*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

**Next:** 2. Ring Distribution

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However, the cyclotron maser [Wu and Lee, 1979] has
become one of the more favored mechanisms. The cyclotron
maser, which is driven by gyroresonance between energetic
electrons and high phase velocity electromagnetic waves, has
the advantage of transferring energy directly from the particles [*Omida and Gurnett, 1982, 1984]*.

One important feature of the cyclotron maser is the requirement
that the gyroresonance condition be relativistically
correct, as originally pointed out by Wu and Lee [1979] and
subsequently discussed in some detail by several authors [Wu
et al., 1982; Omidi and Gurnett, 1982; Melrose et al., 1982;
Dusenbery and Lyons, 1982]. Bearing in mind the dependence
of net growth on the refractive properties of the medium,
Pritchett [1984*a*, *b*], Pritchett and
Strangeway [1985], and Strangeway
[1985] have recently pointed out that relativistic effects
are also important for the wave dispersion near the electron
gyrofrequency. Typically, relativistic corrections are important
*p* / *mc*
/
where
*p* is the characteristic momentum
of the electrons, *m* is the electron rest mass, *c* is the speed of
light,
is the electron plasma frequency, and
is the electron gyrofrequency.

Strangeway [1985] showed that if the electron distribution can be modelled as two separate species and both species contribute to the wave dispersion, then since the higher-energy particles have their own gyrofrequency due to relativistic effects the most unstable mode is trapped between the two gyrofrequencies and is decoupled from the freely propagating branch of the dispersion relation. In one sense, this may be advantageous, since the wave may be reflected at the edges of the auroral arc. This could enhance the growth in a manner analogous to the feedback mechanism of Calvert [1982]. On the other hand, if the unstable mode is indeed on a separate branch of the dispersion relation, some means for mode conversion must be invoked to allow the radiation to escape from the source region.

Before embarking on efforts to calculate mode conversion
rates and ray paths in the complicated plasma region found
on auroral zone field lines, we should confirm that the previous analysis
is indeed relevant to the auroral zone distributions. The condition
*p* / *mc*
/
appears to be readily
satisfied since Calvert [1981] has shown the presence of a
substantial plasma cavity on auroral zone field lines during active times.
The ratio /
is less than 0.1 for a considerable
altitude range along an auroral field line. This value of
/
gives a lower limit on the particle energy of 2.5 keV,
which is about the same as typical field-aligned potential
drops.

It is more difficult to determine if the electrons should be treated as two species. We should clarify here the notion of a "two-species" electron distribution. It is usually assumed in instability theory, including the generation of AKR, that the cold ambient plasma dominates the wave dispersion and the more energetic electrons act as a source of free energy for resonant instabilities. However, if the relative numbers of ambient and energetic electrons becomes comparable, it is likely that the hot electrons will also modify the wave dispersion, in addition to driving instability. In this case, two electron species are present in the wave dispersion relation. It seems reasonable to assume that the electrons be assumed to be "two species" if ambient and hot electrons have roughly equal densities, especially if both species are cold (i.e., no thermal spread) and the hot electrons only have a drift with respect to the ambient electrons. However, we must also consider the effects of temperature. Increasing the temperature of a distribution in essence spreads the gyrofrequency over a range of frequencies, due to relativistic effects. If the spread in gyrofrequency becomes too large, then it is not obvious that the electrons can be characterized as "two species."

The present study is principally motivated by the need to determine when weakly relativistic wave dispersion is applicable to auroral zone electron distributions and hence determine if the electrons can be classified as "two species." A major limitation of the work of Strangeway [1985] is the assumption of a cold ring of electrons as a model for the energetic auroral electrons. This form of distribution automatically results in a "two-species" electron plasma. Furthermore, it is not clear that a ring is a reasonable model for the electron distribution. The distribution functions published, for example, by Croley et al. [1978] show considerable structure. As pointed out by Omidi et al. [1984], the measured distribution contains features such as a single-sided loss cone in the upgoing distribution, a "bump" at near 90° pitch angle and a "hole" in the downgoing distribution. While the "bump" may be reminiscent of the cold ring distribution, the additional features suggest that a more realistic model should be employed. Features that are taken into account in the present analysis are the more nearly spherical nature of the hot electron distribution, and the spread of the distribution in velocity space.

We only qualitatively investigate the modifications to wave dispersion associated with the asymmetry of the distributions introduced by the loss cone and "hole." The results of the simulation study of Pritchett and Strangeway [1985] indicate that an asymmetric distribution has an asymmetric dependence on angle for the growth rate. However, the presence of the unstable mode is still due to relativistic corrections to the wave dispersion, and we consider the inclusion of temperature to be more important.

Several other authors have also considered the effect of
relativistic corrections to wave dispersion for various types of
distribution [Tsai et al., 1981; Wu et al., 1981, 1982; Winglee,
1983; Le Queau et al., 1984*a*, *b*]. In addition, much effort has
recently been directed to the generation of *Z* mode radiation
by the cyclotron maser [Hewitt et al., 1983; Melrose et
al.,
1984; Omidi et al., 1984; Dusenbery and Lyons, 1985;
Omidi and Wu, 1985]. We shall therefore discuss in some detail the
various unstable modes found for the simplified ring distribution
function previously employed by Chu and Hirshfield
[1978] and Strangeway [1985], for the purposes of relating the
present work with other analyses, and as an introduction to
the more involved warm electron dispersion relation.

The structure of the paper is as follows. In section 2 we shall
present solutions of a simple ring distribution, describing the
various modes found. The effects of finite temperature will be
studied in sections 3 and 4. We will find from the analysis of a
warm electron dispersion relation that provided the peak in
the distribution function occurs at a higher momentum than
the thermal spread of the distribution, the most unstable mode
at low wave vector is decoupled from the freely propagating
mode. Section 3 will describe the warm electron dispersion
relation for **k** = 0, where **k** is the wave vector, and section 4
will present results for an approximate dispersion relation for
**k**
0, perpendicular being defined
with respect to the ambient
magnetic field. In section 5 we shall discuss the form of the
distribution function used in the warm electron dispersion
relation. We shall show that for certain plasma parameters,
the model distribution may be a reasonable approximation to
measured distributions. In particular, we will stress that a
large peak in the distribution at low energies does not
necessarily mean that a cold plasma approximation is adequate. We
will summarize the results of the present study in the last
section.

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