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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
/
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