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 been recognized for at least a decade that the earth is an emitter of intense radio waves [Gurnett, 1974] and that this emission is correlated with auroral activity [Kurth et al., 1975]. Since the wavelength of the emission is of the order of 1 km, the descriptive characterization of the emission as "auroral kilometric radiation" (AKR) has become common. The general source region for AKR is known to lie between geocentric altitudes of the order of 1.5-3 R in the auroral regions. In terms of plasma properties this is a very interesting region. Plasmas of both magnetospheric and ionospheric origin are present, and the relative concentrations depend strongly on altitude. The existence of a quasi-static electric field parallel to the magnetic field together with the convergent nature of the geomagnetic field leads to several strong nonthermal features in the particle distributions. Upgoing plasma sheet electrons which have been reflected due to the mirror effect exhibit a loss cone distribution, while downgoing plasma sheet electrons are accelerated by the electric field to form beams of a few keV energy. The parallel electric field also serves to retard electrons of ionospheric origin. Using data from the Hawkeye spacecraft, Calvert  has shown the existence of a large plasma cavity with density less than 1 cm from 1.8 R to 3 R at 70° 3° invariant magnetic latitude. As a result of this density depletion the ratio / of plasma frequency to electron cyclotron frequency is less than 0.1 for 1.4-3 R, with a minimum value of the order of 0.03 occurring for 1.8-2 R.
Recently, Pritchett [1984a, b] has pointed out that this characteristic combination of keV electron energies and / < 0.1 can lead to significant modification of plasma wave dispersion near the electron cyclotron frequency. In the cold plasma description, the free-space branch of the R-X mode has a cutoff
Here is the nonrelativistic gyrofrequency, = |e| B/ mc. Because of relativistic effects, however, the cutoff decreases as the electron energy increases. For a Maxwellian plasma the cutoff drops below , for (v/ c) > (2/ 3)(/ ) , where v (T/ m) is the electron thermal speed [Pritchett, 1984b]. The same effect was observed earlier by Wu et al.  and Winglee  in calculations involving an unstable double loss cone distribution. This relativistic shift of the R-X mode cutoff means that it is possible to satisfy the relativistic gyroresonance condition
with k = 0. Here is the usual relativistic factor (1-v/ c )-. In general, (1) defines an ellipse in velocity space, but for k = 0 and < , the ellipse reduces to a circle centered at the origin with radius c(1- / ). In contrast, with > > , as required by cold plasma theory, a minimum value of k
is necessary in order to satisfy (1). It should thus be possible to produce electromagnetic radiation perpendicular to the magnetic field via the electron cyclotron maser instability [Bekefi, 1966] in relativistic plasmas with electron speeds satisfying (v/ c) / ). Using electromagnetic particle simulations, Prichett [1984a, b] observed that for simple ring and shell electron distributions not only was radiation emitted at 90°, but it was also the most intense emission. Thus the radiation pattern was drastically different from what would be expected on the basis of cold plasma theory.
The cyclotron maser instability has been one of the more popular mechanisms for explaining the generation of AKR [Wu and Lee, 1979]. A number of groups have investigated the linear growth rates that could be expected from this process. Some of the calculations [Omidi and Gurnett, 1982; Melrose et al., 1982; Dusenberry and Lyons, 1982; Omidi et al.,1982] used measured electron distributions such as those obtained by the S3-3 spacecraft [Croley et al., 1978], while others employed various model distributions [Lee et al., 1980; Wu et al., 1981, 1982; Hewitt et al., 1982; Wong et al., 1982; Winglee, 1983; LeQueau et al., 1984a, b; Prichett, 1984b; Strangeway, 1985] The analyses using the measured distributions were all based on cold plasma thoery, while none of the relativistic model calculations claimed to incorporate a realistic auroral model. The quantitative implications of relativistic dispresion for the generation of AKR are thus still uncertain. The present work represents an attempt to combine a quasi-realistic auroral model with particle simulations in order to address this question. Wagner at al., [1983, 1984] have previously performed particle simulations to study the cyclotron maser instability driven by double and single loss cone distributions. Their minimum value of / = 0.2, however, is only marginally relevant to AKR generation in the plasma cavity obseved by Calvert , and their choice of a propagation angle 45° to the magnetic field excludes the observance of the relativistic dispersion effects which are the object of the present study. With their choice of parameters, only linear (rather than exponential) growth of the total electromagnetic field was observed.
Our auroral model is based on the analytic work of Chiu and Schulz , who applied the principle of quasi-neutrality to calculate the mutually consistent electrostatic potential and particle distributions along an auroral field line. We shall employ a simplified model in which only the electron distribution is treated explicitly and the potential is arbitrarily assumed to vary linearly with the magnetic field. The electron populations are taken to be a collisionless anisotropic magnetospheric distribution and a component extracted or backscattered from the ionosphere. Because of the presence of the acceleration ellipse and the loss cone, there are sharp demarcations in velocity space for the various particle populations. These sharp boundaries produce steep velocity-space gradients in the distribution functions, with the result that the cyclotron maser instability can be excited strongly. As input to the model we use the data of Calvert  for the total number density as a function of altitude. It is then possible to fit relative abundances of the primary auroral electrons and backscattered secondaries.
A series of electromagnetic particle simulations is then performed using as input the parameters of the auroral model at various altitudes between 1.5 R and 2.8 R. The simulations are performed in a number of stages. Initially, the radiation generation is studied in a pure magnetospheric plasma and for emission at 90° to the magnetic field. Except at the highest altitudes the radiation is produced in a narrow band just below kc/ = 1 and with / 0.99. The linear growth rates as a function of altitude are consistent with theoretical predictions based on the symmetric Dory-Guest-Harris (DGH) distribution [Dory et al., 1965]. The instability leads to strong perpendicular diffusion in velocity space, and at saturation the positive f/p slope has been eliminated. The efficiency of the radiation generation descreases markedly with increasing altitude. The radiation is produced almost entirely in the extraordinary mode; the ordinary mode component is of the order of 1% or less.
The next stage is to consider the angular dependence of the radiation emission, still for a pure magnetospheric electron population. Both one- and two- dimensional simulations indicate that the maximum growth rates occur for emission a few degrees away from normal and directed toward higher altitudes. There is, however, a nonnegligible component of the radiation directed toward lower altitudes. The final stage is to include the secondary electron population. At altitudes below ~ 1.7 R this is the dominant component, and the maser cyclotron instability is strongly quenched. At higher altitudes the secondary electrons become unimportant. The radiation is the most intense at altitudes of 1.75-2.0 R, corresponding to a frequency range of 200-300 kHz. In this region, temporal growth rates of the order of 2 10 are found, and efficiencies of the order 1/2-1% of the intial kinetic energy of the magnetospheric electrons are realized.
The specific outline of the paper is as follows. In section 2 we describe the auroral zone model, including the distribution functions for the primary and secondary electron populations and the variation of the total number density with altitude. Section 3 contains a brief description of the simulations model as well as of some of the diagnostic procedures to be employed. The simulation results are presented and discussed in section 4. Section 5 presents a summary and discussion of the implications for the application of the cyclotron maser instability to the generation of AKR.
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