A Simulation Study of Kilometric Radiation Generation

P. L. Pritchett and R. J. Strangeway

Institute of Geophysics and Planetary Physics,
University of California at Los Angeles

Department of Physics,
University of California at Los Angeles


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


Contents

Title and Abstract
1. Introduction
2. Auroral Zone Model
3. Simulation Methods
4. Simulation Results
5. Conclusions
References


Abstract

      Relativistic modifications to wave dispersion near the electron gyrofrequency are known to be important in a low-density, energetic plasma. Previous investigations based on simple model distribution functions have suggested that these relativistic dispersion effects may substantially alter the predictions of the cyclotron maser theory as applied to the generation of auroral kilometric radiation (AKR). The present work combines a model for the high-altitude auroral zone with local particle simulations in order to assess the quantitative implications of relativistic dispersion for the generation of AKR. Electron distribution functions, which include a loss cone and a "hole" at lower energies produced by the parallel electric field, are determined along an auroral field line. Number densities of both energetic auroral electrons and low-energy backscattered electrons are obtained from a fit to the observed density-depleted region on an auroral field line. It is found that the hot magnetospheric component is the dominant species at altitudes greater than 1.7 R geocentric. These auroral zone parameters and distributions are used as input to one- and two-dimensional particle simulations. The simulations show that the electric field results in more free energy in the hot electron population at the lower altitudes. This is counteracted, however, by the increased density of backscattered and secondary electrons at these altitudes, which tends to quench the maser instability. The most intense radiation is found to occur between 1.75 and 2.0 R, corresponding to AKR frequencies in the range 200-300 kHz. The peak in the radiation is very close to 90° to the ambient magnetic field, with an offset of a few degrees in the direction of the loss cone. The dominant emission is in the extraordinary mode with a wave frequency / 0.99. A weak ( ~ 1%) ordinary mode component is produced in association with the extraordinary mode due to a relativistic coupling between perpendicular wave electric fields and parallel currents.

1. Introduction

      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 [1981] 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. [1981] and Winglee [1983] 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 theory, 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 [1981], and their choice of a propagation angle 45° to the magnetic field excludes the observance of 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 [1978], 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 [1981] 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.

2. Auroral Zone Model

      There are three main features to be determine in the auroral zone model. These are the total number density as a function of altitude, the choice of electron populations and the velocity-space distribution function for each, and the relative abundance of each component as a function of altitude. While measurements exist regarding the first two questions, there are no direct observations of the low-energy electron population on auroral field lines, and thus the relative abundances are not well determined.

      We shall use the results of Calvert [1981] for the total number density as a function of altitude along an auroral zone field line. In his work, Calvert deduced the presence of an auroral density-cavity using wave data from the Hawkeye spacecraft. This density depletion produces a minimum in the plasma frequency/gyrofrequency ratio (/ ) at a geocentric distance of 1.8 R. The variation of / with altitude is shown in Figure 1 [after Strangeway, 1985]. In addition, we have shown the variation of the gyrofrequency along the field line as a function of altitude, assuming a dipole field with an invariant latitude of 70°.

Fig. 1. Variation of the electron gyrofrequency , the plasma frequency/gyrofrequency ratio / , and the characteristic perpendicular momentum p as a function of geocentric distance along an auroral field line [from Strangeway, 1985].

      To determine the electron distributions and abundances, we follow the work of Chiu and Schulz [1978]. We include two distinct electron populations in our model. The first is the primary energetic component, whose source is taken to be a downward propagating half-Maxwellian distribution at the magnetospheric (high altitude) end of the field line. This is refered to as the "magnetopheric" component. The other is a secondary component which is assumed to evolve from a Maxwellian distribution at the ionospheric (low altitude) end of the field line. This component arises as a result of backscattering of the primary electrons, and so it can have relatively high temperature. While it is not truly ionospheric in origin, we shall for purposes of distinction refer to it as the "ionospheric" component. We shall ignore any low-temperature (~ 1 eV) electrons which come from the ionosphere, as these electrons are excluded from the auroral field line by the parallel electric field. We assume the electric field to be present over a range from 1.3 R to 3.3 R geocentric distance. We do not consider explicit ion distributions but simply assume that quasi-neutrality is maintained.

