*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.

1. Introduction

2. Dispersion Relation

3. Plasma Parameter Dependence

4. Wave Vector Dependence

5. Discussion On Group Velocity Variation

6. The Applicability of the Ring Distribution

7. Conclusions

References

Several features of the radiation present major
problems for a theoretical understanding of the
generation of the radio noise. First, the noise is
intense; Gurnett [1974] has calculated that the peak
wave power is of the order of 10 W, some 1% of the
power estimated to be deposited in the ionosphere by
auroral electrons during a substorm. The average power
level was found to be ~ 10 W [Gallagher and Gurnett,
1979]. Second, the wave is primarily observed in the
cold plasma *R-X* mode. This has been deduced from direct
polarization measurements on the Voyager spacecraft
[Kaiser et al., 1978] and also inferred through ray
path calculations [Green et al., 1977]. In addition,
spacecraft have flown directly through the generation
region for AKR, which is presumed to be below altitudes
of 1-2 R on auroral zone field lines. Data from ISIS
[Benson and Calvert, 1979; Calvert, 1981a; Benson,
1982] support the inference that AKR is generated in
the *R-X* mode through the observation of a low-frequency
cutoff for AKR, which is strongly tied to the local
gyrofrequency. Recently, Mellott et al. [1984] have
used DE-1 data to show that AKR is usually observed to
be in the *R-X* mode near the presumed generation
region.

Taking into account these properties of AKR, one of the
more popular mechanisms for the generation of AKR is the
cyclotron maser instability [Wu and Lee, 1979]. This
instability is driven by the direct gyroresonance of
energetic electrons with the *R-X* mode. Direct
gyroresonance was first proposed by Melrose [1976] to
explain the generation of Jovian decametric radiation
(DAM). Since Melrose used a drifting bi-Maxwellian to
describe the energetic particle distributions, he found
that significant thermal anisotropies were required to
drive the *R-X* mode unstable through gyroresonance.
Typically, *v*
*v*
*c* ,
where *v*
and
*v*
are the
perpendicular and parallel thermal velocities,
respectively, and *c* is the speed of light.

It was shown by Wu and Lee [1979] that gyroresonance
can generate waves more efficiently, provided there is
a positive slope of the energetic particle
distribution as a function of perpendicular velocity.
These authors applied their theory to the generation of
AKR and found that coupling to the *R-X* mode was
enhanced by assuming that the plasma frequency in the
generation region was significantly lower than the
gyrofrequency. Additionally, for relativistic
gyroresonance, resonant particles describe an ellipse
in velocity space. If the ellipse lies fully within the
loss cone, for example, then no damping terms are
introduced on integrating over all resonant particles.
The low plasma frequency allows the *R-X* mode cutoff to
approach the gyrofrequency, which in turn allows lower-energy
electrons to resonate with the waves.

Recently, several authors [Wu et al., 1982; Omidi and Gurnett, 1982; Dusenbery and Lyons, 1982] have investigated the generation of AKR by the cyclotron maser instability, using particle data to determine the sources of free energy in the plasma. While a particular frequency may have large growth at one location on an auroral zone field line, Omidi and Gurnett [1982] have shown that this wave can be damped as the wave propagates along the field and enters different plasma regimes. The effect of the parallel electric field presumed to exist on auroral field lines [Croley et al., 1978] was included by Wu et al. [1982], who addressed the effects introduced by assuming that only energetic electrons determined the wave propagation characteristics. Generation by both the primary downgoing electrons and the loss cone in the reflected and backscattered electrons was investigated by Dusenbery and Lyons [1982]. One aspect of all the work cited here is the sensitivity of the growth rate to the parameters chosen to describe the plasma, in terms of both the energetic electrons and the presumed cold electrons.

It should be pointed out that while the S3-3
observations reported by Croley et al. [1978] show
population inversions associated with both the loss
cone and field-aligned acceleration of the energetic
electrons, the low-energy electron population has not
been measured directly on auroral field lines. Wave
measurements from the Hawkeye spacecraft have been used
by Calvert [1981*b*] to show the presence of an auroral
density cavity, but again this does not allow us to
directly determine the properties of the low-energy electron
population.

