*J. Geophys. Res., 91*, 3152 - 3166, 1986.

(Received August 8, 1985;
revised November 8, 1985;
accepted December 2, 1985.)

Copyright 1986 by the American Geophysical Union.

Paper number 5A8825

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We have employed two different plasma dispersion relations
in the present study. The first was the simple ring distribution.
This distribution has the advantage of allowing us to explore
plasma parameter space with relative ease. In addition, we can
investigate the dependence on wave vector of the normal
modes of the plasma. We have found using the ring distribution
that there are three modes with frequencies close to the
electron gyrofrequency which can be unstable in a weakly
relativistic electron plasma. At very low plasma frequencies, typically
/
*p* /
*mc*, the most unstable perpendicular
mode is the high phase velocity "trapped" mode. The mode is
trapped since the wave frequency lies between the cold
electron and hot electron gyrofrequencies and the mode is
decoupled from the freely propagating *R-X* branch when a cold
background plasma is present. Similarly to Pritchett [1984*b*]
, we find that the growth rate maximizes for this mode near
*k**c* /
.
Since we have not included temperature in the
ring distribution, we do not obtain the damping at low wave
vectors reported by Pritchett.

Using a cold plasma approximation for the dispersion relation
, Omidi and Wu [1985] have stated that the transition
from unstable *R-X* mode waves to unstable *Z* mode waves occurs at
/
= 0.1. In their study, Omidi and Wu used a
measured distribution function to obtain the growth rates.
However, since we include the effects of the hot electrons on
the wave dispersion, we find that the transition from larger
growth rate *Z* mode depends on the ring momentum also.

The ring distribution function is an extremely simplified
distribution function. The distribution almost certainly overestimates
growth rates, especially for modes with some finite *k*.
In addition, it is not clear that the wave dispersion is necessarily
modified by the presence of hot electrons in the measured
distributions, where thermal velocities can be large. We have
therefore studied the dispersion using a shell distribution
which is a spherically symmetric Dory-Guest-Harris distribution
. The form of the DGH distribution was chosen so as to
enable us to use the analysis of Shkarofsky [1966] as a basis
for the present study. Importantly, we found that provided
*j* > 1, i.e., *p* >
*p*, where
*p* is the momentum for which the
DGH distribution has a peak, the most unstable mode lies on
a "trapped" branch of the dispersion relation.

That the unstable mode for *j* > 1 is separate from the *R-X*
mode suggests that for most cases, the dispersion analysis
using the ring distribution may in fact be adequate. However,
the results of the warm "shell" distribution indicate that the
growth rates from the ring can be inaccurate. The effect of
temperature is to restrict the range of instabilities (see also
Winglee [1983]). Growth occurs only if the hot electron density
is large and the thermal spread of the energetic particles is
low enough.

Interestingly, once the thermal spread becomes large, *j* = 1,
the dispersive properties of the waves can change. We have
found that for some plasma parameters the unstable mode can
couple directly to the freely propagating *R-X* mode for *j* = 1.
When *j* > 1, the modes can approach one another closely, but
they do not couple directly. To show this, we used an approximate
form of the conductivity tensor which assumed that
*c**k* /
1.

Lastly, we discussed whether or not the current analysis can
be applied to auroral zone distributions. Figure 8
shows that for a DGH distribution there is considerable variation in the
appearance of the distribution function. It appears reasonable
to assume that *j* > 1 for those distributions that have been
accelerated through parallel electric fields. Also, the magnetic
mirror force causes the distribution to curve in momentum
space, giving a shell-like appearance. Most importantly, a
large peak in the distribution function at low energy does not
imply that a cold plasma approximation is adequate, because
higher-energy particles occupy a larger volume in momentum
space than lower-energy particles and so contribute more to
the integrals used to evaluate the conductivity tensor. While
more data are now available [Menietti and Burch, 1985], the
relative amount of cold (or cool) plasma on an auroral zone
field line is still an open issue.

In conclusion, we have described in some detail the wave
dispersion associated with weakly relativistic electron distributions. If, as we suggest here, auroral electron distributions
can be modelled by two species with different gyrofrequencies
due to relativistic effects, then a cold plasma approximation is
probably not adequate for describing the dispersion of wave
modes near the gyrofrequency. Our study shows that the
modified dispersion can have two offsetting effects when considering
the generation of AKR. First, the separation of the unstable
mode at low wave vectors from the free space branch of
the *R-X* mode may result in enhanced growth due to reflection
of the wave at the edges of the auroral cavity, as discussed
by Strangeway [1985]. Restricting the altitude range over
which the unstable mode can propagate may also remove the
problem of "detuning" the instability by refractive effects [see
Omidi and Gurnett, 1982, 1984]. In addition, the trapped mode
can be driven unstable by gradients in the phase density of
both downgoing and upgoing electrons, since
<
. On the
other hand, the separation of the unstable mode introduces
the requirement of an additional conversion process in any
mechanism for producing AKR. It is important that future
studies investigate these more global aspects of the wave

*Acknowledgments*. The author would like to thank P. L. Pritchett
and R. M. Winglee for many useful discussions. This work was
supported by the NASA Solar Terrestrial Theory Program under grant
NAGW-78 and the Air Force under grant F19628-85-K-0027.

The Editor thanks D. Le Queau and another referee for their assistance
in evaluating this paper.

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