J. Geophys. Res., 91, 3152 - 3166, 1986.
(Received August 8, 1985; revised November 8, 1985; accepted December 2, 1985.)
Copyright 1986 by the American Geophysical Union.
Paper number 5A8825
The effect of including relativistic corrections in the dispersion relation for a low-density plasma consisting of moderately energetic electrons (energy ~ 2.5 keV) is investigated. Such a low-density plasma is presumed to exist at those altitudes on auroral zone field lines where auroral kilometric radiation is generated. Two different types of dispersion relation are employed for the purpose of studying the wave dispersion. The simpler of these is a "ring" distribution, in which both ambient and hot electrons are assumed to be cold (i.e., have no thermal spread associated with them). Using this dispersion relation, we find that for sufficiently low densities, the most unstable mode is a "trapped" mode that is decoupled from the freely propagating R-X mode found in a cold plasma. Since the dispersion relation neglects the effects of temperature, we also study wave dispersion using a spherically symmetric Dory-Guest-Harris (DGH) distribution. This "shell" distribution shows that provided the peak momentum of the DGH distribution is larger than the thermal spread, the most unstable mode at low wave vectors is still the trapped mode. Our analysis indicates that the trapped mode is usually the most unstable mode and the freely propagating R-X mode is not driven unstable, when the wave dispersion is assumed to be given by a weakly relativistic dispersion relation. We show that relativistic effects may be important for typical auroral electron distributions, and that there is insufficient evidence from particle data to justify the usual assumption that the cold plasma approximation is adequate when describing wave dispersion near the gyrofrequency on auroral zone field lines.
Many mechanisms have been proposed for the generation of auroral kilometric radiation (AKR). Historically, one of the earliest mechanisms was the linear mode conversion of electrostatic waves, originally proposed by Oya  to explain Jovian decametric radiation (DAM). Subsequently, various nonlinear mode conversion schemes were proposed [e.g., Jones, 1977; Roux and Pellat, 1979; Palmadesso et al., 1976; Grabbe et al., 1980]. Recently, Buti and Lakhina  proposed that nonlinear interactions between whistler solitons and upper hybrid waves can generate AKR.
However, the cyclotron maser [Wu and Lee, 1979] has become one of the more favored mechanisms. The cyclotron maser, which is driven by gyroresonance between energetic electrons and high phase velocity electromagnetic waves, has the advantage of transferring energy directly from the particles [Omida and Gurnett, 1982, 1984].
One important feature of the cyclotron maser is the requirement that the gyroresonance condition be relativistically correct, as originally pointed out by Wu and Lee  and subsequently discussed in some detail by several authors [Wu et al., 1982; Omidi and Gurnett, 1982; Melrose et al., 1982; Dusenbery and Lyons, 1982]. Bearing in mind the dependence of net growth on the refractive properties of the medium, Pritchett [1984a, b], Pritchett and Strangeway , and Strangeway  have recently pointed out that relativistic effects are also important for the wave dispersion near the electron gyrofrequency. Typically, relativistic corrections are important p / mc / where p is the characteristic momentum of the electrons, m is the electron rest mass, c is the speed of light, is the electron plasma frequency, and is the electron gyrofrequency.
Strangeway  showed that if the electron distribution can be modelled as two separate species and both species contribute to the wave dispersion, then since the higher-energy particles have their own gyrofrequency due to relativistic effects the most unstable mode is trapped between the two gyrofrequencies and is decoupled from the freely propagating branch of the dispersion relation. In one sense, this may be advantageous, since the wave may be reflected at the edges of the auroral arc. This could enhance the growth in a manner analogous to the feedback mechanism of Calvert . On the other hand, if the unstable mode is indeed on a separate branch of the dispersion relation, some means for mode conversion must be invoked to allow the radiation to escape from the source region.
Before embarking on efforts to calculate mode conversion rates and ray paths in the complicated plasma region found on auroral zone field lines, we should confirm that the previous analysis is indeed relevant to the auroral zone distributions. The condition p / mc / appears to be readily satisfied since Calvert  has shown the presence of a substantial plasma cavity on auroral zone field lines during active times. The ratio / is less than 0.1 for a considerable altitude range along an auroral field line. This value of / gives a lower limit on the particle energy of 2.5 keV, which is about the same as typical field-aligned potential drops.
It is more difficult to determine if the electrons should be treated as two species. We should clarify here the notion of a "two-species" electron distribution. It is usually assumed in instability theory, including the generation of AKR, that the cold ambient plasma dominates the wave dispersion and the more energetic electrons act as a source of free energy for resonant instabilities. However, if the relative numbers of ambient and energetic electrons becomes comparable, it is likely that the hot electrons will also modify the wave dispersion, in addition to driving instability. In this case, two electron species are present in the wave dispersion relation. It seems reasonable to assume that the electrons be assumed to be "two species" if ambient and hot electrons have roughly equal densities, especially if both species are cold (i.e., no thermal spread) and the hot electrons only have a drift with respect to the ambient electrons. However, we must also consider the effects of temperature. Increasing the temperature of a distribution in essence spreads the gyrofrequency over a range of frequencies, due to relativistic effects. If the spread in gyrofrequency becomes too large, then it is not obvious that the electrons can be characterized as "two species."
The present study is principally motivated by the need to determine when weakly relativistic wave dispersion is applicable to auroral zone electron distributions and hence determine if the electrons can be classified as "two species." A major limitation of the work of Strangeway  is the assumption of a cold ring of electrons as a model for the energetic auroral electrons. This form of distribution automatically results in a "two-species" electron plasma. Furthermore, it is not clear that a ring is a reasonable model for the electron distribution. The distribution functions published, for example, by Croley et al.  show considerable structure. As pointed out by Omidi et al. , the measured distribution contains features such as a single-sided loss cone in the upgoing distribution, a "bump" at near 90° pitch angle and a "hole" in the downgoing distribution. While the "bump" may be reminiscent of the cold ring distribution, the additional features suggest that a more realistic model should be employed. Features that are taken into account in the present analysis are the more nearly spherical nature of the hot electron distribution, and the spread of the distribution in velocity space.
