Geoscience Reference
In-Depth Information
densityof verticalvelocityvarianceincreasedwith depthforthe first coupleoftur-
bulenceclusters(toabout4mfromtheice)butthatforgreaterdepths(8-26m),the
peaks occurred at roughly the same wave number. We were also aware that Busch
and Panofsky (1968) had shown that velocity variance spectra measured in the at-
mospheric boundary layer had the following characteristics: (i) wavelengths at the
maximuminthelogarithmic(area-preserving)spectraofverticalvelocityincreased
linearly with heightup to about50m, and moreslowly beyond;(ii) the dimension-
less frequency,
f
m
=
V
where
n
is frequency at the spectral maximum and
V
is
mean wind speed, scaled with Monin-Obukhovsimilarity in the surface layer (dis-
cussed in Chapter 4); (iii) there was a relativelyuniformshape to the normalized
w
spectra,whentheabscissavalueswerescaledby
f
m
andordinatevaluesby
u
2
nz
/
∗
0
;and
(iv)thelongitudinal
spectradid
not
showsimilarlypredictablebehavior.Forthe
neutrally stratified surface layer, this means that the wave number at the maximum
inthearea-preserving
w
spectrum
(
u
)
(
k
max
=
f
m
/
z
)
isinverselyproportionalto
z
hence
to
z
. The fact that this dependenceweakened for heights greater than about
50m led Busch and Panofsky to speculate that the connection between
λ
sl
=
κ
λ
and
k
max
mightpersistbeyondthesurfacelayer.
We reasoned (McPhee and Smith 1976) that if the increase in
λ
with distance
from the boundary reached some limit comparable to
times the surface layer
thickness (a few meters in the IOBL), then our observations of
k
max
behavior at
the various levels would be consistent with
κ
k
max
, as suggested by Busch
and Panofsky. Experiments since (described in Chapter 5) have corroborated this
view. The proportionality constant appears to be about 0.85. The vertical line in
Fig. 3.9 thus suggests that the master turbulent length scale at 20m during the 6-h
eventshownwasabout3.2m.Ifshearproductionanddissipationbalance,(3.5)then
impliesthattheReynoldsstress,
λ
=
c
λ
/
10
−
4
m
2
s
−
2
,inagree-
mentwithdirectmeasurements(seeFig.5.3).Notethatthereisenoughinformation
fromthe
w
spectrumbyitself to estimatethe eddyviscosity at20m:
K
2
/
3
, wasabout5
τ
=(
λε
)
×
1
/
3
4
/
3
,
≈
ε
λ
022m
2
s
−
1
.
Theisotropyconditionindicatedbythe4/3separationbetween
w
and
u
spectraas
shownbythe ISWdata(Fig.3.9)is notalwayspresentin theIOBL measurements,
particularlyatlevelscloserto thesurface.DuringtheSHEBA project,forexample,
average normalized
w
spectra were remarkably similar at four levels ranging from
4 to 16m from the ice undersurface (see Fig. 3.6 of McPhee 2004). However, for
the two upper TICs, the
u
spectrum was consistently more energetic in the inertial
subrange(asindicatedbythe
w
spectrum).We attributedthistoalackofhorizontal
homogeneity in the underice surface, as TKE advected from a prominent pressure
ridge keel, often about 100m “upstream” from the turbulence mast, spread verti-
cally. In this case, we found that the gradient of TKE flux played a significant role
in the TKE equation, and developed an alternative (but closely related) method for
estimatingthe magnitudeofstress fromthe
w
spectrumbyconsideringthe produc-
tionratherthandissipationofTKE,resultingina simplerelation
whichis about0
.
=
c
γ
γ
∗
2
2
/
3
u
∗
(3.12)
