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

R. J. Strangeway

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

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.


      1. Introduction
      2. Dispersion Relation
      3. Plasma Parameter Dependence
      4. Wave Vector Dependence
      5. Discussion On Group Velocity Variation
      6. The Applicability of the Ring Distribution
      7. Conclusions


      Relativistic modifications to wave dispersion near the electron gyrofrequency are known to be important for sufficiently energetic electrons. In general, if the characteristic momentum of the energetic particles (p / mc) is greater than the ratio of the plasma frequency to the gyrofrequency ( / ) wave propagation can be markedly different than the classical cold plasma limit. We investigate the wave dispersion in a weakly relativistic electron plasma, specifically in the context of the generation of auroral kilometric radiation (AKR). To do this, we have used a simple electron distribution consisting of a ring of energetic electrons at constant perpendicular momentum together with a stationary cold component. It is found that for such an electron distribution, where the two electron components have different gyrofrequencies due to relativistic effects, a new wave mode is introduced. This wave is trapped between the two electron gyrofrequencies. We find that the group velocity for such a wave is small; convective growth lengths for the wave are calculated to be of the order of 1 km. Consequently, ray propagation in a simple model for the density cavity present on auroral zone field lines is studied. We find that the ray paths tend to bring waves to altitudes where the growth rate maximizes; at these altitudes the waves propagate primarily across the ambient magnetic field. This could result in substantial amplification of the wave, since damping is minimized, and the waves may undergo multiple transitions across the auroral arc. However, the wave is trapped and must undergo mode conversion to escape from the auroral arc. Since the wave polarization is found to be very similar to the R-X mode polarization, we deduce that coupling to the R-X mode at the edge of the auroral arc may be reasonably efficient.

1. Introduction

      In recent years the earth has been observed to be a significant emitter of radio noise. In particular, intense radio waves with free space wavelengths of the order of kilometers have been observed to be emanating from above the auroral oval [Gurnett, 1974]. Since the wave noise is known to be associated with auroral activity [Kurth et al., 1975], the wave emission is commonly referred to as auroral kilometric radiation (AKR).

      Several features of the radiation present major problems for a theoretical understanding of the generation of the radio noise. First, the noise is intense; Gurnett [1974] has calculated that the peak wave power is of the order of 10 W, some 1% of the power estimated to be deposited in the ionosphere by auroral electrons during a substorm. The average power level was found to be ~ 10 W [Gallagher and Gurnett, 1979]. Second, the wave is primarily observed in the cold plasma R-X mode. This has been deduced from direct polarization measurements on the Voyager spacecraft [Kaiser et al., 1978] and also inferred through ray path calculations [Green et al., 1977]. In addition, spacecraft have flown directly through the generation region for AKR, which is presumed to be below altitudes of 1-2 R on auroral zone field lines. Data from ISIS [Benson and Calvert, 1979; Calvert, 1981a; Benson, 1982] support the inference that AKR is generated in the R-X mode through the observation of a low-frequency cutoff for AKR, which is strongly tied to the local gyrofrequency. Recently, Mellott et al. [1984] have used DE-1 data to show that AKR is usually observed to be in the R-X mode near the presumed generation region.

      Taking into account these properties of AKR, one of the more popular mechanisms for the generation of AKR is the cyclotron maser instability [Wu and Lee, 1979]. This instability is driven by the direct gyroresonance of energetic electrons with the R-X mode. Direct gyroresonance was first proposed by Melrose [1976] to explain the generation of Jovian decametric radiation (DAM). Since Melrose used a drifting bi-Maxwellian to describe the energetic particle distributions, he found that significant thermal anisotropies were required to drive the R-X mode unstable through gyroresonance. Typically, v v c , where v and v are the perpendicular and parallel thermal velocities, respectively, and c is the speed of light.

      It was shown by Wu and Lee [1979] that gyroresonance can generate waves more efficiently, provided there is a positive slope of the energetic particle distribution as a function of perpendicular velocity. These authors applied their theory to the generation of AKR and found that coupling to the R-X mode was enhanced by assuming that the plasma frequency in the generation region was significantly lower than the gyrofrequency. Additionally, for relativistic gyroresonance, resonant particles describe an ellipse in velocity space. If the ellipse lies fully within the loss cone, for example, then no damping terms are introduced on integrating over all resonant particles. The low plasma frequency allows the R-X mode cutoff to approach the gyrofrequency, which in turn allows lower-energy electrons to resonate with the waves.

      Recently, several authors [Wu et al., 1982; Omidi and Gurnett, 1982; Dusenbery and Lyons, 1982] have investigated the generation of AKR by the cyclotron maser instability, using particle data to determine the sources of free energy in the plasma. While a particular frequency may have large growth at one location on an auroral zone field line, Omidi and Gurnett [1982] have shown that this wave can be damped as the wave propagates along the field and enters different plasma regimes. The effect of the parallel electric field presumed to exist on auroral field lines [Croley et al., 1978] was included by Wu et al. [1982], who addressed the effects introduced by assuming that only energetic electrons determined the wave propagation characteristics. Generation by both the primary downgoing electrons and the loss cone in the reflected and backscattered electrons was investigated by Dusenbery and Lyons [1982]. One aspect of all the work cited here is the sensitivity of the growth rate to the parameters chosen to describe the plasma, in terms of both the energetic electrons and the presumed cold electrons.