      As discussed in detail by Chiu and Schulz [1978], these electron distributions are restricted to different regions of momentum space because of the effects of electric field acceleration and magnetic mirroring. We shall briefly review the work of Chiu and Schulz in the context of the present study.

      The particles conserve energy and magnetic moment:

where p is the electron momentum, m is the electron mass, e is the magnitude of the electron charge, is the local electrostatic potential, and B is the local magnetic field strangeth. The subscripts and || are used to denote perpendicular and parallel with respect to the ambient magnetic field, p = p + p . These conservation equations can be rewritten to give the characteristic boundaries in momentum space due to the accelerating electric field and the atmosphere loss cone. The electric field results in an elliptical boundary

where B denotes the magnetic field strength at the magnetospheric end of the field line where = 0. Only particles with momenta greater than the momentum given by this curve can reached an altitude characterized by B and and have originated in the magnetosphere. All other electrons are trapped by the electric field. Loss at the atmosphere results in a loss cone hyperbola

where the subscript I denotes the ionospheric end of the field line. All electrons with a momentum less than this boundary curve come from the ionosphere, although the actual source may be backscatter of the primary auroral electrons.

      Chiu and Schulz pointed out that the curves given above only yield the particle accessibility boundaries provided d / dB 0. If this is not the case, it is possible that local electrostatic mirrors exist. For this purpose of simplicity we shall assume that d / dB = 0, i.e.,

With this assumption the p = 0 intercepts of the loss cone hyperbola and the acceleration ellipse are both given by

We can rewrite (5) as p = p I B/ BI, where p I is p at the ionospheric end of the field line. The momentum plotted in Figure 1 corresponds to the p = 0 intercept given in (5). The momentum has been normalized to mc. we have chosen p I= 0.2mc, corresponding to e = 10 keV.

      Having determine the boundaries that result from mapping of particle momentum from one location to another along a field line, we must also determine how the particle distribution function is mapped along the field line. Liouville's theorem states that the phase space density is a constant along a particle trajectory, i.e.,

with

The subscripts 1 and 2 denote different locations on the field line. We shall assume that the magnetospheric electrons have a Maxwellian distribution at B = B. Consequently, the distribution function as a function of position along the field line is given by

where p is the magnetospheric electron thermal momentum. Similarly, an ionospheric Maxwellian distribution when mapped up the field line results in a local distribution given by

where p I is the ionospheric electron thermal momentum. The expressions (8) and (9) for the electron distribution functions are valid only within the kinematically allowed regions for each species as determined from the boundary curves (2) and (3). Outside the allowed regions the distribution functions vanish. Note that because of these kinematically forbidden regions the constants n0 and n0I, in (8) and (9) do not represent the actual total number densities of magnetospheric electrons at B = B or of ionospheric electrons at B = BI . The expressions for the number densities will be given in (10) and (11).

      The mapping of the electron distribution functions along an auroral field line is illustrated in Figure 2. The figure shows contours of constant phase space density, together with the boundaries in momentum space due to the loss cone and the parallel electric field. The boundaries are given by the thicker lines. We have shown three locations, one corresponding to the ionospheric end of the field line where = I, an intermediate location where = 0.2I and the magnetospheric end of the field line where = 0. We have assumed that the magnetic field can be modeled by a dipole field with an invariant latitude of 70°, that the electric field lies between 1.3 and 3.3 R, and that the potential scales linearly with the magnetic field magnitude.

Fig. 2. The electron distribution function at three altitudes along an auroral field line. Contours of constant phase density are shown for the magnetospheric electron population (solid circles) and for the ionospheric electron population (dotted circles).

      The solid circles in Figure 2 show the first 15 e-foldings of the phase density for the magnetospheric electron population. The contours are calculated with respect to the p = 0 value of the distribution function at R = 3.3 R. As a consequence of Liouville's theorem the phase density is also equal to this value at the p = 0 intercept of the acceleration ellipse for all locations along the field line. We have chosen a thermal energy of 1 keV for these electrons, which is 0.1 of the total parallel potential, and it is apparent that the total number density of the magnetospheric electrons will not vary greatly as a function of position. The number density will only show a marked decrease at lower altitudes for much lower temperatures.

