Pages 1067-1073


W. Lotko and A. V. Streltsov

Thayer School of Engineering, Dartmouth College, Hanover, NH 03755, USA, E-mail:


Multipoint ``satellite'' diagnostics from a magnetically incompressible, two-fluid simulation model for resonant, dispersive Alfvén waves standing on auroral field lines suggest a causal link between shear Alfvén resonances, usually identified on the basis of ground-based and equatorial magnetospheric signatures, and some of the large-amplitude, kilometer-scale electric fields, typically measured by polar-orbiting satellites in the lower magnetosphere. It is also shown that one documented DE 1 electric field event exhibits the waveform, impedance, and phase signatures of a dispersive field line resonances.


The Earth's magnetosphere resonates through a continuum of frequencies determined by the length of the geomagnetic field lines and the shear wave propagation speed along them. These standing oscillations, known as field line resonances (FLRs), are the geomagnetic analogues to resonances of a tied string. The existence of FLRs was confirmed more than two decades ago with the application of ideal MHD theory to ground-based observations of high-latitude, toroidal micropulsations [see Samson, 1991 for a review]. Low-inclination magnetospheric satellites have since surveyed their harmonic behavior, compressibility, and statistical occurrence probabilities [Anderson et al., 1990; Zhu and Kivelson, 1991; Cao et al., 1994], but, to date, no definitive observations have been reported of FLRs on low-altitude, polar-orbiting satellites, which must traverse the auroral L-shells where resonances are usually found. This data gap is largely due to difficulties in sorting out the spatiotemporal signatures of north-south localized, ULF waves on a rapidly moving, observational platform.

Measurements of FLRs at low-altitude are of interest for at least two reasons. First, the two-fluid MHD theory of dispersive FLRs [ Streltsov and Lotko, 1995, 1996a; hereafter SL1,2] predicts especially large perpendicular electric fields at relatively low altitude. Second, the largest amplitude electric fields found in near-Earth space, up to 1 V/m, are observed in this same region at auroral latitudes [Bennett et al., 1983; Weimer and Gurnett, 1993].

Are FLRs and the observed low-altitude fields related? Large-scale structure in observed electric fields (100 km transverse) is often associated with convection and broad inverted V precipitation structures [Gurnett, 1972]. Smaller-scale features, representing fine structure near the edges of inverted V regions, as well as isolated structures, are especially relevant to dispersive FLRs. The small-scale structures are often associated with field-aligned electron acceleration and narrow, discrete auroral arcs [Kletzing et al., 1983; Boehm et al., 1990] and with black aurora [Marklund et al., 1995]. The small-scale fields have been identified variously as electrostatic shocks [Torbert and Mozer, 1978], Alfvén waves [Lysak, 1990 and references therein], Alfvén vortices [Chmyrev et al., 1988] and shocks [Mishin and Förster, 1995], standing Alfvén waves [Dubinin et al., 1990], and simply intense or large-amplitude electric fields [Weimer and Gurnett, 1993; Aikio et al., 1996; Karlsson and Marklund, 1996]. Although these distinctions serve to organize the data, and are justified on the basis of specific observed features, the proliferation in nomenclature may obscure a possible common origin, which we will show may often be closely connected with dispersive FLRs.

The SL2 model describes magnetically incompressible, kilometer-scale and larger structure in linear dispersive FLRs in a uniform magnetic field, with realistic parallel and perpendicular variations in Alfvén speed modeled via density variations. The model may be used to generate synthetic data for comparison with actual data at arbitrary points along a flux tube. Wave dispersion, which arises when the transverse width of the FLR approaches the ion/ion acoustic gyroradius at high altitude [Hasegawa, 1976] or the electron inertial length at low altitude [Goertz, 1984] is also included (see also Lysak and Carlson, [1981]). Dispersion leads to substantial wave parallel electric fields in low-altitude sections of the standing wave structure [Wei et al., 1994], where field-aligned electron acceleration is also expected. electric field. Dispersion is activated when weakly dissipative, contracting FLRs form on steep transverse density gradients [ Streltsov and Lotko, 1996b] or when small-scale features accompany the nonlinear development of the FLR [Rankin et al., 1993a,b].

Using synthetic satellite measurements as a guide, we will demonstrate that a suitably instrumented, low-altitude, polar orbiter provides an advantageous platform for measuring the small-scale structure of FLRs--structure that is difficult to resolve either on the ground or in a low-inclination orbit--and that the synthetic data reproduces key features of the observations. This simulation model, used in conjunction with satellite data, offers a potentially powerful new diagnostic tool for exploring the consequences of FLRs on auroral processes and, conversely, for using low-altitude measurements to diagnose FLRs.


