Geoscience Reference
In-Depth Information
atmosphericsurfacelayerextentwillberoughly30timesthatoftheocean.Athigh
latitudes,theoceansurfacelayerisoftenconfinedto within2-4mofthe boundary,
and, for example, the counterpart of the standard 10-m measurement height in the
atmospherewouldbeonly30cmfromtheinterfaceinthe ocean.
4.1.1 Mixing Length in the Neutral Surface Layer
As discussed in Chapter 3, one of the most useful simplifications for treating
exchange of momentum in turbulent geophysical flows is the concept of eddy
viscosity, which relates the turbulent momentum flux to mean current shear in the
vertical
w
u
=
−
K
U
z
L
2
T
−
1
.Ifweassumethatareasonablescalevelocityinthe
surface layer is
u
∗
0
, then it is immediately obvious from (4.1) that mixing length
there is
where
K
hasunits
[
K
]=
λ
sl
=
κ
z
(where for the present
z
is taken to be the positive distance from
theboundary).
4.1.2 The Law of the Wall and Surface Roughness Length
Integration of (4.1) leads to the logarithmic current (wind) profile in the neutrally
stratifiedsurfacelayer
u
∗
0
κ
u
∗
0
κ
log
z
z
0
U
(
z
)=
+
=
log
z
const
(4.3)
where
z
0
isinessenceanintegrationconstant,takentobealengthscaleindicativeof
theroughnessof theboundary.Equation(4.3)is oftentermedthe “law ofthe wall”
(LOW)andtothedegreethatitaccuratelydescribesthewindprofile,measurements
attwolevelswithinthesurfacelayeraresufficienttoestimatethesurfacestressand
roughness.
In laboratory flows,
z
0
is found to be roughly 1/30th the size of the roughness
elements on the surface. However, the undersurface of sea ice comprises a rather
broad spectrum of roughness scales, with estimates ranging from several centime-
tersormoreinold,highlydeformedseaice(e.g.,thewesternWeddellSea[McPhee
2008,in press]) and in the marginalice zonewhere floes tendto breakinto smaller
pieces with large edge area (McPhee et al. 1987), to sub-millimeter scales un-
der young ice in the eastern Weddell (McPhee et al. 1999). Undeformed seasonal
fast ice may be
hydraulically smooth
(e.g., Crawford et al. 1999) in which case,
the undersurface roughness loses its dependence on the physical properties of the
boundary, and instead depends only on friction velocity and molecular viscosity,
i.e.,
z
0
s
=
ν
u
0
e
−
2
(Hinze1975).
