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[
S
w
]
q
2
=
0
q
1
[
k
]
[
ε
]
whichaftersolvingfor
q
1
and
q
2
yields
k
5
/
3
S
w
ε
=
constant
2
/
3
Thus in that part of the area-preserving spectrum (Fig. 3.9) where the above as-
sumptions hold, we would expect the spectra to fall off with a
3 slope (in the
log-logrepresentation),asindicatedbythetriangle.Byconvention,theequationfor
thevertical(cross-stream)spectrumisexpressedas
−
2
/
4
α
3
ε
/
3
k
−
5
/
3
2
S
w
(
k
)=
(3.10)
where
α
ε
istheKolmogorovconstantforthealong-streamspectrum,
S
u
.Itisfound
from laboratory and atmospheric studies to have a value equal to about 0.51 when
k
is the angular wave number (e.g., Edson et al. 1991). The 4/3 factor in (3.10)
comesfromtheoreticalconsiderationsofhomogeneous,isotropicturbulencewhere
it is possible to relate the one-dimensional along-stream and cross-stream spectra
to the total (three-dimensional) energy spectrum (Batchelor 1967). In addition to
derivingthe“minusfive-thirds”spectralshapefortheinertialsubrangeinagreement
withthedimensionalanalysis,thetheoryprovidesarelationshipbetweenthecross-
stream
(
v
,
w
)
and along-stream
(
u
)
spectra (measured in the direction aligned with
themeanvelocity):
S
u
−
1
2
k
∂
S
u
∂
S
w
=
S
v
=
(3.11)
k
fromwhichitfollowsthatinthe
3sloperegion,thecross-streamspectrashould
be 4/3 the magnitude of the along-stream spectrum. We stressed above that IOBL
shear flows are clearly anisotropic at large scales, and obviously from Fig. 3.9,
S
w
and
S
v
differwidelyfromeachotherandfrom
S
u
atsmallwavenumbers.Yetatwave
numbers around 3m
−
1
−
5
/
, they are about equal, and are roughly 4/3
largerthan
S
u
(indicatedbythe separatedashedlinenearthe triangle),thusatthese
scalestheturbulenceappearstobemoreisotropicthannot.Theseparationbetween
along- and cross-stream spectra is often taken to mark the inertial subrange. Note
that any point in the
(
log
10
k
∼
0
.
5
)
−
2
/
3 range of the
w
spectrum in Fig. 3.9 suffices to estimate
ε
, e.g., the horizontal line intersects at the spectrum at
k
=
1 with ordinate about
10
−
7
Wkg
−
1
.
log
10
(
kS
w
)
≈−
4
.
8,whichfrom(3.10)provides
ε
≈
1
.
1
×
3 log-log behavior, it lends
confidence that we have made measurements at scales small enough to capture
mostof thecovarianceof thelarge-scaleeddies.Atscaleswherethe turbulenceap-
proachesisotropy,the covariancecontributionis verysmall. This can be confirmed
If area-preserving velocity spectra exhibit the
−
2
/