Published in Journal of Geophysical Research, volume 87, number 10, pages 8199-8214, October 1, 1982

A FEEDBACK MODEL FOR THE SOURCE OF AURORAL KILOMETRIC RADIATION

W. Calvert
Department of Physics and Astronomy,
The University of Iowa,
Iowa City, Iowa 52242

Abstract. The self-excited oscillations which result from positive wave feedback are believed to produce the discrete spectrum of auroral kilometric radiation. This would require a closed wave feedback path inside the source, having a length equal to a multiple of the wavelength, and it yields an emission model which is quite analogous to an optical laser oscillator. The end reflections for this feedback path, which would be comparable to those produced by the laser mirrors, are attributed to partial wave reflections at the source boundaries. However, in order to compensate for the wave refraction inside the source, the boundary reflection surfaces must converge with altitude, and this implies that the most likely auroral kilometric radiation source would be a thin, local density enhancement, since the refractive index contours at its boundaries would be expected to slope inward. The ISEE observations of multiple spectral components, which are attributed to separate oscillations at different altitudes in the same enhancement, suggest a source thickness as small as 25 km and an internal wave growth threshold of roughly 40 dB, rather than the 70-120 dB previously believed necessary to account for auroral kilometric radiation without feedback. But more significantly, the feedback model accounts for numerous aspects of the auroral kilometric radiation behavior, predicts emission at the wave growth saturation level, and leads to the conclusion that auroral kilometric radiation originates at many compact sites, each emitting a nearly monochromatic wave.

Introduction

Auroral kilometric radiation (abbreviated "AKR") consists of intense natural wave emissions, at frequencies of 50-700 kHz, which originate in the auroral zone in conjunction with both energetic electrons and discrete, visible auroral arcs [Gurnett, 1974]. Emitting typically up to 10⁷ W at times of much auroral activity [Gallagher and Gurnett, 1979], AKR is the strongest of the earth's radio emissions and presumably the counterpart of those from other planets. As such, it has received keen experimental and theoretical attention to determine its properties and to establish the basic mechanism for its production. The proposed new model, which accounts for the previously unexplained discrete emission spectrum of AKR, carries this a step further and provides new insight into the AKR emission process.

AKR Properties

Unobservable from the ground, AKR was discovered with the first distant satellites having suitable wave receivers: Electron 2 [Benediktov et al., 1965] and OGO I [Dunckel et al., 1970]. The auroral origin of AKR was established by Gurnett [1974] and confirmed by a number of subsequent studies [Kurth et al., 1975; Alexander and Kaiser, 1976; Gurnett and Green, 1978; Benson and Calvert, 1979]. The average spectral properties were studied by Kaiser and Alexander [1977a], who showed that most of the AKR originates in the evening sector, being strongest and most prevalent around 2100 MLT (magnetic local time). They also demonstrated the strong dependence on auroral activity and the occurrence with magnetospheric substorms [see also Kaiser and Alexander, 1977b, and Voots et al., 1977]. The association with energetic auroral electrons, cleverly deduced by Gurnett [1974], has since become firmly established [Benson and Calvert, 1979; Green et al., 1979; Benson et al., 1980]. These electrons comprise the so-called "inverted V" events, spanning up to a degree of latitude and containing fluxes of 10⁷ to 10⁹ el/cm² sr s at 1-10 keV [Frank and Ackerson, 1971].

Being pivotal to the theories, the wave mode of AKR and its source altitude were matters of some controversy. Evidence from the Hawkeye satellite [Gurnett and Green, 1978] and the Voyager spacecraft [Kaiser et al., 1978] already favored the extraordinary wave mode before that was clearly confirmed by the ISIS I observations at the source [Benson and Calvert, 1979; Calvert, 1981a]. These observations (at 450-700 kHz) indicated that AKR originates roughly perpendicular to the magnetic field at frequencies up to a few percent above that of the local extraordinary wave cutoff, which, because of the low plasma densities which accompany AKR, always lies quite close to the local electron cyclotron frequency. This suggested a vertically extended source, between 1.3 and 3.3 R_E (earth radii, geocentric), with each frequency originating from a different altitude. Although this differs from certain of the source locations determined by the lunar occultation of AKR at RAE 2 [Alexander and Kaiser, 1976], there is reason to believe that propagation effects confused those measurements [Alexander et al., 1978], and a distributed source at the local cyclotron frequency would be consistent with all the other source localization studies [Kurth et al., 1975; Green et al., 1977; Gurnett and Green, 1978; Gallagher and Gurnett, 1979].

Although most of the AKR can be attributed to extraordinary emissions at the cyclotron fundamental, there is some evidence from the Hawkeye and ISEE I wave observations for accompanying Z mode and ordinary emissions, both perhaps 20 dB or more weaker than the primary signals [Calvert, 1980; Anderson et al., 1981]. There is also some evidence for accompanying harmonic emissions, in which certain of the spectral features of AKR observed with ISEE I and ISEE 2 were found to be replicated at twice t e frequency [Anderson, 19821. These cases seem to support the previous reports of possible harmonics based on ISIS I [Benson and Calvert, 1979; Benson, 1982]. However, it remains unclear whether the observed harmonic signals should be attributed to a separate source at twice the local cyclotron frequency or to a harmonic distortion of the intense AKR source signals at the cyclotron fundamental and, similarly, whether the ordinary emissions ever exist independently of the primary extraordinary radiation. The new model would suggest that both could be produced as byproducts.

Environment

The ISIS I observations [Benson and Calvert, 19791 revealed quite low plasma densities in the vicinity of the AKR source and suggested that the plasma-cyclotron frequency ratio (fN/fH) had to be less than 0.2 for its generation. A subsequent Hawkeye analysis [Calvert, 1981b] showed that such low densities sometimes extended to above 3 R_E and formed an enormous auroral plasma cavity, 6^o wide and centered on 70^o invariant magnetic latitude, which accompanies AKR and seems to control its emission spectrum. A recent observation of the cavity with ISEE I (unpublished) indicated that the cavity was continuous throughout the evening sector, from 1800 to 2400 MLT, and exhibited a nearly constant plasma density of only 0.2 cm^-3 (f_N = 4 kHz).

Inside the cavity and seemingly associated with the AKR source, quite thin density enhancements were sometimes observed with ISIS I (see Figure 3 of Benson and Calvert [19791). Frequently less than 100 km across and thus easy to miss with the ISIS I sounder, some of these enhancements were more than an order of magnitude density increase. Such enhancements, which have largely been ignored in the previous AKR work, play a special role in the feedback model presented below.

Discrete Spectra

Most of the earlier observations of AKR were performed with multichannel or sweep frequency wave receivers having relatively coarse frequency resolution. However, the ISEE I satellite was equipped with a broadband data channel for the reception of continuous 40 kHz segments of the spectrum, and it revealed the discrete spectral structure of AKR [Gurnett et al., 1979]. An example of this structure, with many overlaid, narrowband, drifting components, is shown in Figure 1.

Figure 1. An ISEE I spectrogram showing the discrete spectral components of AKR [after Gurnett et al., 1979].

The spectral bandwidth of an individual emission component in the AKR spectrum is usually less than 1 kHz, and most of them drift toward higher frequencies at rates sometimes exceeding a few kilohertz per second [Gurnett and Anderson, 1981; Morioka et al., 1981]. However, some of the discrete spectral components were found to drift toward lower frequencies, and some were occasionally found to reverse their drift direction gradually over a minute or so (e.g., at 0550 UT in Figure 1). Although rarely as strict harmonics, the components sometimes also occur as multiplets, with a regular spacing of roughly 10 kHz. The drifts in the AKR spectrum have been attributed to propagating disturbances, traveling usually downward at roughly the ion-acoustic speed and generating AKR at the increasing cyclotron frequency they encounter [Gurnett and Anderson, 1981]. The model below would account for the drifts differently, since it predicts an emission frequency which depends on the product of the source thickness and the wave refractive index inside the source, as will be discussed later.

Apart from the assumption that the action of the traveling disturbance was of limited spatial extent, no previous theory accounts for the discreteness of the AKR spectrum. Although related to this question, the theory of Grabbe [1982; Grabbe et al., 1980] pertains instead to an ion wave modulation of the spectrum rather than to its discreteness.