Bearing in mind the uncertainty in determining the
amount of cold plasma present, it is possible that the
wave propagation characteristics may be determined by
the more energetic electrons, and several authors [Tsai
et al., 1981; Wu et al., 1981, 1982; Winglee, 1983;
LeQueau et al., 1984*a*, *b*] have studied the
modifications of the wave properties by including
energetic electrons in the wave dispersion relation.

Recently, Pritchett [1984*a*, *b*] has investigated
instabilities associated with the mildly relativistic
magnetized plasma, using both simulations and linear
analysis. The types of particle distribution
functions employed in his simulation were both "ring"
and "shell" distributions. In general, a ring distribution
is characterized by a ring of energetic
particles with constant perpendicular momentum (where
perpendicular is defined with respect to the ambient
magnetic field), whereas a shell distribution is
typically given by a spherical shell of particles in
momentum space. In the simulations these distributions
can also have a thermal speed associated with
them. Pritchett concluded that when *p* / *m**c* ~ /
(where *p* / *m**c* is the characteristic energetic particle
momentum and / is the ratio of the electron plasma
frequency to the electron gyrofrequency), the plasma
can be unstable to waves propagating normal to the
ambient field direction. When only hot electrons are
present, the unstable mode couples directly to the *R-X
*mode, but inclusion of a thermal electron component can
decouple the two modes.

Relativistic effects consequently result in the electrons being essentially a two-species distribution, with the new mode lying between the gyrofrequencies associated with each particle species. Since the difference between the two gyrofrequencies is small for typical parameters associated with auroral electrons, one might expect the wave group velocity to be small. Hence it would be of interest to study the wave characteristics in some detail. In this paper we shall investigate the wave modes introduced by relativistic effects. We shall particularly emphasize the variation of the wave modes as a function of the parameters which describe the plasma and further study some of the properties of the waves themselves, such as polarization and group velocity.

Since the group velocity of the unstable waves associated with a ring distribution will be shown to have considerable variation, we shall study this wave property in more detail. We shall show, using a simple model for the auroral density cavity, that the convective properties of the waves may have important implications for the generation of AKR. Not only is the magnitude of the group velocity important, but also the direction of ray propagation. The former results in short convective growth lengths, whereas the latter can effectively focus the waves at an altitude most suited for growth due to the bending of the ray paths.

The structure of the paper is as follows. In the next section we shall briefly describe the dispersion relation used in the present analysis, together with a summary of the plasma parameters and wave diagnostics used. In section 3 we shall present solutions of the wave dispersion relation as a function of the plasma parameters, while in section 4 we shall describe some of the wave properties as a function of wave vector. The implications of the group velocity variations as applied to the generation region for AKR will be discussed in section 5. In section 6 we investigate in detail the applicability of the present analysis to auroral electron distributions. The results of the study will be summarized in the final section.

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 [1984*b*] and Winglee [1983], who
also considered a Dory-Guest-Harris (DGH)
distribution [Dory et al., 1965].
Pritchett [1984*b*]
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. [1984*b*] 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
[1984*a*, *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* / *m**c*, *m* is the
rest mass of the electrons,

= [(1 + *p* / *m**c* )],
*B* is the ambient magnetic field, and the other symbols
have their

usual meaning;

*p* / *m**c* 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*-*k**z*-*k**x*)], then
the waves must satisfy the dispersion relation

with

where

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.

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
.

Having shown how the instability of a ring distribution
varies as a function of the plasma parameters, we shall
now
explore the dependence on the wave vector. In the
previous section we considered the limit *k*
= *k*
= 0.
In this section we shall fix the plasma parameters and
vary the wave vector direction and magnitude. As in the
previous section, the ring electrons shall be assumed to
have a momentum
*p* /
*m**c* = 0.1.
Since the growth rate is near maximum for
/
=
*p* /
*m**c* ,
we shall choose
/
= 0.01.
Finally, we shall somewhat arbitrarily assume that
*n* /
*n* = 0.75.

Fig. 3a. Plot of wave frequency as a function of wave vector. The plasma parameters are held fixed, while the wave vector magnitude and direction are varied.

Fig. 3Solutions of the dispersion relation with the plasma parameters as specified above are shown in Figures 3b. Plot of growth rate as a function of wave vector. The growth rate corresponds to the frequencies shown in Figure 3a.

