Geoscience Reference
In-Depth Information
5
Ψ
m
0
−
5
−
10
15
−
0
1
2
3
4
5
ζ
Fig. 2.12 The dependence of
Ψ
m
on
ζ
under stable conditions, as determined in Cheng and Brutsaert
(2005) from experimental wind profile data over grass (
z
0
=
0
.
0219 m,
d
0
=
0
.
110m) in
Kansas in October, 1999. The solid curve represents Equation (2.59) and the dashed curve
represents Equation (2.58).
indicates that, although the
φ
h
(
ζ
) data points for temperature exhibit more scatter, Equation
(2.60) can represent these points practically as well as the
φ
m
(
ζ
) data points for wind
speed; this suggests that it is safe to assume that under stable conditions the ASL similarity
functions for sensible heat and for momentum are the same. Moreover, experimental and
theoretical evidence by Dias and Brutsaert (1996) supports the turbulence similarity of
scalars under stable conditions. Thus the Reynolds analogy appears to be valid and one can
put
) for a stably stratified ASL.
For unstable conditions, Kader and Yaglom (1990) used a more fundamental approach;
they reasoned, and were able to support with experimental evidence, that the surface layer
can be subdivided into three sublayers, namely a dynamic, a dynamic-convective and a
convective sublayer, for each of which they derived simple power laws to describe the
turbulence. However, the resulting
φ
-functions cover only certain ranges, corresponding
to these sublayers. Again, to cover the entire
ζ
-range, an interpolation formulation should
be developed; accordingly, Brutsaert (1992; 1999) combined the functional behavior of
φ
m
and
φ
h
in each sublayer, and proposed the following expressions
φ
m
(
ζ
)
=
φ
h
(
ζ
)
=
φ
v
(
ζ
) and
m
(
ζ
)
=
h
(
ζ
)
=
v
(
ζ
φ
m
(
ζ
)
=
(
a
+
by
4
/
3
)
/
(
a
+
y
)
for
y
≤
b
−
3
(2.61)
for
y
>
b
−
3
φ
m
(
ζ
)
=
1
.
0
and
φ
h
(
ζ
)
=
(
c
+
dy
n
)
/
(
c
+
y
n
)
(2.62)
in which
y
=−
ζ
=−
(
z
−
d
0
)
/
L
, and
a
,
b
,
c
,
d
and
n
are constants. After considering
available data collections, the constants were assigned the following values
a
=
0
.
33
,
b
=
0
.
41
,
c
=
0
.
33,
d
=
0
.
057 and
n
=
0
.
78. Figure 2.13 shows these
φ
-functions. The
corresponding stability correction functions can be obtained in integral form by means of
