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large enough to generate collisional EIC waves in the upper E region at wave-
lengths of 10-20 m. The most favorable altitude predicted by theory (about 150
km) is consistent with the observations, and the measured frequency range of
50-80 Hz is near the gyrofrequencies of the E-region ion constituents. How-
ever, 3 m ion cyclotron waves can be directly excited only by unreasonably
large field-aligned drifts (greater than the electron thermal speed). In addition,
the theory predicts that for minimum threshold,
k
007 (aspect angle of
89.6
◦
), whereas radar measurements indicate that aspect angles are as small as
80
◦
||
/
k
=
0
.
k
=
||
/
.
17). The threshold velocity is much larger at these small aspect
angles. It is possible that some nonlinear mechanism could generate coherent
(resonant) 3m waves from linearly unstable 20 m waves. In fact, Satyanarayana
and Chaturvedi (1985) suggested that ion resonance broadening could provide
such a cascade to short wavelengths. However, ion resonance broadening would
generate broad spectra (spectral width much greater than
(
k
0
i
) at 3m and, there-
fore, cannot explain the radar data. Seyler and Providakes (1987) performed a
numerical simulator that reproduced the type 3 echoes, but considerable work
remains before we will finally understand these echoes.
10.6.4 Nonlinear Theories
The analytic (turbulence-based) gradient drift instability theory of Sudan and
co-workers has already been discussed in Chapter 4, as have the simulation
results of McDonald and co-workers. The theory should be applicable to the
auroral case, although pure primary gradient drift waves are not as important
at high altitudes as they are at the equator and at midlatitudes. In this approach
the 3m waves responsible for 50MHz backscatter are due to a cascade of energy
fromunstable long-wavelengthwaves to stable short-wavelengthmodes. Because
of the interest in electron heating at high latitudes we pursue the case of the
nonlinear two-stream wave here.
Sudan (1983a, b) has suggested that momentum transfer between electrons and
short-wavelength waves leads to an anomalous cross-field diffusion coefficient
D
∗
. Since
D
∗
is proportional to the collision frequency, there is a corresponding
anomalous electron collision frequency
ν
∗
that increases with the wave ampli-
tude. The nonlinear dispersion relation can be obtained by replacing
ν
e
with
ν
∗
. The nonlinear two-stream oscillation frequency and growth rate are then
given by
k
·
V
D
NL
r
ω
=
+
+
∗
=
kC
s
(10.23)
1
k
2
C
s
2
+
∗
NL
NL
r
γ
=
ω
−
(10.24)
ν
i
(
1
+
+
∗
)
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