Published in Journal of Geophsical Research, volume 100, pages 14,887-14,894, August 1, 1995.
AN EXPLANATION FOR AURORAL STRUCTURE AND THE
TRIGGERING OF AURORAL KILOMETRIC RADIATION
W. Calvert
Department of Physics and Astronomy,
The University of Iowa,
Iowa City, Iowa
Abstract. Auroral electron precipitation can be attributed to energetic electrons which have been scattered into the loss cone by the cyclotron maser instability, whereupon the structure of the diffuse aurora can then be attributed to enhanced scattering in density depletions as a result of the increased wave gain of the cyclotron maser instability at lower densities. The resulting electron precipitation, moreover, also presumably enhances the density depletions in which it occurs, thus causing arc-shaped density depletions as a result of particle drifts which then account for the formation of homogeneous diffuse arcs prior to substorm expansion. Discrete arcs, on the other hand, have previously been attributed to radio lasing caused by wave feedback inside such depletions, and the onset of discrete arcs at substorm onset can then be attributed to the onset of radio lasing within these arc-shaped density depletions. Substorm onset should therefore be delayed less during the arrival of stronger incoming waves being amplified by the cyclotron maser instability to produce the diffuse aurora prior to substorm expansion, thus accounting for the observed triggering of auroral kilometric radiation (AKR) by external waves and also suggesting that the similar triggering of adjacent arcs by the stray field of the resulting AKR then accounts for the subsequent latitudinal expansion of the discrete aurora during a substorm.
On closed auroral field lines, energetic electrons in the loss cone for precipitation into the ionosphere obviously precipitate into the ionosphere to cause the aurora, whereas those outside the loss cone mirror in the Earth's magnetic field and do not cause the aurora. The key to understanding the aurora is therefore understanding how energetic electrons end up in the loss cone.
As discussed by Whipple [1977] and Chiu and Schulz [1978], the loss cone becomes enlarged in the presence of upward electric fields which accelerate electrons downward into the ionosphere, and this has been used to measure electric fields in the high-altitude auroral zone [Evans, 1974; Fennell et al., 1981]. These electric fields are thus found to span the few degrees of latitude occupied by inverted V electrons and to exhibit field-aligned potentials of a few kilovolts at altitudes of 2000 to 15,000 km above the auroral zone [Reiff et al., 1994]. Outside thin regions corresponding to discrete auroral arcs, the accompanying electrons are characterized by approximately isotropic velocity distributions with empty or nearly empty bidirectional loss cones, whereas inside these regions these loss cones become filled or partially filled by energetic electrons of about the same energy as the peak energy outside the loss cone [P. H. Reiff, private communication, 1994].
Although frequently cited as evidence for localized electron acceleration, these observations clearly do not support this point of view, since the observed electric fields are not localized as predicted, and the predicted acceleration into the empty loss cones from which they are measured obviously does not occur where this model predicts that it should. Moreover, measurements of auroral electrons also fail to support this point of view, since they frequently show multiple energies and broad energy distributions which are clearly impossible by this mechanism [Whalen and Daly, 1979; Hoffman and Lin, 1981]. On the other hand, if localized acceleration cannot account for the electrons in the loss cone which cause the aurora, this then leaves scattering by waves, as originally proposed by Kennel and Petschek [1966], Parks et al. [1972], Kropotkin [1972], Whalen and Daly [1979], and others. Bryant [1981] has also shown that such scattering can account for the narrow energy peaks which are traditionally attributed to electron acceleration.
It is therefore proposed that auroral electron precipitation results from the scattering of energetic electrons into the loss cone by the loss-cone-driven cyclotron maser instability of Wu and Lee [1979], since this instability automatically causes the required scattering and presumably produces the AKR which accompanies the discrete aurora during substorms [Voots et al., 1977; Kaiser and Alexander, 1977; Benson and Akasofu, 1984; Calvert, 1985c, Huff et al., 1988; Baker et al., 1990]. It has also previously been suggested that the discrete aurora results from such pitch angle scattering during the generation of AKR by radio lasing [Calvert, 1987], and the purpose of this paper is to extend this theory to account for the diffuse aurora, substorm onset, and the triggering of AKR by external waves.
