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value where
x
3 and then follows a power law for larger
x
. The value of the
power law index for
E
≈
is the same as the value that would be mea-
sured by a one-dimensional cut through the turbulent plasma using devices that
measure either
(
x
)
=
xI
(
x
)
2
since the one-dimensional measurement spec-
trum integrates out one power of
k
and has the same shape as
xI
2
or
(δ
n
/
n
)
(δ
E
)
(
x
)
(proportional
to
kI
k
)
. The power law regime occurs only for large
S
and occurs in regime II in
Fig. 4.37, where the eddy growth rate dominates the linear growth, correspond-
ing to the inertial subrange in neutral turbulence theory (Kolmogorov, 1941). It
is interesting that the same one-dimensional spectral form is predicted for this
plasma case as for the three-dimensional neutral fluid turbulence in the inertial
subrange. For finite
S
, the spectrum becomes very steep at large
k
, corresponding
to the viscous subrange in neutral turbulence and, in the present case, is due to
diffusive damping. For
S
infinite or negative, the fluid theory breaks down due
to the excitation of short-wavelength waves, which require a kinetic description.
There are a number of ways to check this theory. First, the numerical simula-
tions may be tested against the analytic expression (4.59). The numerical results
of McDonald et al. (1974, 1975) and Keskinen et al. (1979) seem to disagree
with the prediction since they report
I
k
∝
k
−
3
.
5
rather than
k
−
11
/
3
. This problem
can be reconciled as shown in Fig. 4.38. The data points in this figure come from
the simulation work reported by Keskinen et al. (1979). The curved line is a fit to
the analytic calculation of Sudan (1983) described above for
k
d
=
15
k
s
, where
k
s
is the wave number corresponding to the physical size of the grid. The fit is
quite good but it must be realized that
k
c
, the outer scale for the process, must
be considerably smaller than
k
s
to yield such a steep slope in the range of
k
space
plotted. For smaller values of
k
(not plotted), Sudan's calculations yield a power
spectrum of
k
−
2
.
67
. Presumably,
if
the simulations and theory are in agreement
and
if
the simulation occurred in a larger “box,” the calculated power spectra
would yield a
k
−
2
.
67
two-dimensional power law at small
k
. However, such a
result has not yet been obtained in any simulation and the comparison shown in
Fig. 4.38 must remain somewhat suspect.
4.8.4 Nonlinear Studies of Farley-Buneman (FB) Waves
Linear theory gives conditions for the onset of an instability and characteris-
tics of the initial growing waves but cannot explain saturation or provide the
amplitude and spectral characteristics of developed turbulence. A number of
theoretical models have been developed to explain saturated FB waves. Hamza
and St.-Maurice (1993a, b) proposed a strongly turbulent mode-coupling the-
ory based upon a two-fluid model. Another approach, developed by Albert and
Sudan (1991), Sahr and Farley (1995), Otani and Oppenheim (1998), Dimant
(2000), and Otani and Oppenheim (2006), uses a truncated three-wave mode-
coupling dynamic model to explain instability saturation. None of these theories
provides a fully consistent quantitative description of nonlinear saturation of
E-region instabilities.
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