      It should be pointed out that while the S3-3 observations reported by Croley et al. [1978] show population inversions associated with both the loss cone and field-aligned acceleration of the energetic electrons, the low-energy electron population has not been measured directly on auroral field lines. Wave measurements from the Hawkeye spacecraft have been used by Calvert [1981b] to show the presence of an auroral density cavity, but again this does not allow us to directly determine the properties of the low-energy electron population.

      Bearing in mind the uncertainty in determining the amount of cold plasma present, it is possible that the wave propagation characteristics may be determined by the more energetic electrons, and several authors [Tsai et al., 1981; Wu et al., 1981, 1982; Winglee, 1983; LeQueau et al., 1984a, b] have studied the modifications of the wave properties by including energetic electrons in the wave dispersion relation.

      Recently, Pritchett [1984a, b] has investigated instabilities associated with the mildly relativistic magnetized plasma, using both simulations and linear analysis. The types of particle distribution functions employed in his simulation were both "ring" and "shell" distributions. In general, a ring distribution is characterized by a ring of energetic particles with constant perpendicular momentum (where perpendicular is defined with respect to the ambient magnetic field), whereas a shell distribution is typically given by a spherical shell of particles in momentum space. In the simulations these distributions can also have a thermal speed associated with them. Pritchett concluded that when p / mc ~ / (where p / mc is the characteristic energetic particle momentum and / is the ratio of the electron plasma frequency to the electron gyrofrequency), the plasma can be unstable to waves propagating normal to the ambient field direction. When only hot electrons are present, the unstable mode couples directly to the R-X mode, but inclusion of a thermal electron component can decouple the two modes.

      Relativistic effects consequently result in the electrons being essentially a two-species distribution, with the new mode lying between the gyrofrequencies associated with each particle species. Since the difference between the two gyrofrequencies is small for typical parameters associated with auroral electrons, one might expect the wave group velocity to be small. Hence it would be of interest to study the wave characteristics in some detail. In this paper we shall investigate the wave modes introduced by relativistic effects. We shall particularly emphasize the variation of the wave modes as a function of the parameters which describe the plasma and further study some of the properties of the waves themselves, such as polarization and group velocity.

      Since the group velocity of the unstable waves associated with a ring distribution will be shown to have considerable variation, we shall study this wave property in more detail. We shall show, using a simple model for the auroral density cavity, that the convective properties of the waves may have important implications for the generation of AKR. Not only is the magnitude of the group velocity important, but also the direction of ray propagation. The former results in short convective growth lengths, whereas the latter can effectively focus the waves at an altitude most suited for growth due to the bending of the ray paths.

      The structure of the paper is as follows. In the next section we shall briefly describe the dispersion relation used in the present analysis, together with a summary of the plasma parameters and wave diagnostics used. In section 3 we shall present solutions of the wave dispersion relation as a function of the plasma parameters, while in section 4 we shall describe some of the wave properties as a function of wave vector. The implications of the group velocity variations as applied to the generation region for AKR will be discussed in section 5. In section 6 we investigate in detail the applicability of the present analysis to auroral electron distributions. The results of the study will be summarized in the final section.

2. Dispersion Relation

      In this paper we are principally interested in the modifications of wave dispersion due to relativistic effects and in the dependence of the wave instabilities on the parameters which describe the plasma. Bearing this in mind, we shall use the simplest distribution function which incorporates the effects under consideration, i.e., a source of free energy and sufficient energetic electrons to modify wave dispersion. The distribution function employed will hence be the delta function ring distribution given by

where (p) is the Dirac delta function; p is the perpendicular particle momentum defined with respect to the ambient magnetic field and p is the parallel component; p is the momentum of the ring particles; and n and n are the cold and hot electron number densities, respectively.

      The distribution function as given by (1) is defined as particle density in momentum phase space, rather than velocity phase space as is usually the case, to ensure that relativistic effects are included correctly. The delta function ring distribution was originally investigated by Chu and Hirshfield [1978], who described the instabilities associated with such a distribution in terms of azimuthal and axial bunching of electrons by the wave fields. These authors noted that for parallel propagation, azimuthal bunching is primarily associated with high phase velocity waves. The delta function ring distribution has been further studied by Pritchett [1984b] and Winglee [1983], who also considered a Dory-Guest-Harris (DGH) distribution [Dory et al., 1965]. Pritchett [1984b] carried out simulations which indicate that the ring distribution tends to overestimate the growth rates of the linear stage of the instability for waves propagating at angles significantly away from the normal to the ambient field. Presumably this is due to effects not included in such a simple distribution function, such as cyclotron damping by warm electrons. However, the main thrust of the present work is to indicate those properties of the modified wave dispersion which may be of significance for the generation of AKR. Moreover, near 90° propagation, thermal effects are less likely to be important.

