Wave Dispersion and Ray Propagation in a Weakly Relativistic Electron Plasma: Implications for the Generation of Auroral Kilometric Radiation

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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3. Plasma Parameter Dependence

      As mentioned in the introduction, we only expect relativistic effects to be important for some range of plasma parameters. As a first step in determining the parameter ranges for which relativistic corrections are sufficient to modify the wave dispersion, we shall assume that there are only hot ring electrons. The solutions of the dispersion relation are shown in Figures 1a and 1b. Figure 1a shows contours of constant wave frequency normalized to the electron gyrofrequency plotted as a function of ring momentum and / assuming k = k = 0; Figure 1b shows the corresponding normalized growth rate times 10. In Figure 1b the contour interval is 0.2, with a thick contour every 1.0. The maximum growth rate in Figure 1b is ~ 5 x 10. For reasons of clarity the contours in Figure 1a are spaced at two different intervals. For > 1.02 the contour interval is 0.002, whereas below = 1.02 the contour interval is 0.001.

Fig. 1a.     Contour plot of wave frequency for a ring distribution. Contours of normalized frequency are plotted as a function of ring momentum (p / mc) and normalized plasma frequency squared ( / ). Contours are spaced every 0.002 for / > 1.02 with thick contours at intervals of 0.02 . For / < 1.02 the spacing is .00l between thin contours.
Fig. 1b. Contour plot of growth rate for a ring distribution. The growth rate corresponds to the frequencies shown in Figure1a. The contours are spaced at every 2 x 10.

      As can be seen from Figure 1b, instability only occurs for large ring momentum and small gyrofrequencies. For sufficiently small ring momentum the wave frequency is given by the cold plasma frequency R-X mode cutoff, as shown in Figure 1a. The slight kink in the frequency contours plotted in Figure 1a occurs at the transition from stable to unstable modes, as given by the = 0 contour in Figure lb. Both plasma parameters are plotted using a logarithmic scale in the figures, and the transition is a straight line. This line corresponds very well to the limit given by Pritchett [1984b], p / mc = / 2 . The wave frequency equals the gyrofrequency along the line p / mc = / , which was also determined by Pritchett. As can be seen from Figures 1a and 1b, the growth rate at k = 0 is maximum for a ring distribution when = . However, this is coincidental: equation (10) of Pritchett [1984b] shows that the growth rate is a maximum for p / mc = / , which also corresponds to = . On the other hand, the growth rate is maximum for p / mc = (3/2) / for a shell distribution (see equation (19) of Pritchett [1984b]), which is a three-dimensional distribution, whereas a ring is essentially two-dimensional.

      The maximum value of ring momentum used in Figures 1a and lb is 0.1, which corresponds to 2.5-keV electrons. Classically, these electrons would not be considered to be relativistic, and so relativistic corrections are essentially first-order effects. For higher values of momentum the condition for maximum growth might be expected to depart from the linearity displayed in Figure 1b. However, typical auroral electron energies are not observed to be in excess of a few keV [Croley et al., 1978], and so we shall restrict our analysis in the rest of the paper to the case of p / mc = 0.1.

Fig. 2a. Plot of wave frequency for a ring distribution plus a cold background plasma. The frequency is shown as function of normalized plasma frequency squared ( / ) and the ratio of hot to total number density (n / n) using a three-dimensional representation.
Fig. 2b. Plot of growth rate for a ring distribution plus a cold background plasma. The growth rate corresponds to the frequencies shown in Figure 2a.
      Having chosen a particular value of ring momentum, we can now vary a different plasma parameter. Figures 2a and 2b show the variation of frequency and growth rate as a function of / and the fraction of hot ring electrons, n / n. Rather than use a contour plot, we have chosen to display the solutions using a three-dimensional representation. The cusp at n / n = 1 in Figure 2a marks the transition from unstable to stable solutions. In the previous figures we showed the stable solutions which coupled to the R-X mode. As pointed out by Pritchett [1984b], the transition into the unstable regime occurs when the R-X mode and the Bernstein mode merge, and the stable solution shown in Figure 2a corresponds to the Bernstein mode.

      It is apparent from Figure 2a that the unstable mode can no longer couple to the R-X mode once some cold plasma is introduced to the plasma dispersion relation. Since neither electron distribution has any thermal spread associated with it, both cold electrons and hot ring electrons can be considered to be separate particle species, each with its own gyrofrequency. Except for the case n = n , all solutions lie in the frequency range / < < where is the relativistic gamma for the ring electrons.

      The growth rate as plotted in Figure 2b shows that the inclusion of cold plasma removes the restriction on the range of instability. Unstable solutions are found for quite large values of the electron plasma frequency. In addition, although the growth rate does decrease as the fraction of hot electrons is decreased, there is some instability even for n / n = 0.01. It should be noted, however, that the frequency is very close to the relativistic gyrofrequency, and thermal effects may introduce significant damping, which cannot be included using a delta function distribution. The growth rate as given in Figure 2b should be interpreted as an upper limit, especially for low ring electron number densities. However, the growth rate is still moderately large ~ 5 x 10 .

      Summarizing the results of the plasma parameter study, we have found that for a plasma that can be characterized by two electron distributions there exists an unstable mode which lies between the gyrofrequencies of the two electron species. The growth maximizes near p / mc = / and also for n ~ n. Typical values of the growth rate are of the order of ~ 10 .

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