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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In this section we shall obtain solutions of a simple dispersion relation for a plasma with a cold ambient plasma and an energetic ring of particles at constant perpendicular momentum. At present we shall not include any temperature in the particle distribution functions as we are primarily interested in elucidating some of the more general characteristics of the dispersion relation for a weakly relativistic plasma. The distribution function employed is given by
where p and p are the parallel and perpendicular momenta defined with respect to the ambient magnetic field, and n and n are the ambient (cold) and hot (ring) electron number densities, respectively.
The dispersion relation for such a distribution has been given by Pritchett [1984b] and Strangeway . When deriving the dispersion relation, we have retained the relativistic terms. Several authors [Wu and Lee, 1979; Wu et al., 1982; Omidi and Gurnett, 1982; Melrose et al., 1982; Dusenbery and Lyons, 1982] have emphasized the importance of retaining the relativistic terms in the condition for gyroresonance. However, as pointed out by Chu and Hirshfield , for a sufficiently low density plasma where the plasma frequency is much smaller than the electron gyrofrequency, relativistic effects are also important for the wave dispersion. The changes introduced to the wave dispersion and their implications for the generation of auroral kilometric radiation have been discussed by Pritchett [1984a,b] and Strangeway . p> Chu and Hirshfield  and also Winglee  (using a more sophisticated Dory-Guest-Harris [Dory et al., 1965] distribution) have studied wave dispersion for parallel propagating waves, and Pritchett [1984a, b] has presented dispersion curves for perpendicularly propagating waves. However, it is useful to consider arbitrary propagation directions in order to obtain a more complete understanding of the wave dispersion in a weakly relativistic plasma. Strangeway  has presented some results of such an analysis, but in his work he was principally interested in studying the group velocity variation as a function of propagation direction for a specific mode. In this section we shall extend the work of Strangeway to emphasize the relationship between the various modes present in the plasma.
Fig. 1 Wave dispersion surfaces for the ring dispersion. Four models are present when both a hot and cold electron component are included.
Solutions of the dispersion relation for a ring distribution are presented in Figure 1. Since we are primarily interested in instabilities near the electron gyrofrequency, we have only considered modes near this frequency. There are four dispersion surfaces shown in Figure 1. These surfaces have been plotted as a function of parallel and perpendicular wave vector normalized to a characteristic distance c / where c is the speed of light and is the cold electron gyrofrequency ( = eB / mc, where e is the magnitude of the electron charge, B is the ambient magnetic field strength and m is the electron mass). When obtaining the solutions shown in the figure, we have assumed that the hot electrons are 75% of the total electron density, the ring momentum is 0.1mc (corresponding to an energy of ~2.5 keV), and the ratio of plasma frequency to gyrofrequency squared / ) is 0.005. The plasma frequency is the total electron plasma frequency, = 4ne / m, where nt, is the total electron number density. A plasma for which / < p / mc has been described as an underdense plasma by Strangeway , for reasons we will outline below. The frequency range used in plotting all four dispersion surfaces is the same, 0.985 < / < 1.005, where / is the real part of the wave frequency ( = + i , where is the growth rate).
The simplest surface to describe is the surface plotted in the upper left of the figure. This corresponds to the cold plasma R-X mode, somewhat modified by relativistic effects. The surface has been truncated at / = 1.005, although the mode continues to higher frequencies, where the surface asymptotically approaches the light cone. The R-X mode cutoff in this case is above the cold electron gyrofrequency.
Moving to lower frequencies, that is clockwise around the figure from the upper left-hand corner, the next mode encountered shows a more complicated structure. At high wave vectors, the frequency lies just above the cold electron gyrofrequency, and we shall refer to this mode as the "cold intermediate" mode for kc / > 1, where there is sudden change in the dispersion surface. Below this value of wave vector (i.e., waves with phase velocities greater than the speed of light) the mode is unstable. The growth rates for this mode and the other unstable modes present are given in Figure 2, and we shall discuss the growth rates in more detail when describing Figure 2. The unstable mode in the upper right of Figure 1 is the mode found in the simulations of Pritchett [1984a, b] and discussed at some length by Strangeway . Since the mode is decoupled from the freely propagating R-X mode, we shall refer to this unstable mode as the "trapped" mode. It will be noted that the group velocity for the unstable mode, given by the slope of the dispersion surface , is small. This is generally the case for this mode. For most values of wave vector, except near the transition, the group velocity is positive. In an overdense plasma where / > p / mc, the parallel group velocity is usually anomalous, the dispersion surface having negative slope, as pointed out by Strangeway .
