Excitation of high-frequency electromagnetic waves by energetic electrons with a loss cone distribution in a field-aligned potential drop

The electron cyclotron maser instability (CMI) driven by momentum space anisotropy, {partial_derivative}f/{partial_derivative}p{perpendicular} > 0, has been invoked to explain many aspects, such as the modes of propagation, harmonic emissions, and the source characteristics of the auroral kilometric radiation (AKR). Recent satellite observations of AKR sources indicate that the source regions are often imbedded within the auroral acceleration region characterized by the presence of a field-aligned potential drop. In this paper the authors investigate the excitation of the fundamental extraordinary mode radiation due to the accelerated electrons. The momentum space distribution of these energetic electrons is modeled by a realistic upward loss cone as modified by the presence of a parallel potential drop below the observation point. On the basis of linear growth rate calculations the authors present the emission characteristics, such as the frequency spectrum and the emission angular distribution as functions of the plasma parameters. They will discuss the implication of their results on the generation of the AKR from the edges of the auroral density cavities. 31 refs., 12 figs., 1 tab.

. The Dory-Guest-Harris model distribution commonly used as an effective loss cone distribution for studying the electron cyclotron maser instability. Persson [1966] showed that a quasi-neutral plasma equilibrium can be established by a quasistatic potential drop along a converging magnetic field. If such conditions exist inside the auroral acceleration regions, the field-aligned potential drops can modify the loss cone boundaries and lead to widening of the loss cones [Croley et al., 1978;Reiffet al., 1988]. Wong et al. [1982] and Wu et al., [1982] pointed out that enhanced wave growths can result from both the depletion of the background plasma as well as the modified distribution caused by the presence of a parallel electric field. They demonstrated that the CMI growth rates can increase by about a factor of 3 as a result of a relatively large field-aligned potential drop (--•7.5 keV) present below the source location. Their calculations indicate that growth rates of the order of 3//•ce "• 10 -3 can be obtained when the energetic electrons dominate the plasma electron content and their dielectric effects are taken into consideration. Thus these results are compatible with the observations that AKR sources are located well within the auroral density cavity regions. However, the apparent successes of the existing CMI models with a dominant energetic electron population tend to overshadow the possibility of radio emissions being generated near the edges of the density cavities or wherever the cold background plasma may remain to be the primary plasma electron component. Persoon et al. [1988] analyzed the data from the plasma wave instrument (PWI) on the Dynamics Explorer (DE) 1 satellite and showed that the AKR and Z mode radiation are highly correlated with the density cavity structures in the auroral zone. AKR was found to occur inside as well as poleward of a density cavity in 97% of the 74 cavity intervals surveyed, while the Z mode radiation has a 78% occurrence rate. Furthermore, detailed analyses of three auroral events by Persoon et al. [1988] indicate that the energetic precipitating electrons (> 1 keV) are insignificant compared to all the electron precipitation observed in the polar cap and poleward of the density cavities. On average, they constitute less than 25% of all the auroral electrons with only a slightly higher percentage concentration inside the cavity regions. The recent Viking observations reported by Hilgers et al. [1992] also suggest that the energetic precipitating electrons may account for no more than a few percent of the auroral electrons near the edges of auroral density cavities. We investigate in this paper the radio emissions excited by a tenuous population of energetic electrons with a realistic loss cone distribution. Our main objective is to study heuristically the effects of a quasi-static electric potential drop that may be present in the source region. Our results presented below show that the radio emissions generated from the e?dges of auroral density cavities can have intensities comparable to those excited near the centers of the cavities. Although both the ordinary (O) and extraordinary (X) mode waves can be excited, we concentrate in this paper only on the fundamental X mode waves, which is known to be the dominant mode to be excited by the CMI.