Wave Instability

Although a number of alternate mechanisms have been proposed for the generation of AKR [see Grabbe, 1981], those involving the direct amplification of waves by the Doppler-shifted cyclotron resonance instability seem to be most consistent with the observations and also best adaptable to the new feedback source model to provide the wave gain which is required. In this instability, originally proposed by Melrose [1976] to explain both the AKR and the Jovian decametric radio emissions, the rotating wave electric field is resonant with the cyclotron motion of energetic auroral electrons, presumably the inverted V electrons found to accompany AKR. Wu and Lee [19791 adopted the same basic instability, pointing out that relativistic effects are important in spite of the relatively low electron energy, that waves at large angles to the magnetic field could be amplified, and that the free energy for wave growth could be supplied by an upward traveling electron loss cone rather than by the downward beam that Melrose [1976] had assumed. They also emphasized the need for a low plasma-cyclotron frequency ratio (f N/fH), since that would reduce the Doppler shift (in the electron reference frame) which is needed for resonance. This accounted for the generation of AKR within a region of reduced density and yielded excellent agreement with the observation that the ratios must always be less than 0.2 at the source [Benson and Calvert, 1979; Calvert, 1981b]. They went on to calculate the wave growth rates for analytical loss cone models, producing results which suggested that the spatial growth rates for the extraordinary mode could sometimes exceed a few decibels per kilometer (deduced from Figures 2, 3, and 10 of their subsequent paper [Lee and Wu, 1980]).

Previous Model

Heretofore, it has been assumed that the instability provides sufficient wave growth to amplify some ambient noise up to the levels which are observed in AKR. In other words, the AKR source was previously modeled by a distributed, open-loop wave amplifier. Although somewhat simpler than the new model, that model cannot account for the observed discrete emission spectrum, nor has it been sufficiently developed to allow for the wave refraction at the source and the consequent limitation of the amplification path length that would produce.

The peak signals for AKR at 250 kHz reported by Gurnett [1974] and extrapolated to a geocentric distance of 25 R_E were 10^-14 W/m² Hz. This implies roughly 10^-10 W/m² Hz at the boundary of a 1000-km source; that is compatible with other ISEE I observations in the source vicinity. On the assumption that the ambient noise is 10^-20 to 10^-18 W/m² Hz [Brown, 1973], a gain of 80-100 dB would then be required to account for the peak AKR signals. A smaller source would imply a proportionately larger gain (e.g., 100-120 dB for a source 100 km deep), and a gain of around 70 dB in a 1000 km source would be required to account for the more typical AKR signals reported by Gallagher and Gurnett [1979].

Concept

The new source model calls for the occurrence of self-excited wave oscillations. As before, the wave is amplified as it propagates across the source, presumably by the cyclotron resonance wave instability mentioned above. However, in the new model, part of that amplified wave is routed back to its origin with the appropriate amplitude, direction, and phase to replenish itself. This produces a closed-loop oscillator capable of generating spontaneous emissions without any external excitation. In other words, the source of AKR is visualized as a natural laser oscillator, albeit considerably larger and at a very much lower frequency than its man-made counterpart.

Although the new model requires less gain (possibly 40 dB or even less), the most significant difference is that it produces markedly different behavior. An oscillator yields discrete frequencies from only those isolated locations where the feedback phase and wave direction are correct, and hence it quite naturally accounts for the discrete spectrum of AKR. It would be expected to produce wave amplitudes large enough to saturate the growth process, and hence it yields predictable emission levels, relatively independent of the wave growth process required to produce them. It would also be expected to exhibit a threshold for the wave growth, below which the emissions would disappear, plus a variety of other behavior associated with positive feedback, like the quenching of one oscillation by another.

A likely candidate for producing the wave feedback in the model would be partial wave reflections at the source boundaries, where the plasma density is assumed to decrease abruptly. This would produce shallow, closed feedback paths inside the source, oriented roughly perpendicular to the magnetic field gradient and quite analogous to those produced in an optical laser by its partially reflecting end mirrors. Such a path, illustrated in Figure 2, would support oscillations, provided its length were an integral multiple of the wavelength and the internal gain were sufficient to overcome the end losses (as indicated by G = 1/(rho₁rho₂) in the figure).

Figure 2. A potential AKR feedback path produced by partial wave reflections at the abrupt opposite boundaries of a density enhancement. The reflection coefficients rho₁ and rho₂ at these boundaries would determine the threshold gain G required for oscillation and the generation of AKR.

Preview

The next section is devoted to the behavior of oscillators and how that could affect the production of AKR. The following section deals with the formation of closed feedback paths and an analysis of the partial-reflection feedback model to determine the location and efficiency of those paths. Then certain of the ISEE I wave observations will be analyzed to illustrate a few of the features attributed to feedback and to show how they may be used to deduce some of the source properties.

OSCILLATOR BEHAVIOR

In many respects, the behavior of the new AKR source model is independent of the exact nature of the feedback mechanism. It depends, instead, primarily on a few simple properties of the feedback circuit, like its gain G, its feedback ratio rho, and its phase shift phi. Such aspects will be discussed now, on the assumption that the feedback can be characterized by such properties, and a discussion of how this feedback might be generated is postponed to the following section.

Schematic

The behavior of a feedback system will be analyzed by using the schematic diagram in Figure 3, which is identical to that used to represent an electronic oscillator circuit. The signals, which are indicated by the arrows in this diagram and would have been the voltages or currents in an electronic circuit, represent propagating waves in the AKR model. The output wave, which constitutes the radiating AKR, is labeled A, and the feedback, which equals rho A, is labeled F. The signal N represents the ambient system noise or perhaps some other signal of external origin.

Figure 3. The schematic diagram for a feedback oscillator with gain G and feedback rho. The output signal is designated A, the feedback is F = rho times A, and N represents an external noise input signal. When oscillations occur, a saturation process, indicated by S, always adjusts the gain so that 1/Go - rho = N_o/A_o at the oscillator's resonant frequency, and the phase vector diagram to the right indicates the response N/A at an adjacent frequency where the feedback phase shift is phi.

Strictly speaking, the wave signals in this diagram should be considered to represent spectral amplitudes such that their squares are proportional to the wave power flux in a small frequency interval delta f. The triangle in the schematic represents the system gain G, presumably provided by the cyclotron-resonant wave instability, and the circle represents the feedback mechanism. The latter is characterized by the feedback ratio rho and also by the loop phase shift phi (including that through the amplifier), or else by an equivalent feedback delay, t_rho = phi/(2 pi f), where f is the wave frequency. The basic equation for the feedback system in Figure 3, which is applicable once all the transients have settled down, is

(1)

which merely states that the superimposed input signals (N + F) are amplified by the factor G to become the output signal.

Threshold

Neglecting the noise, equation (1) would be satisfied by

(2)

which represents the threshold for oscillations to occur in a noiseless feedback system. It indicates simply that the feedback signal is just sufficient to replenish itself. It also implies, since rho is actually a vector, that the net loop phase shift must also be zero or a multiple of two pi radians:

(3)

where m is an integer, sometimes called the mode number. Equations (2) and (3) are the classic equations for a feedback oscillator, known as the Barkhausen criteria [see Millman and Halkias, 1972], which state that oscillations occur when the loop gain is unity (equation (2)) and the feedback is in phase (equation (3)).

The existence of an oscillation threshold would readily explain the sporadic nature of AKR, where it seems to turn on and off in a matter of minutes or less. This, plus the saturation effects discussed below, might also explain why the AKR observed with ISEE I almost always occurs at the same intense level of roughly 10^-14 W/m² Hz (at 5-10 R_E) and only infrequently at the lower, but easily detectable, levels of 10^-17 to 10^-16 W/m² Hz. An oscillation threshold would also help to explain how AKR could be externally stimulated by the much weaker type III solar radio bursts [Calvert, 1981c], since a small change of either the gain or the feedback could then exert a substantial influence. (This, in fact, was the initial concept which inspired the new model.)

Saturation

Some of the most interesting properties of an oscillator stem from the fact that the growing oscillations produced when rho times G exceeds unity cannot continue indefinitely. The wave amplitude must ultimately saturate and thereby reduce either rho or G until the growth ceases. By this process, which is represented by the dashed signal labeled S in Figure 3, the oscillator itself assures that equation (2) is satisfied and thus that the Barkhausen criteria apply.

In simple terms, the oscillator saturates when it runs out of something. In an electronic oscillator circuit, saturation frequently occurs when the peak-to-peak amplitude of the signal equals the supply voltage and the transistor has "run out" of enough voltage to generate a larger signal. In this case, the waveform becomes truncated at the supply voltage, and that is the mechanism by which the effective gain is reduced until the Barkhausen criteria are satisfied. (Well designed circuits are often more graceful than this, but the principle is the same.)

