*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

As mentioned in the introduction, we only expect
relativistic effects to be important for some range of
plasma parameters. As a first step in determining the
parameter ranges for which relativistic corrections are
sufficient to modify the wave dispersion, we shall
assume that there are only hot ring electrons. The
solutions of the dispersion relation are shown in
Figures 1*a* and 1*b*. Figure 1*a* shows contours of constant
wave frequency normalized to the electron
gyrofrequency plotted as a function of ring momentum
and /
assuming *k* =
*k* = 0;
Figure 1*b* shows the
corresponding normalized growth rate times
10. In
Figure 1*b* the contour interval is 0.2, with a thick
contour every 1.0. The maximum growth rate in Figure 1*b*
is ~ 5 x 10. For reasons of clarity the contours in
Figure 1*a* are spaced at two different intervals. For
> 1.02 the contour interval is 0.002, whereas below
= 1.02 the contour interval is 0.001.

Fig. 1a. Contour plot of wave frequency for a ring distribution. Contours of normalized frequency are plotted as a function of ring momentum (p/mc) and normalized plasma frequency squared ( / ). Contours are spaced every 0.002 for / > 1.02 with thick contours at intervals of 0.02 . For / < 1.02 the spacing is .00l between thin contours.

Fig. 1b. Contour plot of growth rate for a ring distribution. The growth rate corresponds to the frequencies shown in Figure1a. The contours are spaced at every 2 x 10.

As can be seen from Figure 1*b*, instability only occurs
for large ring momentum and small gyrofrequencies. For
sufficiently small ring momentum the wave frequency
is given by the cold plasma frequency *R-X* mode cutoff,
as shown in Figure 1*a*. The slight kink in the frequency
contours plotted in Figure 1*a* occurs at the transition
from stable to unstable modes, as given by the = 0
contour in Figure l*b*. Both plasma parameters are
plotted using a logarithmic scale in the figures, and
the transition is a straight line. This line corresponds
very well to the limit given by Pritchett
[1984*b*], *p* /
*m**c* =
/
2
.
The wave frequency equals
the gyrofrequency along the line
*p* /
*m**c* =
/
,
which was also determined by Pritchett. As can be seen
from Figures 1*a* and
1*b*, the growth rate at *k* = 0 is maximum for a ring
distribution when =
.
However, this is
coincidental: equation (10) of Pritchett [1984*b*] shows
that the growth rate is a maximum for
*p* /
*m**c* =
/
, which also corresponds to
=
.
On the other
hand, the growth rate is maximum for
*p* /
*m**c* =
(3/2)
/
for a shell distribution (see equation
(19) of Pritchett [1984*b*]), which is a
three-dimensional distribution, whereas a ring is
essentially two-dimensional.

The maximum value of ring momentum used in Figures 1*a
*and l*b* is 0.1, which corresponds to 2.5-keV electrons.
Classically, these electrons would not be considered
to be relativistic, and so relativistic corrections
are essentially first-order effects. For higher values
of momentum the condition for maximum growth might be
expected to depart from the linearity displayed in
Figure 1*b*. However, typical auroral electron energies
are not observed to be in excess of a few keV [Croley
et al., 1978], and so we shall restrict our analysis in
the rest of the paper to the case of
*p* /
*m**c* =
0.1.

Fig. 2a. Plot of wave frequency for a ring distribution plus a cold background plasma. The frequency is shown as function of normalized plasma frequency squared ( / ) and the ratio of hot to total number density (n/n) using a three-dimensional representation.

Fig. 2Having chosen a particular value of ring momentum, we can now vary a different plasma parameter. Figures 2b. Plot of growth rate for a ring distribution plus a cold background plasma. The growth rate corresponds to the frequencies shown in Figure 2a.

It is apparent from Figure 2*a* that the unstable mode
can no longer couple to the *R-X* mode once some cold
plasma is introduced to the plasma dispersion relation.
Since neither electron distribution has any thermal
spread associated with it, both cold electrons and hot
ring electrons can be considered to be separate
particle species, each with its own gyrofrequency.
Except for the case
*n* =
*n* ,
all solutions lie in the
frequency range
/
<
<
where
is the relativistic gamma for the ring electrons.

The growth rate as plotted in Figure 2*b* shows that the
inclusion of cold plasma removes the restriction on the
range of instability. Unstable solutions are found for
quite large values of the electron plasma frequency. In
addition, although the growth rate does decrease as the
fraction of hot electrons is decreased, there is some
instability even for
*n* /
*n* = 0.01.
It should be noted,
however, that the frequency is very close to the
relativistic gyrofrequency, and thermal effects may
introduce significant damping, which cannot be
included using a
delta function distribution. The growth rate as given
in Figure 2*b* should be interpreted as an upper limit,
especially for low ring electron number densities.
However, the growth rate is still moderately large
~ 5 x 10
.

Summarizing the results of the plasma parameter study,
we have found that for a plasma that can be
characterized by two electron distributions there
exists an unstable mode which lies between the
gyrofrequencies of the two electron species. The growth
maximizes near
*p* /
*m**c* =
/
and also for
*n* ~
*n*.
Typical values of the growth rate are of the order of
~
10
.

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