One aspect of the wave dispersion which we shall
discuss in more detail in the next section is the group
velocity variation. The total range in frequency in
Figure 3*a*,
,
0.002
,
while the range in wave vector,
*k*,
/ *c*.
The group velocity is consequently small;
/
*k*,
0.002*c*. Additionally, while
/
*k*
is positive,
/
*k*
< 0 for the parameters shown in
Figure 3*a*.

The growth rate plotted in
Figure 3*b* shows little
variation for *kc* /
< 0.1. As the wave phase velocity
decreases, aproaching the speed of light, the growth
rate decreases, until a stable mode is encountered close *kc* /
= 1.
There is a slight difference in where the
transition is encountered as a function of *k*
and *k*
.
With the particular choice of system parameters the
growth is maximum for small *k*. Typical growth rates from
Figure 3*b* are
/
0.002, which on combination with
the estimate for group velocity gives convective growth
lengths of the order *c* /
less than the free space
wavelength which is a few kilometers.

As discussed in the previous section, the unstable mode
is on a separate branch of the dispersion relation when
some cold plasma is present, and it does not couple
directly to the *R-X* mode. If this mode is ultimately
responsible for the generation of AKR, then some
coupling to a freely propagating mode must occur. While
the mechanism for mode conversion beyond the scope of
the present work, we can indicate some
features of the mode which imply that mode conversion
may be feasible. At the limit *k*
= 0 or
*k* = 0 the
dispersion relation given by (2), (3), and (4)
separates into two equations. In the cold plasma limit,
these result in the *R-X* mode and the *L-O* mode. On
including a hot ring component, the new mode is just an
additional root to the *R-X* dispersion relation.
Consequently, the polarization of the new wave mode is
the same as the *R-X* mode at the limits *k*
= 0 or
*k* = 0.

Fig. 4. Fraction of wave electric field that is right-hand circularly polarized. The polarization is plotted for the solutions shown in Figures 3When the wave vector is neither parallel nor perpendicular, the dispersion relation is no longer separable. It is hence not obvious what the polarization of the new mode should be. In Figure 4 we show the wave polarization for the unstable mode plotted in Figures 3aand 3b.

Fig 5. Ratio of longitudinal to transverse electric field magnitude. This ratio is plotted for the solutions in Figures 3In Figure 5 we show the ratio of longitudinal wave electric field magnitude to the transverse electric field magnitude. Near the speed of light the wave is almost completely transverse, as expected. The wave is least electromagnetic nearaand 3b.

Fig. 6a. Plot of the wave frequency for a low-density plasma. This figure is similar in format to Figure 3a, but / = 10, i.e.,p/mc> / , which we refer to as an underdense plasma.

Fig. 6However the wave frequency and growth rate variation shown in Figures 3b. Plot of the growth rate for a low density plasma. The growth rate corresponds to the frequencies shown in Figure 6a.

To summarize the properties of the unstable mode as a
function of wave vector, we find that the polarization
is very similar to the cold plasma *R-X* mode. Usually,
the wave frequency is a maximum for large
*k* (
/
*k*
> 0), but the dependence on *k*
depends on whether the plasma is underdense or overdense. If
*p* /
*m**c* >
/
(underdense),
/
*k*
> 0, whereas
/
*k*
< 0 for
*p* /
*m**c*
/
(overdense).
The growth rate is also dependent on the relative density of the plasma.
For an underdense plasma,
is a maximum near
*k**c* /
= 1,
whereas is a maximum for *k* = 0 in an overdense
plasma. We shall address the transition from underdense
to overdense plasma in more detail in the next
section.

The group velocity variation discussed in the previous section is strongly dependent on the plasma system parameters, and some implications for the generation of AKR can be formed. As an initial attempt to study the group velocity variation in the context of auroral zone phenomena we shall present a simple model for the plasma parameters associated with an auroral zone field line.

Fig. 7. Model of the auroral density cavity. The electron gyrofrequency, plasma frequency-gyrofrequency ratio, and characteristic perpendicualr momentum are plotted as a function of geocentric distance along a field line. The derivation of these parameters is described in more detail in the text.Figure 7 shows how the plasma parameters are assumed to vary as a function of geocentric distance from 1.3 to 3.3

We have not included the variation of
*n* /
*n*
as a function of altitude, since this parameter is even more
uncertain than the other parameters presented here.
Given the plasma frequency variation shown in Figure 7,
the total number density is less than unity between 1.8 and 3
*R*
geocentric distance. It is consequently not
unreasonable to assume that the energetic electrons
contribute significantly to the total number density.