The time required to achieve saturation in an oscillator depends upon the excess loop gain. This time can be estimated by considering equation (1) to be a recursion equation, giving on the left a new value for A = A + delta A, which is achieved during a feedback delay period delta t = t_rho. This yields (still neglecting the noise N)

(4)

for the characteristic amplitude e-folding time (i.e., the growth time constant). Equation (4) implies that the growth toward saturation (as well as the response to any subsequent changes) should be faster when the excess loop gain is greater. It also implies a short saturation time for the AKR oscillations pictured in Figure 2, where the source thickness may be only a few tens of kilometers and the feedback delay t, would consequently be only a few hundred microseconds. For instance, an excess loop gain of only 10% (0.8 dB) should produce saturation in a few milliseconds, whereas AKR frequently lasts for hours, and its discrete spectral components can often last for a number of minutes. It should thus be expected that an AKR feedback oscillator should be saturated most of the time, provided the wave amplification and feedback are sufficiently stable.

Frequency and Bandwidth

The frequency of a saturated noiseless oscillator is determined by equation (3), and that would imply an emission spectrum which is infinitely narrow (that is, a delta function of the frequency). In order to examine the spectral bandwidth produced by an actual oscillator, it is sometimes necessary to include the noise signal, N in Figure 3. In doing so, the oscillator becomes a narrow-band regenerative amplifier of the sort used to explain the behavior of a laser [see Ross, 1969]. In fact, with this approach the difference between an oscillator and an amplifier becomes a quantitative matter, depending upon the value of rho compared with the ratio of the noise to the saturated output signal.

The gain of a saturated noisy oscillator (or that of a saturated regenerative amplifier) will adjust itself to the value which satisfies equation (1), namely,

(5)

where A_o is the saturated output amplitude, assumed to occur at a frequency f_o, where the feedback phase shift is zero and the noise is N_o. For a quiet system, where N_o is much less than rho times A_o, this becomes equation (2), representing an oscillator, whereas for a very noisy system in which the reverse is true, it yields the straightforward amplifier gain equation A_o = G_o N_o. The amplitude at adjacent frequencies is then given by

(6)

(again from equation (1)), and this is illustrated by the phase vector diagram in Figure 3. Using the trigonometric law of cosines and equation (5),

(7)

where phi is the feedback phase shift at the new frequency.

The spectral bandwidth, which will be designated (delta f)_1/2, is measured at the so-called half-power points, where A² = A_o² /2. The corresponding feedback phase shift, which will be designated phi_1/2, is then given by

(8)

where N = N_o is assumed to be independent of the frequency. For a relatively quiet system (where N_o much less than rho times A_o G_o will be quite close to 1/rho, so that phi_1/2 will be relatively small and approximately equal to

(9)

For the m wavelength feedback loop envisaged for the AKR model,

(10)

where delta f = f - fo is the frequency offset from the resonant frequency of the oscillator.

The full spectral bandwidth between adjacent half-power points attributable to the noise is therefore

(11)

When the noise is low, equation (11) predicts extremely narrow bandwidths, which amounts to simply the well-known assertion that oscillators produce discrete emissions. For AKR, where N_o/A_o has been estimated to be between -70 and -120 dB and the model below also suggests that m ~ 50 and G ~ 100, this would imply a noise-dominated bandwidth of only 0.03 to 10 Hz. This is considerably less than the 1 kHz bandwidth for AKR reported by Gurnett and Anderson [1981] and shown in Figure 4. The oscillator model thus readily accounts for a narrow spectrum, and, in fact, it raises the question of why the observed spectrum should be so broad.

Figure 4. An expanded spectrogram of the AKR multiplet in Figure I at 0550 UT, which exhibits a spacing of 10 kHz (nW = 15 km) and a bandwidth of roughly 1 kHz. The frequency shifts at 8, 25, and 36 s in the top panel could also indicate the possible quenching of different collocated oscillation modes.

On the other hand, N_o/A_o is not well known, and it could also be larger than these estimates. However, this cannot account for the observed AKR bandwidth, simply because the oscillator model would break down first. For instance, with m = 50 and (delta f)/f = 1/500, equations (5), (8), and (10) would imply that Ao = 3.7 G_oN_o (and also that rho times G_o = 0.73). This means that the saturated oscillations (A_o) would be only 10 dB stronger than the amplified noise without feedback (G N_o) and such oscillations, at the isolated locations where the constructive feedback occurs, would not be expected to dominate the amplified emissions produced elsewhere.

A possible explanation for the observed bandwidth would be temporal fluctuations of the gain, the feedback, or the saturation amplitude. Each such fluctuation would introduce a transient in the oscillator waveform envelope, and the spectrum of those transients would be imposed on the emitted signal. The width of that spectrum should then be roughly the reciprocal of the growth time constant in equation (4):

(12)

assuming that the gain G changed abruptly by delta G. Since t_rho is expected to be around 200 microseconds, a fractional gain change of 20% in less than a millisecond would be sufficient to account for the observed bandwidth. Although probably unmeasurable by current techniques, a change comparable to this in the auroral electron flux might not be considered surprising.

Alternatively, a broadened bandwidth might be attributed to Doppler shifts at the end points of the feedback path or, equivalently, to fluctuations of the oscillator's resonant frequency. There is already some evidence for this in the ion cyclotron modulation of AKR discovered by R. R. Anderson [private communication, 1982] and interpreted differently by Grabbe [1982], since an ion density wave at the source boundary might well alter the feedback path enough to impress its frequency on the emitted signal.

Quenching

The saturation of an oscillator also accounts for some more subtle aspects of its behavior, among which is the quenching of one oscillation by another. This occurs when a system is capable of oscillating at two different frequencies, and the saturation of one reduces the gain for the other to the extent that the latter falls below its threshold. As a result, only the first will persist, while the other will die away. Should matters subsequently change so that the second mode exceeds its threshold, then it can grow to saturation and cause the first to die away. Like rival siblings, one oscillation mode usually dominates and suppresses all the others.

There is some evidence for quenching in the ISEE I observations of the AKR spectrum, such as the apparent frequency shift at 05:48:25 UT and 503-506 kHz in the upper panel of Figure 4 and also possibly at 05:48:08 UT and 508 kHz or at 0548:36 UT and 507 kHz in the same record. Notice how a separate oscillation at a somewhat lower frequency seems to appear sporadically for a few seconds, and then at 05:48:26 UT it appears to assert itself and quench the upper oscillation entirely. Since the two major components, spaced by 10 kHz in Figure 4 and drifting slowly upward when the quenching occurred, will be attributed feedback paths in the same source that differ by a whole wavelength, the quenching behavior must be attributed to some more subtle aspect of the source. One possibility is that the quenching occurs between the shallow, deep, and compound feedback paths discussed in the next section under "path location," all having a similar length but differing slightly in their average refractive index. In this case, it might be no accident that the quenching occurs near the drift reversal (which occurs at 05:48:43 and 507 kHz in Figure 4), since that's where these three paths should coalesce (see below, under 'drifts' in the 'discussion' section).

AKR saturation

The unavoidable saturation of an oscillator always produces emissions at a large amplitude, and that tends to yield high efficiency. This ubiquitous feature of an oscillator might well answer the question posed by Gurnett [1974] of how AKR acquires such a large fraction of the incident electron energy, sometimes possibly up to 1%. Provided the saturation occurs by depleting the electron free energy, it would be reasonable to expect that the change of free energy necessary for saturation might be appreciable compared to the free energy itself. Since the free energy consumed by the oscillator appears as AKR, it might then be expected that the AKR could carry away a sizeable fraction of the total free energy. Although the same argument might apply to an amplifier, it is mandatory for an oscillator simply because saturation always occurs.

In the loss cone instability of Wu and Lee [1979], presuming that's the oscillator's source of gain, the free energy equals that of the missing electrons. At 2 R_E, where the peak AKR is produced, the loss cone angle should be 24^o and its solid angle would be 4% of the electron energy sphere. This means, for an electron flux of 10⁷ to 10⁹ el/cm² sr s in the auroral inverted V's, that the upcoming free power should be l0-10³ W/km² keV. Thus the total free power might be as large as 10⁸ to 10¹⁰ W for a 1-keV average energy and an auroral zone area of 10⁷ km². The portion of this converted into AKR would then be the fractional area covered by oscillators times their average efficiency. There is clearly enough free energy to account for the 10⁷ W reported by Gallagher and Gurnett [1979], with as little as 1% coverage and 10% efficiency. Although there is still a problem explaining the 10⁹ W reported by Gurnett [1974] (unless one could accept 10% coverage and 100% efficiency), even that might be accommodated by assuming a larger flux, a greater average energy, or an enhanced loss cone caused by a parallel electric field.