A comparison between SL2 simulation results and electric field data from the S3-3 satellite [Torbert and Mozer, 1978] is shown in Figure 1. The color inset is an instantaneous snapshot from a 2D, time-dependent simulation of a weakly radiative, third harmonic FLR that has formed on a density boundary layer, nominally located near an L=9 flux tube. The simulated FLR is stimulated by a two-order-of-magnitude weaker oscillation on the right simulation boundary. The term ``radiative'' refers to the dispersive relaxation in the observed wavetrain and is distinguished by SL1 from a ``nonradiative'' FLR in which the energy stored in the resonance is confined more closely to the resonant L-shell. Naturally occurring resonances usually exhibit a finite bandwidth in frequency although relatively narrow spectral peaks are not uncommon. Here, a monochromatic, third harmonic, 30 mHz resonance has been modeled to illustrate the behavior of the resonance at different points along an auroral flux tube.

Fig. 1 Top right. Electric field measured by the S3-3 satellite [Torbert and Mozer, 1978]. Color panel: "Virtual satellite" trajectories (pink dashed lines) in a simulated third harmonic FLR. Virtual satellite "measurements" of the low-altitude, N-S electric field (bottom right) and the magnetospheric radial electric field (top left).

The trajectories of low-altitude and near-equatorial ``virtual satellites'' traversing the resonance layer are indicated by pink dashed lines in the color panel. The fields evolve in time during the virtual satellites' transits. Synthetic ``measurements'' of the north-south (N-S) electric field on a satellite moving across the resonance at 6.8 km/s at 2000 km altitude and the radial electric field on a magnetospheric satellite moving at 2.3 km/s at 0.5 RE off-equator are also shown. The phase and amplitude of the simulated resonance have been chosen to yield approximate correspondence with the measured N-S electric field in the lower right data expansion. The simulation is uniform in the east-west direction and does not model the east-west electric field.

The results of SL2 indicate that, with increasing harmonic number, the width of a dispersive FLR decreases while its eigenfrequency increases. We interpret the S3-3 event as one with small-scale structure indicative of narrow, nonfundamental FLRs with time variation sufficiently slow that the fast moving satellite is basically measuring Doppler-shifted spatial structure. In choosing an odd harmonic we effectively assume the field-line displacement of the generator exhibits even symmetry about the geomagnetic equator. The synthetic magnetospheric signal near the equator appears as a classical toroidal pulsation in the (radial) electric field due to the much longer satellite dwell time in the resonance (many wave cycles), a consequence of the slower speed of the higher altitude orbiter, combined with a broadening of the resonance layer due to the divergence of geomagnetic flux tubes with increasing altitude (artificially introduced in the synthetic data as described SL2). Examples of intense odd harmonic toroidal pulsations, resembling the one in the upper left panel of Figure 1, have been observed at intermediate altitudes by DE-1 (Figure 18/19 of Cahill et al. [1986]) and near the magnetic equator by AMPTE CCE (Plate 2 of Anderson et al. [1990]).

The vast difference in electric amplitude evident at high and low altitude in Figure 1 may be understood qualitatively as a consequence of two cooperative effects of Alfvén waves: (1) physical optics, wherein the power flow in the resonance per cycle E2NSdAB/vp|| is approximately constant along the flux tube ( ENS is the N-S electric field, dAB  B-1is the local cross-sectional area of the flux tube--artificially introduced in the synthetic data, and vp||is the parallel phase velocity--the Alfvén speed vA to lowest order, which varies up to a factor of 50 from high to low altitude), and (2) Alfvén wave dispersion, especially in the low-altitude inertial regime where the electron inertial length becomes larger than the resonance width, thereby sustaining a large parallel electric field and a correspondingly large polarization ENS to maintain approximate charge neutrality. These effects may explain why it has been so difficult to find observational evidence in the equatorial magnetosphere for the large-amplitude electric fields measured at low altitude: The large-amplitude electric signals observed at low altitude do not map quasi-statically to high altitude.


It is well-known that the phase difference between ENS and BEW (the east-west magnetic deflection) /2 is in an ideal, classical FLR, both temporally and spatially along the ambient magnetic field. (The finite conductivity of the ionosphere modifies this phase relation in detail [Knudsen et al., 1992].) When the FLR becomes dispersive, this phase shift may persist only near the resonant flux surface where the field amplitude is large and where the transverse mode structure becomes evanescent. The transverse structure of a nonradiative, dispersive FLR resembles that of a simple nondispersive FLR [e.g. Greenwald and Walker, 1980], but its scale size is much narrower; the transverse structure of a numerical, radiative, dispersive FLR, like the one shown in Figure 1, is oscillatory in space on one side of the resonance. When the FLR is strongly dispersive, whether radiative or nonradiative, the wave impedance, µ0ENW/BEW, becomes considerably larger than µ0vA at low altitude, giving the impression that the structure is locally electrostatic.