      The dotted circles show the ionospheric electron phase space density. In this case the contours are normalized to the p = 0 value at 1.3 R. We have again assumed a temperature of 1-keV for these electrons, indicating that this distribution is mainly produced by the backscattering of the primary auroral electrons. It is apparent that even for a 1-keV temperature the electric field excludes a large amount of these particles. At R = 2.054 R the phase space density has descreased by eight e-foldings at p = 0.

      Having determined how the phase space density maps along an auroral zone field line, we can calculate the local number densities associated with the magnetospheric and ionospheric electron distributions. First, we define

Apart from normalization, these are the same as the integrals defined by Chiu and Schulz [1978]. We have normalized the integrals so that E(0) = 1. In addition, E(x) and E(x) are directly related to the complex complementary error function w(z) as defined, for example, by Abramowitz and Stegun [1964]:

Next, we let

where p, is a thermal momentum, which can either be p or pI depending on the particle species. It should be noted that and * are constant; the altitude dependence is incorporated into b and b*. Another important point concerning the variables defined above is that both and * depend only on the ratio eI/ p; neither temperature nor potential appears separately. This is due to the assumption that the potential scales linearly with the magnetic field strength and that the distribution functions are Maxwellian.

      For the magnetospheric electrons,

where denotes evaluated with p. As b increases, correponding to a movement to lower altitudes, the first term in (10) increases, whereas the other two terms decrease. In general, the magnetospheric electron population is expected to show an initial decrease followed by a subsequent increase as a function of decreasing altitude. Furthermore, n will only show a large increase at lower altitudes if b is large. For the parameters used in Figure 2, we expect an increase in number density of about a factor of 2 at the ionospheric end of the field line.

      In contrast, the number density of the ionosphere electrons is expected to be dominated by the deceleration due to the parallel electric field,

with computed using p. It should be noted that * - = 2me/ p. We have written (11) so as to bring out the relationship between n and n. The term multiplied by exp (-* + ) in (11) is functionally the same as (10). This term corresponds to that part of the ionosphere electron population that would lie outside the loss cone at the magnetospheric end of the field line. The dominant term (11) is the first term, as expected. As previously deduced, the ionosphere electron population will only be dominant at the lower altitudes. For these secondary electrons to be dominant at the magnetospheric end of the field line,

      Having determined the functional dependence of the electron number density, as given as given by (10) and (11), we can compare different predictions with the observed number density [Calvert, 1981]. There are four free parameters we can vary when attempting to fit the data. These are the temperatures of the two electrons populations, together with the factors n0 and n0 in equations (10) and (11). The latter two parameters can also be specified by fixing the number densities of the ionospheric and magnetospheric electrons at their respective ends of the field line.

      One such fit is shown in Figure 3. The data of Calvert [1981] show a rapid decrease in number density with increasing altitude down to a value of 1 cm at 1.8 R. Above this altitude "the plasma density within the cavity is roughly uniform with altitude." We have chosen a density of 0.5 cm in the cavity to produce the curve in Figure 3, but this value could reasonably be changed by 50%. The rapid falloff in total density at the low altitude arises from the sharp decrease of the ionospheric population with increasing altitude. The variation in the density of the magnetospheric component is much less dramatic, and this component by itself provides a very good fit to the constant in the cavity. The transition from a majority ionospheric component to a majority magnetosphreic component occurs at 1.7 R. If the cavity density were taken to be as low as 0.25 cm . The transition point would increase only to 1.8 R. For the fit shown in Figure 3 the temperatures of the magnetospheric and ionospheric electrons were 0.92 and 0.82 keV, respectively, whereas the total potential drop was assumed to be 10 kV.

Fig. 3. The electron density as a function of geocentric distance along an auroral field line. The curve labeled "data" gives the observed variation [Calvert, 1981]. The curve "fit," which is the sum of n and n given by equations (10) and (11), is best fit to this observed density.