Knowing the available free power and assuming an efficiency, it is a simple matter to estimate the saturation amplitude for AKR. Apart from a possible factor imposed by the oscillator geometry, the power flux of the saturated wave should then be 1-100 W/km² for an oscillator with 10% efficiency driven by 1-keV loss cone electrons. This yields a wave electric field inside the source (given by 377 P where P is the power flux) of 20-200 mV/m. A source signal of this amplitude is quite consistent with the observed spectral power flux of AKR and the 25 km source size deduced below, to wit: For the 1-kHz bandwidth of a discrete emission like that in Figure 4, the source spectral power flux should be 3 x 10^-8 to 3 x 10^-6 W/m² Hz, and that should produce a signal of roughly 10^-15 to 10^-13 W/m² Hz at 25 R_E. Considering that the discrete components seem to occupy 10% of the spectrum, this would account for the 10^-16 to 10^-14 W/m² Hz typically observed at the spectral peak of AKR [Kaiser and Alexander, 1977a].

The saturation concept could also explain how the extraordinary cyclotron instability manages to win over all other auroral instabilities and gain access to the bulk of the electron free energy. Since an oscillation always implies saturation, an instability with sufficient feedback has a tremendous advantage. Its amplitudes are automatically as large as possible, whereas those of an open-loop instability would have to grow from some ambient background level. The saturated oscillation thus has an opportunity to deplete the free energy before its competitors, so to speak, have a chance. In other words, the existence of feedback for the extraordinary wave mode could account for the production of radiation in that mode, despite the competing instabilities which presumably lack feedback.

Summary

Although it remains to be demonstrated that feedback is feasible, the oscillator model is already an attractive candidate to account for the behavior of AKR. First of all, it accounts for the emissions at discrete frequencies. In fact, it predicts a bandwidth substantially less than that which is observed, and this suggests that the rapid temporal fluctuations to broaden that bandwidth must exist at the source. Secondly, it predicts an oscillation threshold and prompt saturation once that threshold is reached. This accounts for the sporadic nature of AKR and its propensity to occur at a roughly constant level. The saturation also accounts for the high efficiency of AKR generation, and the wave amplitudes it implies, based on a loss cone free energy source, are quite consistent with the observed AKR amplitudes. Finally, there is even a hope that it will account for other details of AKR, such as the apparent quenching of one oscillation mode by another.

FEEDBACK MODEL

In order to study the extraordinary wave paths which could produce feedback, it is necessary to examine first the wave refraction in the source region near cutoff. It will then be shown that the path closure which is essential for oscillations would most likely occur inside density enhancements, provided their walls are sufficiently abrupt to produce partial wave reflections. Based on this model, the reflection coefficients, and hence the required wave growth inside the source, will be calculated as a function of the source parameters.

Refractive Index

The plasma frequency f_N in the AKR source region is expected to be substantially less than the cyclotron frequency f_H, with their ratio f_N/f_H being somewhere between 0.02 and 0.2 [Calvert, 1981b]. Furthermore, since a low f_N/f_H is mandatory for the extraordinary-mode cyclotron resonance at low electron energies, such a range is unavoidable if AKR is produced by that instability. Under such conditions, the extraordinary-mode wave refractive index n obeys the following approximate formula, a variant of that previously used in the ISIS I AKR studies [Calvert, 1981a]:

(13)

where theta is the wave normal angle with respect to the magnetic field direction, and

(14)

is a parameter which is approximately proportional to the frequency offset, equal to delta f = f - f_H, between the wave frequency f and the local cyclotron frequency f_H:

(15)

The "xi" parameter, which equals zero at cyclotron resonance (f = f_H) and unity at the extraordinary (f_N² = f (f - f_H)), will also be used to measure vertical distance in the AKR source region. Quite near the source, where the magnetic field variation is approximately linear,

(16)

where (grad B)/ B is the log gradient of the magnetic field strength and z is the distance measured along that gradient from the level for cyclotron resonance at the frequency f. This gives

(17)

and shows that xi is proportional to the distance, as long as f_N is reasonably constant (i.e., z much less than N/(grad N)). The reciprocal of the proportionality factor will be designated H_R and referred to as the refraction scale height. For f near f_H and B proportional to r^-3, this refraction scale height is

(18)

where r is the geocentric distance. At the base of the AKR source region ® = 1.3 R_E) the value of HR is roughly 100 km, whereas at the top ® = 3.3 R_E) it is almost 300 km, since f_N/f_H = 0.2 at both locations. However, since f_N/f_H exhibits a minimum of perhaps 0.02 near r = 2 R_E, the value for H_R in the heart of the AKR source region could be as small as a few kilometers. Such very small values for the refraction scale height would imply that the extraordinary wave refraction occurs only quite near the cyclotron resonance level and thus that the potential feedback paths would be similarly confined.

Although it may sometimes be questionable, the refractive index in equation (13) will be further approximated by its value for theta = pi / 2:

(19)

even though the actual wave angle (to be redefined shortly by using alpha and gamma) will still be considered to vary. This is equivalent to assuming that the refractive index is isotropic at large wave angles, and that not only simplifies the resulting equations, but it also avoids the complex problem of anisotropic propagation. It should be valid as long as xi is much greater than (1 + cos²(theta))/2, and a 10% error arises only when xi is greater than 3 or theta is less than 63^o. Equation (19) is plotted in Figure 5, along with the wave group velocity deduced from the same formula.

Figure 5. The approximate refractive index n and group velocity u for perpendicular extraordinary waves near the cyclotron frequency fH at low plasma densities. For a constant plasma frequency f_N and a linear f_H, the parameter xi is nearly proportional to the distance above the cyclotron resonance level (at which f = f_H) with a scale factor of H_R = r f_N² / 3 f_H², where r is the geocentric distance.

Wave Paths

In the previous analysis of AKR wave propagation [Calvert, 1981a], it was assumed that the magnetic gradient, rather than that of the plasma density, had the dominant influence. Except at the end points of the feedback path, this assumption will be retained, and that would require a smooth and relatively constant density inside the source. This produces wave ray paths which curve upward and soon become straight only a few refraction scale heights above their reflection levels.

Figure 6. The wave path, approximated by a circular arc and tangential straight segments. The arc radius R represents the path curvature at the reflection point, where xi = xi_o. The wave angle a and its terminal value gamma are measured with respect to the surfaces of constant magnetic field strength. This approximation underestimates the actual lateral distance (schematically shown dotted) and could be improved for small theta by using a gamma satisfying Snell's law at every point (shown dashed).

As illustrated in Figure 6, the ray path will be approximated by a circular arc of radius R which joins to tangential straight segments at a greater altitude. The path is symmetrical with respect to the grad B direction at the reflection point, where the value of xi will be designated xi_o. The new wave angle alpha, which is identical to that of the ray path because of the isotropic approximation, will be measured relative to the constant-B surfaces, and its terminal value, which equals the inclination angle of the straight line segments, will be designated by gamma. These angles are the complement of theta plus delta, where delta is the angle between B and grad B. The equations for the curved section of the ray path, parametric in alpha,which is less than gamma, would thus be

(20)

where chi is the distance from the reflection point measured perpendicular to the grad B axis in the same units as xi (i.e., with the scale factor H_R). The equation for the straight segments is then

(21)

where equation (20) was used to evaluate where the curved and straight segments must join. Equations (20) and (21) specify a shape for the ray path (xi versus chi) in terms of the parameters xi_o, R/H_R, and gamma.

The quantity R in these equations is the ray path radius of curvature at the reflection point. It can be shown (for isotropic propagation) that this radius of curvature equals the reciprocal log gradient of the refractive index:

(22)

evaluated at xi = xi_o. With equation (19), this becomes

(23)

since grad xi = 1/H_R. The angle gamma, on the other hand, may be deduced from a simple application of Snell's law and the condition that alpha is zero at xi = xi_o:

(24)

where n and n_o are the refractive indices at xi and xi_o, respectively. Again by using equation (19), this yields

(25)

Equations (23) and (25) could be used to eliminate two of the three path parameters to produce ray path equations which depend only upon a single parameter (e.g., xi_o). In order to strictly retain the arc-and-straight line model for the ray path, it would be necessary to evaluate equation (25) for infinite xi. As illustrated in Figure 5, this would yield a model which underestimates chi (for a given xi_o and gamma), since the ray path curvature actually decreases with altitude. A somewhat better approximation is obtained by considering gamma to always be the local wave angle, retaining equation (25) for every point of the path, and using equations (21) and (23) in the form

(26)

This ray path model will be used below to evaluate the vertical position of the feedback path, from which the source refractive index and the end point reflection coefficients can be estimated. Although of doubtful accuracy for large values of gamma, it should suffice for the current purposes and yield at the least the correct qualitative behavior.

