*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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In this section we shall obtain solutions of a simple dispersion relation for a plasma with a cold ambient plasma and an energetic ring of particles at constant perpendicular momentum. At present we shall not include any temperature in the particle distribution functions as we are primarily interested in elucidating some of the more general characteristics of the dispersion relation for a weakly relativistic plasma. The distribution function employed is given by

where *p* and
*p*
are the parallel and perpendicular momenta
defined with respect to the ambient magnetic field, and
*n* and
*n* are the
ambient (cold) and hot (ring) electron number densities,
respectively.

The dispersion relation for such a distribution has been
given by Pritchett [1984*b*] and Strangeway
[1985]. When
deriving the dispersion relation, we have retained the relativistic
terms. Several authors [Wu and Lee, 1979; Wu et al.,
1982;
Omidi and Gurnett, 1982; Melrose et al.,
1982; Dusenbery and
Lyons, 1982] have emphasized the importance of retaining the
relativistic terms in the condition for gyroresonance. However,
as pointed out by Chu and Hirshfield [1978], for a sufficiently
low density plasma where the plasma frequency is much
smaller than the electron gyrofrequency, relativistic effects are
also important for the wave dispersion. The changes introduced
to the wave dispersion and their implications for the
generation of auroral kilometric radiation have been discussed
by Pritchett [1984*a*,*b*] and Strangeway
[1985].
p>
Chu and Hirshfield [1978] and also Winglee
[1983] (using a
more sophisticated Dory-Guest-Harris [Dory et al., 1965]
distribution) have studied wave dispersion for parallel propagating
waves, and Pritchett [1984*a*, *b*] has presented dispersion
curves for perpendicularly propagating waves. However, it is
useful to consider arbitrary propagation directions in order to
obtain a more complete understanding of the wave dispersion
in a weakly relativistic plasma. Strangeway [1985] has
presented some results of such an analysis, but in his work he was
principally interested in studying the group velocity variation
as a function of propagation direction for a specific mode. In
this section we shall extend the work of Strangeway to emphasize
the relationship between the various modes present in the
plasma.

Fig. 1 Wave dispersion surfaces for the ring dispersion. Four models are present when both a hot and cold electron component are included.

Solutions of the dispersion relation for a ring distribution are presented in Figure 1.
Since we are primarily interested in instabilities near the
electron gyrofrequency, we have only considered modes near
this frequency. There are four dispersion surfaces shown in
Figure 1. These surfaces have been
plotted as a function of parallel and perpendicular wave
vector normalized to a characteristic distance *c* /
where *c* is the speed of light and
is the cold electron gyrofrequency
( =
*eB* / *mc*, where *e* is the magnitude of the electron charge,
*B* is the ambient magnetic field strength and *m* is the electron
mass). When obtaining the solutions shown in the figure, we
have assumed that the hot electrons are 75% of the total
electron density, the ring momentum is 0.1*mc* (corresponding to an energy
of ~2.5 keV), and the ratio of plasma frequency to gyrofrequency squared
/
) is 0.005.
The plasma frequency is the total electron plasma frequency,
=
4*n**e* / *m*,
where *n** _{t}*, is the total electron number density. A plasma for which
/
<

The simplest surface to describe is the surface plotted in the
upper left of the figure. This corresponds to the cold plasma *R-X* mode,
somewhat modified by relativistic effects. The surface has been truncated at
/
= 1.005,
although the mode continues to higher frequencies, where the surface
asymptotically approaches the light cone. The *R-X* mode cutoff in
this case is above the cold electron gyrofrequency.

Moving to lower frequencies, that is clockwise around the
figure from the upper left-hand corner, the next mode
encountered shows a more complicated structure. At high wave
vectors, the frequency lies just above the cold electron gyrofrequency,
and we shall refer to this mode as the "cold intermediate" mode for *kc* /
> 1, where
there is sudden change in the dispersion surface. Below this value of wave
vector (i.e., waves with phase velocities greater than the speed
of light) the mode is unstable. The growth rates for this mode
and the other unstable modes present are given in
Figure 2,
and we shall discuss the growth rates in more detail when
describing Figure 2.
The unstable mode in the upper right of
Figure 1 is the mode
found in the simulations of Pritchett
[1984*a*, *b*]
and discussed at some length by Strangeway
[1985]. Since the mode is decoupled from the freely propagating
*R-X* mode, we shall refer to this unstable mode as the
"trapped" mode. It will be noted that the group velocity for
the unstable mode, given by the slope of the dispersion surface
, is small. This is generally the case for this mode. For
most values of wave vector, except near the transition, the
group velocity is positive. In an overdense plasma where
/
>
*p* / *mc*,
the parallel group velocity is usually anomalous,
the dispersion surface having negative slope, as pointed
out by Strangeway [1985].

