*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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In this paper we have presented the results of a linear
stability analysis for waves in a weakly relativistic
electron plasma. We have been primarily interested in
the variation of the wave modes as a function of the
plasma system parameters and also in the dependence of
wave properties such as the group velocity on the
wave vector. We have shown that relativistic effects become important for
*p* /
*m**c* >
/
,
and the *R-X* mode cutoff can then lie below the cold electron
gyrofrequency. This enables perpendicularly propagating
waves to become unstable.

We have further shown that when both hot and cold electrons
are present, the range of instability is no longer confined to
*p* /
*m**c* >
/
.
When the plasma distribution function can be characterized by two
plasma components, a new wave mode is present, and this
mode lies between the gyrofrequencies of each
electron species. We have shown that this wave mode has
similar polarization to the cold plasma *R-X* mode,
although it is decoupled from this mode when cold
electrons are present. The wave frequency has little
variation as a function of the wave vector, resulting
in the small group velocities and short convective
growth lengths of the order of 1 km.

We have explored the variation of the group velocity in
the context of the generation of auroral kilometric
radiation. Using a very simple model for the density
cavity present on auroral field lines [Calvert, 1981*b*],
we have found that the ray propagation is such that ray
paths tend to direct the waves toward those altitudes
where the growth maximizes, since the wave propagates
more and more obliquely to the magnetic field until the wave propagates
perpendicularly. In addition to maximizing the growth rate as determined
in the present analysis, this will reduce damping due to gyroresonance
with the thermal electrons. Other effects such as feedback, as discussed
by Calvert [1982], may also be enhanced by the bending of ray
paths.

If the new mode is indeed responsible for the generation of AKR, the tendency to propagate perpendicular to the magnetic field and hence possibly carry out mulitple bounces across the auroral cavity may be an essential feature of the generation mechanism. The new mode is usually trapped and can only escape from the generation region through mode conversion. Mode conversion will probably occur near the edge of the auroral arc where the flux of energetic electrons decreases.

We have briefly discussed mode conversion to the *Z* mode in this paper,
using rather simple estimates for reflection coefficients. Although the
results of Mellot et al. [1984] show the presence of *L-O*
mode waves, we have not considered the possibility of mode conversion to the
*L-O* mode. Because of the presence of a stop band between the unstable
and the *R-X* mode, some form of "tunnelling" must be invoked to couple to
the freely propagating *R-X* mode. On the other hand, Pritchett
[1984*b*] has pointed out that for the propagation angles < 80° the *R-X* mode can be driven unstable by a DGH distribution.

We have shown that the delta function ring distribution may be a reasonable approximation for the electron distribution in the auroral density cavity for the purposes of studying wave dispersion. Nevertheless, it is apparent that a more sophisticated analysis is required to better address the effects on wave dispersion and ray propagation of temperature and nonuniformity in the plasma.

*Acknowledgements.* The author would like to thank
P.L. Pritchett and M. Ashour-Abdalla for many helpful discussions. This
work was supported by NASA Solar Terrestrial Theory Program under grant NAGW-78.

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