Model Distribution Function
As mentioned before, the DGH model distribution function, devised for stability studies of trapped particles in magnetic mirror fields, has often been adopted by investigations of the electron cyclotron instability. Its popularity arises for the most part from its characteristic momentum space anisotropy with Of/Opñ > 0 and its ease of analytical and numerical implementations. However, as pointed out by Wu et al. [1982], the DGH distribution is unrealistic for modeling the electron loss cone distribution as observed in the auroral zones [Croley et al., 1978;Menietti et al., 1993].
Other existing models of the loss cone are also unrealistic in one way or another [Sharma et al., 1982;Hewitt et al., 1982;Wong et al., 1982;Wu et al., 1982]. In these models the distribution functions are either completely evacuated in the loss cone and become discontinuous at the loss cone bound-aries, contrary to observations, or they fail to correctly represent the changes of the loss cone boundaries by a parallel potential drop.
In the present study we have assumed for simplicity a invariants of the initially field-aligned energetic electrons [Gurgiolo and Burch, 1988;Winglee and Pritchett, 1986]. The width of the loss cone boundary • measures the steepness of the boundary wall and thus the emptiness of the loss cone interior. As shown by in situ particle observations [Croley et al., 1978;Gurgiolo and Burch, 1988;Menietti et al., 1993], the electron loss cones observed in the auroral zones are typically partially filled and have smooth boundaries. Omidi and Gurnett [1984] pointed out that the finite width of the loss cone boundary arises naturally from wave-particle interaction and collision processes along the auroral field lines and in the ionosphere. These scattering processes effectively introduce some "fuzziness" in the mirror points of the precipitating particles, thereby causing a spread in the loss cone boundary. Therefore • can be modeled by the uncertainty or the amount of "fuzziness" in the mirror point locations, •RM/RM, of the reflected electrons after having suffered different amount of scattering.
Then by differentiating (2) with respect to R•, and substituting u 2 for simplicity by (u 2 ) • a 2, the thermal spread of the energetic electrons, we obtain n is given by (12); the normalization factor A, /x0 and b are given by (2)-(4), respectively; and G is the same factor as given by Wu and Qiu [1983]. We note here also that a number of typographical errors contained by Wu and Qiu [1983] have been corrected in our derivation of (13). In addition, we have replaced II h in Q ij and/x r by II c .

We now investigate the conditions in which large CMI growth rates, such as that required (•,/l•ce •-10 -3) for the generation of AKR [Omidi and Gurnett, 1982, 1984], can result from an unstable loss cone distribution given by (la) and (lb). We have calculated the linear temporal growth
rates of the CMI in the limit of small density ratios, nh/nc << 1, by numerically evaluating (13). For the frequencies of interest the rn = _+1 terms in the infinite sums in (13) have the most significant contributions. Thus it is sufficient to numerically calculate the growth rates by keeping only the rn = 0, _+ 1, and _+2 terms. The plasma model parameter sets used in the following growth rate calculations shown in Figures 5-9 are summarized in Table 1

In this case the loss cone angle (equation (2)) is determined only by the magnetic mirror ratio (at R = 1.5Roe in these examples).The resulting growth rate decreases as the wave normal angle increases. Figures 5a and 5b also
indicate that high growth rates, reaching 7/l•ce •-10 -3 for nh/nc • 10 -2, can be attained from high thermal energies of the energetic particles. In addition, the frequency bandwidth of the unstable waves propagating at a given wave normal angle also decreases as a result of higher temperature. Figure 6 are the same as in Figure 5b except that ARMIR M is increased from 0.025 for a relatively empty loss cone with sharp boundaries to 0.63 for a partially filled loss cone with smooth boundaries. Figure 6a shows the case of eA•/rno c2 = 0.0 as in Figure 5b. Even at relatively high Th (10 keV), the larger b as a result of increased ARM/R M clearly leads to reduced growth rates due to the smaller Of/Ou• associated with the larger b. Figure 6b shows for the same ARMIR M the effects of a moderate potential drop, eA•/rno c2 = 0.01 (corresponding to a 5 kV potential drop). As discussed by Ornidi and Gurnett [1982Gurnett [ , 1984, the widened loss cone due to the finite potential drop (equation (2)) encloses a larger portion of a given resonance ellipse, and should result in higher CMI growth rates. However, compared to the zero potential drop case (Figure 6a), only slight increases in growth rates were obtained. Although higher growth rates are possible when nh/nc > 1, Wong et al. [1982] and Wu et al. [1982] have in fact obtained less than a factor of 10 increase in the CMI growth rates when the assumed potential drop below the observation point exceeds 7 kV; whereas typical potential drops observed below the altitude of 2Roe in the auroral zone tend to vary only between 1 and 3 kV [Reiff et al., 1988]. Therefore much larger potential drops are required to yield a significant growth rate enhancement. In the case of Figure 6b it appears that the enhanced growth rates are also suppressed by the slightly larger b due to the finite potential drop (see (3) and Table 1).  Table 1). loss cone (R has been increased from 1.5Roe to 2Roe) and lower T h . Again, there are no dramatic enhancements in the CMI growth rates as a result of the finite potential drop. Finally, we investigate the emission and frequency patterns of the unstable waves as it is important for the interpretation of radio emission observations. In Figures 5-8 the emission pattern is a filled emission cone centered around the background magnetic field, with slightly lower frequencies emanating at smaller wave normal angles. Similar results have been obtained by Wang et al. [1982] and Wu et al. [1982] for the case of n h/nc < 1. However, they have also shown that when the energetic electrons dominate the plasma content, the excited waves will predominantly be in the oblique to normal directions.

