Environmental Engineering Reference
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relatively inaccurate, displaying z 0 u /
z 0 versus h 0 / L would display strong artifi-
cial self-correlation (the uncertain term L, would explicitly appear in the abscissa
and,
implicitly,
in the ordinate). To overcome this problem,
in Fig. 7.3 we
present z 0 u /
z 0 as dependent on a dimensionless stratification parameter (a kind
of Richardson number): Ri
18 32 ) h 0 / U 32 , where the subscripts
refer to the measurement heights. Our (7.8) in combination with (7.10) yields
Ri
=
( g /
32 )(
( h 0 / L ) 14/9 . Then (7.9) rewritten in terms of Ri becomes z 0 u /
+
C Ri 3 / 14 . Data in Fig. 7.3 are consistent with this power law and give C =
1.23
z 0
=
1
±
=
0.05. Eventually, this allows the estimation of the constant in (7.9): C US
3 1 C C U ( h 0 /
z 0 ) 2 / 3 ( h 0 /
z 3 ) 1 / 3 1 / 4
C 7 / 6
z 1 ) 1 / 3
0.05.
In contrast to the traditional assumption of a constant roughness length fully
characterized by the geometric properties of the surface, we have demonstrated its
essential dependence on the hydrostatic stability. Figure 7.4 shows z 0 u /
( h 0 /
=
1.24
±
z 0 versus
h 0 / L after (7.7) and (7.9) with our empirical constants ( C SS =
8.13, C US =
1.24). It
follows that z 0 u monotonically decreases with increasing stability and, in the “mete-
orological interval”
10< h 0 / L <10, varies over more than two orders of magnitude
from 4 z 0 to 10 2 z 0 . Much stronger stability dependence under stable compared to
unstable conditions, revealed in Fig. 7.3, corroborates previous empirical evidence
(Arya, 1975; Joffre, 1982).
Recall that the currently used roughness length formulation neglects this depen-
dence, resulting in strong, systematic overestimations of the surface resistance in
stable stratification. The proposed model makes up for this drawback. It can be
immediately implemented in urban and forest meteorology, largely to improve mod-
elling of the most harmful air pollution episodes typical of very stable stratification
in cities.
Furthermore, in the light of our results, it would be relevant to check whether
the inherent scatter in empirical determinations of the so-called universal constants
of turbulent flows in the past could be reduced once this roughness-stability depen-
dence is taken into account. Besides meteorology the new roughness length model
is applicable to oceanographic and engineering problems which deal with stratified
flows.
Fig. 7.4 Comparison of the
new and traditional
formulations of the roughness
length: the solid line shows
z 0 u /
z 0 versus h 0 / L in the
“meteorological interval”
10 < h 0 / L <10 after our Eqs.
(7.7) and (7.9) with the
empirical coefficients:
C SS
1.24.
The dashed line show the
classical formulation:
z 0 u
=
8.13 and C US
=
/
z 0
=
1
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