      As stated previously, we have assumed that the region of parallel electric field is confined to the range 1.3-3.3 R and furthermore that the potential varies linearly with magnetic field. Given the uncertainty in the electron densities and distribution functions measured on the auroral zone field line, these assumptions are probably not unreasonable. Figure 2 and equation (11) both show that the dominant factor affecting the ionospheric electron density is the exponential decrease due to the parallel electric field. Since the ionospheric distribution remains Maxwellian with constant temperature as a function of altitude and Calvert's data give us an estimate for the number density at 1.3 R, the presence of accelerating electric fields below this altitude need not be included. Additionally, while the assumption of a 10-kV total potential drop may overestimate typical auroral zone potential drops, the decrease with altitude is only dependent on the ratio 2me/ p (~ 10). This ratio was obtained by fitting the densities given by (10) and (11) to Calvert's data and is not a free parameter. If we decrease the assumed total potential drop, we must decrease the temperature by the same factor. Relaxing the constraint that the potential varies linearly with the magnetic field strength may allow us to change the temperature of the ionospheric distribution, but the large decrease in density present in Calvert's data will still result in a decrease in the ionospheric population similar to that shown Figure 3.

      Similarly, changing the assumptions concerning the parallel potential drop will not result in large changes in the magnetospheric electron density. However, it should be noted that while the number densities will not change markedly, the boundaries in momentum space may well change significantly on altering the assumptions concerning the parallel potential. For example, if we assume that the losses to the atmosphere occur at a lower altitude, the loss cone will be narrower. In general, changes of this nature will affect the growth rates of instabilities, not the modifications of the wave dispersion introduced by the presence of hot electrons.

3. Simulation Methods

      In this section we describe the simulation methods used to study the generation of electromagnetic radiation for the auroral zone model of section 2. The simulation code employed is a relativistic electromagnetic particle code [Dawson, 1983]. This code solves the full set of Mawell's equations self-consistently, and the particles are advanced in time using the relativistic Lorentz force equation. In its present version the code allows for two electron populations, one of magnetospheric origin and one of ionospheric origin. Since we are interested in frequencies of order , the ions are treated as a fixed neutralizing background. A series of simulations is performed at various altitudes for the auroral model of section 2. Each simulation is local in that the magnetic field and plasma density are taken to be spatially uniform, and periodic boundary conditions are employed. In addition, each simulation is treated as a pure initial value problem in that no attempt is made to preserve the intial electron distributions. The system is allowed to evolve self-consistently in time from the initial configuration of the model.

      The simulation model just described fails to represent the actual auroral zone configuration in several important respects. There is clearly no source to replenish the electron free energy, and the assumption of periodic boundary conditions implies that the wave energy and interacting electrons are retained in the system rather than being lost. Nevertheless, the model is useful, since it retains the exact wave dispersion and wave-particle interaction and thus reproduces the correct dynamics of the cyclotron maser instability. Furthermore, the limitations associated with the local nature of the model are mitigated considerably by the fact the run time of the simulations is very short. A typical duration is t ~ 2000, which corresponds to ~ 2 ms. Thus the typical distance traveled by an electron parallel to the magnetic field during the entire simulation is only 60 km. Since the simulations are so short, the effect of the dc parallel electric field on the electron dynamics is negligible. The field associated with the potential (4) is of the order of 1 mV/ m for r ~ 2 R. Thus the change in parallel velocity produced by this field during the simulation is only 1% of the initial velocity p/ m. We thus neglected the parallel field in the simulations. The one feature that may be affected by the limitations of the present model is the saturation mechanism of the instability. As we shall see in section 4, saturation results from the perpendicular diffusion in velocity space which removes the initial f/ v > 0 free energy source. In the real AKR source region, new energetic particles are continually flowing down the field line to replenish the initial distribution, and the instability would likely be driven to higher levels. The conversion efficiencies observed in the present simulations are thus probably only lower limits for the real values.

      Most of the simulations were performed using a version of the code with one spatial dimension and three velocity and field components. Although such a configuration is somewhat artificial, it does permit one to study the case where waves can propagate in only a single direction relative to the magnetic field. By performing a series of such runs, one can examine the dependence on the angle of propagations relative to B. In addition, with only one spatial dimension, it is possible to use a large number of particles per Debye cell (nD ~ 100) so that thermal fluctuations are relatively small and weak instabilities can be resolved. In the simulations with two spatial dimensions, computer memory limitations restrict nD ~ 4, and only strong instabilities stand out above the background.

