Geoscience Reference
In-Depth Information
currentsbeyondthesurfacelayer. Inthe southernhemisphere,currentbacksclock-
wise withincreasingdepthto anobserverontheice.
4.2.1 Similarity for Turbulent Stress in the Outer Layer
As suggested by dimensionalanalysis in Section 4.1, dimensionless vector current
shearlikelydependson
z
scaledbytheplanetarylength
u
∗
0
/
f
asindicatedby(4.2).
We usedthissimilaritytoinferthatscalesinthePBLsoftheatmosphereandocean
differedbyafactorofabout30.Hereweextendtheearlieranalysistosimilarityfor
theouterboundarylayer,includingconsiderationofturbulentstress.
Startwiththe steady,horizontallyhomogeneousOBL equation
=
∂τ
∂
if
u
(4.7)
z
andchoosenondimensionalvariablesforstressandverticalcoordinate
τ
u
∗
0
u
∗
0
T
=
z
H
ξ
=
wheretheobviouschoiceforscalingstressistheappliedsurfacestress.Substituting
into(4.7),wehavethenondimensionalmomentumequation
=
∂
T
∂ζ
i
U
(4.8)
provided
fH
u
∗
0
·
u
u
∗
0
U
=
(4.9)
If
H
is the planetary scale
(
u
∗
0
/
f
)
, then obviously the dimensionless velocity is
u
/
u
∗
0
, butherewewill retainthegeneralityimplicitin(4.9).
Forfirst-orderclosure
K
∂
u
τ
=
∂
z
K
∗
∂
U
∂ξ
T
=
K
fH
2
.Ifwehypothesize,following
Ekman,thatthe
z
dependenceofeddyviscosityisnegligibleintheouterlayer,then
differentiationof(4.8)leadstoasecondorderordinarydifferentialequationfor
T
:
wherethenondimensionaleddyviscosityis
K
∗
=
