J. Geophys. Res., 90, 9650-9662, 1985
(Received January 14, 1985; revised June 11, 1985; accepted June 12, 1985)
Copyright 1985 by the American Geophysical Union
Paper number 5A8480
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It has previously been shown that relativistic modifications to wave dispersion in regions of reduced plasma density, such as are frequently present on auroral zone field lines, may significantly alter the instabilities associated with the energetic auroral electrons [Prichett, 1984a, b; Strangeway, 1985]. It was found that the most unstable wave is preferentially generated at 90° to the ambient field direction. Maximum instability at 90° for an X mode polarized wave is primarily due to the inclusion of hot electrons in the wave dispersion relation. It is therefore of some importance to determine the relative aboundances of primary and secondary (or backscattered) auroral electrons. In addition, the previous analyses used electron distribution functions such as ring or shell distributions which are symmetric in parallel momentum. It has been suggested by Wu et al.  that the generation of instabilities at 90° may be due to this assumption of symmetry.
We have consequently carried out both one- and two-dimensional simulations using more realistic functions for the energetic electrons. The distribution functions employed are no longer symmetric in parallel momentum and include the presence of a loss cone together with a "hole" at lower momenta due to the presence of an accelerating electric field on the auroral zone field line. The accelerating electric field has also been employed in determining the relative abundances of hot and background electrons. The analytic work of Chiu and Schulz  has been used to fit particle distributions to the auroral field line number densities obtained by Calvert [ 1981].
We have found that the electric field does not greatly change the number densities associated with magnetospheric electrons but that the backscattered electrons tend to be excluded from the higher altitudes by the electric field. To obtain the fall off in number density observed by Calvert, the backscattered electrons had to have a temperature of the order of 10% of the total parallel potential drop on the auroral zone field lines. For typical potentials of the order a few kilovolts the backscattered and secondary electrons must have a temperature of some few hundreds of electron volts. Even when the temperature is high, we have found that the hot magnetospheric electrons are the primary contributor to the electron number density above some 2 R geocentric distance. The results of the fit to Calvert's data show that both electron distributions have equal number density at 1.7 R.
The simulations in which only hot electrons are included are included are applicable to altitudes above 2R, and these simulations show an increase in growth rate and saturation levels with decreasing altitude. This is due to the parallel electric field accelerating the electrons and increasing the momentum at the peak of the distribution, while the thermal spread about the peak remains roughly constant. A second result from the simulations with only hot electrons is a change in the angular distribution of the instability in comparison with the results from analyses using symmetric distributions. The radiation intensity was found to peak near 95° with a width of 88° - 100°, where angles greater than 90° correspond to propagation up the field line.
For altitudes below 2R, backscattered and secondary electrons are no longer a negligible component of the plasma. At 1.75 R, where the hot electrons contributed 75% to the total number density, saturation levels of about 1% were obtained. Below this altitude the large fraction of background electrons resulted in a decrease in the growth rates and saturation levels. At 1.75R there was a decrease in the angular width of the radiation, and the peak intensity was only a few degreees away from 90°.
An interesting result from the present simulations concerns the generation of O mode AKR. The dominant wave polarization in the simulations has X mode polarization. However, a low-intensity O mode wave is also observed in the simulations. This wave mode has intensity of about 1% of the X mode wave, and the wave amplitude is strongly correlated with the X mode wave. The growth rate of O mode radiation is much too high to be accounted for by a linear process, and wave-wave coupling does not appear to be able to explain the presence of this wave. Our analysis indicates that the O mode wave results from relativistic effects coupling perpendicular (X mode) electric fields into parallel (O mode) currents. The levels of the O mode radiation are consistent with the results of Mellott et el. , who found O mode intensities to be typically 2% of the X mode intensities.
In conclusion, our analysis has shown that the most intense cyclotron maser emission should occur in the altitude range 1.75-2.0 R. At lower altitudes the secondary electrons are dominant, and the instability saturates at a low level. At higher altitudes the energy of the primary electrons is lower, and again the radiation level drops. The simulations indicate that linear growth rates of the order of 2 10 occur in this region and that the conversion of primary electron energy into AKR is about 1%, similar to the level deduced by Gurnett . The cyclotron frequency in this range varies from ~ 300 kHz to ~ 200 kHz, which is also in good agreement with the observed frequency of the most intense AKR [Gurnett et al., 1983].
As a last remark, we point out that the analysis reported here has been of a purely local nature. Other work on AKR [Omidi and Gurnett, 1984] has stressed the nonlocal aspects of generating AKR. Specifically, while a distribution may be locally unstable to a wave at a particular frequency, it is not certain that the associated wave packet will propagate out of the source region without being reabsorbed. A second question arising from nonlocal effects is associated with the modified wave dispersion due to the hot electrons. Strangeway  has shown that the unstable mode may in fact be decoupled from the freely propagating R - X branch. Local simulations such as those presented here cannot readily address these problems.
Acknowledgments. We express our appreciation to M. Ashour-Abdalla, W. Calvert, J, M. Dawson, V. K. Decyk, M. M. Mellott, and R. M. Winglee for helpful discussions. This work was supported by NASA Solar Terrestrial Theory grant NAGW-78 and by National Science Foundation grant ATM 82-18746. The Editor thanks W. Calvert and another referee for their assistance in evaluating this paper.
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