*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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**Previous:** 3. Finite Temperature Distribution

**Top:** Title and Abstract

The results of the analysis of the warm dispersion relation
for **k** = 0 are very similar to those found for the ring, provided
*j* 2. While the
growth rates are different, and sometimes
even negative, we can still identify the modes with those found
for the ring dispersion relation. From our knowledge of the
wave dispersion for the ring we have identified the high-frequency mode with the *R-X* mode, and the low-frequency
mode with the trapped mode. The *R-X* mode is usually
damped or marginally stable, while the trapped mode can be
unstable. However, the identification of the modes is not easy
for *j* = 1, and our labelling of the modes for *j* = 1 in the
preceding section may be suspect.

In this section we shall demonstrate that the presumed
properties of the wave modes are at least approximately
correct for *j* > 1, and further clarify the nature of the modes for
*j* = 1. As shown by Shkarofsky [1966], the conductivity
tensor is very complicated if **k** 0 even for the
Maxwellian. However, we can set
*k*
= 0 for the purposes of discussing wave
dispersion, since it has been shown that nearly perpendicularly
propagating waves are most unstable [Pritchett, 1984*b*;
Strangeway, 1985; Pritchett and Strangeway
, 1985]. This
simplifies the final form of the conductivity tensor and also
separates *O* mode and *X* mode waves in the dispersion relation.
However, the right and left circularly polarized wave fields are
coupled together unless
*k* = 0.

On assuming that the distribution function is given by the
relativistic Maxwellian, the final form of the conductivity
tensor will be similar to that given by Shkarofsky [1966] in his
equation (2). Some differences may be introduced depending
on the approximations made, but the form of the tensor
*T*
as defined by Shkarofsky will be the same. The tensor
*T*
contains an infinite series of Bessel functions and the argument
in the Bessel functions is
= (
*k**c*) /
[( 1 -
*i*)] where
is an integration variable varying from 0 to . We
can use the small argument expansion for the Bessel functions provided
*k**c* /
1.
Obviously, this condition is satisfied
when *k* = 0,
in which case the *X* mode components of the
conductivity tensor reduce to

and

where () is the conductivity tensor for left- and right-hand polarized waves and is obtained from

where *f*(*p*) is the phase space density for the relativistic Maxwellian. We use (5) to obtain the conductivity tensor for higher
*j* values, ().

Rather than let *k*
0, we shall assume
. In this
case, we can rewrite the dispersion relation (6), including finite
*k*, as

where

and

*D*
() is the same as
*D*() as defined in (6), and corresponds to
the right-hand polarized wave. *D*
() can also be obtained
from (6) provided is redefined as
= ( +
) /
. Since we
are seeking solutions of (10), with
and hence 2,
we should use a large
form of (6).
*D*() as defined in (6) is
subject to numerical errors for
1. While large argument expansions of *Z*
(i()) exist, one finds on substituting these
into (6) that the first (*l* + *j*) powers of
exactly cancel,
and round-off errors may be significant. We therefore use the simplest large
form of (6), which is the dispersion relation for a
delta function shell. From Pritchett [1984*b*], we find that

where
= 1 + *j*/
.
We note that for typical parameters, with
and *j*/
0.005, the term containing
( - 1)
10, and
*D*()
nearly equals the value obtained for a purely cold plasma.

For the purposes of exploring the dependence on wave
vector the smallest value of
used will be = 200. In
addition, we shall restrict ourselves to
*k**c* /
1, therefore the highest value of
*k**c* /
5 x 10
1 We are hence
justified in using the dispersion relation (10) to determine wave
dispersion for finite *k*.

Solutions of the approximate dispersion relation are shown
in Figure 6
. We have chosen a high hot electron density
(*n* /
*n* =
0.9) and a low ambient electron temperature (
/
=
10) with a peak momentum at *p*/*mc* 0.1
and /
=
10, We have
used the same convention as that used in
Figures 3,
4, and
5, where the dashed line corresponds to the *R-X*
mode, the dotted line identifies the Bernstein mode, and the
solid line represents the trapped mode. In
Figure 6, the
identification of the modes appears to be correct. The *R-X* mode
phase velocity approaches the speed of light, while the
Bernstein mode and trapped mode have nearly constant frequency.

Fig. 6. Solutions of the approximate shell dispersion relation (10) for finitekand low ambient electron temperature. Similar in format to Figure 3.

For *j* 2, the
maximum growth rate of the trapped mode is
found for *k* =
0, while the maximum for *j* = 1 occurs near
*k**c* /
1. Pritchett
[1984*b*] also reported that the growth rate maximizes near
*k**c* /
1 for the loss cone DGH plus a
cold background, although he found that this occurred for all
*j* values. We do not show it here, but we have determined that
if we decrease the hot number density to
*n* /
*n* = 0.5, the *j* = 1
mode is always damped, while the *j* = 2 mode has maximum growth near
*k**c* /
= 1. Allowing the ambient plasma to have
a temperature therefore introduces some additional
complexity to growth rate dependence on wave vector.

Fig. 7. Solutions of the approximate shell dispersion relation (10) for equal temperature distributions. Similar in format to Figure 3. Note the expanded frequency scale.

For low ambient electron temperature, the dispersive
properties are very similar for all *j* values.
Figure 7 shows that
this is not necessarily the case for other plasma parameters. In
Figure 7, we have assumed that both ambient and hot
electrons have the same temperature. Since the dispersion curves
are more complicated, we have expanded the frequency scale.
For *j* 3, the
dispersion curves are similar to those shown in
Figure 6. However,
the dispersion curves are somewhat different
for *j* = 2 and *j* = 1.

When *j* = 2, the trapped mode approaches the *R-X* mode
and at first sight it appears that the two modes are coupled.
While it is not readily seen in the figure, the modes are still
decoupled because the transition to marginal stability for the
*R-X* mode occurs at a slightly lower value of
*k**c* /
than that for which the trapped mode becomes damped. Nevertheless,
the closeness of the dispersion surfaces suggests that the
modes may couple as a result of refraction and mode
conversion. If the mode conversion can take place, then we might
expect large radiation amplitudes, since the mode is unstable
for a large range in *k*
before coupling to the free space branch near
*k**c* /
0.6. We do not encounter the problem of the
sensitivity of the gyroresonant interaction to wave vector
magnitude and direction in a cold plasma. On the other hand,
it appears to be more typical for the trapped mode to be well
separated from the *R-X* mode.

When *j* = 1, the *R-X* mode and the unstable mode are
coupled in Figure 7.
The modes are coupled since the Bernstein
mode is always coupled to one of the other two modes. For
this reason, no Bernstein mode is shown for *j* = 1. At low k
,
the Bernstein mode couples with the *R-X* mode, while the
trapped mode lies on the branch cut with
0 from below.
At high *k*
, the Bernstein mode couples with the trapped mode.

The analysis with finite *k*
shows that for *j* 2 the trapped
mode is the unstable mode, and while the trapped mode can
approach the *R-X* mode, it never couples directly with the free
space branch. For *j* = l, the trapped mode may be unstable,
but for some plasma parameters, the unstable mode can
couple directly to the free space branch. It appears then that
the *j* = l DGH can be a special case. We will discuss this
point further in the next section.

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