      By using a delta function distribution function we not only neglect thermal effects, but also no longer generate instabilities through gyroresonance, which is the usual assumption for the cyclotron maser instability. Indeed, the instability driven by a ring distribution is a fluid-type instability, analogous to classical instabilities such as the Buneman instability [Buneman, 1958], where the free energy for instability is due to relative drift of two particle species. In this particular case there is no net flow, but the gyrational energy of the particles is available for wave growth. When the number densities of each species are comparable, we shall show that the frequency lies near the middle of the unstable frequency range. If the two electron populations have low temperatures, there will be very few electrons available for gyroresonance, and the present analysis may be more applicable. However, the auroral electron distribution function usually displays considerable structure, and whether or not the wave dispersion for such a plasma results in fluidlike or resonant instabilities is not easily determined. LeQueau et al. [1984b] have addressed the transition from one instability regime to another in some detail.

      A second criticism must be raised concerning the use of a ring distribution when modeling auroral electrons. When Wu et al. [1982] discussed the modification of the R-X mode cutoff due to relativistic effects, they noted that this might be an artifact of what they called a double loss cone distribution. The double loss cone is symmetric about p = 0, with a widened loss cone due to parallel electric fields. In an extreme case this distribution could be modeled by a ring distribution. However, double loss cones are not observed in the particle data. On the other hand, the data do show a shell-like distribution in addition to the single loss cone feature. The simulations of Pritchett [1984a, b] show that for 90° propagating waves the wave modes for the shell and the ring distribution are similar. While the ring distribution may significantly overestimate the growth rates due to the neglect of damping, the distribution is at least useful in determining some of the properties of the wave mode introduced by relativistic modifications. We shall address the applicability of the ring distribution to auroral electron distributions in more detail in section 6.

      With (1) defining the electron distribution, we can define a set of parameters which characterize the plasma:

/     the square of the ratio of the electron plasma frequency over the electron gyrofrequency, where
                    = 4e(n + n / ) / m and = eB / mc, m is the rest mass of the electrons,
                    = [(1 + p / mc )], B is the ambient magnetic field, and the other symbols have their
                    usual meaning;
p / mc       characteristic normalized ring momentum;
n / n         ratio of hot ring electrons to the total number density.

      If we assume a first-order electromagnetic wave perturbation of the form E(r) = E exp [-i(t-kz-kx)], then the waves must satisfy the dispersion relation

(, k) = 0             (2)


with k = (k, 0, k), indicates summation over species, and is the conductivity tensor for each particle species derived from the linearized Vlasov equation using standard techniques [e.g., Clemmow and Dougherty, 1969]. It should be noted that the ions are assumed to be fixed, and the summation is only carried out for the hot and cold electrons. The conductivity tensor for a ring distribution can be easily calculated from the dielectric tensor given by Pritchett [1984b] (his equation (6)). That is,

where J = J( ) is the Bessel function of the first kind of order n, with n an integer, = kp / m, and n is the number density for the particle species.

      We can consequently parameterize the wave perturbation by its complex frequency ( = + i) and its wave vector (k, k ). Growth of an instability corresponds to positive . Also, for convenience, although as used above is a complex quantity, when presenting solutions of the dispersion relation, shall correspond to the real part of the frequency. In addition , we shall use the wave group velocity (v), the ratio kE / k E = E / E , and the ratio E - iE / 2 E = E / E as wave diagnostics. The latter two quantities give information on the wave polarization, E / E being the ratio of longitudinal to transverse electric fields and E / E giving the fraction of perpendicular electric field that is right-hand circularly polarized.

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 .

4. Wave Vector Dependence

      Having shown how the instability of a ring distribution varies as a function of the plasma parameters, we shall now explore the dependence on the wave vector. In the previous section we considered the limit k = k = 0. In this section we shall fix the plasma parameters and vary the wave vector direction and magnitude. As in the previous section, the ring electrons shall be assumed to have a momentum p / mc = 0.1. Since the growth rate is near maximum for / = p / mc , we shall choose / = 0.01. Finally, we shall somewhat arbitrarily assume that n / n = 0.75.

Fig. 3a. Plot of wave frequency as a function of wave vector. The plasma parameters are held fixed, while the wave vector magnitude and direction are varied.
Fig. 3b. Plot of growth rate as a function of wave vector. The growth rate corresponds to the frequencies shown in Figure 3a.
      Solutions of the dispersion relation with the plasma parameters as specified above are shown in Figures 3a and 3b. The frequency (in Figure 3a) and growth rate (in Figure 3b) are plotted using a three-dimensional representation as a function of normalized wave vector. It should be noted that the wave vector varies logarithmically; the solutions at the lower left- hand corner of the figure correspond to k = k = 0.01 / c, not zero. The maximum value of k is / c, and as can be seen from Figure 3a, where the wave frequency is shown to be very nearly equal to the gyrofrequency, this corresponds to waves propagating at the speed of light.

      One aspect of the wave dispersion which we shall discuss in more detail in the next section is the group velocity variation. The total range in frequency in Figure 3a, , 0.002 , while the range in wave vector, k, / c. The group velocity is consequently small; / k, 0.002c. Additionally, while / k is positive, / k < 0 for the parameters shown in Figure 3a.

