRi
ln z z Ri()
()./ (^0)
(1)
Here z is height and f ( Ri ) and c ( Ri ) are known functions
of the Richardson number which increase as the Richardson
number decreases. The standard deviation of vertical wind
direction plays an important role in air pollution, because it
determines the initial angular spread of a plume in the verti-
cal. If it is large, the pollution spreads rapidly in the vertical.
It turns out that under such conditions, the contaminant also
spreads rapidly sideways, so that the central concentrations
decrease rapidly downstream. If s (^) u is small, there is negli-
gible spreading.
Equation 1 states that the standard deviation of vertical
wind direction does not explicitly depend on the wind speed,
but at a given height, depends only on terrain roughness and
Richardson number. Over rough terrain, vertical spreading is
faster than over smooth terrain. The variation with Richardson
number given in Eq. (1) gives the variation of spreading with
the type of turbulence as indicated in Table 1: greatest verti-
cal spreading with negative Ri with large numerical values,
less spreading in mechanical turbulence ( Ri 0), and negli-
gible spreading on stable temperature stratification with little
wind change in the vertical.
An equation similar to Eq. (1) governs the standard devi-
ation of horizontal wind direction. Generally, this is some-
what larger than s (^) u. For light-wind, stable conditions, we do
not know how to estimate s (^) u. Large s (^) u are often observed,
particularly for Ri 0.25. These cause volume meanders,
and are due to gravity waves or other large-sclae phenomena,
which are not related to the usual predictors.
In summary, then, dispersion of a plume from a continu-
ous elevated source in all directions increases with increasing
roughness, and with increasing convection relative to mechan-
ical turbulence. It would then be particularly strong on a clear
day, with a large lapse rate and a weak wind, particularly weak
in an inversion, and intermediate in mechanical turbulence
a) Ri LARGE (strong wind).
CONVECTION
DOMINANT
b) Ri = 0
MECHANICAL
TURBULENCE
c) Ri > 0.25
NO VERTICAL
TURBULENCE
FIGURE 2 Average vertical spread of effluent from
an elevated source under different meteorological
conditions (schematic).
TABLE 1
Turbulence characteristics with various Richardson numbers
0.24 Ri No vertical mixing
0 Ri 0.25 Mechanical turbulence, weakened by
stratification
Ri 0 Mechanical turbulence only
0.03 Ri 0 Mechanical turbulence and convection but
mixing mostly due to the former
Ri 0.04 Convective mixing dominates mechanical
mixing
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