Figures 6 and 7 illustrate the effects of varying • and A• (equation (3)) on the growth rates. The parameters used in
We should note here that in the case of n h/n c < 1, the outward propagating waves may suffer absorption as they encounter weaker local magnetic fields, which may then result in a net hollow-cone emission pattern in the absence of wave scattering. This can be realized by considering the relativistic wave-particle resonance condition written as (see equation (7) Therefore damping of field-aligned propagating waves can be neglected for a sufficiently cold plasma. On the other hand, Figures 9a and 9b show that multiple frequency peaks can also be excited at the same (oblique) wave normal angle (see the growth rate curves at 65 ø wave normal angle). The frequency separation of the two frequency components is of the order of a few percent of •/c at the source. We note further that the multiple peak feature is present even when u 0 = 0. Similar double frequency peaks have also been obtained by Gurnett [1982, 1984] from loss cone distributions observed in the Earth's auroral zone. Thus an emission characteristics of the loss cone distribution is that the fundamental X mode waves can be excited simultaneously within the field-aligned emission cone and in the oblique directions at slightly higher frequencies.
Let us now discuss briefly the generation of AKR in terms of our growth rate calculations. We pointed out earlier that our calculations assumed small density ratios, n h/n c < 1. However, it is generally known that the parallel electric fields inside auroral density cavities will deplete the lowenergy background plasma such that the energetic electrons usually form the major electron component within the au-roral acceleration region, where strong AKR sources have been observed [Ungstrup et al., 1990;Hilgers et al., 1992]. Therefore the results of our calculations are strictly inapplicable to explaining the generation of AKR from the interior of the auroral density cavities.
The small density ratio approximation may be applicable near the edges of the auroral density cavities. From observations by the Viking satellite at altitudes of 6500-8500 km (i.e., R • 2Roe as in the parameters used for our calculations, see Table 1), Hilgers et al. [1992] determined that n c and n h near the centers of auroral density cavities are comparable and are usually limited to <0.35 cm -3. Outside the cavities, nc increases to 15 cm -3 (•Opc/• c = 0.2, see Table 1) or higher with n h dropping precipitously. These observations indicate that the energetic-to-background electron density ratios near the edges of auroral density cavities may well have been nh/n c --< 10-2, adequate for providing sufficiently high growth rates to explain the generation of AKR (see discussion above).

Conclusions
We have investigated the electron cyclotron maser instability (CMI) driven by a drift-loss cone distribution (equation (1)) such as that exhibited by the accelerated plasma sheet electrons (Figure 2) [Gurgiala and . From this study we conclude that 1. The DGH distibution (Figure 1) does not give a proper representation of an observed loss cone distribution and may, however, be more appropriate for modeling the trapped population on auroral field lines. Moreover, it cannot be readily used to study the effects of a parallel potential drop on the loss cone boundary.
2. For a true loss cone (equations (1 a) and (lb)) the width of the loss cone boundary can be modeled in terms of the "fuzziness" of the mirror point location at low altitudes (equation (3)), larger widths weaken the CMI (Figures 5b and   6a).
3. The presence of a parallel potential drop increases only slightly the CMI growth rates excited by a partially filled loss cone (large ARM/R M) (Figure 6), and it enhances the damping due to the energetic electrons for an evacuated loss cone (small ARM/R M) (Figure 7). 4. A finite u 0 or relatively hot energetic electrons are needed to attain sufficiently large growth rates (7,/•ce • 10 -3 ) (Figure 8).
In the limit that the energetic electrons are tenuous (10 -3 < nn/nc < 1) our linear growth rates calculations indicate that the fundamental X mode excited by the CMI are characterized by the following: 5. A filled emission cone (up to --• 120 ø wide) formed by the waves with frequencies near the R-X cutoff, with lowerfrequency waves being more field-aligned (Figure 5-8). Similar results have been obtained by Wang et al. [1982] and Wu et al. [1982] in the low density ratio limit.
6. Higher-frequency waves can be excited from the same source region (loss cone) at the same wave normal angle as the lower-frequency components, possibly with a reversed angular distribution with frequency ( Figure 9); may be applicable to the fine structure AKR [Gurnett and Anderson, 1979;Benson et al., 1988].
7. These emission characteristics may be relevant to waves generated at edges of the auroral density cavities. The resulting wave intensities may be comparable to those generated near the centers of the density depletion.