Geoscience Reference
In-Depth Information
This was found to hold through the lower tens of meters in the atmosphere, where
itwas assumed(andverifiedbyexperiment)thattheturbulentstresswas nearlythe
sameasthewindstressactingatthesurface,sothatfrictionspeedis
u
0
≈|
2
,
wheretheReynoldsstressismeasurednearthesurface,typicallyatastandardlevel
of10m.
If wind shear depends mainly on distance from the surface, on the local stress,
and on viscosity of the fluid, then straightforward dimensional analysis (e.g.,
Barenblatt 1996) reveals that a dimensionless parameter including the dependent
quantity wind shear and one or more of the independent governing parameters
(
1
/
u
w
|
will be a function of one other dimensionless group formed from the
governingparameterssinceonlytwo haveindependentdimensions.Consequently,
z
,
u
∗
0
,
ν
)
zU
z
u
∗
0
=
Φ
(
zu
0
ν
)=
Φ
(
Re
)
where
Re
isaReynoldsnumberformedwiththefrictionvelocity.Practicallyspeak-
ing, at the high Reynolds numbers typical of nearly all flows in the atmosphere or
ocean, the Reynolds number dependence is minimal, and the
dimensionless wind
shear
(fortheneutrallystable)surfacelayeris
φ
m
=
κ
zU
z
u
∗
0
=
1
(4.1)
κ
(=
Φ
−
1
where
isvonKarman'sconstant,usuallytakentobe0.4.
Note that in this development, there is no consideration of rotation, which
manifestsitselfinthefluidequationsintermsoftheCoriolisparameter,
f
.Ifweac-
ceptthat
)
is unimportant,thenwecanreformulatethedimensionalanalysisabove
bysubstituting
f
for
ν
, wherenowthe governingparametershavedimensions(fol-
lowingthe notationof Barenblatt1996):
ν
T
−
1
.We
again have a dimensionless group including the variable
U
z
that depends on one
other dimensionless group (since there are three postulated governing parameters,
twowith independentdimensions):
LT
−
1
,and
[
z
]=
L
,
[
u
∗
0
]=
[
f
]=
z
U
z
u
∗
0
=
Φ
(
fz
u
0
)=
Φ
(
ξ
)
(4.2)
Note that the dimensionless group on the left is a vector (complex) relation
indicating that shear and surface friction velocity need not be aligned.
is a
dimensionless vertical coordinate, where the scale length is the planetary scale
u
∗
0
/
ξ
f
. A typical range for the planetary scale in polar oceans is 70-100m. For di-
mensionlessshear to be constantas a stipulationfor remainingin the surfacelayer,
ξ
must be small since we observeangular shear relatively close to the interface. In
the IOBL, the surface layer is much thinnerthan in the atmosphere.Assuming that
airstressandwaterstressnearlybalanceinthin,freelydriftingseaice,andprovided
the
nondimensional
surface layer extent is about equal in both fluids (a basic tenet
of PBL similarity discussed below), the ratio of dimensional surface layer extent
will go as the square root of the density ratio, since
ρ
a
u
∗
a
2
≈
ρ
w
u
∗
w
2
. Thus the
