*J. Geophys. Res., 90*, 9675 - 9687, 1985.

(Received August 24, 1984;
revised June 4, 1985;
accepted June 5, 1985.)

Copyright 1985 by the American Geophysical Union.

Paper number 4A8293.

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Title and Abstract

Having shown how the instability of a ring distribution
varies as a function of the plasma parameters, we shall
now
explore the dependence on the wave vector. In the
previous section we considered the limit *k*
= *k*
= 0.
In this section we shall fix the plasma parameters and
vary the wave vector direction and magnitude. As in the
previous section, the ring electrons shall be assumed to
have a momentum
*p* /
*m**c* = 0.1.
Since the growth rate is near maximum for
/
=
*p* /
*m**c* ,
we shall choose
/
= 0.01.
Finally, we shall somewhat arbitrarily assume that
*n* /
*n* = 0.75.

Fig. 3a. Plot of wave frequency as a function of wave vector. The plasma parameters are held fixed, while the wave vector magnitude and direction are varied.

Fig. 3Solutions of the dispersion relation with the plasma parameters as specified above are shown in Figures 3b. Plot of growth rate as a function of wave vector. The growth rate corresponds to the frequencies shown in Figure 3a.

One aspect of the wave dispersion which we shall
discuss in more detail in the next section is the group
velocity variation. The total range in frequency in
Figure 3*a*,
,
0.002
,
while the range in wave vector,
*k*,
/ *c*.
The group velocity is consequently small;
/
*k*,
0.002*c*. Additionally, while
/
*k*
is positive,
/
*k*
< 0 for the parameters shown in
Figure 3*a*.

The growth rate plotted in
Figure 3*b* shows little
variation for *kc* /
< 0.1. As the wave phase velocity
decreases, aproaching the speed of light, the growth
rate decreases, until a stable mode is encountered close *kc* /
= 1.
There is a slight difference in where the
transition is encountered as a function of *k*
and *k*
.
With the particular choice of system parameters the
growth is maximum for small *k*. Typical growth rates from
Figure 3*b* are
/
0.002, which on combination with
the estimate for group velocity gives convective growth
lengths of the order *c* /
less than the free space
wavelength which is a few kilometers.

As discussed in the previous section, the unstable mode
is on a separate branch of the dispersion relation when
some cold plasma is present, and it does not couple
directly to the *R-X* mode. If this mode is ultimately
responsible for the generation of AKR, then some
coupling to a freely propagating mode must occur. While
the mechanism for mode conversion beyond the scope of
the present work, we can indicate some
features of the mode which imply that mode conversion
may be feasible. At the limit *k*
= 0 or
*k* = 0 the
dispersion relation given by (2), (3), and (4)
separates into two equations. In the cold plasma limit,
these result in the *R-X* mode and the *L-O* mode. On
including a hot ring component, the new mode is just an
additional root to the *R-X* dispersion relation.
Consequently, the polarization of the new wave mode is
the same as the *R-X* mode at the limits *k*
= 0 or
*k* = 0.

Fig. 4. Fraction of wave electric field that is right-hand circularly polarized. The polarization is plotted for the solutions shown in Figures 3When the wave vector is neither parallel nor perpendicular, the dispersion relation is no longer separable. It is hence not obvious what the polarization of the new mode should be. In Figure 4 we show the wave polarization for the unstable mode plotted in Figures 3aand 3b.

Fig 5. Ratio of longitudinal to transverse electric field magnitude. This ratio is plotted for the solutions in Figures 3In Figure 5 we show the ratio of longitudinal wave electric field magnitude to the transverse electric field magnitude. Near the speed of light the wave is almost completely transverse, as expected. The wave is least electromagnetic nearaand 3b.

Fig. 6a. Plot of the wave frequency for a low-density plasma. This figure is similar in format to Figure 3a, but / = 10, i.e.,p/mc> / , which we refer to as an underdense plasma.

Fig. 6However the wave frequency and growth rate variation shown in Figures 3b. Plot of the growth rate for a low density plasma. The growth rate corresponds to the frequencies shown in Figure 6a.

To summarize the properties of the unstable mode as a
function of wave vector, we find that the polarization
is very similar to the cold plasma *R-X* mode. Usually,
the wave frequency is a maximum for large
*k* (
/
*k*
> 0), but the dependence on *k*
depends on whether the plasma is underdense or overdense. If
*p* /
*m**c* >
/
(underdense),
/
*k*
> 0, whereas
/
*k*
< 0 for
*p* /
*m**c*
/
(overdense).
The growth rate is also dependent on the relative density of the plasma.
For an underdense plasma,
is a maximum near
*k**c* /
= 1,
whereas is a maximum for *k* = 0 in an overdense
plasma. We shall address the transition from underdense
to overdense plasma in more detail in the next
section.

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