Geoscience Reference
In-Depth Information
usingtheTaylor-seriesexpansionoftheexponentialstress
∂
U
∂ξ
=
K
∗
=
−
η
∗
T
(
1
−
βµ
∗
η
∗
ξ
)(
1
+
δξ
)
ξ
>
Λ
∗
/
κ
(4.31)
κξ
SofortheentireIOBL,thesimilarityequationsare
T
=
exp
(
δξ
)
(4.32)
and
−
i
δ
exp
(
δξ
)
ξ
≤
ξ
sl
log
ξ
2
−
ξ
sl
2
U
(
ξ
)=
(4.33)
U
E
−
η
κ
a
2
ξ
sl
+(
δ
−
a
)(
ξ
−
ξ
sl
)
−
ξ
ξ
>
ξ
sl
=
βµ
∗
η
∗
=
κ
(
1
−
η
∗
)
1
/
2
,and
where
a
.
Equations(4.32)and(4.33)constitutethesimilaritytheorywithparameters
R
c
η
∗
Λ
∗
,
δ
=(
±
i
/
Λ
∗
)
ξ
sl
=
−
Λ
∗
/
κ
=
0
.
2and
028.
Thedimensionlesssurfacevelocityis
Λ
∗
=
0
.
log
ξ
sl
2
ξ
sl
2
+
δξ
sl
)+
η
∗
κ
a
U
0
−
i
δ
(
1
ξ
0
−
(
a
−
δ
)
ξ
sl
−
(4.34)
fz
0
η
∗
where
ξ
0
=
−
∗
0
.Thegeostrophicdragrelationis
u
V
s
u
∗
0
=
U
0
η
∗
(4.35)
whichasbeforemaybemanipulatedintoaRossby-similarityform,exceptthatnow
theparameters
A
and
B
dependon
µ
∗
V
s
u
∗
0
=
1
κ
[
log
Ro
∗
−
(
µ
∗
)
−
(
µ
∗
)]
A
iB
(4.36)
4.3 IOBL Similarity and the Atmospheric Boundary Layer
4.3.1 Dimensionless Shear
Velocity in the surface layer is related to stress by eddy viscosity that depends on
bothdistancefromtheboundaryandtheObukhovlength
κ
|
z
|
u
∗
0
K
=
(4.37)
1
+
β
|
z
| /
L
0
withnondimensionalform