The results presented in the previous sections have
already suggested that there is a transition in the
wave properties near the region
*p* /
*m**c* =
/
.
We shall explore the variation of frequency, growth rate,
and group velocity near this critical region more
thoroughly here. To reduce the number of variables
which describe the plasma, we shall assume for the time
being that the variation of the ratio
/
as a function of altitude is most significant. By assuming
that the other parameters which describe the plasma
are constant, only the parallel component of wave
vector varies with altitude. We can consequently
trace ray paths to different altitudes by obtaining
solutions of the dispersion relation as a function of
/
and
*k* *c* /
.

It is important to note that in addition to assuming that
*p* /
*m**c* and
*n* /
*n* are both constant,
we are also implicitly assuming that the gyrofrequency does not
vary significantly. This allows us to assume that
*k**c* /
is constant. We shall subsequently show that the
gyrofrequency variation is usually of major importance.
For the present, however, we shall ignore the change in
gyrofrequency as a function of altitude.

Fig. 8a. Plot of wave frequency for a model of an auroral zone field line. Solutions of the ring dispersion relation have been found assuming that only the normalized plasma frequency and parallel wave vector vary as a function of altitude along an auroral zone field line.

Fig. 8Figures 8b. Plot of growth rate for a model of an auroral zone field line. The growth rate corresponds to the frequencies shown in Figure 8a.

A transition in the parallel group velocity is apparent
in Figure 8*a*, for small values of
/
,
/
*k*
> 0, while
/
*k*
< 0 for large values of
/
.
To emphasize this transition, we have plotted contours of
parallel group velocity (*v*
) in
Figure 9. We have restricted the range in
*v*
to
*v*
/ *c*
2 x 10,
and for clarity we have not shown contours of group velocity for
the stable branch. The shaded area in the figure shows that part of the
parameter space for which
*v*
< 0. When
*k* *c* /
0.1, the transition occurs at a constant value of
/
= 0.01, i.e.,
/
=
*p* /
*m**c*.

To emphasize that
*v*
= 0 near
/
=
*p* /
*m**c* for
small values of wave vector, we show contours of
*v*
when
*k* *c* /
= 0.01
in Figure 10. For this figure we allow both
/
and
*p* /
*m**c* to vary.
Since the wave vector is small, the corresponding group velocities are small.
For reference the perpendicular group velocity is roughly of the same order,
*v*
/ *c*
2 x 10
. The
*v*
= 0
contour in Figure 10 is given by
/
=
*p* /
*m**c*.

Fig. 9. Contour plot of parallel group velocity. The contours have been plotted forv/c2 x 10, with a contour interval of 10. The plasma parameters are the same as in Figures 8aand 8b. The shaded area indicates that region for which the parallel group velocity is less than zero.

Fig. 10. Contour plot of parallel group velocity for fixed density ratio. Similar in format to Figure 9. The contour interval is 10 .In summary, Figures 9 and 10 show that for small wave vectors the parallel group velocity goes to zero when

Except where
*p* /
*m**c* ~
/
,
the parameter whose variation as a function of altitude is of primary
importance is most likely to be the gyrofrequency. From
the solutions presented in the previous section
together with the group velocity variation discussed
here we can determine how the wave properties will vary
as a function of altitude. When both hot and cold
electrons are present, we have shown that the unstable
mode is trapped between the two electron
gyrofrequencies. This will determine the range in
altitude over which a wave at a particular frequency
can propagate. With our knowledge of the group velocity
variation we can further infer the values of wave
vector for which the frequency will be a maximum or a
minimum. It should be remembered, however, that we have
not completely explored the parameter space, and so
some caution should be exercised.

Fig. 11. Schematic of the wave properties at different altitudes along the auroral field line. The arrows in the top panel indicate the ray direction for a fixed frequency. The thickening is meant to suggest propagation out of the plane of the figure across the ambient field. The bottom three panels give estimates of the growth rate, probable wave vector directions, and wave vector magnitude. The wave properties have been summarized for both an underdense and an overdense region on the field line.We summarize the expected dependence on altitude of the wave properties in Figure 11. We have chosen two altitude ranges, one centered on 1.9

The bottom three panels show how we expect the wave
vector magnitude (*kc* /
),
wave vector direction (angle), and growth rate
( /
)
to vary given the constraints on the
wave frequency. The numbers on these three panels are
only qualitative, indicating the typical values we
expect. In the plots of wave vector angle we have shown
both the range in angle and the direction of *k*
with respect to the ambient magnetic fields. As we shall
discuss below, the sections drawn with a dashed line
are meant to indicate that the wave may be reflected.

