Geoscience Reference
In-Depth Information
K
∗
=
−
κξ
η
∗
−
βµ
∗
η
∗
ξ
)
−
1
(
1
ξ
>
Λ
∗
/
κ
(4.38)
Forsimilarity,
K
mustmatchtheouter-layereddyviscosityat
z
sl
=
−
η
∗
u
∗
0
Λ
∗
f
.From
κ
this it follows that
, considered an empirical factor in the discussion of the at-
mospheric surface layer, is here an internal parameter in the problem that depends
weaklyonthestability parameter
β
µ
∗
1
R
c
+
κ
µ
∗
Λ
∗
β
=
(
1
−
η
∗
)
(4.39)
We can examine the behavior of
β
in the surface layers as follows. The dimen-
sionlessshearis
φ
(
ζ
)=
κ
|
u
∗
0
∂
z
|
u
z
=
−
κξ
∂
U
∂ξ
=
T
K
∗
=
T
(
1
−
βη
∗
µ
∗
ξ
)
(4.40)
∂
η
∗
and the identity
ζ
=
−
η
∗
µ
∗
ξ
along with the Taylor-series approximation for
T
provide
1
−
δζ
η
∗
µ
∗
φ
(
ζ
)=(
1
+
βζ
)
(4.41)
Since
β
isa functionof
µ
∗
inthesimilarity theory,so is
φ
, whichis plottedfortwo
values
(
µ
∗
=
50
,
100
)
inFig.4.9.Alsoshownaretheempiricalformulasfor
L
0
>
0
from (4.5)and (4.6).For
50, the similarity estimate is virtually indistinguish-
ablefromLettau's(1979)empiricallyfitted function.
µ
∗
=
Surface Layer Dimensionless Shear
4
3.5
3
φ
= 1+4.7
ζ
µ
*
=100
2.5
2
1.5
)
3/4
(dashed)
φ
= (1+5
ζ
µ
*
= 50 (solid)
1
0.5
0
0
0.1
0.2
0.3
0.4
0.5
ζ
= z / L
0
Fig. 4.9
Surface-layer dimensionless shear equation as a function of
z
/
L
0
. Solid curves are from
the IOBL similarity theory for two different values of
µ
∗
. Dashed curves are from Businger et al.
(1971) (upper) and from Lettau (1979) (lower)
