*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.

**Next:** 6. The Applicability of the Ring Distribution

**Previous:** 4. Wave Vector Dependence

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Title and Abstract

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.

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Title and Abstract

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