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byconsideringthe
cross-spectrum
formedbysmoothing,forexample,the complex
productof the
w
Fourier componentsand the complex conjugateof the
T
compo-
nents (Bloomfield 1976). The complex cross-spectrum is characterized by coher-
ence(magnitudesquared,normalized)and thephaseangle,which is thearctangent
oftheimaginarypart(quadraturespectrum)overtherealpart(cospectrum).Ingen-
eral,awavenumberbandwillcontributetoverticalfluxifthecospectrumdominates
(phasenear0 or
2(quadraturedominant)con-
tributelittle.Examplesof
wT
and
wS
cross-spectrafortheLeadExmeasurements
aregivenbyMcPheeandStanton(1996).
π
), whilebandswithphasenear
±
π
/
3.6 Mixing Length, Eddy Viscosity, and the
w
Spectrum
InSection 3.3we introduceda lengthscale ofthe energy-containingeddies,
,and
used it in the expression for shear production given by (3.5). Discussion of IOBL
turbulencescalesisthesubjectofChapter5,buthereweanticipatethoseresultsby
identifying
λ
with the wavenumberat the peak in the area-preserving
w
spectrum,
k
max
,andintroducetheconceptof
eddy viscosity
inaturbulentflow.Eddyviscosity
providesaconceptualmethodforclosingtheturbulenceproblemat“first-order”by
relatingfluxesofmomentumandscalarpropertiesinashearflowtotheirrespective
gradients. By analogy with kinematic viscosity which depends on the products of
velocity (internal energy) and mean free path of the molecules in the fluid, eddy
viscosity is commonly represented by the productof a turbulent velocity scale and
a length scale over which the dominant eddies in a flow are effective at diffusing
momentum(thisisdifferentfromtheactualscaleoftheeddymotions):
λ
K
=
u
τ
λ
In the simplest form of turbulent shear flow near a boundary where buoyancy and
rotation are unimportant, the velocity profile is logarithmic with distance from the
boundary,andthepertinentscalevelocityis
u
∗
0
,thesquarerootofkinematicbound-
ary stress. From this it follows immediately (Section 4.1) that
λ
=
κ
z
where
κ
is
von K´arm´an's constant. If
is known in this type of flow (say from measuring the
spectrum), the magnitude of the turbulentstress is simply
ε
2
/
3
. The prob-
τ
=(
κ
z
ε
)
lem for the IOBL, however, is that the linear dependence of
on
z
is limited to at
most a few meters from the boundary, and determining the scale of turbulence in
the outer part of the boundary layer becomes a central issue in understanding tur-
bulent transfer there. It appears that the inverse of the wave number at the peak in
the
w
spectrum (but not the
u
and
v
spectra) provides a consistent estimate of
λ
,
henceeddyviscosity.MeasuringthisatdiscretelevelsthroughtheentireIOBLthen
provides an important observational constraint on models that purport to simulate
turbulentexchangesinthePBL.
ApplicationofthisconcepttotheIOBLdatesfromthe1972AIDJEXPilotStudy
data(McPheeandSmith1976).Wefoundthatpeaksinthearea-preservingspectral
λ
