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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6. Summary and Conclusions

      We have employed two different plasma dispersion relations in the present study. The first was the simple ring distribution. This distribution has the advantage of allowing us to explore plasma parameter space with relative ease. In addition, we can investigate the dependence on wave vector of the normal modes of the plasma. We have found using the ring distribution that there are three modes with frequencies close to the electron gyrofrequency which can be unstable in a weakly relativistic electron plasma. At very low plasma frequencies, typically / p / mc, the most unstable perpendicular mode is the high phase velocity "trapped" mode. The mode is trapped since the wave frequency lies between the cold electron and hot electron gyrofrequencies and the mode is decoupled from the freely propagating R-X branch when a cold background plasma is present. Similarly to Pritchett [1984b] , we find that the growth rate maximizes for this mode near kc / . Since we have not included temperature in the ring distribution, we do not obtain the damping at low wave vectors reported by Pritchett.

      Using a cold plasma approximation for the dispersion relation , Omidi and Wu [1985] have stated that the transition from unstable R-X mode waves to unstable Z mode waves occurs at / = 0.1. In their study, Omidi and Wu used a measured distribution function to obtain the growth rates. However, since we include the effects of the hot electrons on the wave dispersion, we find that the transition from larger growth rate Z mode depends on the ring momentum also.

      The ring distribution function is an extremely simplified distribution function. The distribution almost certainly overestimates growth rates, especially for modes with some finite k. In addition, it is not clear that the wave dispersion is necessarily modified by the presence of hot electrons in the measured distributions, where thermal velocities can be large. We have therefore studied the dispersion using a shell distribution which is a spherically symmetric Dory-Guest-Harris distribution . The form of the DGH distribution was chosen so as to enable us to use the analysis of Shkarofsky [1966] as a basis for the present study. Importantly, we found that provided j > 1, i.e., p > p, where p is the momentum for which the DGH distribution has a peak, the most unstable mode lies on a "trapped" branch of the dispersion relation.

      That the unstable mode for j > 1 is separate from the R-X mode suggests that for most cases, the dispersion analysis using the ring distribution may in fact be adequate. However, the results of the warm "shell" distribution indicate that the growth rates from the ring can be inaccurate. The effect of temperature is to restrict the range of instabilities (see also Winglee [1983]). Growth occurs only if the hot electron density is large and the thermal spread of the energetic particles is low enough.

      Interestingly, once the thermal spread becomes large, j = 1, the dispersive properties of the waves can change. We have found that for some plasma parameters the unstable mode can couple directly to the freely propagating R-X mode for j = 1. When j > 1, the modes can approach one another closely, but they do not couple directly. To show this, we used an approximate form of the conductivity tensor which assumed that ck / 1.

      Lastly, we discussed whether or not the current analysis can be applied to auroral zone distributions. Figure 8 shows that for a DGH distribution there is considerable variation in the appearance of the distribution function. It appears reasonable to assume that j > 1 for those distributions that have been accelerated through parallel electric fields. Also, the magnetic mirror force causes the distribution to curve in momentum space, giving a shell-like appearance. Most importantly, a large peak in the distribution function at low energy does not imply that a cold plasma approximation is adequate, because higher-energy particles occupy a larger volume in momentum space than lower-energy particles and so contribute more to the integrals used to evaluate the conductivity tensor. While more data are now available [Menietti and Burch, 1985], the relative amount of cold (or cool) plasma on an auroral zone field line is still an open issue.

      In conclusion, we have described in some detail the wave dispersion associated with weakly relativistic electron distributions. If, as we suggest here, auroral electron distributions can be modelled by two species with different gyrofrequencies due to relativistic effects, then a cold plasma approximation is probably not adequate for describing the dispersion of wave modes near the gyrofrequency. Our study shows that the modified dispersion can have two offsetting effects when considering the generation of AKR. First, the separation of the unstable mode at low wave vectors from the free space branch of the R-X mode may result in enhanced growth due to reflection of the wave at the edges of the auroral cavity, as discussed by Strangeway [1985]. Restricting the altitude range over which the unstable mode can propagate may also remove the problem of "detuning" the instability by refractive effects [see Omidi and Gurnett, 1982, 1984]. In addition, the trapped mode can be driven unstable by gradients in the phase density of both downgoing and upgoing electrons, since < . On the other hand, the separation of the unstable mode introduces the requirement of an additional conversion process in any mechanism for producing AKR. It is important that future studies investigate these more global aspects of the wave

      Acknowledgments.    The author would like to thank P. L. Pritchett and R. M. Winglee for many useful discussions. This work was supported by the NASA Solar Terrestrial Theory Program under grant NAGW-78 and the Air Force under grant F19628-85-K-0027.
      The Editor thanks D. Le Queau and another referee for their assistance in evaluating this paper.

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