The loss-cone-driven cyclotron maser instability, which was originally proposed by Wu and Lee [1979] to account for AKR, results from the cyclotron-resonant scattering of energetic electrons into the loss cone which is caused by electron precipitation into the ionosphere. On closed auroral field lines, however, these electrons which are scattered into the loss cone obviously precipitate into the ionosphere, and this instability therefore also causes electron precipitation.
Figure 1. Geometry for electron scattering by a circularly polarized wave electric field.
As shown in Figure, 1 for waves in the extraordinary wave mode at large angles to the magnetic field near wave cutoff in the low-density, high-altitude auroral zone, the wave electric field is approximately circularly polarized perpendicular to the magnetic field in the same sense as electron gyration [Wu and Lee, 1979; Allis et al., 1963, chapter 4]. This then causes electron deceleration perpendicular to the magnetic field for electron phase angles from minus pi radians to zero radians, and it is this deceleration which is not matched by an equal and opposite acceleration of electrons outward from the empty loss cone which causes this instability.
Figure 2. Scattering into the loss cone by the cyclotron maser instability.
A vector diagram for the scattering of electrons into the loss cone by the loss-cone-driven cyclotron maser instability is shown in Figure 2, where
is the scattering angle, alpha is the pitch angle of the loss cone, v is the average electron velocity, delta v = v1 - v2 is the change of velocity, and v1 and v2 are the initial and final velocities. The electron velocity before scattering is therefore
and solving for v12 - v22, with the tangent of delta much less than the tangent of 2 times alpha, then yields
Since m(v12 - v22)/2 is the energy which is lost by an electron, where m is its mass and mv22/2 is the final energy after scattering
where Pwave is the power which is transferred to the wave and Paurora is the power of the electrons which are scattered into the loss cone.
Neglecting absorption and the initial power of the waves, this equation then relates the power of the waves which are emitted by the cyclotron maser instability to the resulting power of the aurora. If AKR is then generated by the loss-cone-driven cyclotron maser instability with an efficiency of 1%, as reported by Gurnett [1974], a scattering angle no greater than 0.7 degrees, for loss cone angles of 20 to 25 degrees, would then be sufficient for electron scattering during the generation of AKR to account for the total power of the aurora.
The discrete spectrum of AKR detected by Gurnett and Anderson [1981] has been attributed to wave feedback inside magnetic field aligned density structures, thereby causing coherent oscillations as a result of the consequent growth to saturation and subsequent quenching of all but a single frequency [Calvert, 1982]. This is then called radio lasing because this is exactly what happens inside optical lasers to produce all of their remarkable properties, as discussed by Verdeyen [1981], and the electron scattering by the resulting laser wave field then accounts for the localized electron precipitation of the discrete aurora [Calvert, 1987].
Although questioned by Pritchett [1986], Pritchett and Winglee [1989], and McKean and Winglee [1991], radio lasing still remains the only explanation for both the observed coherence of AKR in different directions [Baumback et al., 1986] and the incredibly narrow spectral bandwidths which its discrete spectra have been found to exhibit [Baumback and Calvert, 1987]. This lasing was also predicted to occur in density depletions [Calvert, 1987], and these depletions have also apparently now been observed by Hilgers [1992], Hilgers et al. [1992], and Roux et al. [1993].
The multiple, equally spaced, discrete spectra detected by Gurnett and Anderson [1981] are thus attributed to multiple lasers at different altitudes where the round trip distance between reflections is an integral multiple of the wavelength [Calvert, 1982]. The laser length can then be determined from the spectral spacing of these discrete spectra and is thereby found to be about 25 km. According to laser theory [Verdeyen, 1981, chapter 5], this then implies a gaussian width for the laser wave field of only about 4 km, and the strength of this field has also been measured by De Feraudy et al. [1987] and Bahnsen et al. [1989], who reported field strengths of 10 to 50 mV/m at the source of AKR.
The phase shift of an electron during scattering can then be calculated as follows. Although the frequency of a laser is constant with position, the cyclotron frequency varies with altitude and the resulting phase shift then equals the integrated angular frequency difference between the wave and cyclotron frequencies along the electron orbit. For a dipole magnetic field where B is the magnetic field strength and r is the geocentric radial distance, the vertical gradient of the magnetic field strength is -B/r, and this frequency difference then becomes
where fce is the cyclotron frequency, f is the wave frequency, and h is the vertical distance from the height for exact resonance. The phase shift thus reverses at the height of exact resonance and the maximum phase shift for exact resonance at the center of the laser is therefore
where w = 4 km is the laser width along the magnetic field. For 5 keV electrons v = 3 x 107 m/sec, and for f = 250 kHz at r = 1.8 RE where alpha is approximately 20 to 25 degrees, the expected phase shift is therefore only about 3 degrees, thus justifying the assumption of a constant phase during scattering.
