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called
Rossby similarity
hasbeenusedtoquantifytheeffectsdescribedqualitatively
above. Let
V
s
=
be the surface velocity relative to the ocean beyond
theIOBL.Againbydimensionalanalysis
F
(
u
∗
0
,
z
0
,
f
)
U
0
u
∗
0
fz
0
V
s
u
∗
0
=
(4.15)
where the dimensionless grouping of governing parameters is the ratio of the
planetary scale to the boundary roughness scale, which in the atmospheric litera-
ture is often called the surface friction Rossby number,
Ro
∗
(e.g., Blackadar and
Tennekes1968).
We can exploit the simple conceptual model of nondimensional stress as a
complexexponential,combinedwith asurfacelayer in whicheddyviscosityvaries
linearly with
ξ
, to investigate the functional form of (4.15). The integral of (4.11)
from
−
∞
tothebaseofthesurfacelayer,
ξ
sl
=
−
Λ
∗
/
κ
,providesthenondimensional
velocityat thetopoftheEkmanlayer:
e
δ
ξ
sl
U
E
=
−
i
δ
(4.16)
In the surface layer (small
|
ξ
|
) a Taylor-series expansion for the exponential
provides
T
1
+
δξ
and
U
E
−
i
δ
(
1
+
δξ
sl
)
and
∂
U
∂ξ
=
1
−
κξ
(
1
+
δξ
)
from which integration to the boundary
(
ξ
=
ξ
0
=
−
fz
0
/
u
∗
0
)
provides the nondi-
mensionalsurfacevelocity
ln
ξ
sl
1
κ
U
0
=
U
E
+
ξ
0
+
δ
(
ξ
sl
−
ξ
0
)
(4.17)
with
|
ξ
0
||
ξ
sl
|
therealandimaginarycomponentsof
U
0
are
log
u
∗
0
1
κ
log
Λ
∗
κ
√
2
Λ
∗
2
1
κ
Re
(
U
0
)=
fz
0
+
κ
+
Λ
∗
−
1
−
=
[
log
Ro
∗
−
A
]
κ
2
1
κ
κ
√
2
Λ
∗
2
=
−
1
κ
Im
(
U
0
)=
−
Λ
∗
−
B
(4.18)
2
κ
Thesecomplicatedlookingexpressionsthusreducetoaformulaforthe“geostrophic”
drag law for sea ice