We only qualitatively investigate the modifications to wave dispersion associated with the asymmetry of the distributions introduced by the loss cone and "hole." The results of the simulation study of Pritchett and Strangeway  indicate that an asymmetric distribution has an asymmetric dependence on angle for the growth rate. However, the presence of the unstable mode is still due to relativistic corrections to the wave dispersion, and we consider the inclusion of temperature to be more important.
Several other authors have also considered the effect of relativistic corrections to wave dispersion for various types of distribution [Tsai et al., 1981; Wu et al., 1981, 1982; Winglee, 1983; Le Queau et al., 1984a, b]. In addition, much effort has recently been directed to the generation of Z mode radiation by the cyclotron maser [Hewitt et al., 1983; Melrose et al., 1984; Omidi et al., 1984; Dusenbery and Lyons, 1985; Omidi and Wu, 1985]. We shall therefore discuss in some detail the various unstable modes found for the simplified ring distribution function previously employed by Chu and Hirshfield  and Strangeway , for the purposes of relating the present work with other analyses, and as an introduction to the more involved warm electron dispersion relation.
The structure of the paper is as follows. In section 2 we shall present solutions of a simple ring distribution, describing the various modes found. The effects of finite temperature will be studied in sections 3 and 4. We will find from the analysis of a warm electron dispersion relation that provided the peak in the distribution function occurs at a higher momentum than the thermal spread of the distribution, the most unstable mode at low wave vector is decoupled from the freely propagating mode. Section 3 will describe the warm electron dispersion relation for k = 0, where k is the wave vector, and section 4 will present results for an approximate dispersion relation for k 0, perpendicular being defined with respect to the ambient magnetic field. In section 5 we shall discuss the form of the distribution function used in the warm electron dispersion relation. We shall show that for certain plasma parameters, the model distribution may be a reasonable approximation to measured distributions. In particular, we will stress that a large peak in the distribution at low energies does not necessarily mean that a cold plasma approximation is adequate. We will summarize the results of the present study in the last section.
In this section we shall obtain solutions of a simple dispersion relation for a plasma with a cold ambient plasma and an energetic ring of particles at constant perpendicular momentum. At present we shall not include any temperature in the particle distribution functions as we are primarily interested in elucidating some of the more general characteristics of the dispersion relation for a weakly relativistic plasma. The distribution function employed is given by
where p and p are the parallel and perpendicular momenta defined with respect to the ambient magnetic field, and n and n are the ambient (cold) and hot (ring) electron number densities, respectively.
The dispersion relation for such a distribution has been given by Pritchett [1984b] and Strangeway . When deriving the dispersion relation, we have retained the relativistic terms. Several authors [Wu and Lee, 1979; Wu et al., 1982; Omidi and Gurnett, 1982; Melrose et al., 1982; Dusenbery and Lyons, 1982] have emphasized the importance of retaining the relativistic terms in the condition for gyroresonance. However, as pointed out by Chu and Hirshfield , for a sufficiently low density plasma where the plasma frequency is much smaller than the electron gyrofrequency, relativistic effects are also important for the wave dispersion. The changes introduced to the wave dispersion and their implications for the generation of auroral kilometric radiation have been discussed by Pritchett [1984a,b] and Strangeway .
Chu and Hirshfield  and also Winglee  (using a more sophisticated Dory-Guest-Harris [Dory et al., 1965] distribution) have studied wave dispersion for parallel propagating waves, and Pritchett [1984a, b] has presented dispersion curves for perpendicularly propagating waves. However, it is useful to consider arbitrary propagation directions in order to obtain a more complete understanding of the wave dispersion in a weakly relativistic plasma. Strangeway  has presented some results of such an analysis, but in his work he was principally interested in studying the group velocity variation as a function of propagation direction for a specific mode. In this section we shall extend the work of Strangeway to emphasize the relationship between the various modes present in the plasma.
Fig. 1 Wave dispersion surfaces for the ring dispersion. Four models are present when both a hot and cold electron component are included.
Solutions of the dispersion relation for a ring distribution are presented in Figure 1. Since we are primarily interested in instabilities near the electron gyrofrequency, we have only considered modes near this frequency. There are four dispersion surfaces shown in Figure 1. These surfaces have been plotted as a function of parallel and perpendicular wave vector normalized to a characteristic distance c / where c is the speed of light and is the cold electron gyrofrequency ( = eB / mc, where e is the magnitude of the electron charge, B is the ambient magnetic field strength and m is the electron mass). When obtaining the solutions shown in the figure, we have assumed that the hot electrons are 75% of the total electron density, the ring momentum is 0.1mc (corresponding to an energy of ~2.5 keV), and the ratio of plasma frequency to gyrofrequency squared / ) is 0.005. The plasma frequency is the total electron plasma frequency, = 4ne / m, where nt, is the total electron number density. A plasma for which / < p / mc has been described as an underdense plasma by Strangeway , for reasons we will outline below. The frequency range used in plotting all four dispersion surfaces is the same, 0.985 < / < 1.005, where / is the real part of the wave frequency ( = + i , where is the growth rate).
The simplest surface to describe is the surface plotted in the upper left of the figure. This corresponds to the cold plasma R-X mode, somewhat modified by relativistic effects. The surface has been truncated at / = 1.005, although the mode continues to higher frequencies, where the surface asymptotically approaches the light cone. The R-X mode cutoff in this case is above the cold electron gyrofrequency.