For both an underdense and an overdense plasma the wave
frequency approaches the cold electron gyrofrequency
for propagation near 90° with
*ck* /
1.
The group velocity is nearly perpendicular for large
*k*.
For a wave at a particular frequency the wave frequency will approach the cold
electron gyrofrequency at the higher altitudes. The above properties when
are indicated in Figure
11, the only difference between an undersense and
overdense plasma being the estimate for the growth rate at
.
Since propagation is near 90° and damping is minimized,
the ring distribution growth rate is a reasonable estimate.
As shown in Figures 3*b* and
6*b*,
/
is a minimum at large
*k* for an overdense
plasma, whereas growth is near maximum for large
*k* in an underdense plasma.

We have already suggested that a perpendicularly propagating wave may be reflected at the edge of the auroral arc. It would be useful to obtain some quantitative estimates of the efficiency of the mirror. The edge of the auroral arc is likely to be associated with both an increase in cold electron density and a decrease in the hot electron ring momentum, because of the weakening parallel electric field. These two effects will tend to remove the unstable mode, and it is not clear how the wave will refract in such a medium. Additionally, while a wave may be propagating perpendicularly to the magnetic field, the wave vector can be parallel to the edge of the auroral arc.

Bearing these problems in mind, we can at present only
roughly estimate reflection coefficients. Using
arguments similar to those of Calvert [1982], the
reflection coefficient for waves incident perpendicular
to the edge of the auroral arc is given by

when and
are the refractive indices outside and
inside the arc. We shall assume
(*ck*
/
1).
Since the wave polarization for the unstable mode is
generally similar to the *R-X* mode, it might be argued
that the wave will propagate away from the arc in the *Z*
mode. For 90° propagation the *Z* mode is given by

Consequently, when
,
2 -
/
. So for
/
<< 1,
1.414, and *R*
0.17 from (5).
Consequently, only about 3% of the wave power would be
reflected.

It should be pointed out that the estimate is a "worst case" estimate, and the reflection coefficient may be substantially higher. This calculation does emphasize the necessity of a more sophisticated mode conversion analysis. For the time being we shall assume some reflection does occur at the edge of the arc.

While we have indicated the possibility of reflection
for the underdense plasma through the dashed vectors
for wave propagation angle, we have not shown similar
reflection for the overdense case. Because
/
*k*
<
0 for an overdense plasma, the wave frequency
approaches the hot electron gyrofrequency for *k*
*c* /
1. Since there is no freedom in choosing the wave
vector direction with respect to the gradient along the
field, unlike perpendicular propagation as discussed
above, we may be more confident in our estimate of
reflection coefficients at the low-altitude cutoff for
the unstable mode.

Again through polarization considerations the mode is
likely to couple to the *Z* mode. For parallel
propagation,

At the low-frequency cutoff,
/
, i.e.,
/
1 -
*p*
/
2*m*
*c*. Consequently,

1 + (2
*m*
*c* /
*p*
)

For an overdense plasma,
3. When
1,
*R* 0.27,
and the reflected power is approximately 7%.

This figure is somewhat higher than for perpendicular
reflection, and it might be argued that reflection
can occur at
both high- and low-frequency cutoffs in an overdense plasma. However,
*k*
0
for the low-frequency cutoff
in an underdense plasma, and reflection is likely to
be much more efficient. We have consequently assumed
that an underdense plasma regime is more favorable to
multiple reflections and possible feedback such as
discussed by Calvert [1982].

One important feature of the wave properties shown in
Figure 11 is the altitude range over which we expect a
wave at a particular frequency to exist. Since the wave
is confined to the altitudes where
/
<
<
,
the altitude range must be less than the distance over
which the change in magnetic field
*B* < (
- 1)*B*. In
Figure 11, corresponding to the parameters shown in
Figure 7, the altitude range is at most 0.005 *R*
and can be as small as 0.002 *R*
.
The scale length along the
field line is consequently a few tens of kilometers.
While this is at least 2 orders of magnitude larger
than the Debye length (some hundreds of meters), the
plasma may be quite structured at this scale length.
Moreover, the wavelength of the waves can be comparable
to this scale length, and this could result in
significant changes in the wave dispersion.