      The growth rate plotted in Figure 3b shows little variation for kc / < 0.1. As the wave phase velocity decreases, aproaching the speed of light, the growth rate decreases, until a stable mode is encountered close kc / = 1. There is a slight difference in where the transition is encountered as a function of k and k . With the particular choice of system parameters the growth is maximum for small k. Typical growth rates from Figure 3b are / 0.002, which on combination with the estimate for group velocity gives convective growth lengths of the order c / less than the free space wavelength which is a few kilometers.

      As discussed in the previous section, the unstable mode is on a separate branch of the dispersion relation when some cold plasma is present, and it does not couple directly to the R-X mode. If this mode is ultimately responsible for the generation of AKR, then some coupling to a freely propagating mode must occur. While the mechanism for mode conversion beyond the scope of the present work, we can indicate some features of the mode which imply that mode conversion may be feasible. At the limit k = 0 or k = 0 the dispersion relation given by (2), (3), and (4) separates into two equations. In the cold plasma limit, these result in the R-X mode and the L-O mode. On including a hot ring component, the new mode is just an additional root to the R-X dispersion relation. Consequently, the polarization of the new wave mode is the same as the R-X mode at the limits k = 0 or k = 0.

Fig. 4. Fraction of wave electric field that is right-hand circularly polarized. The polarization is plotted for the solutions shown in Figures 3a and 3b.
      When the wave vector is neither parallel nor perpendicular, the dispersion relation is no longer separable. It is hence not obvious what the polarization of the new mode should be. In Figure 4 we show the wave polarization for the unstable mode plotted in Figures 3a and 3b. E / E is the fraction of perpendicular wave electric field magnitude that is right-hand circularly (RHC) polarized. Since the polarization is defined with respect to the ambient magnetic field direction, the fraction of RHC polarized field decreases as the perpendicular phase velocity approaches the speed of light. This is similar to the cold plasma wave polarization, since the mode becomes transverse as the phase speed approaches the speed of light for a cold plasma wave. For parallel propagation the wave can still be RHC, but for perpendicular polarization the mode eventually becomes plane polarized.

Fig 5. Ratio of longitudinal to transverse electric field magnitude. This ratio is plotted for the solutions in Figures 3a and 3b.
      In Figure 5 we show the ratio of longitudinal wave electric field magnitude to the transverse electric field magnitude. Near the speed of light the wave is almost completely transverse, as expected. The wave is least electromagnetic near k c / = 0.1, but throughout the wave vector range considered the mode displays similar polarization characteristics to the cold plasma R-X mode. This indicates that in an inhomogeneous medium such as the region of intense electron precipitation above an auroral arc, mode coupling between the unstable trapped mode and the freely propagating R-X mode should be possible. A detailed analysis of the mode conversion is required to adequately address this point, in particular studying aspects of mode conversion such as the ratio of L-O to R-X mode produced as a consequence of the coupling.

Fig. 6a. Plot of the wave frequency for a low-density plasma. This figure is similar in format to Figure 3a, but / = 10, i.e., p / mc > / , which we refer to as an underdense plasma.
Fig. 6b. Plot of the growth rate for a low density plasma. The growth rate corresponds to the frequencies shown in Figure 6a.
      However the wave frequency and growth rate variation shown in Figures 3a and 3b are not necessarily the same for all plasma parameters. Figures 6a and 6b show solutions of the ring dispersion relation for / = 10 as a function of wave vector. All other parameters are the same as in Figures 3a and 3b. Since / < p / mc, we shall refer to the plasma as being underdense. In contrast to Figure 3b, where the growth rate was maximum for k = k = 0.01c / 0, the growth rate in Figure 6b maximizes near kc / = 1. The wave frequency of the unstable mode in Figure 6a maximizes near kc / 1. While the corresponding group velocity will still be small, the ray direction is more nearly parallel to the wave direction.

      To summarize the properties of the unstable mode as a function of wave vector, we find that the polarization is very similar to the cold plasma R-X mode. Usually, the wave frequency is a maximum for large k ( / k > 0), but the dependence on k depends on whether the plasma is underdense or overdense. If p / mc > / (underdense), / k > 0, whereas / k < 0 for p / mc / (overdense). The growth rate is also dependent on the relative density of the plasma. For an underdense plasma, is a maximum near kc / = 1, whereas is a maximum for k = 0 in an overdense plasma. We shall address the transition from underdense to overdense plasma in more detail in the next section.

5. Discussion On Group Velocity Variation

      The group velocity variation discussed in the previous section is strongly dependent on the plasma system parameters, and some implications for the generation of AKR can be formed. As an initial attempt to study the group velocity variation in the context of auroral zone phenomena we shall present a simple model for the plasma parameters associated with an auroral zone field line.