Moving to lower frequencies, that is clockwise around the figure from the upper left-hand corner, the next mode encountered shows a more complicated structure. At high wave vectors, the frequency lies just above the cold electron gyrofrequency, and we shall refer to this mode as the "cold intermediate" mode for kc / > 1, where there is sudden change in the dispersion surface. Below this value of wave vector (i.e., waves with phase velocities greater than the speed of light) the mode is unstable. The growth rates for this mode and the other unstable modes present are given in Figure 2, and we shall discuss the growth rates in more detail when describing Figure 2. The unstable mode in the upper right of Figure 1 is the mode found in the simulations of Pritchett [1984a, b] and discussed at some length by Strangeway . Since the mode is decoupled from the freely propagating R-X mode, we shall refer to this unstable mode as the "trapped" mode. It will be noted that the group velocity for the unstable mode, given by the slope of the dispersion surface , is small. This is generally the case for this mode. For most values of wave vector, except near the transition, the group velocity is positive. In an overdense plasma where / > p / mc, the parallel group velocity is usually anomalous, the dispersion surface having negative slope, as pointed out by Strangeway .
Fig. 2. Concatenated dispersion surfaces with their corresponding growth rates. The four dispersion surfaces shown in Figure 1 have been merged into a single display. We have truncated the surface identified with the "cold intermediate" mode for reasons of clarity. An unstable mode is present where two stable modes coalesce, and the corresponding growth rates are shown by the contour plot.
The next surface, which is plotted in the lower right-hand corner of Figure 1, shall be referred to as a "hot intermediate" mode. The frequency of this mode is near the hot electron gyrofrequency. Upon comparison with the dispersion surface in the upper right of the figure, it can be seen that the dispersion surface is the same in both plots for small kc / i.e., the "trapped" mode. Since we are using delta function particle distributions, complex solutions of the dispersion relation come in complex conjugate pairs. Unstable solutions are found where two different dispersion surfaces coalesce.
The last dispersion surface plotted in Figure 1 corresponds to the Z mode and is shown in the lower left-hand corner of the figure. The dispersion surface has been truncated at the low-frequency limit of the plot. As the wave frequency approaches the ring electron gyrofrequency (= 0.995 ), the Z mode leaves the light cone. At larger wave vectors two unstable branches are found, one predominantly propagating parallel to the magnetic field, the other propagating perpendicular to it. It is interesting to note that the Z mode is not unstable for all propagation angles. The mode at large k corresponds to the Weibel-type instability described by Chu and Hirshfield  and Winglee , while the mode at large k corresponds to the unstable Z mode discussed by Pritchett [1984b].
Having described the different dispersion surfaces in some detail, we now merge them into a single plot as shown in Figure 2. The parts of the dispersion surfaces corresponding to the unstable modes are the regions for which different dispersion surfaces are colocated. It will be noted that for reasons of clarity we have not plotted all of the dispersion surface for the "cold intermediate" mode. At the right-hand side of the figure we have plotted contours of growth rate for the unstable modes. As mentioned before, there are three unstable branches. The Weibel-type instability appears to have the larger growth rate, although Landau damping by thermal electrons may reduce this growth rate. For perpendicular propagation, the largest growth rate occurs on the trapped branch. We have not shown it here, but even though the growth rate is increasing for kc / = 2, the maximum for the Z mode branch is less than that for the trapped branch.
The growth rates for different branches of the dispersion relation are sensitive to the plasma parameters, and it is not certain that the fastest growing mode for oblique propagation will always be the low wave vector trapped mode. On neglecting the effects of hot electrons on the wave dispersion, Omidi and Wu  have stated that the transition from larger growth rate R-X mode waves to larger growth rate Z mode waves occurs at / = 0.1. We do not present the results here, but we have found the transition from the trapped mode to the perpendicular Z mode also depends on the relative density of hot electrons and the energy of the hot electrons as well as the plasma frequency-gyrofrequency ratio. For example, when n / n = 0.75, the transition occurs at / pmc. Since we have neglected the effects of temperature, the calculated growth rates from the ring distribution are probably in error, and we shall not give a more detailed investigation of the growth rates for the different modes in the present study. Nevertheless, it appears that the inclusion of the relativistic corrections to the wave dispersion allows other modes to be more unstable than the Z mode over a larger range of / .
The inclusion of temperature in the dispersion relation will almost certainly modify the growth rates of the various instabilities. In addition, temperature may also modify the dispersive properties of the plasma. Specifically, if the temperature of both energetic and ambient electrons is high enough, it is not clear that the plasma could be modeled as two separate species. We shall therefore investigate the effects of temperature in the next section.
Using simple arguments, Strangeway  determined that for distributions like those observed on the polar-orbiting S3-3 spacecraft, the ring distribution given by (1) may be adequate provided there are roughly equal densities of energetic and ambient electrons. Since the discussion was somewhat qualitative, we shall study the effects of temperature in more detail here. The distribution function we shall employ is the spherically symmetric DGH distribution.
where is the Lorentz factor equal to (1 + p / m2 c2 ), K() is the second-order MacDonald function, and is a parameter specifying the thermal spread of the distribution. As , (2) reduces to
where (v) is the gamma function. We therefore identify with with an inverse temperature, = 2mc / p, where p is a thermal momentum, the exponential term in (3) being equivalent to exp (-p / p).
When j 0, the DGH distribution function has a maximum at
As , the peak occurs at p jp. In the previous sections we used p / mc = 0.1 as a representative ring momentum. For the peak to occur . For the peak to occur at the same momentum in (3) we require / j 200.
When j = 0 the distribution function given by (2) reduces to the relativistic Maxwellian. Shkarofsky  has derived the conductivity tensor for electromagnetic waves for the Maxwellian . If we denote the conductivity tensor for f(p) as , then
since the derivation of the conductivity tensor required integration over momentum space and the momentum only enters through the Lorentz factor in (2). We can therefore use the results of Shkarofsky to evaluate the conductivity tensor for the distribution given by (2).