      In the one-dimensional simulations, waves can propagate in both the +x and -x directions. Thus two angles of propagation are allowed, and 180° - . These "right" and "left" traveling waves are separated during postprocessing by using the observed frequency spectrum and the stored time history of the fields. It is possible, for example, to determine a growth rate for propagation at 85° even though the simulation is dominated by a stronger instability for propagation at 95° (see the discussion in section 4.2). In the results presented here the system length is L = 256, where the grid spacing is the unit of length. The inverse plasma frequency is the unit of time. The total number of electrons in the simulations is 25,600. A typical resolution in k is c(k)/ = 0.034.

      In order to check the one-dimensional simulations, we have also performed a run with two spatial dimensions using the parameters of the auroral model at an altitude of 1.5 R. The propagation vector k now is taked to lie in the xy plane. The modes are then denoted by (m, n), where m and n are integers and k = 2m/ L, k = 2n/ L. The system is L L = 256 64 , so that the resolution along the x axis is the same as the one-dimensional runs. If the magnetic field were directed along the y axis, then the entire range 0 would be present in the simulation. However, for kc/ ~ 1 the minimum deviation from 90° that could be resolved on the grid would be 8.1°. In order to reduce this value, we orient the magnetic field in the yx plane as indicated in Figure 4. The propagation angle is then given by

With = 20°, for example, the minimum deviation from 90 ° is reduced to 2.8°. With such a configuration, only propagation in the range 90° - to 90° + is permitted. Previous simulation studies of radiation generation by the cyclotron maser instability [Pritchett, 1984], however, and the one-dimensional simulations to be discussed subsequently have shown that the emission is dominated by perpendicular and near-perpendicular propagation. Thus elimination of the predominantly parallel modes should not be a significant omission. With n = 4 the total number of electrons in the two-dimensional is 65,536.

Fig. 4. Coordinate system used in the two-dimensional particle simulations.

4. Simulation Results

      We now present the simulation results for various sets of parameters determined from the auroral model discussed in section 2. The choice of parameters corresponds to altitudes of approximately 1.5 R, 1.75 R, 2.0 R, 2.5 R, and 2.8 R. We have chosen a magnetospheric electron temperature of 1 keV (p/ mc = 0.063), which is quite close to the value 0.92 keV obtained in the fit to the density distribution. This value remains fixed in all the simulations. The remaining parameters are given in Table 1. The resulting magnetospheric electron distribution in momentum space is shown in Figures 5a and 6a for 1.5 R and 2.25 R, respectively. At the lower altitude the loss cone is very wide (52°), while at the higher altitude it is considerably narrower (21°). In addition to the loss cone for upgoing electrons, the acceleration ellipse on the downgoing side provides an additional region of f/ p > 0.

Fig. 5. Electron momentum-space distributions in a two-dimensional simulation with 1.5 R auroral parameters. (a) Initial distribution (t = 0). (b) Distribution at saturation (t = 850). (c) Contours of f(p, p) at saturation. The contour values range from 0.056fmax for outermost dotted contour to 0.944fmax for the innermost solid contour. The spacing between countours is linear.

Fig. 6. Same as Figure 5 except for a one-dimensional simulation with = 90° and 2.5 R auroral parameters. The saturation plots in Figures 6b and 6c are for t = 2400.

      The simulation results are discussed in the following sequence. In section 4.1 we illustrate the altitude dependence of the cyclotron maser instability for the case of a pure magnetospheric electron population for emission exactly perpendicular to the magnetic field. In section 4.2 we consider the angular dependence of the radiation emission, again for a pure magnetospheric population. In section 4.3 we consider the effect of the ionospheric component in the auroral electron distribution.

4.1 Perpendicular Emission

      In the simulations with parameters corresponding to altitudes of 1.5 R and 2.0 R there is a strong instability observed in mode 28 of the electromagnetic fields. This mode has kc/ = 0.962. Figure 7 shows for 1.5 R the time histories of the total tranverse electric energy (Figure 7a) and the extraordinary mode amplitude squared |E| for mode 28 alone (Figure 7b). The exponential growth stage is quite apparent in both plots. The linear growth rate computed for mode 28 from this plot is / = 7.0 10. At 2.0 R the corresponding linear growth rate is / = 2.4 10. The peak percentage of electron kinetic energy converted into radiation for the two altitudes is 6.5% and 1.1%, respectively. The corresponding values of B/ B are 1.6 10 and 2.5 10. When the altitude is increased to 2.5 R, the bandwidth of the instability in k broadens so that modes 26-28 (kc/ = 0.89-0.96) all show the instability. The field energies for these modes are within a factor of 3 of each other, and the average growth rate is / = 2.1 10. The energy conversion efficiency is now only 0.4%. When the altitude is further increased to 2.8 R, the instability becomes even weaker (thus making determination of growth rates difficult) and is no longer concentrated near kc/ ~ 1. The energy conversion efficiency is well below 0.1%.