Fig. 7. Model of the auroral density cavity. The electron gyrofrequency, plasma frequency-gyrofrequency ratio, and characteristic perpendicualr momentum are plotted as a function of geocentric distance along a field line. The derivation of these parameters is described in more detail in the text.
      Figure 7 shows how the plasma parameters are assumed to vary as a function of geocentric distance from 1.3 to 3.3 R The electron gyrofrequency ( ) has been found assuming a dipole field for an invariant latitude of 70°. The ratio / is based on the results of Calvert [1981b], and we have assumed that / = 0.3 at distances of 1.3 and 3.3 R, with a minimum value of less than 0.03 near 2 R. The characteristic perpendicular momentum (p / mc) has been found assuming that the accelerating electric potential varies linearly with the magnetic field strength and is zero at 3.3 R. The corresponding perpendicular momentum is then obtained from the ellipse produced in momentum space due to the magnetic mirror force [Chiu and Schulz, 1978]. The maximum value of p / mc is assumed to be 0.2, corresponding to 10-keV electrons. It should be noted that p / mc does not go to zero when the accelerating potential goes to zero because of the widened loss cone.

      We have not included the variation of n / n as a function of altitude, since this parameter is even more uncertain than the other parameters presented here. Given the plasma frequency variation shown in Figure 7, the total number density is less than unity between 1.8 and 3 R geocentric distance. It is consequently not unreasonable to assume that the energetic electrons contribute significantly to the total number density.

      The results presented in the previous sections have already suggested that there is a transition in the wave properties near the region p / mc = / . We shall explore the variation of frequency, growth rate, and group velocity near this critical region more thoroughly here. To reduce the number of variables which describe the plasma, we shall assume for the time being that the variation of the ratio / as a function of altitude is most significant. By assuming that the other parameters which describe the plasma are constant, only the parallel component of wave vector varies with altitude. We can consequently trace ray paths to different altitudes by obtaining solutions of the dispersion relation as a function of / and k c / .

      It is important to note that in addition to assuming that p / mc and n / n are both constant, we are also implicitly assuming that the gyrofrequency does not vary significantly. This allows us to assume that kc / is constant. We shall subsequently show that the gyrofrequency variation is usually of major importance. For the present, however, we shall ignore the change in gyrofrequency as a function of altitude.

Fig. 8a. Plot of wave frequency for a model of an auroral zone field line. Solutions of the ring dispersion relation have been found assuming that only the normalized plasma frequency and parallel wave vector vary as a function of altitude along an auroral zone field line.
Fig. 8b. Plot of growth rate for a model of an auroral zone field line. The growth rate corresponds to the frequencies shown in Figure 8a.
      Figures 8a and 8b show the frequency and growth rate of the unstable mode as a function of / and k c / . At high k , k c / ~ 1 the mode becomes stable, as indicated by the sudden change in frequency in Figure 8a and the rapid decrease in growth rate in Figure 8b. The growth rate has a maximum for most of the range in k c / at / = 0.007. When only hot electrons were included in the dispersion relation , the maximum occured at a slighly higher value.

      A transition in the parallel group velocity is apparent in Figure 8a, for small values of / , / k > 0, while / k < 0 for large values of / . To emphasize this transition, we have plotted contours of parallel group velocity (v ) in Figure 9. We have restricted the range in v to v / c 2 x 10, and for clarity we have not shown contours of group velocity for the stable branch. The shaded area in the figure shows that part of the parameter space for which v < 0. When k c / 0.1, the transition occurs at a constant value of / = 0.01, i.e., / = p / mc.

      To emphasize that v = 0 near / = p / mc for small values of wave vector, we show contours of v when k c / = 0.01 in Figure 10. For this figure we allow both / and p / mc to vary. Since the wave vector is small, the corresponding group velocities are small. For reference the perpendicular group velocity is roughly of the same order, v / c 2 x 10 . The v = 0 contour in Figure 10 is given by / = p / mc.

Fig. 9. Contour plot of parallel group velocity. The contours have been plotted for v / c 2 x 10, with a contour interval of 10. The plasma parameters are the same as in Figures 8a and 8b. The shaded area indicates that region for which the parallel group velocity is less than zero.
Fig. 10. Contour plot of parallel group velocity for fixed density ratio. Similar in format to Figure 9. The contour interval is 10 .
      In summary, Figures 9 and 10 show that for small wave vectors the parallel group velocity goes to zero when p / mc / . We had already inferred the presence of this transition from results presented in the previous section, where we introduced the notion of underdense and overdense plasma regions. Without carrying out a full ray tracing analysis, we can still make some inferences on the properties of both underdense and overdense plasma regions in the context of a more general model of the auroral field line as outlined in Figure 7, where we can no longer assume that any plasma parameter can be taken to be constant.

      Except where p / mc ~ / , the parameter whose variation as a function of altitude is of primary importance is most likely to be the gyrofrequency. From the solutions presented in the previous section together with the group velocity variation discussed here we can determine how the wave properties will vary as a function of altitude. When both hot and cold electrons are present, we have shown that the unstable mode is trapped between the two electron gyrofrequencies. This will determine the range in altitude over which a wave at a particular frequency can propagate. With our knowledge of the group velocity variation we can further infer the values of wave vector for which the frequency will be a maximum or a minimum. It should be remembered, however, that we have not completely explored the parameter space, and so some caution should be exercised.