One advantage then of using the distribution function (2) is the ability to relate the conductivity tensor (5) to the previous work of Shkarofsky. A second advantage concerns the application to energetic electron distributions as observed on polar-orbiting spacecraft such as S3-3 and DE-1. In general the observed distributions show features reminiscent of both the ring distribution (1) and the spherically symmetric "shell" distribution (2). The electron data often show a peak in the phase space density at 90° pitch angles with the peak momentum in the keV range, similar to a ring distribution. On the other hand, the distributions are much more shell-like at higher momenta. Furthermore, theoretically we expect the combined effect of an accelerating field-aligned potential drop and the magnetic mirror force to produce an oblate spheroidal boundary in momentum space [Chiu and Schulz, 1978]. At low altitudes, the boundary is almost spherical, and so a shell-like distribution may be a reasonable approximation to the theoretically predicted distribution function, apart from the presence of the loss cone. It has been shown [Croley et al., 1978] that the theoretical model of Chiu and Schulz can be related to the observed distribution, bearing in mind the smoothing inherent in the observations due to both plasma instabilities and instrumental effects.
Strangeway  has argued that the dispersion properties of both ring and shell distributions are likely to be similar. However, the ring distribution is not easily related to the observations. For this reason, a shell-like distribution function is preferable. Unfortunately, including relativistic terms results in a rather unwieldy form of the conductivity tensor for arbitrary wave vectors, even for the Maxwellian [Shkarofsky, 1966]. Hence, the ring distribution is of use for studying oblique propagation. On the other hand, the shell distribution relation is simplified considerably by assuming k = 0. Since we would like to compare both ring and shell distributions, we shall therefore assume that k = 0, which allows us to identify the wave modes and explore their dependence on various plasma parameters. In section 4 we shall address the dependence on wave vector and justify our identification of the wave modes.
From R. J. Strangeway (unpublished manuscript, 1986) where the derivation is described in detail, we find that the dispersion relation for R-X polarized waves is given by
when k = 0, where indicates summation of electron species, is the plasma frequency for a particular species, is the inverse temperature for that species, j is the power of the shell distribution, = ( - ) / , and Z is the plasma dispersion function [Fried and Conte, 1961].
As discussed by R. J. Strangeway (unpublished manuscript, 1986), the introduction of the square root in the argument for the Z function in (6) results in a branch cut from = 0 to infinity. If the branch cut is taken to = +, then the analytic continuation of the function is such that the physically allowable solutions lie on one sheet, which we refer to as the lower sheet. For the lower sheet Im [( )] > 0. The value of D() on the upper sheet can be shown to be simply related to the value on the lower sheet through the analytic continuation of the Z function; hence
except on the branch cut, where the subscripts u and l refer to the upper and lower sheets, respectively, and angle brackets indicate complex conjugation. The relationship (7) indicates that solutions to (6) come in complex conjugate pairs, except when Re () > 0 and Im () = 0 . The restriction of solutions to the lower sheet ensures that a Maxwellian is stable.
Solutions of (6) are plotted in Figure 3 assuming both hot and ambient population are present. The hot population can have different values of j, while the ambient population has j = 0, i.e., Maxwellian. Frequency is plotted in the left-hand column of Figure 3, with the corresponding growth rate in the right-hand column for different values of j. As indicated by the heading to the figure, we fix / j = 200, i.e., p / mc 0.1, the temperatures of the ambient and hot electron distributions are the same, as are the number densities. The numbers to the right of each panel give the j parameter for the hot electron population, with j = corresponding to a delta function. When j = 0, both populations are Maxwellian and we use ph = 200 for the hot electron population. For reference with this and subsequent figures, if we define = / j, then = j, j 0: = , j = 0.
Fig. 3. Solutions of the k = 0 shell dispersion relation (6) for variable /. The left-handed column shows frequency while the right column shows the corresponding growth rates. The integers inside and to the right of each panel give the j parameter of the DGH distribution. For j >1, / j = 200. When j = 0, both distributions are Maxwellian and we set = 200. For the prupose of identifying modes, the solid line corresponds to the "trapped" mode, the dashed line corresponds to the R-X mode, and the dotted line corresponds to a Bernstein mode.
For the delta function we find that there are two modes. The higher- frequency mode, as indicated by the dashed line, increases in frequency as / is increased. The frequency of this mode is greater than the gyrofrequency. The corresponding growth rate, shown in the right-hand column, is zero, and the mode is marginally stable. We therefore identify this mode with the R-X mode in Figure 1. The second mode, shown by the solid line, lies below the gyrofrequency and has a positive growth rate. We identify this mode with the unstable trapped mode in Figure 1. In this and subsequent figures, we use dashed lines to represent the R-X mode, while solid lines are used to indicate the "trapped" mode. For the high j values this identification is unambiguous, the trapped mode is the lower-frequency mode, and this mode can be unstable. However, as we shall show later, the identification of the modes becomes more difficult for low j values.
A third mode is also plotted in Figure 3. For the purposes of identifying the mode, we consider the solutions shown in the bottom panels for j = 0. These solutions correspond to a single Maxwellian. At low / only one mode (the R-X mode) is shown, and that mode is damped. As / is increased, the mode becomes marginally stable and a second mode is also found. This has been called a "Bernstein" mode by Pritchett [1984b], and we indicate this mode by dotted lines. The Bernstein mode shown in Figure 3 is not present when j = . This is because there is no branch cut for the delta function dispersion relation. When j , both the R-X mode and the Bernstein mode lie on the branch cut described earlier, and therefore are only marginally stable for > . On the lower sheet the R-X mode is found as 0 from above, while the Bernstein mode is found for 0 from below. As the frequency decreases so that , both modes merge, and the solutions are found for complex conjugate frequencies, one on each sheet. We only show the solution for the lower sheet and that solution is damped for j = 0.
When j is large, i.e., low-temperature distributions, the solutions are very similar to thej = case, apart from the additional Bernstein mode. However, including temperature restricts the range in instability for the trapped mode. At low / the mode is damped as pointed out by Winglee  for a loss cone type distribution. For lower j values, the trapped mode is damped for all values of / .
One last point to be made from Figure 3 concerns the form of the solutions for j = 1. The R-X mode looks very similar. but the trapped mode behavior is rather different than that found for j > 1. Unlike the other cases, the trapped mode couples to the Bernstein mode at high / . At low / . the mode we have called the trapped mode lies on the branch cut with > and 0 from below. It appears then that the labelling of the modes is not so obvious for j = 1.