Bearing in mind the assumptions inherent in the results
presented here, the convective properties of the waves
driven unstable by a ring distribution yield some
interesting implications for the generation of AKR.
Specifically, the dependence of frequency on wave
vector is such that the waves tend to propagate to
those altitudes where *k*
is large, and growth is maximized when
*p* /
*m**c* >
/
.
Also for altitudes where
*p* /
*m**c* >
/
the waves may be able to perform
multiple transitions of the auroral cavity, thus
enhancing their growth. We note that the above
condition for reflection restricts the region of
maximum growth to those altitudes which yield a
frequency bandwidth consistent with observations, 100
kHz *f* 500 kHz.

One of the principal advantages obtained by using the delta function ring distribution is the simple form of the resultant dispersion relation. This simplicity allows us to carry out extensive parameter studies. While the ring distribution may be considered to be a modification of cold plasma theory which takes into account the presence of hot electrons, it is not clear that the dispersion relation can be arbitrarily applied to the auroral plasma. In this section we shall address this point in more detail, using the results presented in the previous sections as a basis for the discussion.

The two principal results of the analysis are, first, that for
*p*
/
*m*
*c*
/
relativistic corrections are significant, and second, if the electrons can be
considered as two separate species, then the unstable
mode is trapped between the two species
gyrofrequencies. We must hence determine when the
auroral electron distribution function can be said to
satisfy both of these conditions, in which case the
ring distribution may be a reasonable approximation for
the purposes of investigating wave dispersion.

We must reiterate that the main goal for the present analysis is to study wave dispersion. Obviously, the growth rate for instability will depend strongly on the detailed form of the distribution function. However, the wave dispersion is controlled more by the collective properties of particle species, as embodied by the integration over all momentum space when calculating the conductivity tensor for a plasma.

Another point concerning the integration over momentum
space is that at a particular momentum, particles with
pitch angles near 90° occupy a larger solid angle than
particles near 0° or 180°, and so contribute more to the
integrals. For this reason we find that the ring and
shell dispersion relation for **k** = 0 are very similar
[see Pritchett, 1984*b*]. The dispersion relation for the
shell distribution for finite wave vector is not as
tractable as the ring, since the integrals cannot be
readily separated into parts when relativistic
corrections are included. The ring distribution
relation is hence used to extend the analysis to finite
wave vector.

We must now determine whether the ring or shell distributions are applicable to auroral electron distributions. The electron distributions as measured on S3-3 show three basic features, as pointed out by Omidi et al. [1984]. These are a widened loss cone in the upgoing particles, a "hole" at low energies, and a "bump" near 90° pitch angles. We also note that at higher energies the distributions are close to spherical symmetry apart from the loss cone feature. For the purposes of determining wave dispersion the loss cone is probably not significant (again invoking solid angle arguments). We are then left with a spherically symmetric distribution at higher energies, i.e., a shell, with a bump near 90°, i.e., a ring.

It appears then that both a cylindrically symmetric "ring" distribution and a spherically symmetric "shell" distribution can be applied to the auroral electron distribution. However, we have made one further assumption, that is, that the hot electrons have no thermal spread. While the measured distributions show quite large temperatures, the approximation is still reasonable provided we can characterize the ambient and energetic electrons as two species with different gyrofrequencies due to relativistic effects. Unfortunately, the bulk of the low-energy ambient plasma (if present) is presumably below the energy range of an electron detector such as that flown on the S3-3 spacecraft. Nevertheless, we can determine some limits for the applicability of our analysis.

A useful distribution for modeling a spherically symmetric distribution with a peak at some finite momentum is the DGH distribution, which can be generalized as

where *n*
is the number density of the particle species,
*p* is the thermal momentum,
(*v*) is the gamma function, and
*p*
is the momentum at which *f*(*p*) has a maximum. For the
purposes of calculating a dispersion relation,
*p* /
*p*
is usually set to some integer value.

From (8) the average momentum squared for the distribution is given by

where <*w*> is the characteristic energy of the
distribution.

