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 are principally interested in the
modifications of wave dispersion due to relativistic
effects and in the dependence of the wave instabilities
on the parameters which describe the plasma. Bearing
this in mind, we shall use the simplest distribution
function which incorporates the effects under
consideration, i.e., a source of free energy and
sufficient energetic electrons to modify wave
dispersion. The distribution function employed will
hence be the delta function ring distribution given by
where 
(p)
is the Dirac delta function; p
is the
perpendicular particle momentum defined with respect to
the ambient magnetic field and p
is the parallel
component; p
is the momentum of the ring particles;
and n
and n
are the cold and hot electron number
densities, respectively.
The distribution function as given by (1) is defined as particle density in momentum phase space, rather than velocity phase space as is usually the case, to ensure that relativistic effects are included correctly. The delta function ring distribution was originally investigated by Chu and Hirshfield [1978], who described the instabilities associated with such a distribution in terms of azimuthal and axial bunching of electrons by the wave fields. These authors noted that for parallel propagation, azimuthal bunching is primarily associated with high phase velocity waves. The delta function ring distribution has been further studied by Pritchett [1984b] and Winglee [1983], who also considered a Dory-Guest-Harris (DGH) distribution [Dory et al., 1965]. Pritchett [1984b] carried out simulations which indicate that the ring distribution tends to overestimate the growth rates of the linear stage of the instability for waves propagating at angles significantly away from the normal to the ambient field. Presumably this is due to effects not included in such a simple distribution function, such as cyclotron damping by warm electrons. However, the main thrust of the present work is to indicate those properties of the modified wave dispersion which may be of significance for the generation of AKR. Moreover, near 90° propagation, thermal effects are less likely to be important.
By using a delta function distribution function we not only neglect thermal effects, but also no longer generate instabilities through gyroresonance, which is the usual assumption for the cyclotron maser instability. Indeed, the instability driven by a ring distribution is a fluid-type instability, analogous to classical instabilities such as the Buneman instability [Buneman, 1958], where the free energy for instability is due to relative drift of two particle species. In this particular case there is no net flow, but the gyrational energy of the particles is available for wave growth. When the number densities of each species are comparable, we shall show that the frequency lies near the middle of the unstable frequency range. If the two electron populations have low temperatures, there will be very few electrons available for gyroresonance, and the present analysis may be more applicable. However, the auroral electron distribution function usually displays considerable structure, and whether or not the wave dispersion for such a plasma results in fluidlike or resonant instabilities is not easily determined. LeQueau et al. [1984b] have addressed the transition from one instability regime to another in some detail.
A second criticism must be raised concerning the use of
a ring distribution when modeling auroral electrons.
When Wu et al. [1982] discussed the modification of the
R-X mode cutoff due to relativistic effects, they noted
that this might be an artifact of what they called a
double loss cone distribution. The double loss cone is
symmetric about p = 0, with a widened loss cone due
to parallel electric fields. In an extreme case this
distribution could be modeled by a ring distribution.
However, double loss cones are not observed in the
particle data. On the other hand, the data do show a
shell-like distribution in addition to the single
loss cone feature. The simulations of Pritchett
[1984a, b] show that for 90° propagating waves the wave
modes for the shell and the ring distribution are
similar. While the ring distribution may significantly
overestimate the growth rates due to the neglect of
damping, the distribution is at least useful in
determining some of the properties of the wave mode
introduced by relativistic modifications. We shall
address the applicability of the ring distribution to
auroral electron distributions in more detail in section 6.
With (1) defining the electron distribution, we can define a set of parameters which characterize the plasma:
/ 
the square of the ratio of the
electron plasma frequency over the electron
gyrofrequency, where
= 4
e(n + n / 
) / m
and = eB / mc, m is the
rest mass of the electrons,
= [(1 + p / mc )]
,
B is the ambient magnetic field, and the other symbols
have their
usual meaning;
p / mc characteristic normalized ring momentum;
n / n
ratio of hot ring electrons to the total number
density.
If we assume a first-order electromagnetic wave
perturbation of the form E(r) = E exp [-i(t-kz-kx)], then
the waves must satisfy the dispersion relation
(,
k) = 0
(2)
, 0, k), 
indicates summation over
species, and 
is the conductivity tensor for each
particle species derived from the linearized Vlasov
equation using standard techniques [e.g., Clemmow and
Dougherty, 1969]. It should be noted that the ions are
assumed to be fixed, and the summation is only carried
out for the hot and cold electrons. The conductivity
tensor for a ring distribution can be easily calculated
from the dielectric tensor given by Pritchett [1984b]
(his equation (6)). That is,
= J(
) is the Bessel function of the first
kind of order n, with n an integer, = kp / m, and
n is the number density for the particle species.
We can consequently parameterize the wave perturbation
by its complex frequency ( = 
+ i) and its wave
vector (k, k ). Growth of an instability corresponds
to positive . Also, for convenience, although
as
used above is a complex quantity, when presenting
solutions of the dispersion relation, shall
correspond to the real part of the frequency. In addition
, we shall use the wave group velocity (v
), the
ratio k
E / k 
E = E
/ E
, and the ratio
E
-
iE
/
2
E =
E
/
E
as wave diagnostics. The
latter two quantities give information on the wave
polarization, E / E being the ratio of
longitudinal to transverse electric fields and
E
/ E giving the fraction of perpendicular electric
field that is right-hand circularly polarized.
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