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.
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Several features of the radiation present major problems for a theoretical understanding of the generation of the radio noise. First, the noise is intense; Gurnett  has calculated that the peak wave power is of the order of 10 W, some 1% of the power estimated to be deposited in the ionosphere by auroral electrons during a substorm. The average power level was found to be ~ 10 W [Gallagher and Gurnett, 1979]. Second, the wave is primarily observed in the cold plasma R-X mode. This has been deduced from direct polarization measurements on the Voyager spacecraft [Kaiser et al., 1978] and also inferred through ray path calculations [Green et al., 1977]. In addition, spacecraft have flown directly through the generation region for AKR, which is presumed to be below altitudes of 1-2 R on auroral zone field lines. Data from ISIS [Benson and Calvert, 1979; Calvert, 1981a; Benson, 1982] support the inference that AKR is generated in the R-X mode through the observation of a low-frequency cutoff for AKR, which is strongly tied to the local gyrofrequency. Recently, Mellott et al.  have used DE-1 data to show that AKR is usually observed to be in the R-X mode near the presumed generation region.
Taking into account these properties of AKR, one of the more popular mechanisms for the generation of AKR is the cyclotron maser instability [Wu and Lee, 1979]. This instability is driven by the direct gyroresonance of energetic electrons with the R-X mode. Direct gyroresonance was first proposed by Melrose  to explain the generation of Jovian decametric radiation (DAM). Since Melrose used a drifting bi-Maxwellian to describe the energetic particle distributions, he found that significant thermal anisotropies were required to drive the R-X mode unstable through gyroresonance. Typically, v v c , where v and v are the perpendicular and parallel thermal velocities, respectively, and c is the speed of light.
It was shown by Wu and Lee  that gyroresonance can generate waves more efficiently, provided there is a positive slope of the energetic particle distribution as a function of perpendicular velocity. These authors applied their theory to the generation of AKR and found that coupling to the R-X mode was enhanced by assuming that the plasma frequency in the generation region was significantly lower than the gyrofrequency. Additionally, for relativistic gyroresonance, resonant particles describe an ellipse in velocity space. If the ellipse lies fully within the loss cone, for example, then no damping terms are introduced on integrating over all resonant particles. The low plasma frequency allows the R-X mode cutoff to approach the gyrofrequency, which in turn allows lower-energy electrons to resonate with the waves.
Recently, several authors [Wu et al., 1982; Omidi and Gurnett, 1982; Dusenbery and Lyons, 1982] have investigated the generation of AKR by the cyclotron maser instability, using particle data to determine the sources of free energy in the plasma. While a particular frequency may have large growth at one location on an auroral zone field line, Omidi and Gurnett  have shown that this wave can be damped as the wave propagates along the field and enters different plasma regimes. The effect of the parallel electric field presumed to exist on auroral field lines [Croley et al., 1978] was included by Wu et al. , who addressed the effects introduced by assuming that only energetic electrons determined the wave propagation characteristics. Generation by both the primary downgoing electrons and the loss cone in the reflected and backscattered electrons was investigated by Dusenbery and Lyons . One aspect of all the work cited here is the sensitivity of the growth rate to the parameters chosen to describe the plasma, in terms of both the energetic electrons and the presumed cold electrons.
It should be pointed out that while the S3-3 observations reported by Croley et al.  show population inversions associated with both the loss cone and field-aligned acceleration of the energetic electrons, the low-energy electron population has not been measured directly on auroral field lines. Wave measurements from the Hawkeye spacecraft have been used by Calvert [1981b] to show the presence of an auroral density cavity, but again this does not allow us to directly determine the properties of the low-energy electron population.
Bearing in mind the uncertainty in determining the amount of cold plasma present, it is possible that the wave propagation characteristics may be determined by the more energetic electrons, and several authors [Tsai et al., 1981; Wu et al., 1981, 1982; Winglee, 1983; LeQueau et al., 1984a, b] have studied the modifications of the wave properties by including energetic electrons in the wave dispersion relation.
Recently, Pritchett [1984a, b] has investigated instabilities associated with the mildly relativistic magnetized plasma, using both simulations and linear analysis. The types of particle distribution functions employed in his simulation were both "ring" and "shell" distributions. In general, a ring distribution is characterized by a ring of energetic particles with constant perpendicular momentum (where perpendicular is defined with respect to the ambient magnetic field), whereas a shell distribution is typically given by a spherical shell of particles in momentum space. In the simulations these distributions can also have a thermal speed associated with them. Pritchett concluded that when p / mc ~ / (where p / mc is the characteristic energetic particle momentum and / is the ratio of the electron plasma frequency to the electron gyrofrequency), the plasma can be unstable to waves propagating normal to the ambient field direction. When only hot electrons are present, the unstable mode couples directly to the R-X mode, but inclusion of a thermal electron component can decouple the two modes.
Relativistic effects consequently result in the electrons being essentially a two-species distribution, with the new mode lying between the gyrofrequencies associated with each particle species. Since the difference between the two gyrofrequencies is small for typical parameters associated with auroral electrons, one might expect the wave group velocity to be small. Hence it would be of interest to study the wave characteristics in some detail. In this paper we shall investigate the wave modes introduced by relativistic effects. We shall particularly emphasize the variation of the wave modes as a function of the parameters which describe the plasma and further study some of the properties of the waves themselves, such as polarization and group velocity.
Since the group velocity of the unstable waves associated with a ring distribution will be shown to have considerable variation, we shall study this wave property in more detail. We shall show, using a simple model for the auroral density cavity, that the convective properties of the waves may have important implications for the generation of AKR. Not only is the magnitude of the group velocity important, but also the direction of ray propagation. The former results in short convective growth lengths, whereas the latter can effectively focus the waves at an altitude most suited for growth due to the bending of the ray paths.
The structure of the paper is as follows. In the next section we shall briefly describe the dispersion relation used in the present analysis, together with a summary of the plasma parameters and wave diagnostics used. In section 3 we shall present solutions of the wave dispersion relation as a function of the plasma parameters, while in section 4 we shall describe some of the wave properties as a function of wave vector. The implications of the group velocity variations as applied to the generation region for AKR will be discussed in section 5. In section 6 we investigate in detail the applicability of the present analysis to auroral electron distributions. The results of the study will be summarized in the final section.
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