Fig. 2. Concatenated dispersion surfaces with their corresponding growth rates. The four dispersion surfaces shown in Figure 1 have been merged into a single display. We have truncated the surface identified with the "cold intermediate" mode for reasons of clarity. An unstable mode is present where two stable modes coalesce, and the corresponding growth rates are shown by the contour plot.
The next surface, which is plotted in the lower right-hand corner of Figure 1, shall be referred to as a "hot intermediate" mode. The frequency of this mode is near the hot electron gyrofrequency. Upon comparison with the dispersion surface in the upper right of the figure, it can be seen that the dispersion surface is the same in both plots for small kc / i.e., the "trapped" mode. Since we are using delta function particle distributions, complex solutions of the dispersion relation come in complex conjugate pairs. Unstable solutions are found where two different dispersion surfaces coalesce.
The last dispersion surface plotted in Figure 1 corresponds to the Z mode and is shown in the lower left-hand corner of the figure. The dispersion surface has been truncated at the low-frequency limit of the plot. As the wave frequency approaches the ring electron gyrofrequency (= 0.995 ), the Z mode leaves the light cone. At larger wave vectors two unstable branches are found, one predominantly propagating parallel to the magnetic field, the other propagating perpendicular to it. It is interesting to note that the Z mode is not unstable for all propagation angles. The mode at large k corresponds to the Weibel-type instability described by Chu and Hirshfield  and Winglee , while the mode at large k corresponds to the unstable Z mode discussed by Pritchett [1984b].
Having described the different dispersion surfaces in some detail, we now merge them into a single plot as shown in Figure 2. The parts of the dispersion surfaces corresponding to the unstable modes are the regions for which different dispersion surfaces are colocated. It will be noted that for reasons of clarity we have not plotted all of the dispersion surface for the "cold intermediate" mode. At the right-hand side of the figure we have plotted contours of growth rate for the unstable modes. As mentioned before, there are three unstable branches. The Weibel-type instability appears to have the larger growth rate, although Landau damping by thermal electrons may reduce this growth rate. For perpendicular propagation, the largest growth rate occurs on the trapped branch. We have not shown it here, but even though the growth rate is increasing for kc / = 2, the maximum for the Z mode branch is less than that for the trapped branch.
The growth rates for different branches of the dispersion relation are sensitive to the plasma parameters, and it is not certain that the fastest growing mode for oblique propagation will always be the low wave vector trapped mode. On neglecting the effects of hot electrons on the wave dispersion, Omidi and Wu  have stated that the transition from larger growth rate R-X mode waves to larger growth rate Z mode waves occurs at / = 0.1. We do not present the results here, but we have found the transition from the trapped mode to the perpendicular Z mode also depends on the relative density of hot electrons and the energy of the hot electrons as well as the plasma frequency-gyrofrequency ratio. For example, when n / n = 0.75, the transition occurs at / pmc. Since we have neglected the effects of temperature, the calculated growth rates from the ring distribution are probably in error, and we shall not give a more detailed investigation of the growth rates for the different modes in the present study. Nevertheless, it appears that the inclusion of the relativistic corrections to the wave dispersion allows other modes to be more unstable than the Z mode over a larger range of / .
The inclusion of temperature in the dispersion relation will almost certainly modify the growth rates of the various instabilities. In addition, temperature may also modify the dispersive properties of the plasma. Specifically, if the temperature of both energetic and ambient electrons is high enough, it is not clear that the plasma could be modeled as two separate species. We shall therefore investigate the effects of temperature in the next section.
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