Fig. 11. Schematic of the wave properties at different altitudes along the auroral field line. The arrows in the top panel indicate the ray direction for a fixed frequency. The thickening is meant to suggest propagation out of the plane of the figure across the ambient field. The bottom three panels give estimates of the growth rate, probable wave vector directions, and wave vector magnitude. The wave properties have been summarized for both an underdense and an overdense region on the field line.
      We summarize the expected dependence on altitude of the wave properties in Figure 11. We have chosen two altitude ranges, one centered on 1.9 R corresponding to an underdense plasma region, the second centered on 2.9 R, where the plasma is overdense. The two sloping lines in the top panel of each set show the variation with altitude of the cold and hot electron gyroErequencies. The upper line corresponds to the cold electron gyrofrequency. The thick arrows indicate how the group velocity varies with altitude for a wave of a given frequency. The wave is confined to the altitude where / < < , being the relativistic gamma of the hot electrons. The thickening of the arrows is meant to suggest that the group velocity is turning perpendicular to the ambient magnetic field and pointing out of the plane of the figure.

      The bottom three panels show how we expect the wave vector magnitude (kc / ), wave vector direction (angle), and growth rate ( / ) to vary given the constraints on the wave frequency. The numbers on these three panels are only qualitative, indicating the typical values we expect. In the plots of wave vector angle we have shown both the range in angle and the direction of k with respect to the ambient magnetic fields. As we shall discuss below, the sections drawn with a dashed line are meant to indicate that the wave may be reflected.

      For both an underdense and an overdense plasma the wave frequency approaches the cold electron gyrofrequency for propagation near 90° with ck / 1. The group velocity is nearly perpendicular for large k. For a wave at a particular frequency the wave frequency will approach the cold electron gyrofrequency at the higher altitudes. The above properties when are indicated in Figure 11, the only difference between an undersense and overdense plasma being the estimate for the growth rate at . Since propagation is near 90° and damping is minimized, the ring distribution growth rate is a reasonable estimate. As shown in Figures 3b and 6b, / is a minimum at large k for an overdense plasma, whereas growth is near maximum for large k in an underdense plasma.

      We have already suggested that a perpendicularly propagating wave may be reflected at the edge of the auroral arc. It would be useful to obtain some quantitative estimates of the efficiency of the mirror. The edge of the auroral arc is likely to be associated with both an increase in cold electron density and a decrease in the hot electron ring momentum, because of the weakening parallel electric field. These two effects will tend to remove the unstable mode, and it is not clear how the wave will refract in such a medium. Additionally, while a wave may be propagating perpendicularly to the magnetic field, the wave vector can be parallel to the edge of the auroral arc.

      Bearing these problems in mind, we can at present only roughly estimate reflection coefficients. Using arguments similar to those of Calvert [1982], the reflection coefficient for waves incident perpendicular to the edge of the auroral arc is given by

when and are the refractive indices outside and inside the arc. We shall assume (ck / 1). Since the wave polarization for the unstable mode is generally similar to the R-X mode, it might be argued that the wave will propagate away from the arc in the Z mode. For 90° propagation the Z mode is given by

Consequently, when , 2 - / . So for / << 1, 1.414, and R 0.17 from (5). Consequently, only about 3% of the wave power would be reflected.

      It should be pointed out that the estimate is a "worst case" estimate, and the reflection coefficient may be substantially higher. This calculation does emphasize the necessity of a more sophisticated mode conversion analysis. For the time being we shall assume some reflection does occur at the edge of the arc.

      While we have indicated the possibility of reflection for the underdense plasma through the dashed vectors for wave propagation angle, we have not shown similar reflection for the overdense case. Because / k < 0 for an overdense plasma, the wave frequency approaches the hot electron gyrofrequency for k c / 1. Since there is no freedom in choosing the wave vector direction with respect to the gradient along the field, unlike perpendicular propagation as discussed above, we may be more confident in our estimate of reflection coefficients at the low-altitude cutoff for the unstable mode.

      Again through polarization considerations the mode is likely to couple to the Z mode. For parallel propagation,

      At the low-frequency cutoff, / , i.e., / 1 - p / 2m c. Consequently,

1 + (2 m c / p )

      For an overdense plasma, 3. When 1, R 0.27, and the reflected power is approximately 7%.

      This figure is somewhat higher than for perpendicular reflection, and it might be argued that reflection can occur at both high- and low-frequency cutoffs in an overdense plasma. However, k 0 for the low-frequency cutoff in an underdense plasma, and reflection is likely to be much more efficient. We have consequently assumed that an underdense plasma regime is more favorable to multiple reflections and possible feedback such as discussed by Calvert [1982].

      One important feature of the wave properties shown in Figure 11 is the altitude range over which we expect a wave at a particular frequency to exist. Since the wave is confined to the altitudes where / < < , the altitude range must be less than the distance over which the change in magnetic field B < ( - 1)B. In Figure 11, corresponding to the parameters shown in Figure 7, the altitude range is at most 0.005 R and can be as small as 0.002 R . The scale length along the field line is consequently a few tens of kilometers. While this is at least 2 orders of magnitude larger than the Debye length (some hundreds of meters), the plasma may be quite structured at this scale length. Moreover, the wavelength of the waves can be comparable to this scale length, and this could result in significant changes in the wave dispersion.