We analyzed the largest-magnitude electric field event measured by the DE 1 satellite [Weimer and Gurnett, 1993] to see if its phase, impedance and spectral properties are consistent with the occurrence of a dispersive FLR, similar to ones simulated by SL2. (The poor time resolution of the S3-3 magnetic field data makes it unsuitable for this analysis.) A portion of the measured time series for ENS and the linearly detrended BEW is shown in Figure 2 (lower left panel). The 1 sec data gaps in Weimer and Gurnett's plot of ENS, resulting from satellite-spin demodulation of the single-axis electric field measurement, have been linearly interpolated. The 6 sec oscillation evident in BEW is an artifact of the satellite rotation. At a nominal satellite speed of 6 km/s across L-shells, the distance between opposite electric field spikes, if interpreted entirely as satellite Doppler-shifted spatial structure, is less than 10 km. At first glance, the time scales for variations in the measured electric and magnetic fields seem unrelated.

Fig 2. DE 1 satellite event [Weimer and Gurnett, 1993]. Lower left: Spin demodulated, deconvolved ENS and BEW . Upper left: Fourier spectra of ENS and BEW normalized to their respective values at f= (1/6) Hz, the satellite rotation frequency, where the signals exhibit contamination. Bandpass filter window is shown by the heavy dashed line. Upper right: Bandpass-filtered ENS and BEW with peak values out of phase. Lower right: Replication of ENS from above and Hilber transform of BEWfrom above overlaid.

The spectra of the discrete Fourier transforms of these signals are also shown (Figure 2, upper left), each normalized to its respective value at f = 0.17 Hz, the satellite rotation frequency. The heavy dashed curve, overlaid on both spectra, is a bandpass filter that is subsequently used to eliminate power at and below 0.17 Hz and above about 1 Hz, passing the main electromagnetic power evident at frequencies within the filter window. For reasons discussed below, we provisionally identify the portion of the spectrum within the filter window as that of a dispersive FLR.

The bandpass-filtered time series for ENS (dark trace) and BEW (light trace) are then plotted and overlaid (Figure 2, upper right). The waveforms resemble the nearly nonradiative FLRs studied by SL2 with differences discussed below. Three points are noted: (i) the central peaks in ENS and BEW near 9:59:32 UT are not temporally coincident; (ii) the quasi-periodic oscillations in ENS and BEW, particularly before about 9:59:27 UT, appear to be approximately in phase but with some complicated deviations; and (iii) the ratio ENS/BEW for the peak field values is at least one order of magnitude larger than the nominal average value of vA at the observation altitude of 2900 km; however, this ratio is comparable to the nominal value in the region of quasi-periodic oscillation to the left of the central peak, as estimated below. The small-amplitude magnetic oscillations evident to the right of the peak are near or below the 1.5 nT instrument sensitivity.

We now apply a Hilbert transform to the bandpass-filtered magnetic time series and overlay the result on a replication of the bandpass-filtered electric time series (Figure 2, lower right). The result brings the primary peaks in ENS and Hilbert transformed BEW into temporal coincidence, while shifting the quasi-periodic oscillations (to the left of the peak) into apparent phase quadrature. The effect of a Hilbert transform is to shift by /2 the phase of all harmonics in the Fourier decomposition. The coincidence in peak ENS and Hilbert transformed BEW is consistent with a ULF wave structure, localized in the N-S direction and standing along geomagnetic field lines [Dubinin et al., 1990]; however, the single-point satellite measurement cannot distinguish a primarily temporal signal from that of Doppler-shifted spatial structure. The development of smaller amplitude oscillations, confined primarily to one side of the peak (left side for the event in Figure 2), where ENS is more nearly in phase with BEW , is expected for a spatially Doppler-shifted, dispersive FLR.

The N-S localized character of the apparent field-aligned, standing wave structure, combined with the occurrence of smaller-amplitude electromagnetic fluctuations resembling propagating Alfvén waves to one side of the resonance, is similar to the numerical FLRs studied by SL2. However, the DE-1 event also exhibits at least two features which do not conform in detail to the synthetic dispersive FLRs reported by SL2.