Relativistic effects are important for a particular particle species if

i.e., if

where *n*
is measured in particles per cubic centimeter,
<*w*> is in keV, and
*f*
is the nonrelativistic electron gyrofrequency in megahertz.

We shall use the data published by Mizera and Fennel,
[1977] and Croley et al. [1978] to determine whether or
not (11) is satisfied for measured distributions. From
Croley et al there is a peak in the distribution near
*v* = 1.5 x 10
km/s with *f*(*v*) = 21.5 s
km, while at *v*
= 3 x 10 km/s
*f*(*v*) = 0.0464 s
km. On substitution into (8) this yields
*p*
/
*p*
4, with
*p* /
*m**c*
0.05 and
*n*
0.36 cm.
For this distribution, <*w*> 0.88
keV. At the altitude where the distribution was
measured, 7300 km, the magnetic field was 0.058 G
[Mizera and Fennell, 1977], giving a gyrofrequency of
0.16 MHz. From (11) the density must be
1.1 cm.
Relativistic effects are consequently important for
this distribution.

However, if there is a sufficient amount of low-energy
ambient plasma, these particles will dominate wave
dispersion, and a cold plasma formulation is adequate.
Fortunately, we can obtain a lower limit on the
relative densities for hot and ambient electrons. If
the peak associated with the hot electron distribution
is small in comparison to the phase space density of
the ambient electrons at the peak momentum, the hot
electrons are probably unimportant for wave
dispersion. Noting that (8) reduces to a Maxwellian
when *p*
= 0, we can say that if

then the hot electrons are significant. The subscript a
denotes the ambient Maxwellian plasma. In general we expect
*p* <
*p* , and it can be shown that
*f* /
*f*
is then a monotonically increasing function of
*p* /
*p* provided
*p* /
*p* > 1.5.
The constraint imposed by (12) is consequently less well satisfied when
*p* =
*p* ,
in which case, (12) can be rewritten as

The condition given by (13) tends to zero as
*p* /
*p*
,
i.e., when the temperature tends to zero. This is to be expected,
since at this limit the distribution function is
equivalent to a delta function, and the hot electrons
are distinct from the ambient electrons. On the other
hand, the condition is less easily satisfied as the
temperature increases. The condition given by (13)
therefore reflects the spread in gyrofrequencies
associated with a warmer distribution. For
*p* /
*p*
4,
corresponding to the S3-3 distribution,
*n* /
*n*
0.3.

There is one last constraint which we can apply to the
relative densities. If the ambient particles have too
low a number density, then the hot electrons dominate,
and the electrons are again effectively a
single-particle species. To ensure that there are
sufficient numbers of ambient electrons, we shall
require that the difference in
<*p*>
for each species, and hence the difference in the "average"
gyrofrequency, be comparable to the total
<*p*>, i.e.,

For *p* =
*p* ,
(14) reduces to

Again we find that must be greater than 1.5.
*p* /
*p*
must be greater than 1.5.
For *p* /
*p* = 4,
*n* /
*n*
1.67.

As written, the conditions given by (13) and (15) imply that
For *p* /
*p*
2.5,
otherwise the upper limit given by
(15) is lower than the limit given by (13). However, we
note that the limits are flexible, and this requirement
may be relaxed.

To summarize, we have shown that for the S3-3
distribution published by Croley et al. [1978] the
delta function ring distribution is a reasonable
approximation when determining the wave dispersion provided 0.3
*n* /
*n*
1.67.
In obtaining this condition we have made two assumptions
which should be verified. The first is that, provided the
phase space density of 90° pitch angle particles is high,
the asymmetry in parallel momentum associated with the loss
cone is not significant for wave dispersion. The second
is that the dispersion is similar for both a ring and a
shell. Both assumptions essentially reduce to a single
one, that the near-perpendicular particles dominate the
wave dispersion for waves near gyroresonance. The
simulations of Pritchett [1984*a*, *b*] and Pritchett and
Strangeway [1985] indicate that this may be the case.
The latter work included an asymmetric electron
distribution, in which a single-sided loss cone was
included. The growth rates were altered by the use of
different distributions, but the wave modes were
essentially the same. However, a more thorough analysis
would be useful, addressing this issue and also
providing better estimates than those presented here on
the range of plasma parameters for which the dispersion
relation used in the present study is adequate.

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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