Closure

In order to produce sustained oscillations, the feedback path must form a closed wave circuit. Otherwise, the wave would migrate away from the amplification region rather than stay put and continue to replenish itself. This, in fact, is the essential difference between the current model and that of Wu and Lee [1979], who invoked multiple reflections at the source to obtain a longer amplification length. This difference, however, has a profound influence, since it introduces the oscillator behavior discussed above and thereby significantly alters the nature of the emissions.

Because of the upward refraction and the vertical divergence of the magnetic field, the need for path closure becomes a stringent requirement on the possible auroral structures which might produce feedback. This is illustrated in Figure 7a for an extreme case of strictly field aligned end reflectors and negligible refraction (e.g., for xi much greater than 1). The field divergence angle, which is designated 2 beta in the figure, would be no more than one-half degree for an AKR source which is less than 100 km thick at 2 R_E. However, even this small angle would be sufficient to cause a wave to walk rapidly upward, decreasing its angle to the axis by 2 times beta upon each reflection. It can be shown (by unfolding the path in Figure 7a along the reflecting surfaces and assuming beta is quite small) that the total distance traveled by a wave initially perpendicular to the axis is approximately

(27)

where r is the geocentric distance, the magnetic field was assumed to decrease as 1/r³, and delta xi is the corresponding change in the parameter xi, that is, the height change in units of H_R. For r = 2 R_E, f_N/f_H = 0.02, and (delta xi) = 10, this formula gives D = 660 km. That is, the wave would have undergone scarcely seven reflections in a 100 km source (and turned 4^o during the same 2 ms) before it had moved upward a substantial distance relative to that of the refracting region (i.e., 10 refraction scale heights). In other words, a vertical divergence of the field-aligned end point reflectors would tend to rotate the wave direction upward and rapidly expel the wave, even in the absence of any wave refraction.

Figure 7. (a) An illustration showing that the path closure required for wave feedback is impossible with divergent end reflectors and no wave refraction, but (b) could be achieved with an upward density gradient or converging reflectors.

To make matters worse, the angle change and height increase produced by wave refraction are in the same sense and could well be larger than those produced by the field divergence. This will be demonstrated by considering (somewhat arbitrarily) a wave starting perpendicular to grad B at xi_o = 2 and propagating to xi = 3. According to equation (25), the wave angle would have turned through 24^o, and according to equation (26), the wave would have traveled to chi = 3.5 in the process. Since the refraction scale length H_R is expected to be less than 30 km in much of the AKR source region, such a wave would have been refracted upward by 24^o before it had traveled much more than 100 km horizontally and 30 km vertically. That is, it would have been refracted substantially before its first reflection in a 100 km source and perhaps by enough to destroy the cyclotron resonance upon which its growth depends. In any event, it is clear that a closed path could not occur with upward refraction and divergent end reflectors.

Closure Options

There are, however, the two options illustrated in Figure 7b for producing a closed feedback path: those of reversing either the wave refraction or the divergence of the end reflectors. The former could be accomplished with an upward density gradient sufficient to overcome that of the vertically decreasing magnetic field. This would require a reversal of the refractive index gradient,

(28)

from equation (19), and that would require a reversal of

(29)

from equation (17), since f_H is proportional to B and f_N² is proportional to N, the plasma density. Consequently, a reversal of the refraction sense would require a reversed density scale height (i.e. with N/(grad N) less than the inverse log gradient of the magnetic field, equal to B/(grad B) ~ 2700-7000 km) by the small factor (delta f)/f_H. For instance, with (delta f)/f_H = 10% it would require a downward density scale height less than 270-700 km. Although such strong reversed gradients cannot be ruled out, there is no evidence that they exist in the auroral region (e.g., from the ISEE observations of wave cutoffs), and their occurrence is considered unlikely. This would favor the alternate option that a closed wave feedback path probably requires some sort of vertically converging end reflectors.

Endpoint Convergence

It is important to realize that the need for convergent reflectors implies a need for vertically convergent refractive index surfaces, since they control the propagation of waves, and that an actual convergence of the density contours may not be absolutely necessary. This arises from the strong influence of the magnetic field on the vector combination of gradients in equation (29). Since the magnetic gradient will usually be dominant because of the large factor f_H/(delta f), the orientation of the refractive index surfaces is affected most by the cross-field component of the density gradient (assuming that B is roughly parallel to grad B), and it can be shown that these surfaces will tend to slope upward toward a cross-field density increase. Because of this effect, it is concluded that local density enhancements could provide the needed end point convergence for feedback. At the boundaries of such enhancements the density gradient would be directed inward, whereas that of the magnetic field would be directed approximately downward. The gradient of the refractive index would then be directed outward and upward, and the refractive index contours at the boundaries would converge vertically, even if the enhancements were perfectly straight and uniform with altitude. This is illustrated in Figure 8, which was drawn for a 20% enhancement at a location where the H_R (which is proportional to the density) is 25 km inside the source and 20 km outside. The density change introduces a step in the altitude of the wave cutoff, and its gradient at the boundary (assumed to occur over a constant thickness of I km) produces a convergent slope of the boundary contours. It is worth noting that a density depletion would produce the opposite effect, yield diverging refractive index contours, and hence would be unlikely to produce closed wave feedback paths.

Since grad N in equation (29) is roughly perpendicular to grad B, the convergence half angle (alpha) for the refractive index contours at the boundary of a density enhancement would be given by

(30)

Using equations (15) and (17) and replacing grad N by (delta N)/(delta W), where delta N is the density change at the boundary and delta W is the boundary thickness, gives

(31)

Since (delta N)/N appears in the denominator of this equation, it should be expected that the lesser enhancements should produce the larger convergence angles. Furthermore, the convergence angle should also increase with the normalized source thickness W/H_R and the shape factor ((delta W)/W) but decrease with the normalized height xi = z/H_R. For the case illustrated in Figure 8, beta would be roughly 10^o where the potential feedback path is shown, at xi = 1.4 (or z = 34 km).

Source Model

The most likely source for AKR thus seems to be local enhancements of the plasma density, simply because they provide the end point convergence to close the feedback path. The wave reversals at the end points would then have to occur as a partial wave reflection, provided the boundaries are sufficiently thin compared with the wavelength (delta W less than or equal to lambda) and the refractive index increase across the boundary is sufficiently large. This model conveniently allows the radiation to escape (a problem with some of the earlier theories) by penetrating directly through the source boundary and leaving behind only the small partially reflected portion needed to keep the oscillator going.

Figure 8. A closed feedback path produced inside a density enhancement by partial reflections at its boundaries. This idealized example was drawn for a 20% density increase 20 km across and having boundaries 1 km thick. The refractive index contours would pertain to f = 250 kHz, r = 1.85 R_E, and f_N/f_H = 0.08, and hence to a refraction scale height H_R of 25 km inside the source and 20 km outside. As a result, the n = 0.7 contour (at xi = 1.48) is displaced upward by 7.4 km at the source boundary and exhibits the 10^o convergence angle needed for path closure. Since the refractive index at the path end points increases from 0.64 inside the source to 0.76 outside, the reflection coefficient would be 0.086, and the gain threshold for producing AKR would be 42 dB. This diagram is aligned with the magnetic field strength gradient, neglecting the angle between grad B and the magnetic field direction.

There is good evidence that the requisite enhancements probably exist in the AKR source region, from the ISIS I observations of Benson and Calvert [1979]. Such enhancements, of sometimes an order of magnitude in less than 100 km, were found to occur inside the auroral plasma cavity and quite close to the location from which AKR seemed to originate. Although the largest enhancements observed with ISIS I would have had internal densities too high for the local generation of AKR by the cyclotron instability, the association between AKR and density enhancements was unmistakable. There is also some evidence, from both ISIS I and ISEE 1, for brief excursions of the whistler mode noise to higher frequencies in the vicinity of the AKR source [e.g., James, 1980, Figure 10, at 01:51:30 UT and 0.1 MHZ], and this would also suggest localized density enhancements. The ISEE I cases (unpublished) would tend to indicate plasma frequencies of around 20 kHz at 2 R_E (f_N/f_H = 0.1), and they could well represent an upward extension of the larger enhancements observed with ISIS 1 at lower altitudes.