An electron inside a laser it is thus subjected to an electric field which is rotating in the same direction at approximately the same rate, and for a constant phase, its change in velocity perpendicular to the magnetic field is therefore
where e and m are the electron charge and mass and w is the width of the laser wave field. Substituting this into Equation (1) and averaging over phase angles from minus pi to zero radians then yields
where We = mv2/2e is the electron energy in electron volts. The scattering angle for 5 keV electrons in a 4-km-wide, 50 mV/m laser is therefore about 0.7 degrees, in excellent agreement with that required for electron scattering during the generation of AKR to account for the total power of the aurora.
Although the predicted scattering angle is relatively small, the filled or partially filled loss cones which cause the discrete aurora can then be attributed to scattering by multiple lasers at different altitudes. Based on an estimated 5 kW power for an individual laser and different estimates for the total power of AKR, the observed AKR presumably originates from 8000 to 200,000 lasers, and if most of these occupy a few discrete arcs extending up to 5000 km along the auroral zone, this then suggests the order of a few to 160 overlapping lasers to produce the filled loss cones which cause the discrete aurora. This then also suggests that the scattering inside these lasers does not occur uniformly over their entire length, since when the loss cone becomes filled at one location the gain which is necessary for lasing must then be provided at other locations inside the laser where the loss cone has not yet been completely filled.
The radio lasers causing discrete arcs are assumed to be oriented across arc-shaped density depletions which then define the structure of these arcs. The observed laser length projected into the ionosphere then accounts for the arc widths of a few kilometers which have been reported by Stormer [1955], Kim and Volkman [1963], and others, and this same model also accounts for the thin fine structure of these arcs which has been reported by Elvey [1957], Akasofu [1961], Maggs and Davis [1968], Borovsky et al. [1991], and others, as follows.
As shown in Figure 3, the reflections for feedback thus occur along the edges of these density depletions and the reflected wave is then amplified by the cyclotron maser instability as it travels back across the depletion. For weak feedback, the reflected wave should then start out weak and peak sharply at the opposite end of the laser, thus producing sharp enhanced edges along the sides of the resulting discrete arc. Since the wave gain of the cyclotron maser instability can be as much as a few dB per kilometer [see Pritchett, 1986], this then implies enhanced edges along the sides of these arcs which are only a few hundred meters thick.
Figure 3. Reflection inside a laser causing sharp edges which account for the fine structure of discrete arcs.
Figure 3 was calculated for an idealized laser which is 25 km long and powered by a uniform gain from left to right of 2.4 dB/km. The total wave growth inside this laser is then 60 dB, corresponding to an average reflection coefficient of 3% at both ends. The wave which returns to the left end, which is shown by a dashed line in the middle panel of this figure, is then 30 dB weaker than the amplified wave at the right end, thus producing a reflected wave at the left end which is 60 dB weaker and hence correct to regenerate itself. The wave which is being amplified from left to right thus increases by 60 dB as it crosses the density depletion, thus causing precipitation which increases by a factor of a thousand, since according to Equation (4) this precipitation should be proportional to the wave power divided by the scattering angle and according to Equation (7) the scattering angle is proportional to the wave amplitude.
This model thus explains why this feature occurs along edges of arcs in Figures 1 and 2 of Akasofu [1961], and also goes on to explain its shape, in which one side is sharp and the otherdecreases exponentially, as shown in Figure 2 of Borovsky [1993]. Borovsky's Figure 2 also shows that the e-folding distance on the exponential side is about 2.2 times the apparent fine-structure thickness. The edge in Figure 3 thus corresponds to a fine-structure thickness of about 500 m, and the thinnest fine structure which has been observed, which is only about 100 m thick, would then require a wave gain which is five times greater, or about 12 dB/km. Although this is somewhat greater than expected for the cyclotron maser instability, this model otherwise accounts perfectly for the shape, position, and approximate thickness of this structure.
