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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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.
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.  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.
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.
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.
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