Fig. 2. Concatenated dispersion surfaces with their corresponding growth rates. The four dispersion surfaces shown in Figure 1 have been merged into a single display. We have truncated the surface identified with the "cold intermediate" mode for reasons of clarity. An unstable mode is present where two stable modes coalesce, and the corresponding growth rates are shown by the contour plot.

The next surface, which is plotted in the lower right-hand
corner of Figure 1, shall
be referred to as a "hot intermediate"
mode. The frequency of this mode is near the hot electron
gyrofrequency. Upon comparison with the dispersion surface
in the upper right of the figure, it can be seen that the dispersion
surface is the same in both plots for small
*kc* /
i.e., the "trapped" mode. Since we are using delta function particle
distributions, complex solutions of the dispersion relation
come in complex conjugate pairs. Unstable solutions are
found where two different dispersion surfaces coalesce.

The last dispersion surface plotted in
Figure 1 corresponds
to the *Z* mode and is shown in the lower left-hand corner of
the figure. The dispersion surface has been truncated at the
low-frequency limit of the plot. As the wave frequency
approaches the ring electron gyrofrequency (= 0.995
), the
*Z* mode leaves the light cone. At larger wave vectors two unstable
branches are found, one predominantly propagating parallel
to the magnetic field, the other propagating perpendicular
to it. It is interesting to note that the *Z* mode is not unstable
for all propagation angles. The mode at large
*k* corresponds
to the Weibel-type instability described by Chu and Hirshfield
[1978] and Winglee [1983], while the mode at large
*k*
corresponds to the unstable *Z* mode discussed by Pritchett [1984*b*].

Having described the different dispersion surfaces in some
detail, we now merge them into a single plot as shown in
Figure 2. The
parts of the dispersion surfaces corresponding to
the unstable modes are the regions for which different dispersion
surfaces are colocated. It will be noted that for reasons of
clarity we have not plotted all of the dispersion surface for the
"cold intermediate" mode. At the right-hand side of the figure
we have plotted contours of growth rate for the unstable
modes. As mentioned before, there are three unstable
branches. The Weibel-type instability appears to have the
larger growth rate, although Landau damping by thermal
electrons may reduce this growth rate. For perpendicular propagation,
the largest growth rate occurs on the trapped branch. We have
not shown it here, but even though the growth rate is increasing for
*kc* /
= 2, the maximum for the *Z* mode branch is less than
that for the trapped branch.

The growth rates for different branches of the dispersion
relation are sensitive to the plasma parameters, and it is not
certain that the fastest growing mode for oblique propagation
will always be the low wave vector trapped mode. On neglecting
the effects of hot electrons on the wave dispersion, Omidi
and Wu [1985] have stated that the transition from larger growth rate
*R-X* mode waves to larger growth rate *Z* mode waves occurs at
/
= 0.1.
We do not present the results here, but we have found the transition from the
trapped mode to the perpendicular *Z* mode also depends on the relative
density of hot electrons and the energy of the hot electrons as
well as the plasma frequency-gyrofrequency ratio. For example, when
*n* /
*n*
= 0.75, the transition occurs at
/
*p**mc*.
Since we have neglected the effects of temperature, the calculated
growth rates from the ring distribution are probably in
error, and we shall not give a more detailed investigation of
the growth rates for the different modes in the present study.
Nevertheless, it appears that the inclusion of the relativistic
corrections to the wave dispersion allows other modes to be
more unstable than the *Z* mode over a larger range of
/
.

The inclusion of temperature in the dispersion relation will almost certainly modify the growth rates of the various instabilities. In addition, temperature may also modify the dispersive properties of the plasma. Specifically, if the temperature of both energetic and ambient electrons is high enough, it is not clear that the plasma could be modeled as two separate species. We shall therefore investigate the effects of temperature in the next section.

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