Fig. 7. Time history of (a) the total tranverse electric energy E and (b) the extraordinary mode energy | E | for mode 28 in a one-dimensional simulation with = 90° and 1.5 R auroral parameters. The electric energies are normalized to the initial electron kinetic energy (K. E.).

      The electron distributions at saturation for 1.5 R and 2.5 R are shown in Figures 5b, 5c, 6b, and 6c. As discussed by LeQueau et al. [1984b], the diffusion produced by the resonant interaction of strongly superluminous waves (/ k c 1) with weakly relativistic electrons occurs almost entirely in p, with p remaining constant. This feature is apparent in Figures 5 and 6. At saturation the positive gradients in pthat drove the instability have diffused away.

      The strong electromagnetic radiation at 90° to the magnetic field observed in the present auroral zone simulations has the extraordinary mode polarization, and the real frequency lies just below (Re / 0.99). These features are just what were found in linear theory analyses of relativistic distributions [Prichett, 1984b; Strangeway, 1985]. These analyses showed that the growth rates for exactly perpendicular emissions were not too sensitive to the exact nature of the gradient in p space. For example, delta function ring, shell, and DGH distributions produced simular results. It is thus reasonable to compare the present simulation results with the case of a relativistic DGH distribution, for which the dielectric elements can be evaluated analytically in the semirelavistic approximation [Wu et al., 1981; Tsai et al., 1981; Winglee, 1983; LeQueau et al., 1984a]. This distribution has the form

Here l is an integer which determines the peak perpendicular momentum p = lm as well as the relative thermal spread about this maximum. To fit the present auroral model, we take p equal to this peak value, so that p = l m = (2l ) p. With p/ mc = 0.063 we find l = 3.1, 2.0, 1.3, and 0.7 for 1.5 R, 1.75 R, 2.0 R, and 2.5 R, respectively. These values are then rounded to l = 3, 2, 1 and 1. The resulting mazimum growth rates (in units of 10 ) for extraordinary mode are 7.2 for 1.5 R, 4.0 for 1.75 R, 2.0 for 2.0 R, and 2.6 for 2.5 R. the corresponding values of kc/ are 0.965, 0.982, 0.985, and 0.947. These results are in excellent agreement with the simulations.

      It is also of interest to consider the ordinary mode. Using the DGH distribution and auroral zone parameters described above, we find that the O mode is unstable for k = 0 only in a very narrow region near kc/ = 0.99. The peak growth rates are 2.2 10 , 2 10 , and 9 10 for 1.5 R, 2.0 R, and 2.5 R. these are factors of 30-100 smaller than the extraordinary growth rates and are so small that ordinary mode growth would not be observed in the simulations. Nevertheless, weak O mode instabilities are indeed observed. In Figure 8 we compare the time histories of the electric field amplitude squared in the 1.5 R simulation for the extraordinary and ordinary polarizations for mode 28. The ordinary mode has been multiplied by a factor of 124. Apart from this difference in strength, the growth of the O mode lags slightly behind that of the X mode but has a similar growth rate. This observed growth rate is clearly inconsistent with the linear theory predictions for the O mode, and the generation must result from some nonlinear mechanism.

Fig. 8. Time history of the tranverse electric field amplitudes squared for mode 28 in a one-dimensional simulation with = 90° and 1.5 R auroral parameters. The solid curve shows the extraordinary mode | E | , while the dashed curve shows the ordinary mode | E | multiplied by a factor of 124. The field amplitudes have been averaged over half a gyroperiod.

