Geoscience Reference
In-Depth Information
w
b
λ
u
∗
0
3
R
f
=
<
R
c
≈
0
.
2
⇒
λ
max
<
R
c
κ
L
(4.23)
andwe haveestablishedlimitsfor
λ
max
:
→
Λ
∗
u
∗
0
/
→
∞
λ
f
for
L
(4.24)
max
0
+
→
→
λ
R
c
κ
L
for
L
max
A simple expression with these asymptotes is half the harmonicmean of the limits
in(4.24):
2
λ
max
=
η
∗
Λ
∗
u
∗
0
/
f
(4.25)
where
1
−
1
/
2
1
−
1
/
2
+
Λ
∗
u
∗
0
κ
+
Λ
∗
µ
∗
κ
η
∗
=
=
(4.26)
R
c
fL
R
c
µ
∗
isastabilityparameterthatrepresentstheratiooftheplanetarylengthscaletothe
Obukhovlength.Itisnowpossibletore-evaluatethemasterlengthscale
H
,since
u
∗
0
2
2
f
2
H
2
Λ
∗
=
Λ
∗
K
fH
2
=
η
∗
K
∗
=
(4.27)
where the last equality follows from the stipulation that all flows within the class
beingconsidered(neutralandstablystratifiedPBLs)aresimilar.Thusthesimilarity
scalesare
/
Length:
η
∗
u
∗
0
f
Velocity:
u
∗
0
/
η
∗
2
(
η
∗
)
/
Eddyviscosity:
u
∗
0
f
(4.28)
Kinematicstress:
u
∗
0
u
∗
0
wherebothvelocityandkinematicstressscalesarevector(complex)quantities.
FortheIOBLstabilizedbypositivebuoyancyfluxattheboundary,weanticipate
that
V
s
=
where
L
0
is the Obukhov length based on boundary
fluxes,andthatthenondimensionalrelationwillbeoftheform
F
(
u
∗
0
,
z
0
,
f
,
L
0
)
U
0
u
∗
0
V
s
u
∗
0
=
u
∗
0
fL
0
U
0
=
fz
0
,
=
U
0
(
Ro
∗
,
µ
∗
)
(4.29)
A typicalaveragevalueof frictionspeed forperennialsea ice in the Arcticis about
7mms
−
1
. For a specified basal melt rate, the boundary buoyancyflux may be cal-
culated following the formulas developed in Chapter 6. It turns out that for typical
sea-ice parameters, with
u
∗
0
=
7mms
−
1
, the magnitude of
µ
∗
is about the same
as melt rate expressed in centimeters per day (Fig. 4.7). In Fig. 4.8, the dimen-
sional stress is plotted for the same surface stress,
10
−
5
m
2
s
−
2
,
u
∗
0
2
τ
=
=
4
.
9
×
but three differentvalues of
µ
∗
: 0 (neutral), 5, and 25. The dimensionless stress is,
of course, the same in all cases (Fig. 4.3) because the scaled boundary layers are
