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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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 R The electron gyrofrequency ( ) has been found assuming a dipole field for an invariant latitude of 70°. The ratio / is based on the results of Calvert [1981b], and we have assumed that / = 0.3 at distances of 1.3 and 3.3 R, with a minimum value of less than 0.03 near 2 R. The characteristic perpendicular momentum (p / mc) has been found assuming that the accelerating electric potential varies linearly with the magnetic field strength and is zero at 3.3 R. The corresponding perpendicular momentum is then obtained from the ellipse produced in momentum space due to the magnetic mirror force [Chiu and Schulz, 1978]. The maximum value of p / mc is assumed to be 0.2, corresponding to 10-keV electrons. It should be noted that p / mc does not go to zero when the accelerating potential goes to zero because of the widened loss cone.
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 / mc = / . 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 / mc and n / n are both constant, we are also implicitly assuming that the gyrofrequency does not vary significantly. This allows us to assume that kc / 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. 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.Figures 8a and 8b show the frequency and growth rate of the unstable mode as a function of / and k c / . At high k , k c / ~ 1 the mode becomes stable, as indicated by the sudden change in frequency in Figure 8a and the rapid decrease in growth rate in Figure 8b. The growth rate has a maximum for most of the range in k c / at / = 0.007. When only hot electrons were included in the dispersion relation , the maximum occured at a slighly higher value.
A transition in the parallel group velocity is apparent in Figure 8a, 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 / mc.
To emphasize that v = 0 near / = p / mc 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 / mc 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 / mc.
Fig. 9. Contour plot of parallel group velocity. The contours have been plotted for v / c 2 x 10, with a contour interval of 10. The plasma parameters are the same as in Figures 8a and 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 p / mc / . We had already inferred the presence of this transition from results presented in the previous section, where we introduced the notion of underdense and overdense plasma regions. Without carrying out a full ray tracing analysis, we can still make some inferences on the properties of both underdense and overdense plasma regions in the context of a more general model of the auroral field line as outlined in Figure 7, where we can no longer assume that any plasma parameter can be taken to be constant.
Except where p / mc ~ / , 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 R corresponding to an underdense plasma region, the second centered on 2.9 R, where the plasma is overdense. The two sloping lines in the top panel of each set show the variation with altitude of the cold and hot electron gyroErequencies. The upper line corresponds to the cold electron gyrofrequency. The thick arrows indicate how the group velocity varies with altitude for a wave of a given frequency. The wave is confined to the altitude where / < < , being the relativistic gamma of the hot electrons. The thickening of the arrows is meant to suggest that the group velocity is turning perpendicular to the ambient magnetic field and pointing out of the plane of the figure.
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 3b and 6b, / 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 , 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
At the low-frequency cutoff, / , i.e., / 1 - p / 2m 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 .
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 / mc > / . Also for altitudes where p / mc > / 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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