University of California at Los Angeles

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

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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 ( n_{D}**

**
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
**

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

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