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tionships from experimental data (steps 3 and 4). This has been an active ield of
research since the pioneering paper of Businger et al. (
1971
). Here we present and
use only one set of commonly used lux-gradient relationships (the dependence of the
dimensionless gradients
φ
on
z/L
): the so-called Businger-Dyer relationships (Dyer
and Hicks,
1970
). Although they are widely used, alternative functions are available
(reviews in, e.g., Högström,
1996
; Wilson,
2001
; Foken
2006
). The Businger-Dyer
lux-gradient relationships are (see also
Figure 3.16a
):
−
12
/
z
L
=
z
L
=
z
L
= −
z
L
φ
φ
φ
116
h
e
x
z
L
for
≤
0
−
14
/
z
L
z
L
φ
=
−
116
m
(3.21)
z
L
=
z
L
=
φ
z
L
=+
z
L
φ
φ
15
h
e
x
z
L
for
≥
0
z
L
z
L
φ
m
=+
15
A few things are noteworthy:
φ
-functions for scalars (temperature, humidity and the gen-
eral scalar
x
) are different from that of momentum. For stable conditions however, they
are identical. Note that the question whether the lux-gradient relationships (and other
similarity relationships) are
really
identical for temperature and other scalars is still an
active ield or research.
For neutral conditions (
•
For unstable conditions the
•
z/L
= 0) both the unstable and stable formulations tend to a value
of one, that is, at
z/L
= 0 both formulations match.
For unstable conditions, the value of the coeficient that multiplies
•
z/L
(that is: 16) can be
interpreted as follows. Close to neutral conditions (small -
z/L
) the lux-gradient relation-
ships differ little from a value of one and vary approximately linearly with -
z/L
(based
on a Taylor series expansion, see also
Figure 3.16a
). On the other hand, for very unstable
conditions (large -
z/L
) they vary as (-16
z/L
)
α
(with
α
is equal to -1/2 for scalars and -1/4
for momentum). The point where the situation is no longer 'close to neutral' but tends to
become 'very unstable' is where -
z/L
is equal to 1/16. At that point the term involving
z/L
has magnitude one.
The expressions are well-deined over a limited range of
•
z/L
only due to limitations in the
range of stabilities available in the data sets. Foken (
2006
) gives a range of
−< ≤
z
L
2
0
z
L
and 0
≤<
1
for the expressions given in Eq. (
3.21
).
•
The coeficients 16 and 5 occurring in Eq. (
3.21
) seem very irm, but they are the out-
come of a set of experimental data, and hence are only a best-it. See Högström (
1988
)
for an extensive overview of alternative expressions. He suggests that variations in
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