On the Applicability of Relativistic Dispersion to Auroral Zone Electron Distributions

J. Geophys. Res., 91, 3152 - 3166, 1986.
(Received August 8, 1985; revised November 8, 1985; accepted December 2, 1985.)
Copyright 1986 by the American Geophysical Union.
Paper number 5A8825

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4. Extension to Finite Perpendicular Wave Vector

      The results of the analysis of the warm dispersion relation for k = 0 are very similar to those found for the ring, provided j 2. While the growth rates are different, and sometimes even negative, we can still identify the modes with those found for the ring dispersion relation. From our knowledge of the wave dispersion for the ring we have identified the high-frequency mode with the R-X mode, and the low-frequency mode with the trapped mode. The R-X mode is usually damped or marginally stable, while the trapped mode can be unstable. However, the identification of the modes is not easy for j = 1, and our labelling of the modes for j = 1 in the preceding section may be suspect.

      In this section we shall demonstrate that the presumed properties of the wave modes are at least approximately correct for j > 1, and further clarify the nature of the modes for j = 1. As shown by Shkarofsky [1966], the conductivity tensor is very complicated if k 0 even for the Maxwellian. However, we can set k = 0 for the purposes of discussing wave dispersion, since it has been shown that nearly perpendicularly propagating waves are most unstable [Pritchett, 1984b; Strangeway, 1985; Pritchett and Strangeway , 1985]. This simplifies the final form of the conductivity tensor and also separates O mode and X mode waves in the dispersion relation. However, the right and left circularly polarized wave fields are coupled together unless k = 0.

      On assuming that the distribution function is given by the relativistic Maxwellian, the final form of the conductivity tensor will be similar to that given by Shkarofsky [1966] in his equation (2). Some differences may be introduced depending on the approximations made, but the form of the tensor T as defined by Shkarofsky will be the same. The tensor T contains an infinite series of Bessel functions and the argument in the Bessel functions is = ( kc) / [( 1 - i)] where is an integration variable varying from 0 to . We can use the small argument expansion for the Bessel functions provided kc / 1. Obviously, this condition is satisfied when k = 0, in which case the X mode components of the conductivity tensor reduce to


where () is the conductivity tensor for left- and right-hand polarized waves and is obtained from

where f(p) is the phase space density for the relativistic Maxwellian. We use (5) to obtain the conductivity tensor for higher j values, ().

      Rather than let k 0, we shall assume . In this case, we can rewrite the dispersion relation (6), including finite k, as



      D () is the same as D() as defined in (6), and corresponds to the right-hand polarized wave. D () can also be obtained from (6) provided is redefined as = ( + ) / . Since we are seeking solutions of (10), with and hence 2, we should use a large form of (6). D() as defined in (6) is subject to numerical errors for 1. While large argument expansions of Z (i()) exist, one finds on substituting these into (6) that the first (l + j) powers of exactly cancel, and round-off errors may be significant. We therefore use the simplest large form of (6), which is the dispersion relation for a delta function shell. From Pritchett [1984b], we find that

where = 1 + j/ . We note that for typical parameters, with and j/ 0.005, the term containing ( - 1) 10, and D() nearly equals the value obtained for a purely cold plasma.

      For the purposes of exploring the dependence on wave vector the smallest value of used will be = 200. In addition, we shall restrict ourselves to kc / 1, therefore the highest value of kc / 5 x 10 1 We are hence justified in using the dispersion relation (10) to determine wave dispersion for finite k.

      Solutions of the approximate dispersion relation are shown in Figure 6 . We have chosen a high hot electron density (n / n = 0.9) and a low ambient electron temperature ( / = 10) with a peak momentum at p/mc 0.1 and / = 10, We have used the same convention as that used in Figures 3, 4, and 5, where the dashed line corresponds to the R-X mode, the dotted line identifies the Bernstein mode, and the solid line represents the trapped mode. In Figure 6, the identification of the modes appears to be correct. The R-X mode phase velocity approaches the speed of light, while the Bernstein mode and trapped mode have nearly constant frequency.

Fig. 6. Solutions of the approximate shell dispersion relation (10) for finite k and low ambient electron temperature. Similar in format to Figure 3.

      For j 2, the maximum growth rate of the trapped mode is found for k = 0, while the maximum for j = 1 occurs near kc / 1. Pritchett [1984b] also reported that the growth rate maximizes near kc / 1 for the loss cone DGH plus a cold background, although he found that this occurred for all j values. We do not show it here, but we have determined that if we decrease the hot number density to n / n = 0.5, the j = 1 mode is always damped, while the j = 2 mode has maximum growth near kc / = 1. Allowing the ambient plasma to have a temperature therefore introduces some additional complexity to growth rate dependence on wave vector.

Fig. 7. Solutions of the approximate shell dispersion relation (10) for equal temperature distributions. Similar in format to Figure 3. Note the expanded frequency scale.

      For low ambient electron temperature, the dispersive properties are very similar for all j values. Figure 7 shows that this is not necessarily the case for other plasma parameters. In Figure 7, we have assumed that both ambient and hot electrons have the same temperature. Since the dispersion curves are more complicated, we have expanded the frequency scale. For j 3, the dispersion curves are similar to those shown in Figure 6. However, the dispersion curves are somewhat different for j = 2 and j = 1.

      When j = 2, the trapped mode approaches the R-X mode and at first sight it appears that the two modes are coupled. While it is not readily seen in the figure, the modes are still decoupled because the transition to marginal stability for the R-X mode occurs at a slightly lower value of kc / than that for which the trapped mode becomes damped. Nevertheless, the closeness of the dispersion surfaces suggests that the modes may couple as a result of refraction and mode conversion. If the mode conversion can take place, then we might expect large radiation amplitudes, since the mode is unstable for a large range in k before coupling to the free space branch near kc / 0.6. We do not encounter the problem of the sensitivity of the gyroresonant interaction to wave vector magnitude and direction in a cold plasma. On the other hand, it appears to be more typical for the trapped mode to be well separated from the R-X mode.

      When j = 1, the R-X mode and the unstable mode are coupled in Figure 7. The modes are coupled since the Bernstein mode is always coupled to one of the other two modes. For this reason, no Bernstein mode is shown for j = 1. At low k , the Bernstein mode couples with the R-X mode, while the trapped mode lies on the branch cut with 0 from below. At high k , the Bernstein mode couples with the trapped mode.

      The analysis with finite k shows that for j 2 the trapped mode is the unstable mode, and while the trapped mode can approach the R-X mode, it never couples directly with the free space branch. For j = l, the trapped mode may be unstable, but for some plasma parameters, the unstable mode can couple directly to the free space branch. It appears then that the j = l DGH can be a special case. We will discuss this point further in the next section.

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