      To help identify the nonlinear mechanism, the simulations were repeated with a number of changes. To check the possibility of wave-wave coupling, the extraordinary fields E and B for mode 28 were arbitrarily set to zero at t = 560 (which is during the linear growth stage of the instability) and maintained at zero. The ordinary fields E and B for mode 28 showed no immediate change in growth rate. This result rules out a direct wave-wave coupling. The oscillating E field (which is perpendicular to B) in mode 28 does, however, produce corresponding oscillations in the x and z components of the electron velocities. The resulting J and J currents are observed to grow at the same rate as the extraordinary fields. Mode 28 of the electron current also contains a parallel component J. This parallel componenet has a more complicated frequency spectrum than J and J. In addition to a component near , there is also a low-frequency componenet with < 0.01 . The growth of J lags slightly behind that of J and J. This parallel current appears to arise from the relativistic coupling between all three velocity components of a particle in the presence of electric and magnetic fields. Thus with B in the y direction and an E field, the equation of motion for the parallel velocity is

where q is the particle charge. Note that the coupling is of the order of (v/ c). This parallel current is able to excite the ordinary mode fields. If the mode 28 parallel current is set to zero in the integration of Maxwell's equations in the simulation (again at t = 560), the growth of the E and B fields stops. It thus appears that excitation of the R-X mode by the cyclotron maser instability should be accompanied by a weak ( ~ 1% for the weakly relavistic particles involved in the generation of AKR) ordinary mode component. This effect may be the explanation for the observation of Mellott et al. [1984] that in DE 1 observations the strength of the ordinary mode AKR if found to be roughly 2% of the extraordinary mode AKR even though the absolute power fluxes vary considerably.

4.2 Angular Dependence

      So far we have considered only the case of exactly perpendicular emission, = 90°. Now we consider the case of oblique propagation. we will discuss a series of one-dimensional (1D) simulations with different values of for 1.5 R and 2.5 R and also a two-dimensional (2D) simulation for 1.5 R.

      As discussed in section 3, a single 1D simulation contains wave propagation as both and 180° - . An example of decomposition into the two direction is given in Figure 9, which shows the time histories of the tranverse electric amplitude squared for mode 28 for 95° and 85° in a 1.5 R simulation. The = 95° component dominates the instability, but a rough value can be estimated for the growth rate of the weaker 85° component. The resulting growth rates as a function of are given Table 2. It is apparent that the maximum growth rate occurs at angles ~ 5% away from normal and directed toward higher altitudes. For downward propagating waves there is still growth, but the growth rates fall off rapidly away from 90°. In addition to the larger growth rates for the upward propagating cases, the saturation level of the radiation is increased compared to that at 90°. This feature is illustrated in Figure 10 for the 2.5 R simulations.

Fig. 9. Time history of the tranverse electric field amplitude squared for mode 28 in a one-dimensional simulation with oblique propagation and 1.5 R auroral parameters. The different growth rates for propagation at 85° and 95° to the magnetic field are apparent.

Fig. 10. Time history of total tranverse electric energy E in one-dimensional simulations with 2.5 R auroral parameters for various values of .

      We have also performed a 2D simulation in order to determine the angular dependence when all modes are simultaneously present. we have chosen the case of 1.5 R in order to have the largest possible growth so that the required computer time does not become too large. (The present run required 1.6 hours on a CRAY-1 computer.) The magnetic field is oriented in the yz plane as indicated in Figure 4, with the angle = 20°. The development and saturation of the maser instability in the 2D simulation is very similar to that in the 1D simulations. The peak growth rate is / = 6.5 10 for = 95.6°, which is within 10% of the corresponding 1D value in Table 2. The maximum percentage of electron kinetic energy converted into radiation is 4.0%. The corresponding values for the 1D runs at 90°, 95°, and 100° were 6.5%, 6.1%, and 4.4%. It thus appears that the 1D simulations somewhat overstate the efficiency of the generation mechanism.

      The angular dependence of the radiation emission in the 2D simulation is illustrated in Figure 11, which shows the square of the tranverse electric field as a function of k and k. This plot was obtained from a spectral analysis over the time period t = 400-1000. (The entire run lasted until t = 1200.) The data are displayed so that k > 0 corresponds to a downward component. As is apparent, the radiation occurs at an essentially constant of kc/ 0.96, where k = k + k. The most intense radiation occurs in the range 87° 96°. Appoximately 70% of the radiation is emitted in this range. For < 87° the emission falls off abruptly, while for > 96° it decreases less rapidly. The radiation with upward parallel component is the dominant one (constituting 75% of the total emission), but there is significant emission (25%) with downward parallel component.