This new interpretation also resolves the apparent discrepancy of arc width observations by different observers, since when the feedback is stronger this effect should diminish and produce sharp-edged arcs occupying the full width of the density depletions. It also suggests that the AKR radio lasers tend to be high-gain, low-feedback lasers, in contrast with optical lasers which are low-gain, high-feedback systems because of the relatively low gain of optical stimulated emission, thereby further suggesting that the feedback for lasing in density depletions could be produced by wave diffraction as the amplified beam, which is refracted upward by the magnetic field strength gradient inside these lasers, is reflected upward and out of the laser at its ends [see Calvert, 1982]. This then also explains how the radiation escapes, and it can be shown that such feedback should be sufficient for lasing in steep-walled density depletions of roughly 35%.
Once lasing begins inside a density depletion, the resulting electron precipitation should then also further reduce the density, thus accounting for the development of organized arcs in which the lasing expands to higher and lower altitudes [Calvert, 1987], and thereby also explaining why AKR tends to expand from a single frequency during substorms, as observed by Kaiser and Alexander [1977].
Using all-sky-camera photographs from Canada, Alaska, and Siberia, Akasofu 1963, 1964] found that the aurora typically begins as diffuse patches and glow which then develop into homogeneous diffuse arcs just prior to substorm expansion. This is then followed by a sudden local brightening of a single diffuse arc in the midnight sector which then marks substorm onset, as defined by Rostoker et al. [1980, 1987]. Akasofu also noted that accompanying diffuse arcs tend to remain faint and diffuse until after this initial arc has become fully developed, whereupon these arcs then suddenly brighten "as if a new arc is suddenly formed in the midnight sky" [Akasofu, 1964, p. 276].
In order to account for this behavior, the diffuse aurora prior to substorm expansion is attributed to open-loop amplification by the cyclotron maser instability, and its structure to the increased wave gain of this instability at lower densities, since this should then cause stronger amplified waves in density depletions as shown in Figure 4, and hence somewhat fuzzy localized diffuse electron precipitation. Moreover, since this precipitation should also gradually enhance these depletions, this presumably also causes arc-shaped density depletions by particle drifts which then account for the structure of homogeneous diffuse arcs. The initial irregular structure of the diffuse aurora is therefore attributed to enhanced precipitation in ambient density depletions and its subsequent development into homogeneous diffuse arcs is attributed to this precipitation gradually enhancing the density depletions in which it occurs.
This should then cause arc-shaped density depletions in which the wave gain and feedback gradually increase, and the sudden local brightening of a homogeneous diffuse arc at substorm onset can then be accounted for by the onset of radio lasing inside these density depletions. Substorm onset is thus attributed to the onset of radio lasing inside density depletions which are produced by the diffuse aurora, and this then explains why it occurs locally within a pre-existing homogeneous diffuse arc and also why this diffuse arc then develops into the first discrete arc of substorm expansion.
As shown in Figure 1 of Melrose et al. [1984], the growth rate for the cyclotron maser instability increases by about a factor of 2 as the plasma-to-cyclotron frequency ratio decreases from about 0.3 to 0.2 and the plasma density therefore also decreases by about a factor of 2. The growth rate is therefore approximately inversely proportional to density, and the total wave growth, which could be as much as 100 dB according to the original Wu and Lee [1979] theory, thus increases by about an order of magnitude as the density decreases by only 10%. Quite small density changes are therefore sufficient to account for the structure of the diffuse aurora and the diffuse patches and glow which initiate the aurora probably correspond to ambient density depletions of only a few percent.
Although it is not obvious how electrons migrate toward the loss cone to be precipitated by the cyclotron maser instability, if these electrons are being precipitated and not replaced it is obvious that the average electron density, which is what matters to the cyclotron maser instability, should decrease at a rate which is determined by the precipitation flux, phi, at the foot of the field line, as follows:
where V is the field line volume, A is its footprint area perpendicular to the magnetic field in the ionosphere, rho sub zero is the average density, phi sub zero is the initial precipitation flux, and it has been assumed that the growth rate increases inversely proportional to density. The time required for the average density to decrease by a factor of rho sub zero over rho is then
where V/A for dipole field lines is given by
Since the integral in Equation (10) equals 0.25 for a density decrease by a factor of 0.65, the time which is required for an initial precipitation flux of 10^7 cm^-2 s^-1 to reduce an initial density of 1 cm^-3 by 35% at 70 degrees invariant magnetic latitude where L = 8.5 is therefore about an hour, which is clearly not inconsistent with the apparent time scale of substorm onset.
