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Then the 24-hour mean lux can be constructed either by irst multiplying the diffu-
sion coeficient and gradient for each time interval, and by subsequent averaging, or
the order can be reversed:
24
24
24
24
∂
∂
θ θ
∂
∂
″
∂
∂
θ
∂
∂
θ
″
24
24
24
″
″
correct:
H
=−
ρ
c
KK
+
+
=−
ρ
c
K
+
K
p
h
h
p
h
h
z
z
z
z
24
24
24
24
″
∂
∂
θ θ
∂
∂
∂
∂
θ
24
24
24
″
H
incorrect:
=−
ρ
c
KK
+
+
=−
ρ
c
K
p
h
h
p
h
z
z
z
(3.54)
The difference between the two methods is the covariance term occurring in the cor-
rect method, which is missed when the averaging is performed irst. Thus, whenever
there is a correlation between the variables being multiplied (in this case a negative
correlation), the averaging operation should be postponed as long as possible.
19
Mahrt (
1987
,
1996
) has shown that similar arguments hold for spatial averaging
of luxes and gradients in grid boxes of atmospheric models: if two regions with
contrasting sensible heat lux (one positive, one negative) are present within one grid
box, the mean vertical temperature gradient in the grid box may be stable, whereas
the mean lux would be positive.
To conclude, we return to the calculation of luxes from mean gradients. It is clear
that the use of similarity relationships developed in
Section 3.5.5
using 24-hour mean
data is invalid. In the example shown here the problem is very obvious because there
is a clear discrepancy between the direction of the mean lux and the mean gradient.
But using 24-hour mean data is equally dangerous if such a discrepancy is not directly
clear. For example, the wind speed will always increase with height, both in the case
of 30-minute averaged wind speeds and in the case of 24-hour mean winds. Still the
use of the 24-hour mean wind speed to determine the 24-hour mean aerodynamic
resistance is questionable: the mean stability does not correctly represent the effect of
stability on the mean resistance.
3.7 Summary
Because this chapter involves both complex concepts and a heavy load of equations,
a summary in the form of a concept map is appropriate (see
Figure 3.22
).
We put so much emphasis on turbulence for two reasons. First, turbulence is
important because it is the transport mechanism that takes care of the transport of
19
Note that this rule holds equally for averaging over time periods other than 24 hours (e.g., either shorter or longer
time periods).
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