Fig. 11. The tranverse electric energy density | E(k, k)| for a two-dimensional simulation with 1.5 R auroral parameters.

      The close agreement between this 2D simulation and the 1D cases discussed earlier indicates that the restriction to one spatial dimension does not lead to artificial results and gives one confidence in the applicability of the 1D results to the auroral region.

4.3 Dependence on Background Electron Population

      Our discussion so far has ignored the ionospheric electron population. At the lower altitudes ( ~ 1.5 R) this is a severe omission. The density fit in Figure 3 indicates that at 1.5 R this secondary component is strongly dominant, constituting about 90% of the total electron population. By 2.0 R the secondary component has dropped to 7%, however. Thus our previous results at 2.0 R and higher will be essentially unaffected by the ionospheric component. The transition between a majority ionospheric population and a majority magnetospheric population occurs at about 1.7 R.

      To provide a theoretical estimate for the effect of this background population on the emission at 90°, we use the DGH model discussed in section 4.1 modified to include a cold background component. With the presence of a second electron species there is a new branch of the extraordinary mode lying between the relativistic cyclotron frequency / and [Prichett, 1984a, b; Strangeway, 1985]. For emission at 90° the instability lies on this new branch rather than on the conventional one with > . For the parameters at 1.5 R, a 50%-50% mixture of energetic and cold electrons reduces the maximum growth rate to 3.0 10 from 7.2 10. The more realistic case of 10% energetic electrons and 90% cold electrons gives a growth rate of 7.4 10 . Despite this order-of-magnitude reduction in from the pure energetic case, the real part of the frequency and the wave number at maximum temporal growth change only slightly. Re (/ ) decreases from 0.9950 to 0.9905, while ck/ increases from 0.965 to 1.035. The group velocity and hence the convective growth length are much more sensitive to the presence of the cold component. [Prichett, 1984b; Strangway, 1985].

      We have performed a 1D simulation at = 90° with the 1.5 R parameters for the case of a 50% magnetospheric population and 50% ionospheric component with temperature of 1 keV. The growth rate is reduced to 2.5 10 , in good agreement with the theoretical prediction. The decrease in the saturation level is much more dramatic. The percentage of the magnetospheric electron kinetic energy converted into radiation is now only 0.27%, compared to 6.5% for the pure magnetospheric case. A further increase of the background percentage to the realistic level of 90% would render the instability unobservable in the present simulations. It thus appears that the dominant secondary electron population at 1.5 R leads to only minor radiation generation via the cyclotron maser instability.

      It thus seems likely that we need a majority magnetospheric population to get significant cyclotron maser radiation. We thus choose 1.75 R, where the magnetospheric component is about 75% of the total population, for our final set of simulations. For emission at 90% the DGH model predicts a maximum growth rate of 2.4 10 . The value observed in the simulation is 2.6 10. At saturation, 0.86% of the magnetospheric kinetic energy has been converted into radiation, and the ratio B/ B = 2.6 10. Figure 12 shows the electron distributions in momentum space, both initially and at saturation. Although the magnetospheric component still exhibits a positive slope in p for small p at saturation (Figure 12b), the total distribution decreases essentially monotonically in p (Figure 12d). Additional simulations at 92.5° and 95° produce maximum growth rates of 2.7 10 and 0.9 10 and efficiencies of 1.05% and 0.33%. At 89° the growth rate has dropped to 1.8 10 , and at 87.5° the instability is no longer apparent. Thus the dominant emission occurs in the range 89°-94°. This is slightly more concentrated near 90° than were the previous results for the pure magnetospheric distribution at 1.5 R and 2.5 R.

Fig. 12. Electron momentum-space distributions in a one-dimensional simulation with 1.75 R auroral parameters and 75% magnetospheric electrons, 25% ionospheric electrons. (a) Initial distribution (t = 0). (b) Magnetospheric component saturation (t = 1600). (c) Total distribution at saturation. (d) Contours of f (p, p) at saturation.

5. Conclusions

      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. [Pritchett, 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. [1982] 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 [1978] 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. [1984], 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 [1974]. 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 [1985] 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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