## Wave Dispersion and Ray Propagation in a Weakly Relativistic Electron Plasma: Implications for the Generation of Auroral Kilometric Radiation

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.

### 4. Wave Vector Dependence

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 / mc = 0.1. Since the growth rate is near maximum for / = p / mc , 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. 3b. Plot of growth rate as a function of wave vector. The growth rate corresponds to the frequencies shown in Figure 3a.
Solutions of the dispersion relation with the plasma parameters as specified above are shown in Figures 3a and 3b. The frequency (in Figure 3a) and growth rate (in Figure 3b) are plotted using a three-dimensional representation as a function of normalized wave vector. It should be noted that the wave vector varies logarithmically; the solutions at the lower left- hand corner of the figure correspond to k = k = 0.01 / c, not zero. The maximum value of k is / c, and as can be seen from Figure 3a, where the wave frequency is shown to be very nearly equal to the gyrofrequency, this corresponds to waves propagating at the speed of light.

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 3a, , 0.002 , while the range in wave vector, k, / c. The group velocity is consequently small; / k, 0.002c. Additionally, while / k is positive, / k < 0 for the parameters shown in Figure 3a.

The growth rate plotted in Figure 3b 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 3b 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 3a and 3b.
When 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 3a and 3b. E / E is the fraction of perpendicular wave electric field magnitude that is right-hand circularly (RHC) polarized. Since the polarization is defined with respect to the ambient magnetic field direction, the fraction of RHC polarized field decreases as the perpendicular phase velocity approaches the speed of light. This is similar to the cold plasma wave polarization, since the mode becomes transverse as the phase speed approaches the speed of light for a cold plasma wave. For parallel propagation the wave can still be RHC, but for perpendicular polarization the mode eventually becomes plane polarized.

Fig 5. Ratio of longitudinal to transverse electric field magnitude. This ratio is plotted for the solutions in Figures 3a and 3b.
In 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 near k c / = 0.1, but throughout the wave vector range considered the mode displays similar polarization characteristics to the cold plasma R-X mode. This indicates that in an inhomogeneous medium such as the region of intense electron precipitation above an auroral arc, mode coupling between the unstable trapped mode and the freely propagating R-X mode should be possible. A detailed analysis of the mode conversion is required to adequately address this point, in particular studying aspects of mode conversion such as the ratio of L-O to R-X mode produced as a consequence of the coupling.

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. 6b. Plot of the growth rate for a low density plasma. The growth rate corresponds to the frequencies shown in Figure 6a.
However the wave frequency and growth rate variation shown in Figures 3a and 3b are not necessarily the same for all plasma parameters. Figures 6a and 6b show solutions of the ring dispersion relation for / = 10 as a function of wave vector. All other parameters are the same as in Figures 3a and 3b. Since / < p / mc, we shall refer to the plasma as being underdense. In contrast to Figure 3b, where the growth rate was maximum for k = k = 0.01c / 0, the growth rate in Figure 6b maximizes near kc / = 1. The wave frequency of the unstable mode in Figure 6a maximizes near kc / 1. While the corresponding group velocity will still be small, the ray direction is more nearly parallel to the wave direction.

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 / mc > / (underdense), / k > 0, whereas / k < 0 for p / mc / (overdense). The growth rate is also dependent on the relative density of the plasma. For an underdense plasma, is a maximum near kc / = 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.

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