Before investigating the dependence on other plasma parameters we shall summarize the properties of the wave modes. The highest-frequency mode has been identified with the R-X mode, with the implication that this mode approaches the light cone as wave vector increases. The mode is either stable or damped and when marginally stable ( > ), the R-X mode lies on the branch cut with 0 from above. This mode is present even when j = . As frequency decreases, the R-X mode couples to a second mode which lies on the branch cut with 0 from below. We have identified this mode with a Bernstein mode, which implies that the frequency of this mode is nearly constant for all wave vectors. The Bernstein mode and the R-X mode merge and are damped for < . The last mode found is the "trapped" mode. This mode usually has the lowest frequency, and can be unstable. By implication, this mode cannot propagate for finite wave vector. In Figure 3, the trapped mode is separate from both the R-X and Bernstein modes, except for j = 1. We shall verify the inferences made concerning the dispersion properties of the waves in section 4.
In Figure 4 we fix / and vary the number density ratio. We have again assumed equal temperature electron populations. At low hot electron densities, the three modes are very distinct for j 2. The R-X mode and Bernstein are both marginally stable, merging to form a damped mode at sufficiently high n / n (the Bernstein mode is again not present for j = ). The trapped mode is damped at low n / n and becomes unstable at high n / n. As might be expected, the required density ratio for instability decreases as j increases.
Fig. 4. Solutions of the k= 0 shell dispersion relation (6) for variable n / n. Similar to Figure 3.
For j = 1 the trapped mode behavior shown in Figure 4 is again very different. For the particular choice of parameters, the trapped mode is always marginally stable, lying on the branch cut. On the other hand, the R-X mode merges with the Bernstein mode, as found for j > 1, but the R-X mode can also be unstable for sufficiently high n / n.
The parameter / is varied in Figure 5, the other parameters being held fixed. If / is large, then the background plasma has a low temperature in comparison to the hot electrons. When this is the case, the three modes are very distinct for 1 j < , the high-frequency R-X mode and the Bernstein mode are both marginally stable, while the low- frequency trapped mode is unstable provided the energetic electrons are sufficiently peaked. As the ambient electron temperature is increased (i.e., / decreases), the three modes approach one another. At sufficiently low energetic electron temperatures, the low-frequency mode is again unstable, while the R-X mode is damped when j 2. For j = 1, the R-X mode also becomes damped. However, the trapped mode shows a much more complicated structure. Near / = 1, the trapped mode changes from being strongly damped ( / < -0.005, below the lower limit of the plots) to marginally stable. At the same time the frequency varies considerably, reaching a maximum as 0 for the trapped mode. When the frequency of the trapped mode maximizes, both the Bernstein mode and the R-X mode merge and form a single damped mode.
Fig. 5. Solutions of the k=0 shell dispersion relation (6) for variable / . Similar to Figure 3.
The results of the analysis of the warm dispersion relation for k = 0 are very similar to those found for the ring, provided j 2. While the growth rates are different, and sometimes even negative, we can still identify the modes with those found for the ring dispersion relation. From our knowledge of the wave dispersion for the ring we have identified the high-frequency mode with the R-X mode, and the low-frequency mode with the trapped mode. The R-X mode is usually damped or marginally stable, while the trapped mode can be unstable. However, the identification of the modes is not easy for j = 1, and our labelling of the modes for j = 1 in the preceding section may be suspect.
In this section we shall demonstrate that the presumed properties of the wave modes are at least approximately correct for j > 1, and further clarify the nature of the modes for j = 1. As shown by Shkarofsky , the conductivity tensor is very complicated if k 0 even for the Maxwellian. However, we can set k = 0 for the purposes of discussing wave dispersion, since it has been shown that nearly perpendicularly propagating waves are most unstable [Pritchett, 1984b; Strangeway, 1985; Pritchett and Strangeway , 1985]. This simplifies the final form of the conductivity tensor and also separates O mode and X mode waves in the dispersion relation. However, the right and left circularly polarized wave fields are coupled together unless k = 0.
On assuming that the distribution function is given by the relativistic Maxwellian, the final form of the conductivity tensor will be similar to that given by Shkarofsky  in his equation (2). Some differences may be introduced depending on the approximations made, but the form of the tensor T as defined by Shkarofsky will be the same. The tensor T contains an infinite series of Bessel functions and the argument in the Bessel functions is = ( kc) / [( 1 - i)] where is an integration variable varying from 0 to . We can use the small argument expansion for the Bessel functions provided kc / 1. Obviously, this condition is satisfied when k = 0, in which case the X mode components of the conductivity tensor reduce to
where () is the conductivity tensor for left- and right-hand polarized waves and is obtained from
where f(p) is the phase space density for the relativistic Maxwellian. We use (5) to obtain the conductivity tensor for higher j values, ().
Rather than let k 0, we shall assume . In this case, we can rewrite the dispersion relation (6), including finite k, as
D () is the same as D() as defined in (6), and corresponds to the right-hand polarized wave. D () can also be obtained from (6) provided is redefined as = ( + ) / . Since we are seeking solutions of (10), with and hence 2, we should use a large form of (6). D() as defined in (6) is subject to numerical errors for 1. While large argument expansions of Z (i()) exist, one finds on substituting these into (6) that the first (l + j) powers of exactly cancel, and round-off errors may be significant. We therefore use the simplest large form of (6), which is the dispersion relation for a delta function shell. From Pritchett [1984b], we find that
where = 1 + j/ . We note that for typical parameters, with and j/ 0.005, the term containing ( - 1) 10, and D() nearly equals the value obtained for a purely cold plasma.
For the purposes of exploring the dependence on wave vector the smallest value of used will be = 200. In addition, we shall restrict ourselves to kc / 1, therefore the highest value of kc / 5 x 10 1 We are hence justified in using the dispersion relation (10) to determine wave dispersion for finite k.