Figure 4. Enhanced amplification by the cyclotron maser instability causing a diffuse arc.
Since the incoming waves which are amplified by the cyclotron maser instability to produce the diffuse aurora are incoherent and broadband, the particle signature of the diffuse aurora should be a slight change in pitch angle at the edge of the loss cone over a broad energy range. This then accounts for the broad energy spectrum of the diffuse aurora detected by McIlwain [1960], whereas resonant scattering by radio lasers then accounts for the narrow energy spectrum of discrete aurora also measured by McIlwain [1960] and others.
In the laser model, waves traveling in both directions at slight angles to the magnetic perpendicular should cause scattering into both loss cones, whereas in the open-loop model for the diffuse aurora in which incoming waves are amplified during reflection, stronger scattering should occur into the upward loss cone after reflection. For the current purposes, however, it doesn't matter which, since on closed auroral field lines this should cause precipitation into the auroral zone in one hemisphere or the other.
It has long been recognized that seemingly stochastic internal processes affect the actual timing of substorm onset, as discussed by Rostoker et al. [1987]. This model therefore accounts for these stochastic processes by requiring that substorm expansion occur in two stages, first as particle injection which supplies a reservoir of energetic electrons on auroral field lines and then as discrete electron precipitation caused by radio lasing which then dumps these electrons into the ionosphere during substorm expansion. Although this model does not explain electron injection, it thereby predicts that substorm onset should occur with a delay which depends upon the amplitude of the incoming waves being amplified by the cyclotron maser instability to produce the diffuse aurora and this then accounts for the observed riggering of AKR by external waves [Calvert, 1981].
Figure 5. Schematic spectrogram showing AKR being triggered by an incoming type III solar radio burst.
During this triggering AKR begins during a kilometric type III solar radio burst as shown in Figure 5, usually starting at a single frequency near the center of the AKR frequency range and often lasting long after the type III burst has completely disappeared. Such triggering is therefore clearly not an artifact of amplification and it has been verified that it must be caused by the incoming waves, since it also occurs for type II bursts having no other known geophysical effects [Calvert, 1985b]. AKR triggering has also been verified by Farrell and Gurnett [1985] and Farrell et al. [1986], and similar triggering also seems to occur at Jupiter [Calvert 1985a], although this has been debated on the grounds that the narrow beaming and rapid rotation of the corresponding decametric radio emissions might be confused with triggering [Desch and Kaiser, 1985, Calvert, 1985d].
Since AKR is expected to be a reliable indicator of the aurora, this then implies external triggering of the aurora by incoming waves. The observed triggering therefore contradicts all previous theories based on electron acceleration, since there is no known way electron acceleration can be affected by incoming waves, and it becomes important to explain how AKR can be triggered by such waves.
The triggering of both AKR and the aurora, however, can be understood as shown in Figure 6, which shows schematic radio spectrograms with and without triggering, along with sketches of the density depletion in which lasing occurs at substorm onset. In the top panel, weak incoming ambient noise gradually enhances this depletion, taking perhaps an hour or more before lasing occurs. In the bottom panel, on the other hand, where these incoming waves increase by 40 to 60 dB during a type III burst, the predicted diffuse electron precipitation should increase by two to three orders of magnitude, thus collapsing hours into minutes and thereby accounting for the prompt onsets of AKR during the observed AKR triggering events.
Figure 6. Triggering attributed to advancing substorm onset as a result of stronger incoming waves.
The triggering of AKR is therefore attributed to the incoming waves causing stronger diffuse electron precipitation and therefore advancing substorm onset. Moreover, since these waves are amplified by the cyclotron maser instability to produce this effect, this then also explains how such weak waves can have such a dramatic effect.
In order to account for the subsequent latitudinal expansion of the aurora, it is assumed that the resulting AKR then triggers adjacent discrete arcs by the same process. Substorm expansion can then be attributed to one discrete arc triggering another up to the limit of closed field lines, presumably starting somewhere near the low latitude limit for the aurora in order to account for the well-known poleward expansion of the aurora during a substorm.
Figure 7. Proposed triggering of adjacent arcs during substorm expansion.
It is thus suggested that part of the emitted AKR returns to the cyclotron resonance level in order to trigger adjacent arcs as shown in Figure 7. This then also explains Akasofu's [1964] observation of diffuse arcs at substorm onset which tend to remain faint and diffuse until after the first discrete arc of substorm expansion has become fully developed.
