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(b) the dominance and square-wave nature of kilometer-scale waves, (c) the
observation of vertically propagating two-stream waves (perpendicular to the
current), (d) the apparent constant phase velocity of type 1 waves at any angle
to the current, and (e) details of the observed wave number spectra.
4.8 Nonlinear Theories of Electrojet Instabilities
4.8.1 Two-Step Theories for Secondary Waves
Although strictly speaking not a nonlinear theory, the discussion by Sudan et al.
(1973) provides a conceptual framework from which several properties of the
fully developed turbulence can be understood. The basic idea is that “primary”
waves reach sufficiently large amplitudes that the perturbation electric fields
and density gradients are themselves large enough to drive “secondary” waves
unstable. For example, referring to the observational data from the CONDOR
flights in Fig. 4.29, the primary
B
will
drive two-stream waves in the vertical direction. Indeed, the rocket electric field
measurements show that the perturbation electric fields can be of the order of
(
δ
E
is horizontal, so if it is large enough,
δ
E
/
C
s
B
. This is illustrated in Fig. 4.35a, in which the horizontal electric field
fluctuation strength has been divided by
B
and plotted as a function of altitude for
the upleg and downleg of the rocket flight. The dotted lines show the threshold
velocity for a secondary two-stream instability given by
C
s
(
1
+
)
. The first
thing to note is that between 100 and 105 km the observed perturbation electric
field is sufficiently strong to generate a vertical two-stream instability. The region
where these waves can be generated agrees remarkably well with the regionwhere
vertical up- and downgoing two-stream waves were detected simultaneously by
the Jicamarca radar. It is safe to say that vertically propagating two-streamwaves
are created by large-amplitude horizontal electric fields associated with long-
wavelength gradient drift waves.
Quantitatively, one can evaluate the required amplitude of the primary waves
as follows. As summarized by Fejer and Kelley (1980), the oscillation frequency
and growth rate of vertically propagating secondary waves are given by
1
+
)
2
A
sin
ω
r
(
k
s
)
=−
k
s
(
e
/ν
e
)
V
D
/(
1
+
)
δ
(4.51)
e
k
p
V
D
/
2
2
sin
γ
k
s
=
(/
2
e
2
A
2
1
+
)
−
/ν
/ (
1
+
)
(
2
δ)
k
s
C
s
ω
k
s
2
+
(
1
/ν
i
)
−
.
(4.52)
where
k
s
and
k
p
are the wave numbers of the secondary vertically propagating
and primary horizontally propagating waves, respectively. In this expression,
A
is the amplitude (
is its phase
ω
p
t
k
p
x
.
δ
n
/
n
)
of the primary wave and
δ
−
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