Solutions of the approximate dispersion relation are shown in Figure 6 . We have chosen a high hot electron density (n / n = 0.9) and a low ambient electron temperature ( / = 10) with a peak momentum at p/mc 0.1 and / = 10, We have used the same convention as that used in Figures 3, 4, and 5, where the dashed line corresponds to the R-X mode, the dotted line identifies the Bernstein mode, and the solid line represents the trapped mode. In Figure 6, the identification of the modes appears to be correct. The R-X mode phase velocity approaches the speed of light, while the Bernstein mode and trapped mode have nearly constant frequency.
Fig. 6. Solutions of the approximate shell dispersion relation (10) for finite k and low ambient electron temperature. Similar in format to Figure 3.
For j 2, the maximum growth rate of the trapped mode is found for k = 0, while the maximum for j = 1 occurs near kc / 1. Pritchett [1984b] also reported that the growth rate maximizes near kc / 1 for the loss cone DGH plus a cold background, although he found that this occurred for all j values. We do not show it here, but we have determined that if we decrease the hot number density to n / n = 0.5, the j = 1 mode is always damped, while the j = 2 mode has maximum growth near kc / = 1. Allowing the ambient plasma to have a temperature therefore introduces some additional complexity to growth rate dependence on wave vector.
Fig. 7. Solutions of the approximate shell dispersion relation (10) for equal temperature distributions. Similar in format to Figure 3. Note the expanded frequency scale.
For low ambient electron temperature, the dispersive properties are very similar for all j values. Figure 7 shows that this is not necessarily the case for other plasma parameters. In Figure 7, we have assumed that both ambient and hot electrons have the same temperature. Since the dispersion curves are more complicated, we have expanded the frequency scale. For j 3, the dispersion curves are similar to those shown in Figure 6. However, the dispersion curves are somewhat different for j = 2 and j = 1.
When j = 2, the trapped mode approaches the R-X mode and at first sight it appears that the two modes are coupled. While it is not readily seen in the figure, the modes are still decoupled because the transition to marginal stability for the R-X mode occurs at a slightly lower value of kc / than that for which the trapped mode becomes damped. Nevertheless, the closeness of the dispersion surfaces suggests that the modes may couple as a result of refraction and mode conversion. If the mode conversion can take place, then we might expect large radiation amplitudes, since the mode is unstable for a large range in k before coupling to the free space branch near kc / 0.6. We do not encounter the problem of the sensitivity of the gyroresonant interaction to wave vector magnitude and direction in a cold plasma. On the other hand, it appears to be more typical for the trapped mode to be well separated from the R-X mode.
When j = 1, the R-X mode and the unstable mode are coupled in Figure 7. The modes are coupled since the Bernstein mode is always coupled to one of the other two modes. For this reason, no Bernstein mode is shown for j = 1. At low k , the Bernstein mode couples with the R-X mode, while the trapped mode lies on the branch cut with 0 from below. At high k , the Bernstein mode couples with the trapped mode.
The analysis with finite k shows that for j 2 the trapped mode is the unstable mode, and while the trapped mode can approach the R-X mode, it never couples directly with the free space branch. For j = l, the trapped mode may be unstable, but for some plasma parameters, the unstable mode can couple directly to the free space branch. It appears then that the j = l DGH can be a special case. We will discuss this point further in the next section.
In the introduction to this paper we discussed the importance of relativistic effects on wave dispersion in regions of low plasma density, such as the auroral plasma cavity [Calvert, 1981]. We have emphasized the effect of the hot electrons on the wave dispersion, using both a ring distribution and a shell distribution. The latter showed that the unstable mode for low wave vectors was trapped even when temperature is included, provided that the thermal velocity was lower than the momentum for which the energetic electrons have a peak in their phase space density, i.e., j > 1. In this section we shall address the applicability of the analysis presented in preceding sections to the auroral zone electron distributions.
There are three basic assumptions to be questioned. First, is it reasonable to assume that auroral electron distributions fall into the parameter ranges considered here? In other words, can we justify the assumption that 0.1 n / n 10, and that 0.1 / 10, as used in Figures 3, 4, 5, 6, and 7? The range in / is adequate, because of the presence of the auroral cavity [Calvert, 1981]. Second, what effect does the asymmetry of the observed~distributions have on our analysis? Lastly, how important is the assumption that k = 0?
To determine if the range in plasma parameters is reasonable, we show four theoretical phase space density plots in Figure 8. Strangeway  has considered the data published by Croley et al.  and shown that j 4 may be a reasonable value for a DGH fit to the energetic auroral electrons. The upper two plots in Figure 8 show the phase space density of two j = 4 distributions which are just unstable, one with low / , the other with high / . The corresponding solutions of the shell dispersion relation are shown in Figure 5 , and also given as headers to the phase space density plots. The lower two plots correspond to two weakly unstable modes for different j values shown in Figure 4, one with j = 1, the other with j = 5.
Fig. 8. Representative distribution functions for which relativistic modifications are important. The plasma parameters together with the corresponding solutions of the shell dispersion relation for k = 0 are given as headers to each phase space density plot. The dashed lines give the phase density of each electron species, with the sum being given by the solid line. When obtaining solutions of the dispersion relation, we have assumed that / = 0.01, and as can be seen from the figure / j = 200 (p/mc = 0.1).
Figure 8 shows that there can be considerable variability in the shape of the distribution function, but the mode with frequency below the gyrofrequency is unstable. As discussed before, this mode will be trapped for all save the j = 1 distribution function. One significant difference between the j = 1 distribution function and the other three is the presence of a minimum at p = 0 for the former while the latter have a maximum at p = 0. All the j > 1 distributions have double maxima.