Evidence for the ability of AKR to reach the cyclotron resonance level on adjacent field lines is provided by Figure 8 of Gurnett and Anderson [1981] which shows one AKR discrete component triggering others with different spectral spacings and frequency drifts. Since these different spacings and drifts imply a different density structure for these triggered signals, this then suggests that they are triggered on different field lines, and hence that the emitted AKR can reach the cyclotron resonance level at adjacent locations.
The AKR produced by a discrete arc should also be quite intense, having an expected flux of 3 x 10-5 W/m2 at a distance of 100 km from each of a thousand or more 5 kW lasers having the 9 degree beamwidth which is predicted by laser theory. On the other hand, the type III bursts which appear to trigger discrete arcs at substorm onset have spectral fluxes of only about 10-16 W/m2Hz, and hence integrated fluxes of only about 2.5 x 10-11 W/m2 over the entire 250 kHz bandwidth of AKR. The wave signal which appears necessary to trigger adjacent arcs is thus expected to be a million or more times less than the AKR from a single arc, thus suggesting that it could result either by the weak scattering of AKR at higher altitudes, as shown in Figure 7, or by the mode conversion of AKR in the ordinary wave mode, which, although roughly 17 dB weaker that the AKR in the extraordinary mode [Mellott et al., 1984], would not be refracted upward as it escapes and should therefore be more able to reach the cyclotron resonance level on adjacent field lines.
This paper introduces a number of important new ideas which account for the structure and behavior of the aurora and explain the observed triggering of AKR by external waves. Thus accounting for major aspects of the aurora and not being contradicted by anything nearly as fundamental as the conflicting evidence for localized electron acceleration, this therefore constitutes a new theory of the aurora which cannot be dismissed lightly.
This theory is based on the two new concepts that (1) the density on auroral field lines determines the pattern of auroral electron precipitation and that (2) the resulting precipitation significantly reduces the density on these field lines. Its main points are then as follows.
Auroral electron precipitation is attributed to energetic electrons which are scattered into the loss cone by the loss-cone-driven cyclotron maser instability. The diffuse aurora is then attributed to the open-loop amplification of incoming waves and the discrete aurora to closed-loop radio lasing, thereby explaining both kinds of aurora and accounting for their difference.
The structure of the diffuse aurora can then be attributed to enhanced scattering in density depletions as a result of the increased wave gain of the cyclotron maser instability at lower densities, and that of the discrete aurora to radio lasing caused by wave feedback within these depletions. This then accounts for the structure of both kinds of aurora, and the fine structure which has been found to occur along the edges of discrete arcs can also be accounted for by wave growth and abrupt reflection inside the resulting radio lasers.
The resulting precipitation also presumably enhances the density depletions in which it occurs, thus causing arc-shaped depletions as a result of particle drifts which then account for the formation auroral arcs. Homogeneous diffuse arcs, which then develop from random ambient depletions, thereby presumably cause the initial density depletions in which radio lasing occurs at substorm onset.
Substorm onset is therefore attributed to the onset of lasing in density depletions caused by the diffuse aurora prior to substorm expansion, and since this should occur with a delay which depends upon the amplitude of the incoming waves which are being amplified to produce the diffuse aurora, this then accounts for the observed triggering of AKR, whereupon the subsequent latitudinal expansion of the aurora can also be accounted for by the similar triggering of adjacent discrete arcs by the resulting AKR.
The observed AKR triggering, which clearly cannot be accounted for by previous theories based on electron acceleration, has thus turned out to be the key to understanding the aurora, and until this observation is refuted or otherwise explained, no such theories can be considered to adequately account for the observed properties of the aurora.
Acknowledgements. This paper is dedicated to the memory of my good friend C. K. Goertz, who was one of the few who appreciated the significance of these ideas. Useful discussions with D. D. Wallace, B. A. Wahlen, J. Stadsness, J. E. Borovsky, R. F. Benson, and P. H. Reiff are also acknowledged, as are the services of A. J. Erk. This work was supported in part by NSF Grant ATM-93-12013.
The Editor thanks D. B. Melrose and M. L. Kaiser for their assistance in evaluating this paper.
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(Received November 16, 1994; revised January 27, 1995; accepted February 7, 1995.)
Copyright 1995 by the American Geophysical Union.
Paper number 95JA00523.
0148-0227/95/95JA-00523$5.00