Geoscience Reference
In-Depth Information
As we argued for stress, in anisotropic, flux-carrying turbulence a tendency
toward isotropy at smaller scales - beginning in the inertial range, say - should
be evidenced by a spherically averaged flux cospectrum
Co
i
(κ)
that decreases
more rapidly with wavenumber than the energy and scalar spectra. As we dis-
cussed in
Part II
, both the streamwise
(i
3
)
components
of the one-dimensional forms of this cospectrum are observed to be nonzero in
the ABL.
Lumley and Panofsky
(
1964
) predicted that
Co
3
behaves in the inertial
range as
=
1
)
and vertical
(i
=
∂
∂z
1
/
3
κ
−
7
/
3
.
Co
3
∼
(15.92)
This behavior was observed in the one-dimensional cospectrum of
u
3
and
θ
measured in the Kansas experiments (
Wyngaard and Coté
,
1972
). Those authors
predicted that the cospectrum of
u
1
and
θ
behaves as
∂
∂z
∂U
∂z
κ
−
3
,
Co
1
∼
(15.93)
but the slope of the observed one-dimensional spectrum was
−
α
∂
∂z
∂U
∂z
(
1
−
α)/
3
κ
−
(
7
+
2
α)/
3
,
Co
1
∼
(15.94)
with
α
a free parameter. With
α
2
.
55,
which agrees well with the measurements. Thus the observed behavior of these
cospectra is also consistent with the notion of an approach to isotropy at smaller
scales.
=
1
/
3 this yields an inertial-range slope of
−
15.6 Spectra in the plane
15.6.1 The concept
Although they are necessarily inhomogeneous in the vertical, boundary-layer flows
can approach homogeneity in the horizontal plane. If so, then from numerically
simulated fields, for example, we can calculate spectra as a function of thewavenum-
ber vector
κ
h
(κ
1
,κ
2
)
in the horizontal plane; the vertical coordinate is a
parameter, as in
Eq. (15.32)
. Unlike one-dimensional wavenumber spectra, these
two-dimensional spectra vanish at zero wavenumber and so directly indicate the
horizontal spatial scale of eddies contributing to them.
Let's designate the spectrum that integrates over the horizontal plane to the scalar
variance as
φ
(
2
)
:
=
∞
φ
(
2
)
(
κ
h
)dκ
1
dκ
2
=
c
2
.
(15.95)
−∞