In passing, we note that the distinction between j = 1 and j > 1 may explain the difference in the wave dispersion. It can be shown from (3) that when both ambient and hot electrons are present there will always be a maximum in the phase space density at p = 0 for j > 1. If there is a second maximum at higher momentum, then we might expect to be able to characterize the electrons as two electron species with their own gyrotrequencies. On the other hand, if j = 1, then there can be a minimum at p = 0, and the electrons may be considered to be a single electron species. While there need not necessarily be a minimum at p = 0 for j = 1, that there can be a minimum suggests that j = 1 is a special case, and the wave dispersion can be different than for j > 1.
Another important point concerning the shell distribution is well demonstrated by the phase space density plot shown at upper right in Figure 8 . For this distribution, both ambient and hot electrons have equal number densities, even though the maximum at p = 0 is nearly two orders of magnitude larger than the maximum near p = 0.1mc. This is due to the spherical symmetry of the distribution. On integrating over momentum space, the phase space density must be multiplied by p , weighting the integral to higher moments. This suggests that the presence of a large peak at low momenta need not preclude relativistic effects modifying wave dispersion.
We can attempt to compare the model distribution function with the observed distributions. Unfortunately, the data published by Croley et al.  only measure phase space density down to v 10 km/s, i.e., p/mc 0.033. More recently, Menietti and Burch  have published electron data from the DE-1 spacecraft. Their Figure 3 shows electron phase space density for an altitude of 13,200 km at an invariant latitude of 67.9°. There is strong evidence in the measured distribution for acceleration by a parallel electric field, and this distribution might be a likely source of AKR. The measurements of Menietti and Burch go to much lower energy 1 eV, i.e., p/mc 0.005. Their data indicate that the phase space density may indeed be at least two orders of magnitude larger than the peak at larger momenta. However, Menietti and Burch point out that some of the intense low-energy fluxes are due to photoelectrons. It appears then that distribution functions such as those shown on the right of Figure 8 may reasonably represent the observed distributions, at least in terms of the corresponding densities and thermal widths.
We now come to the second issue, in that the distributions shown in Figure 8 are assumed to be spherically symmetric in phase space. One of the more striking features of the observed distributions is the loss cone in the reflected auroral electrons. In their original exposition of the cyclotron maser instability as a source for AKR, Wu and Lee  used the loss cone for driving the instability. Pritchett and Strangeway  have shown that asymmetric distributions containing a single-sided loss cone plus a hole at low energies are most unstable to waves propagating with angles of 90°-95° (typically) with respect to the magnetic field, when the loss cone is only present in the distribution for pitch angles > 90°. The shift from purely perpendicular waves is due to the asymmetry of the distribution.
In terms of the growth rate, the asymmetry is therefore important. However, it is not clear that an asymmetric distribution will have significantly different wave dispersion than a symmetric distribution. For the purpose of discussion, we shall assume that the phase space density is given by
This distribution is a function of p only for all pitch angles less than - . It can be shown that the conductivity tensor for such a distribution contains terms such as
where J is the Bessel function of the first kind of order n, with argument , = k p / m , and the summation is carried out over all n. The magnetic field defines the parallel or z direction and k defines the x direction. We have neglected the terms introduced by the discontinuity in f(p) at = - . In terms of cold plasma dispersion, the parallel electric field corresponds to an O mode wave, while the perpendicular fields give X mode waves. If and vanish in the conductivity tensor, then the X mode and O mode dispersion relations are separable. This occurs if k = 0 end f(p) is a symmetric function in p ( is of a form similar to with nJ / being replaced by -iJJ' ). To determine the importance of asymmetry , we shall assume k = 0 and 1, in which case the n = 1 term is dominant in (13a)and (13b) and
We can therefore estimate that
when p is the characteristic momentum of the distribution. For the present study, kp / m 0.1. More importantly, (15) depends strongly on . For example, if = 35°, then 0.02kp / m.
The asymmetry of the observed distribution is therefore not likely to significantly alter the dispersive properties of the plasma near 90°, since the X mode remains decoupled from the O mode. However, the asymmetry may result in moving the region of maximum growth away from 90°, and we must determine the range in k for which the shell dispersion relation is still valid.
We must therefore address our last assumption, k = 0. From a mathematical point of view, this was a useful assumption , since the conductivity tensor (5) could be calculated analytically. However, the inhomogeneity of the auroral field line suggests that even if the most unstable mode is found for k = 0, refractive effects will result in k 0. In addition, we have already noted that an asymmetric distribution will probably have k 0 for maximum growth.
One way of estimating the range of validity for assuming k = 0 is by inspection of the resonance condition given by the denominator in (13a) and (13b), - k p / m - n. The assumption k = 0 ensures that the correction due to the Lorentz factor is much larger than the Doppler shift. This will still be the case provided ck / < p / 2mc where p is some characteristic momentum, for example, a thermal momentum. However, as we discuss below, the constraint on k is more likely to be ck / p / 2mc where p is the peak momentum of the DGH distribution.
The resonance condition describes an ellipse in momentum space, and as pointed out by Omidi et al. , the ellipse encloses the origin provided < n . The equation for the resonance ellipse can be written as
Because we have used momentum rather than velocity to determine the resonance condition, we find that the major axis of the ellipse lies on the p axis. For the present analysis where , and ck , the center of the resonance ellipse is given by p / mc ck / . Since the resonance ellipse encloses the origin for the trapped mode (for which < ), a major part of the resonance ellipse lies in regions of momentum space between the peaks of the distribution function at p = 0 and p = p if we restrict the center of the resonance ellipse to p p / 2, i.e., ck / p / 2mc.
Obviously, the condition given above does not ensure the presence of instability, but is only indicative of the range in k for which instability is likely. Nevertheless, we note that for ck / 1, and p / mc 0.1, we might expect instability provided k / k 0.05, i.e., 87° 93°. The range in propagation angles is consistent with the results of Pritchett and Strangeway , bearing in mind the asymmetry of the distribution used in the simulation.