Although other possibilities exist, the end point partial reflections which complete the feedback path could occur as shown in Figure 8, at perpendicular incidence to the boundary refractive index contours. The particular wave path which accomplishes this will be determined shortly. In this case it happens to lie centered on xi_o = 1.32 (z = 33 km), and its radius of curvature at that point is R = H_R = 25 km, according to equation (23). For a given value of beta, this feedback path is unique, since the ray path curvature decreases with attitude, and this would give end point wave angles which are too large at the lower altitudes (xi_o less than 1.32) and too small at the higher altitudes (xi_o greater than 1.32). The convergence angle and the source thickness (here 10^o and 20 km) thus determine the altitude for closure of the feedback path and, thereby, through equation (15), the exact frequency at which feedback could occur. For instance, with f_N/f_H = 0.08 (which would be appropriate for Figure 8 at r = 1.85 R_E, where f_H = 250 kHz), the frequency offset would be only 0.8% or 4 kHz. This geometrical selection of the closed wave feedback path also determines the refractive index range encountered by the wave, as well as its change at the boundary. For the feedback path in Figure 8, where these can be read directly from the labeled contours, the internal refractive index varied between 0.62 and 0.64 inside the source, and it increased to 0.76 just across the boundary.

The feedback geometry also determines the wave angles traversed inside the source, as well as the initial angle for the emitted wave. Moreover, by fixing the height above cutoff at which the emission occurs, it establishes the initial refractive index and hence, by Snell's law, the distant radiation direction. For instance, the escaping AKR in Figure 8 originates at an angle of 80' with respect to grad B, and at a height where the refractive index just outside the source is 0.76. Provided the density is reasonably uniform over the escape path, this would imply a distant angle of 48^o, and that would be consistent with the observed AKR emission directions [Green et al., 1977; Calvert, 1981d]. Along with the electron distribution, the wave angles inside the source will control the wave growth rate at different points of the feedback path, and their variation would provide the off-perpendicular wave directions which are needed for growth. Figure 8 would suggest that these internal angles should be skewed with respect to the magnetic perpendicular, favoring upward directions on the equatorward transit and downward directions on the poleward transit, since the magnetic field at the AKR source is inclined equatorward of the magnetic gradient. This could produce an asymmetric emission pattern, depending upon whether the free energy is in the upgoing loss cone or in a downcoming beam. The skewing, however, is inaccurately emphasized in Figure 8, because the expected field-alignment of the density enhancement was neglected, and a more precise treatment will be required to determine how strong the emission asymmetry should be.

Partial Reflections

Provided the boundary is sufficiently abrupt to be treated as a discontinuity, having a refractive index of n₁ inside and n₂ outside, the reflection coefficient rho_i for perpendicular incidence would be

(32)

[see Budden, 1961, equation (8.34)]. For example, the reflection coefficient for Figure 8 would then be 0.086. Since the reflection coefficient specifies the ratio of the reflected to incident amplitudes, only 0.7% of the wave energy would be turned around at the boundary and 99.3% would escape. The partial reflection loss is therefore quite severe, amounting to 21 dB at each end point of the path. Nonetheless, if the other losses can be neglected and the wave growth inside the source is at least 42 dB, then this would be sufficient to produce oscillations.

The reflection coefficient for different situations is shown in Figure 9, which was calculated from equation (32) by using the refractive index from equation (19). Also shown is a scale on the right which indicates the total loop reflection loss, assuming the source is symmetrical, which should equal the minimum internal wave growth G required for oscillations. Since (grad N)/N could be near unity in a strong enhancement, the reflection coefficient could well exceed 10% for xi less than 2, and this would imply a total feedback loss of less than 40 dB. For smaller values of xi, a weaker enhancement could produce comparable reflections, down to xi = 1. I for a 10% enhancement. The optimum location for strong wave feedback thus occurs low in the source, at frequencies as close as possible to the wave cutoff (where xi = 1.0).

Figure 9.. Etraordinary-wave partial reflection coefficient rho for an abrupt density decrease, assuming that the density drops from N₁ inside the source to N₂ = N - (delta N) outside. The corresponding threshold gain G for AKR oscillations is shown at the right.

Uniqueness

The model would produce suitable closed feedback paths and strong partial reflections only near the extraordinary cutoff and hence near the cyclotron fundamental. This would account for the apparent absence of separate harmonic emissions, despite the predictions of wave growth at those frequencies [Lee et al., 1981, simply because the partial reflections would probably be too feeble to produce oscillations so far above cutoff. For instance, the minimum possible refractive index at f = 2 f_H, according to equation (19), should lie between 0.98 and 0.9998 in the AKR source region (where f_N/f_H is less than 0.2 and greater than 0.02), and hence the maximum reflection coefficient would be 10^-2 to 10^-4. Under the best circumstances, the required growth would then have to be 80-160 dB, and that would imply that separate harmonic oscillations would not be likely to occur. A similar argument would account for the apparent absence of separate emissions in the ordinary wave mode near the cyclotron fundamental. On the other hand, this would not preclude the generation of harmonic and ordinary emissions as by-products of the fundamental extraordinary oscillation. The former could be produced in the normal manner by a distortion of the fundamental waveform, whereas the latter could be produced by the polarization mismatch for the escaping wave at the source boundary. It is believed that such byproducts will probably account for most of the harmonic and ordinary signals which have been observed. The observed Z mode signals are a separate matter, however, since the Z mode refractive index varies considerably just below the upper hybrid resonance, and that could produce separate Z mode oscillations.

Path Position

As illustrated by Figure 8, closure of the feedback path is determined by the internal wave refraction and the convergence angle, and it occurs at the particular altitude where these two exactly cancel. For perpendicular endpoint reflections, that cancellation occurs when gamma equals beta and W/H_R = 2 chi, where W is the source thickness. Furthermore, since the wave angle gamma is a monotonically decreasing function of xi (for xi_o less than xi and greater than 1), there should always be a unique solution for a given beta and W/H_R. This solution is illustrated in Figure 10, which was calculated from equations (25) and (26) by solving the former for xi and treating xi_o as a variable parameter. Although this diagram is inaccurate for the larger angles (because of the approximations in equation (19)), it should still represent the general nature of the solution. The lower panel is an enlargement of the dashed portion in the upper panel, and it is the part probably most pertinent to AKR. The vertical axis represents simultaneously the frequency offset (delta f) = xi f_N²/f_H) and the vertical displacement (z = xi H_R) of the path end points. The axis on the right gives the corresponding end point refractive index, according to equation (19).

Figure 10. The vertical position of the feedback endpoints, xi = x/H_R, for an AKR source of thickness W and convergence half angle beta. H_R is the refraction scale height, and the corresponding end point refractive index is shown on the right.

For a given source thickness and vertical position, there can be two convergence angles which produce feedback. These correspond to the low and high waves of shortwave ionospheric reflection just beyond the skip zone, and their occurrence is a more-or-less general property of oblique reflection by a refractive index gradient. In this case they correspond to the shallow and deep wave paths which would return to their original level at the same distance. For instance, in Figure 10 at W/H_R = 1.47 and xi = 1.54, the two wave angles would be 10^o and 60^o, since the curves for those two angles cross at that point. This raises the possibility of a compound feedback path, traversing the shallow path in one direction and the deep path in the other, and this would require a source convergence angle which equals the average of the two angles, or 35^o in this case. It is believed that such compound paths could participate in the production of AKR, along with certain other more complex paths involving multiple transits of the source.

Figure 10 also exhibits a caustic focus at which the large and small angles coalesce, and below which no solution exists. This is the counterpart of the shortwave skip zone, and it implies that a thicker source must necessarily involve larger values for xi and, consequently, internal refractive indices which are closer to unity. It also implies a minimum xi for a varying beta and constant W/H_R and hence presents a possible explanation for the frequency drift reversal in Figure 4, as will be discussed below.

Summary

In this section it has been shown that wave feedback is feasible but that it requires vertically converging end reflectors at the opposite boundaries of the source to compensate for internal wave refraction and produce a closed path. This led to the conclusion that a likely density structure for the source of AKR would be a local enhancement, since the refractive index contours at its boundaries would be convergent, as pictured in Figure 8. Provided its boundaries are abrupt, this same structure would produce partial reflections at the path end points, for which the reflection coefficient could be as large as 10%. This would imply that an internal wave growth of around 40 dB should be sufficient to produce oscillations.