      Bearing in mind the assumptions inherent in the results presented here, the convective properties of the waves driven unstable by a ring distribution yield some interesting implications for the generation of AKR. Specifically, the dependence of frequency on wave vector is such that the waves tend to propagate to those altitudes where k is large, and growth is maximized when p / mc > / . Also for altitudes where p / mc > / the waves may be able to perform multiple transitions of the auroral cavity, thus enhancing their growth. We note that the above condition for reflection restricts the region of maximum growth to those altitudes which yield a frequency bandwidth consistent with observations, 100 kHz f 500 kHz.

6. The Applicability of the Ring Distribution

      One of the principal advantages obtained by using the delta function ring distribution is the simple form of the resultant dispersion relation. This simplicity allows us to carry out extensive parameter studies. While the ring distribution may be considered to be a modification of cold plasma theory which takes into account the presence of hot electrons, it is not clear that the dispersion relation can be arbitrarily applied to the auroral plasma. In this section we shall address this point in more detail, using the results presented in the previous sections as a basis for the discussion.

      The two principal results of the analysis are, first, that for p / m c / relativistic corrections are significant, and second, if the electrons can be considered as two separate species, then the unstable mode is trapped between the two species gyrofrequencies. We must hence determine when the auroral electron distribution function can be said to satisfy both of these conditions, in which case the ring distribution may be a reasonable approximation for the purposes of investigating wave dispersion.

      We must reiterate that the main goal for the present analysis is to study wave dispersion. Obviously, the growth rate for instability will depend strongly on the detailed form of the distribution function. However, the wave dispersion is controlled more by the collective properties of particle species, as embodied by the integration over all momentum space when calculating the conductivity tensor for a plasma.

      Another point concerning the integration over momentum space is that at a particular momentum, particles with pitch angles near 90° occupy a larger solid angle than particles near 0° or 180°, and so contribute more to the integrals. For this reason we find that the ring and shell dispersion relation for k = 0 are very similar [see Pritchett, 1984b]. The dispersion relation for the shell distribution for finite wave vector is not as tractable as the ring, since the integrals cannot be readily separated into parts when relativistic corrections are included. The ring distribution relation is hence used to extend the analysis to finite wave vector.

      We must now determine whether the ring or shell distributions are applicable to auroral electron distributions. The electron distributions as measured on S3-3 show three basic features, as pointed out by Omidi et al. [1984]. These are a widened loss cone in the upgoing particles, a "hole" at low energies, and a "bump" near 90° pitch angles. We also note that at higher energies the distributions are close to spherical symmetry apart from the loss cone feature. For the purposes of determining wave dispersion the loss cone is probably not significant (again invoking solid angle arguments). We are then left with a spherically symmetric distribution at higher energies, i.e., a shell, with a bump near 90°, i.e., a ring.

      It appears then that both a cylindrically symmetric "ring" distribution and a spherically symmetric "shell" distribution can be applied to the auroral electron distribution. However, we have made one further assumption, that is, that the hot electrons have no thermal spread. While the measured distributions show quite large temperatures, the approximation is still reasonable provided we can characterize the ambient and energetic electrons as two species with different gyrofrequencies due to relativistic effects. Unfortunately, the bulk of the low-energy ambient plasma (if present) is presumably below the energy range of an electron detector such as that flown on the S3-3 spacecraft. Nevertheless, we can determine some limits for the applicability of our analysis.

      A useful distribution for modeling a spherically symmetric distribution with a peak at some finite momentum is the DGH distribution, which can be generalized as

where n is the number density of the particle species, p is the thermal momentum, (v) is the gamma function, and p is the momentum at which f(p) has a maximum. For the purposes of calculating a dispersion relation, p / p is usually set to some integer value.

      From (8) the average momentum squared for the distribution is given by

where <w> is the characteristic energy of the distribution.

      Relativistic effects are important for a particular particle species if

i.e., if

where n is measured in particles per cubic centimeter, <w> is in keV, and f is the nonrelativistic electron gyrofrequency in megahertz.

      We shall use the data published by Mizera and Fennel, [1977] and Croley et al. [1978] to determine whether or not (11) is satisfied for measured distributions. From Croley et al there is a peak in the distribution near v = 1.5 x 10 km/s with f(v) = 21.5 s km, while at v = 3 x 10 km/s f(v) = 0.0464 s km. On substitution into (8) this yields p / p 4, with p / mc 0.05 and n 0.36 cm. For this distribution, <w> 0.88 keV. At the altitude where the distribution was measured, 7300 km, the magnetic field was 0.058 G [Mizera and Fennell, 1977], giving a gyrofrequency of 0.16 MHz. From (11) the density must be 1.1 cm. Relativistic effects are consequently important for this distribution.