While the growth rate is likely to be sensitive to the size of k , the dispersive properties of the plasma are less dependent on k , as we found for the effect of asymmetry. To show this, we consider the conductivity tensor for the relativistic Maxwellian, as given by Shkarofsky . Employing similar arguments to those used when discussing the effects of asymmetry, we find that
is an integration variable, varying from 0 . If we assume ck and that ck / then R and
For the spherical shell distribution, we can replace by mc / p = 100 when estimating the coupling between X and O mode waves due to modifications of the conductivity tensor. In addition, ck / 1, hence / < 0.01k / k.
In summary, we have shown that the DGH distribution is useful in determining the nature of the wave dispersion in a plasma containing both hot and cool electrons. While the growth rate will depend on such features as the asymmetry of the distribution and the value of k, we have found that the dispersive properties of the plasma are not radically altered for propagation within a few degrees of perpendicular. We have also estimated that the dispersive properties are essentially unchanged even if the distribution has a 35° loss cone. Lastly, we have argued that the published auroral electron distribution may be consistent with distributions such as those shown on the right of Figure 8.
As a last remark on the applicability of the present analysis to auroral electron distributions, we note that the presence of a "trapped" mode appears to be a general consequence of any "two-species" electron distribution. For example, Winglee  found such a mode was present using a "loss cone" type of distribution and a cold ambient plasma. The possible existence of the trapped mode may be important for auroral electron distributions since this mode occurs for < and can therefore be driven unstable by features in both the upgoing and downgoing distribution, similar to the Z mode as discussed by Omidi et al. .
We have employed two different plasma dispersion relations in the present study. The first was the simple ring distribution. This distribution has the advantage of allowing us to explore plasma parameter space with relative ease. In addition, we can investigate the dependence on wave vector of the normal modes of the plasma. We have found using the ring distribution that there are three modes with frequencies close to the electron gyrofrequency which can be unstable in a weakly relativistic electron plasma. At very low plasma frequencies, typically / p / mc, the most unstable perpendicular mode is the high phase velocity "trapped" mode. The mode is trapped since the wave frequency lies between the cold electron and hot electron gyrofrequencies and the mode is decoupled from the freely propagating R-X branch when a cold background plasma is present. Similarly to Pritchett [1984b] , we find that the growth rate maximizes for this mode near kc / . Since we have not included temperature in the ring distribution, we do not obtain the damping at low wave vectors reported by Pritchett.
Using a cold plasma approximation for the dispersion relation , Omidi and Wu  have stated that the transition from unstable R-X mode waves to unstable Z mode waves occurs at / = 0.1. In their study, Omidi and Wu used a measured distribution function to obtain the growth rates. However, since we include the effects of the hot electrons on the wave dispersion, we find that the transition from larger growth rate Z mode depends on the ring momentum also.
The ring distribution function is an extremely simplified distribution function. The distribution almost certainly overestimates growth rates, especially for modes with some finite k. In addition, it is not clear that the wave dispersion is necessarily modified by the presence of hot electrons in the measured distributions, where thermal velocities can be large. We have therefore studied the dispersion using a shell distribution which is a spherically symmetric Dory-Guest-Harris distribution . The form of the DGH distribution was chosen so as to enable us to use the analysis of Shkarofsky  as a basis for the present study. Importantly, we found that provided j > 1, i.e., p > p, where p is the momentum for which the DGH distribution has a peak, the most unstable mode lies on a "trapped" branch of the dispersion relation.
That the unstable mode for j > 1 is separate from the R-X mode suggests that for most cases, the dispersion analysis using the ring distribution may in fact be adequate. However, the results of the warm "shell" distribution indicate that the growth rates from the ring can be inaccurate. The effect of temperature is to restrict the range of instabilities (see also Winglee ). Growth occurs only if the hot electron density is large and the thermal spread of the energetic particles is low enough.
Interestingly, once the thermal spread becomes large, j = 1, the dispersive properties of the waves can change. We have found that for some plasma parameters the unstable mode can couple directly to the freely propagating R-X mode for j = 1. When j > 1, the modes can approach one another closely, but they do not couple directly. To show this, we used an approximate form of the conductivity tensor which assumed that ck / 1.
Lastly, we discussed whether or not the current analysis can be applied to auroral zone distributions. Figure 8 shows that for a DGH distribution there is considerable variation in the appearance of the distribution function. It appears reasonable to assume that j > 1 for those distributions that have been accelerated through parallel electric fields. Also, the magnetic mirror force causes the distribution to curve in momentum space, giving a shell-like appearance. Most importantly, a large peak in the distribution function at low energy does not imply that a cold plasma approximation is adequate, because higher-energy particles occupy a larger volume in momentum space than lower-energy particles and so contribute more to the integrals used to evaluate the conductivity tensor. While more data are now available [Menietti and Burch, 1985], the relative amount of cold (or cool) plasma on an auroral zone field line is still an open issue.
In conclusion, we have described in some detail the wave dispersion associated with weakly relativistic electron distributions. If, as we suggest here, auroral electron distributions can be modelled by two species with different gyrofrequencies due to relativistic effects, then a cold plasma approximation is probably not adequate for describing the dispersion of wave modes near the gyrofrequency. Our study shows that the modified dispersion can have two offsetting effects when considering the generation of AKR. First, the separation of the unstable mode at low wave vectors from the free space branch of the R-X mode may result in enhanced growth due to reflection of the wave at the edges of the auroral cavity, as discussed by Strangeway . Restricting the altitude range over which the unstable mode can propagate may also remove the problem of "detuning" the instability by refractive effects [see Omidi and Gurnett, 1982, 1984]. In addition, the trapped mode can be driven unstable by gradients in the phase density of both downgoing and upgoing electrons, since < . On the other hand, the separation of the unstable mode introduces the requirement of an additional conversion process in any mechanism for producing AKR. It is important that future studies investigate these more global aspects of the wave dispersion.
Acknowledgments. The author would like to thank P. L. Pritchett
and R. M. Winglee for many useful discussions. This work was
supported by the NASA Solar Terrestrial Theory Program under grant
NAGW-78 and the Air Force under grant F19628-85-K-0027.
The Editor thanks D. Le Queau and another referee for their assistance in evaluating this paper.
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