DISCUSSION

The AKR spectra will now be interpreted on the basis of the new source model. Suffice it to say that the model fits exceedingly well and accounts for much of the observed behavior.

Discreteness

First and foremost, the new model accounts for the discrete components. This is an immediate consequence of oscillations at the source, since those oscillations would require in-phase feedback, and that can occur only at specific locations and frequencies. Once the oscillations occur, they will quickly produce saturation signals with quite narrow spectral bandwidths, limited only by the ambient noise and the temporal stability of the oscillator. As discussed above, the latter probably establishes the bandwidth for AKR, since the predicted noise bandwidths are much too narrow.

The specific locations at which the oscillations could occur are established by the source thickness and the phase delay around the loop in Figure 2. For in-phase feedback, the wave which completes the loop must do so in an integral number of wavelengths. Although the wavelength varies with both the frequency and the refractive index (lambda = c/n f, where c is the speed of light), both of these are determined by path closure and the analysis just completed, with the conclusion that the frequency is always quite near the local cyclotron frequency (delta f much less than f_H) and the refractive index is a slowly varying function of beta, W, and H_R. As a consequence, the wavelength for feedback should increase with altitude more or less inversely with the decreasing cyclotron frequency, and hence the oscillations should occur only at those specific vertical sites where its value is correct.

The previous amplifier model cannot readily account for the discrete AKR components without some mechanism to limit its vertical extent. Although such an amplifier might be expected to function only over a narrow frequency interval near the local cyclotron frequency, the different amplifiers at different altitudes should produce a more or less uniform amplification of the incident ambient noise. In order to account for the discrete AKR with a 1-kHz bandwidth, the amplifiers at certain altitudes would have to be much more efficient than their neighbors (by perhaps 20 dB or more) only a few kilometers away. The oscillator model, of course, serves the purpose of limiting the source height without invoking any vertical structure other than the gradual decrease of the local magnetic field. Although a possible alternative to an oscillator or a vertically thin amplifier would be the occurrence of discrete components in the background noise, there is little reason to expect such preexisting components, nor have they been observed independently of AKR. Besides, this would beg the question of how the discrete spectrum originates in the first place.

Frequency

Since an oscillation requires an integral number of wavelengths around the loop in Figure 2, its frequency should be determined by

(33)

where W is the source thickness, lambda is the wavelength, and m is the number of whole wavelengths both ways across the source. Since the wavelength is c/n f, where c is the speed of light and n is the refractive index, the frequency should be

(34)

This implies that the AKR source might produce a harmonic spectrum, provided the source thickness and refractive index are nearly the same for the different values of m. This harmonic structure is the counterpart of the longitudinal mode structure in a laser, in which the different modes are collocated and sometimes interact. In AKR, however, these modes, which must simultaneously satisfy equation (34) and occur near the local cyclotron frequency, would occur at different altitudes where they could presumably exist quite independently of one another.

Multiplets

Among the most striking aspects found in the AKR spectra are the multiple discrete components with roughly equal spacing, and they will be attributed to oscillations in the same source with different values of m. An example of these multiplets occurs in Figure 1 at 0550 UT and it is shown enlarged in Figure 4. Although they aren't always this conspicuous, multiplets are quite common, and it is usually possible to find at least one potential companion for a slowly drifting discrete component. More typical of the AKR, however, are discrete components drifting rapidly (e.g., those found elsewhere in Figure 1), in which multiplets are suspected but never very obvious.

Both the feedback paths and the wave instability which drives the oscillation must occur near the local cyclotron frequency. This implies that the multiplet components would have to be generated at different altitudes in the source, as depicted by Figure 11. Thus to produce multiplets, the AKR source would have to be elongated vertically, presumably along the magnetic field, and the possible variation of either H_R or the source thickness with altitude could introduce a departure from exact harmonic spacing. Since the cyclotron frequency decreases with altitude, the multiplet components at the lower frequencies would originate at the greater altitudes, and vice versa, each from an altitude where the source thickness is a half-multiple of the wavelength, and hence where the constructive feedback could occur. The frequency spacing for the multiplets, (delta f)_m, should then be given by

(35)

and the mode number by

(36)

assuming that the n and W are reasonably constant between adjacent components.

Figure 11. The generation of AKR multiplets at the different altitudes in a vertically elongated source where the feedback path length differs by whole wavelengths lambda. The multiplet frequency spacing would then indicate the product of the source thickness W and the average refractive index n inside the source, according to equation (35).

Source thickness

According to equation (35), the 10 kHz spacing in Figure 4 (e.g., measured at 05:48:13 in the upper panel) would then indicate that n W = 15 km, and for an internal source refractive index of around 0.6, the source thickness would be roughly 25 km. This source would thus be approximately 25 wavelengths across (since lambda = lambda_o/n = 1 km), and so the mode number (equation (36)) would be some integer near 50. Since the oscillations each occur near the cyclotron frequency, the multiplet spacing also indicates the vertical distance between the adjacent oscillation sites, which would be r (delta f)_m/3 f_H in a dipole field at high latitudes, where r is the geocentric distance. For f_H = 505 kHz and r = 9500 km (which would be appropriate for the multiplet in Figure 4), this vertical spacing should be 63 km. By a similar argument, the bandwidth of the individual components would indicate that the oscillations at each site must be limited to a 6 km height interval. The AKR source suggested by the multiplets in Figure 4 can thus be pictured as a sparse stack of active sites, 25 km front-to-back and 6 km top-to-bottom, spaced 63 km above one another. A dozen such cases have been examined (all with relatively slow drifts but at different frequencies), and they yielded similar results within a factor of 2.

Drifts

The frequency drifts of AKR [Gurnett et al., 19791, which are mostly upward in Figure I but which reverse in Figure 4, clearly indicate that the source must be changing while the AKR is being produced. Gurnett and Anderson [1981] would attribute these drifts to the motion of propagating ion-acoustic disturbances which somehow bring about the generation of AKR at different altitudes where the cyclotron frequency is different. However, the new model suggests an alternative interpretation, since the frequency would then be determined by equation (34) and the value of nW. Although the oscillations still occur near the local cyclotron frequency, their altitudes are now determined by the need for inphase feedback rather than by some external agent. The frequency drift would then represent the history of the nW product during the life of an oscillation site.

The upward drifts characteristically found in AKR would thus require a decreasing refractive index, a decreasing source thickness, or both. Although it would be tempting to claim that a change of thickness accounts for the drifts, the observed drift reversals would then pose a problem, since there is no reason to expect that the source thickness should rebound. The Jikiken observations of these reversals [Morioka et al., 1981; H. Oya, private communication, 1982] suggest that they are relatively common and tend to occur toward the end of a discrete component, after a long upward excursion of perhaps 15-20% frequency increase. This would imply that the drift reversals are not accidental but are probably instead a necessary feature in the later life of some oscillation sites, and this consequently suggests that the variation of refractive index must play some role in the drifts.

Assuming the endpoint refractive index is characteristic of its average over the feedback path, the drift reversals would require a decrease of xi in Figure 10 followed by an increase, since that would produce an excursion through a minimum of the refractive index and hence a excursion through a frequency maximum. Unfortunately, a decreasing thickness would appear to produce only a decreasing xi (even with the change of beta according to equation (31)), and that would only serve to accentuate the upward frequency drift for a contracting source. Similarly, a change of the ambient source plasma density, which could affect H_R via equation (18), would probably not account for the drifts, primarily because the upward drifts would require an unlikely increase of the source density and that, too, would have to rebound to produce a drift reversal. Consequently, if a single change accounts for the drifts, it must be that of the convergence angle beta. In fact, this explanation becomes virtually irresistible once it's realized that beta is a double-valued function of xi in Figure 10 and that its excursion through the caustic focus in that figure would produce the excursion through a minimum of the refractive index that a drift reversal would require.

The convergence angle depends primarily on the density gradient at the source boundary, as expressed by the ((delta W)/W)/((delta N)/N) factor in equation (31), where delta W/W represents the enhancement shape and (delta N)/N represents its relative magnitude. Since a decrease of the gradient would yield a larger beta, the convergence angle should increase as the enhancement dissipates, that is, as (delta N)/N decreases or (delta W)/W increases. With this in mind, the effect of those two factors is shown in Figure 12, in which the left side corresponds to a strong enhancement with a sharp boundary, and the right corresponds to one that is diminished and less distinct. This figure shows the reciprocal of the refractive index, which should be proportional to the frequency according to equation (34), for different values of the source thickness. It reproduces surprisingly well the general behavior of the AKR frequency drifts, and for W less than H_R it exhibits a sufficient relative frequency range to account for that observed with Jikiken. It is thus concluded that the increasing convergence angle in a dissipating enhancement could account for the drifts, apart from any contraction of its thickness or increase of its density.