      However, if there is a sufficient amount of low-energy ambient plasma, these particles will dominate wave dispersion, and a cold plasma formulation is adequate. Fortunately, we can obtain a lower limit on the relative densities for hot and ambient electrons. If the peak associated with the hot electron distribution is small in comparison to the phase space density of the ambient electrons at the peak momentum, the hot electrons are probably unimportant for wave dispersion. Noting that (8) reduces to a Maxwellian when p = 0, we can say that if

then the hot electrons are significant. The subscript a denotes the ambient Maxwellian plasma. In general we expect p < p , and it can be shown that f / f is then a monotonically increasing function of p / p provided p / p > 1.5. The constraint imposed by (12) is consequently less well satisfied when p = p , in which case, (12) can be rewritten as

      The condition given by (13) tends to zero as p / p , i.e., when the temperature tends to zero. This is to be expected, since at this limit the distribution function is equivalent to a delta function, and the hot electrons are distinct from the ambient electrons. On the other hand, the condition is less easily satisfied as the temperature increases. The condition given by (13) therefore reflects the spread in gyrofrequencies associated with a warmer distribution. For p / p 4, corresponding to the S3-3 distribution, n / n 0.3.

      There is one last constraint which we can apply to the relative densities. If the ambient particles have too low a number density, then the hot electrons dominate, and the electrons are again effectively a single-particle species. To ensure that there are sufficient numbers of ambient electrons, we shall require that the difference in <p> for each species, and hence the difference in the "average" gyrofrequency, be comparable to the total <p>, i.e.,

For p = p , (14) reduces to

Again we find that must be greater than 1.5. p / p must be greater than 1.5. For p / p = 4, n / n 1.67.

      As written, the conditions given by (13) and (15) imply that For p / p 2.5, otherwise the upper limit given by (15) is lower than the limit given by (13). However, we note that the limits are flexible, and this requirement may be relaxed.

      To summarize, we have shown that for the S3-3 distribution published by Croley et al. [1978] the delta function ring distribution is a reasonable approximation when determining the wave dispersion provided 0.3 n / n 1.67. In obtaining this condition we have made two assumptions which should be verified. The first is that, provided the phase space density of 90° pitch angle particles is high, the asymmetry in parallel momentum associated with the loss cone is not significant for wave dispersion. The second is that the dispersion is similar for both a ring and a shell. Both assumptions essentially reduce to a single one, that the near-perpendicular particles dominate the wave dispersion for waves near gyroresonance. The simulations of Pritchett [1984a, b] and Pritchett and Strangeway [1985] indicate that this may be the case. The latter work included an asymmetric electron distribution, in which a single-sided loss cone was included. The growth rates were altered by the use of different distributions, but the wave modes were essentially the same. However, a more thorough analysis would be useful, addressing this issue and also providing better estimates than those presented here on the range of plasma parameters for which the dispersion relation used in the present study is adequate.

7. Conclusions

      In this paper we have presented the results of a linear stability analysis for waves in a weakly relativistic electron plasma. We have been primarily interested in the variation of the wave modes as a function of the plasma system parameters and also in the dependence of wave properties such as the group velocity on the wave vector. We have shown that relativistic effects become important for p / mc > / , and the R-X mode cutoff can then lie below the cold electron gyrofrequency. This enables perpendicularly propagating waves to become unstable.

      We have further shown that when both hot and cold electrons are present, the range of instability is no longer confined to p / mc > / . When the plasma distribution function can be characterized by two plasma components, a new wave mode is present, and this mode lies between the gyrofrequencies of each electron species. We have shown that this wave mode has similar polarization to the cold plasma R-X mode, although it is decoupled from this mode when cold electrons are present. The wave frequency has little variation as a function of the wave vector, resulting in the small group velocities and short convective growth lengths of the order of 1 km.

      We have explored the variation of the group velocity in the context of the generation of auroral kilometric radiation. Using a very simple model for the density cavity present on auroral field lines [Calvert, 1981b], we have found that the ray propagation is such that ray paths tend to direct the waves toward those altitudes where the growth maximizes, since the wave propagates more and more obliquely to the magnetic field until the wave propagates perpendicularly. In addition to maximizing the growth rate as determined in the present analysis, this will reduce damping due to gyroresonance with the thermal electrons. Other effects such as feedback, as discussed by Calvert [1982], may also be enhanced by the bending of ray paths.

      If the new mode is indeed responsible for the generation of AKR, the tendency to propagate perpendicular to the magnetic field and hence possibly carry out mulitple bounces across the auroral cavity may be an essential feature of the generation mechanism. The new mode is usually trapped and can only escape from the generation region through mode conversion. Mode conversion will probably occur near the edge of the auroral arc where the flux of energetic electrons decreases.

      We have briefly discussed mode conversion to the Z mode in this paper, using rather simple estimates for reflection coefficients. Although the results of Mellot et al. [1984] show the presence of L-O mode waves, we have not considered the possibility of mode conversion to the L-O mode. Because of the presence of a stop band between the unstable and the R-X mode, some form of "tunnelling" must be invoked to couple to the freely propagating R-X mode. On the other hand, Pritchett [1984b] has pointed out that for the propagation angles < 80° the R-X mode can be driven unstable by a DGH distribution.

      We have shown that the delta function ring distribution may be a reasonable approximation for the electron distribution in the auroral density cavity for the purposes of studying wave dispersion. Nevertheless, it is apparent that a more sophisticated analysis is required to better address the effects on wave dispersion and ray propagation of temperature and nonuniformity in the plasma.

      Acknowledgements.    The author would like to thank P.L. Pritchett and M. Ashour-Abdalla for many helpful discussions. This work was supported by NASA Solar Terrestrial Theory Program under grant NAGW-78.


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