Geoscience Reference
In-Depth Information
in which the subscripts m, h and v refer to momentum, sensible heat and water vapor,
respectively. To be consistent with Equations (2.39) and (2.42), in the dynamic sublayer or
under neutral conditions, when
ζ
1(but
z
−
d
0
z
0
) these
φ
-functions become equal
to unity. It is usually assumed that
φ
v
=
φ
h
, and thus that Reynolds's analogy is valid for
scalar admixtures of the flow.
Equations (2.47)-(2.49) are formulated in terms of the gradients; these are not easy to
determine from field measurements, which more often than not tend to be noisy. To avoid
this problem, Equations (2.47)-(2.49) can be expressed in integral form as follows
u
k
u
2
−
u
1
=
[ln(
ζ
2
/ζ
1
)
−
m
(
ζ
2
)
+
m
(
ζ
1
)]
(2.50)
w
θ
0
ku
∗
θ
1
−
θ
2
=
[ln(
ζ
2
/ζ
1
)
−
h
(
ζ
2
)
+
h
(
ζ
1
)]
(2.51)
w
q
0
ku
∗
q
1
−
q
2
=
[ln(
ζ
2
/ζ
1
)
−
v
(
ζ
2
)
+
v
(
ζ
1
)]
(2.52)
in which each of the
-functions, with its respective subscript, is defined by
ζ
(
ζ
)
=
[1
−
φ
(
x
)]
dx
/
x
(2.53)
0
and
x
is the dummy integration variable. Under neutral conditions, when
|
L
| →∞
and
ζ
→
-functions approach zero and Equations (2.50) and (2.52) redu
ce
to the
logarithmic profiles (2.40) and (2.43). It is also clear that, whenever
u
1
, θ
1
and
q
1
refer to the surface values 0
,θ
s
and
q
s
, the dimensionless height
ζ
1
must be taken as
z
0
/
L
,
z
0h
/
L
and
z
0v
/
L
,
respectively (as can be seen for the analogous neutral case in (2.41)
and (2.44)). In the present case, Equations (2.50), (2.51) and (2.52) assume the form
0, the
ln
z
−
d
0
z
0
−
m
z
−
d
0
L
L
+
m
z
0
u
k
u
=
(2.54)
ln
z
−
d
0
z
0h
−
h
z
−
d
0
L
L
+
h
z
0h
H
ku
∗
ρ
c
p
θ
s
−
θ
=
(2.55)
ln
z
−
v
z
L
+
v
z
ov
E
ku
∗
ρ
−
d
0
z
0v
−
d
0
q
s
−
q
=
(2.56)
L
The profiles described by Equations (2.54) and (2.56) are illustrated as the non-neutral, i.e.
both stable and unstable, curves in Figure 2.9 and 2.10, respectively.
The nature of the “universal” functions, especially
φ
m
and
φ
h
, but less so
φ
v
, has been
the subject of much theoretical and experimental research. One of the earliest forms of these
φ
-functions, intended for near-neutral conditions, i.e. small
|
ζ
|
, was proposed by Monin
and Obukhov (1954) simply by a series expansion and retention of the first term only,
or
φ
=
(1
+
β
s
ζ
), in which
β
s
is a constant. Subsequent experimental investigations have
revealed, however, that this form of
φ
is applicable only under stable conditions, but not
under unstable conditions. It was also observed later on (see Webb, 1970; Kondo
et al.
,
1978) that this form can describe experimental data only over the range 0
≤
ζ
≤
1 with a
β
s
value of the order of 5, but that
φ
remains approximately constant for
ζ>
1. Accordingly, on
