60 ≤ h / L ≤ 30
0 ≤ h / L ≤ 30
60 ≤ h / L ≤ 90
1.0
0.8
0.6
--
z
h
0.4
0.2
0
40 20
0
20
40
40 20
0
20
40
40 20
0
20
40
^
θθ
t
*
Figure 28. Evaluations of FΘ for unstable (60 ≤ ≤ 30), weakly stable (0 ≤ ≤ 30),
and moderately stable (60 ≤ ≤ 90) conditions (after Yamada 1976).
As with the φm and φh or ψm and ψh functions of
FV and FΘ functions that he deduced from plots of data
Monin-Obukhov similarity, Rossby number similarity
from the Wangara experiment in Hay, Australia. (Brut-
saert [1982] offered one of the few discussions of FQ
essentially leads us on a quest to find the nondimen-
sional, universal functions A, B, C and D. Part of this
and D.)
quest requires defining the scales U, V, Θ and Q . In
^ ^ ^
^
Figures 27 and 28 emphasize why it is appropriate
analogy with the Ekman equations (i.e., eq 116), we
to refer to eq 150152 as resistance laws. These figures
might expect U = Ug , V = Vg , Θ = Θ(h) and Q = Q(h) .
^
^
^
^
show that in very unstable conditions FU, FV and FΘ
But in light of frequent baroclinicity (a thermal wind),
are all near zero. This means that the ABL is well mixed;
the wind vector and temperature at h are very near the
Arya and Wyngaard (1975) found that more robust val-
^
^
ues for U and V derive from averaging the geostrophic
values at the surface. That is, there is efficient coupling
wind from the surface to h. Likewise, Yamada (1976)
between the upper air and the surface. In very stable
found that Θ is best taken as the height-averaged poten-
^
conditions, in contrast, U(z), V(z) and Θ(z) vary strongly
^
tial temperature. By extension, I define Q analogously.
with height. In other words, in stable conditions the
Mathematically, we define
transfer of properties from the surface to the top of the
ABL, or vice versa, is inefficient.
h
From his analysis, Yamada (1976) was able to esti-
∫
1
χ =
χ(z) dz
(153)
mate A, B and C. Many others since Yamada have
h
suggested alternate formulations (e.g., Arya 1977,
0
Zilitinkevich 1989a, b). I do not have room here to re-
where χ is U(z), V(z), Ug(z), Vg(z), Θ(z) or Q(z). Thus
view all of these; thus, since Yamada's functions are
(Arya and Wyngaard 1975, Yamada 1976)
the most frequently cited, I will base the remainder of
this section on them. He found
U = Ug = U
^
(154a)
A() = 10.0 8.145 (1 0.008376 )1/3
2
u*
for ≤ 0
V = Vg = V -
^
(157a)
(154b)
fh
= 1.855 0.380
Θ= Θ
^
for 0 ≤ ≤ 35
(155)
(157b)
Q= Q.
^
= 2.94 ( 19.94)1/2
(156)
for 35 ≤
(157c)
After defining U , V and Θ as above, Yamada
^
^ ^
B() = 3.020 (1 3.290 )1/3
(1976) made the classic attempt to find FU, FV and FΘ
for ≤ 0
and, thus, A, B and C. Figures 27 and 28 show the FU,
(158a)
31
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