Figure 12. The expected reciprocal refractive index variation for a diminishing boundary density gradient, computed by iteration by using equation (31) and Figure IO, for sources of different thickness W relative to the refraction scale height H_R. For a drifting AKR spectral component, the abscissa should be proportional to the frequency according to equation (34), whereas the ordinate should increase with time for a dissipating enhancement as its relative density excess (delta N)/N decreases or its shape factor (delta W)/W increases.

Source History

This raises the following picture for the history of an auroral density enhancement and the AKR oscillations it produces. The enhancement, produced initially by some unknown agent, would begin with a large (delta N)/N and sharp boundaries. Although the partial end point reflections should then be strong, the convergence angles would be small and this would produce only shallow feedback paths displaced well above cutoff, where the refractive index is still quite near unity. As a result, the wave instability would presumably produce insufficient gain for oscillations at most altitudes, except at those near the f_N/f_H minimum [see Calvert, 1981b], where the refraction scale height H_R would be least and the convergence angles consequently the greatest (see equation (31)). The AKR oscillations should thus begin near the f_N/f_H minimum, at those particular sites where the feedback is in phase. As the enhancement dissipates (possibly as a consequence of the ongoing AKR), the convergence angles would then increase and the feedback paths would occur closer to the cutoff and hence at a smaller refractive index. The sites with ongoing oscillations would then have to move downward to keep the proper wavelength for in-phase feedback, whereas new oscillations could then be initiated at progressively greater and lesser altitudes, despite the larger H_R which occurs there. This progressive onset, incidentally, might explain the tapered beginning of an AKR sequence noted by Kaiser and Alexander [1977b]. As the sites move downward, their frequency would also be affected by the spatial variations of H_R and W. Subsequently, when the convergence angle reaches 20^o to 40^o and the caustic focus for oblique propagation is encountered, the feedback paths would then begin to move further from the cutoff, and their average refractive index would begin to increase again. As a result, the oscillation sites would rebound upward and their frequency drifts would reverse, again probably at progressively greater and lesser altitudes from the f_N/f_H minimum. Ultimately, as the enhancement dissipates further and its convergence angles continue to increase, the end point reflections would become weaker and the wave geometry inside the source would become less favorable for the instability, until the oscillations finally extinguish. The frequency history of the discrete AKR components thus reflects the history of the dissipating density enhancements from which they originate.

Summary

In this section it was emphasized how feedback oscillations at the AKR source would account for its discrete spectrum. These oscillations can occur only at the specific sites where positive feedback exists, and this accounts for the observed multiplets, more or less harmonically spaced in frequency. It implies that the multiplets originate from different altitudes within the same source enhancement, and their observed spacing would indicate a source thickness of roughly 25 km. The frequency drifts of the AKR discrete components are believed to be a consequence of the changing refractive index inside the source, caused by a dissipating density gradient at its boundary and the consequent increase of the convergence angle that that would cause.

CONCLUSIONS

The new concept that auroral kilometric radiation originates from self-excited wave feedback oscillations accounts for its occurrence as discrete spectral components. These oscillations would occur near the cyclotron frequency at localized sites where the source thickness is a half-multiple of the wavelength and the internal wave growth is sufficient to overcome the end reflection losses. In a vertically elongated source, this could occur simultaneously at different altitudes, since the wavelength should increase as the cyclotron frequency decreases, and this would account for the observed spectral multiplets. In addition, an oscillator would always be expected to saturate, and this would account for the apparent quenching in the ISEE-I high-resolution spectrograms (e.g., in Figure 4). The basic concept, therefore, readily explains much of the puzzling behavior left unexplained by the previous open-loop amplifier model.

The occurrence of the AKR spectral multiplets provided an opportunity to estimate the product of the source thickness and the refractive index inside the source, which was never found to be much larger than a few tens of kilometers. Although this is smaller than the previously assumed wave interaction distances, an oscillator would require substantially less total wave growth than an amplifier (e.g., 40 dB instead of the 70-120 dB previously believed necessary), and that could be provided by a roughly similar spatial wave growth rate of a few decibels per kilometer inside the source. Such wave growth would be quite consistent with the Doppler-shifted cyclotron resonance instability previously proposed by Melrose [1976] and Wu and Lee [1979], in which the extraordinary wave is resonant with the energetic, inverted V, auroral electrons and draws upon the free energy of their loss cone.

The very simple deduction that all oscillators must saturate yields the immediate conclusion that the AKR signals inside the source should almost always occur at the wave growth saturation level. Since such signal levels typically involve a high conversion efficiency, this could account for the high efficiency of the AKR generation process first noted by Gurnett [19741. Although the exact nature of the saturation process remains open to question, if it were the pitch angle scattering of electrons by the AKR wave and the consequent destruction of the loss cone free energy, then the saturation amplitude should be the order of 20-200 mV/m, and that is in rough agreement with the observed AKR amplitudes. Short of saturation, the wave amplitude should quickly drop to the much lower levels of resonantly amplified ambient noise (e.g., a few microvolts per meter in a 1 kHz BW), and that would account for the sporadic nature of AKR and its propensity to always occur at the same full amplitudes.

Local enhancements of the plasma density seem to be the most likely auroral structure for producing the closed wave feedback paths that oscillations require, since they would automatically produce vertically convergent refractive index contours at their opposite boundaries. This vertical convergence is needed to compensate for the upward wave refraction inside the source, and it would produce shallow feedback paths roughly perpendicular to the magnetic field strength gradient, like that illustrated in Figure 8. The end reflection for these paths could then be attributed to partial wave reflections in which perhaps 1% of the wave energy is reflected back and the rest is allowed to escape as AKR. A similar reflection at the opposite end of the path would thus close the loop for 10^-4 of the wave energy, but that would be sufficient for oscillations, provided the internal wave growth exceeds 40 dB.

The wave feedback source model depends critically upon the extraordinary refractive index behavior near cutoff, which would produce closed paths and strong partial reflections only in that wave mode and then only quite near the local cyclotron frequency. This is not only optimum for producing the fundamental extraordinary emissions by the cyclotron instability, but it also accounts for the apparent absence of separate sources in either the ordinary mode at the cyclotron frequency or in the extraordinary mode at the cyclotron harmonics.

The new model for the production of AKR would find immediate application in deducing the source properties from the distant wave observations. For instance, the frequency drifts of the discrete components can now be attributed to the changing product of the source width and refractive index, and the observed drift behavior would suggest that the latter is most important and itself attributable to a decaying density gradient at the source boundary. Also, the multiplet spacing has been used to determine the source thickness in a few cases, and that could be extended to a statistical study with the hope of determining why AKR seems to choose enhancements of a particular size.

It should also find important application in understanding the similar emissions from other planets, like the decametric radiation of Jupiter and the kilometric radiation of Saturn, even though the conditions there may be quite different. It would certainly focus attention on the importance of feedback in establishing much of the emission behavior, and it should stimulate a quest for suitable feedback models in those situations. It also provides some idea of the ingredients needed for strong cyclotron emissions. In addition to electrons with suitable free energy and an appropriate value for f_N/f_H, it would suggest the need for local density enhancements (or perhaps some different feedback-producing structure) as well as a suitable angle between B and grad B so that feedback path traverses the appropriate angles for wave growth. It also would provide, by the existence of an oscillation threshold, a possible explanation for the control of such emissions by seemingly minor changes at the source.

Acknowledgments. This work was supported by NASA grant NGL-16-001-043 and NASA contract NAS5-20093, and it used ISEE wave data processed on equipment provided by the Office of Naval Research. R. R. Anderson, W. S. Kurth, and D. A. Gurnett supplied useful insights and needed encouragement. H. Oya kindly provided the Jikiken observations which were pivotal in explaining the frequency drifts of the discrete AKR components.

The editor thanks H. G. James and M. L. Kaiser for their assistance in evaluating this paper.

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(Received March 4, 1982; revised May 28, 1982; accepted July 15, 1982.)

Copyright 1982 by the American Geophysical Union.
Paper number 2AI053.
0148-0227/82/002A-1053$05.00