First, the high wave impedance near the central peak in Figure 2 requires a N-S resonance width at the observation altitude narrower than the local electron inertial length. In contrast, the SL2 synthetic FLRs exhibit a high wave impedance near 1 RE altitude where the model density profile achieves a local minimum. This local density minimum in the slab simulation model of SL2 is somewhat artificial because the uniform background magnetic field in the model requires a density minimum at 1 REaltitude to produce a presumed (and often observed) peak in the Alfvén speed there. In general, a high impedance plasma response is expected in low-density regions where the electron inertial length becomes relatively large. One statistical study [Persoon et al., 1988] indicates that ultra-low density cavities (cavity density of order 1% ambient) are prevalent at altitudes of order 1 RE ; other studies [e.g. Lundin et al., 1984] show that it is not unusual for order 10% cavities to extend to down to altitudes of 2000 km and lower. Perhaps the difference between the presumed DE1 FLR event and the SL2 synthetic FLRs indicates the presence of a density cavity at the comparatively lower observation altitude.

Second, the dominant (approximately) 4 sec oscillation in the filtered magnetic signal in Figure 2, which is less prevalent in the electric signal, does not appear in the SL2 model FLRs. This behavior suggests to us that the resonance may have formed on a narrow density ramp, with a resonance width comparable to the scale size of the density ramp. Experience in simulating dispersive FLRs shows, in this case, that the FLR radiates some energy as propagating and interfering Alfvén waves into the more uniform plasma region outside the density ramp. This more uniform region would presumably correspond to the region of quasi-periodic oscillation in Figure 2 to the left of the peak field.

Ambient density data are not available for this event. Qualitative features of the density profile that may explain the variance between the DE1 event and the synthetic FLRs of SL2 are estimated here and assessed in terms of typically observed variability in the ambient plasma density. To this end, first note that

ENS/BEW = vA(1+k22)1/2

for a dispersive FLR in the inertial regime; k is the wavenumber in the Fourier decomposition of the north-south wave structure, and = c/pe is the electron inertial length. A spatial Fourier decomposition of the fields is not rigorously valid when the modal wavelengths become comparable to or larger than the scale size for variation in the background plasma parameters; here we make use of the Fourier relation between ENS and BEW and only to evaluate the plausibility of the above interpretation. The relation predicts that BEW should contain less power than ENS when k > 1 owing to the filter effect of the function 1+ k22. The spectra shown in Figure 2, in fact, to exhibit this behavior in the bandpass filter window provisionally identified as the spectral domain representing the FLR.

If we now assume (i) the resonance forms on a steep and localized transverse density gradient, (ii) the step in density occurs over a scale size comparable to that of the presumed Doppler-shifted resonance, of order 10 km transverse at the observation altitude, and (iii) the resonance is strongly dispersive, so that k2>>1, then in the resonance layer,

ENS/BEW vA k n0-1

where n0 represents the background plasma density. Using this relation, the measured peak amplitudes of ENS =718 mV/m and BEW= 9 nT, the dipole value B0 = 19,000 nT for the ambient field strength, and assuming k=2/(vs), where vs = 6km/s is the satellite speed across the resonance layer, and  is the temporal width of the pulse (about 1 sec), we can determine the mean density in the resonance. It ranges from 7 cm-3 for pure O+ to 112 cm-3 for pure H+plasma.

When the resonance width is comparable to the width of the density transition, propagating, interfering dispersive Alfvén waves can escape into the more uniform region outside the density gradient; this radiation process is assumed to be responsible for the lack of phase quadrature between ENS and BEW to the left of the peak oscillation in Figure 2. If the density is sufficiently high in this region, then the escaping dispersive Alfvén waves are only weakly dispersive owing to the smaller electron inertial length to the left of resonance layer. Here we may assume

ENS/BEW vA n0-1/2.

Using this relation, the measured values of ENS 95 mV/m and BEW  6 nT, and the same ambient field strength as above to infer the density, we find values for the ambient density in the region to the left of the resonance layer ranging from 42-672 cm-3 for pure O+ to H+ plasma.

Thus the mean density in the resonance region, where the gradient is presumed to be large and the wave is standing, is estimated to be about 17% of its value in the presumed more uniform region to the left where the dispersive waves appear to propagate. The inferred density value in the uniform region is consistent with characteristic mean values at the satellite altitude of 2900 km, and the drop in density to 17% of ambient over a scale size of order 10 km transverse is also typical of the observed density gradients and cavities measured at somewhat lower altitudes (1700 km) by the Freja satellite [Lundin et al., 1984].

Although the interpretation of low-frequency electromagnetic signals measured on orbiting platforms always exhibits spatiotemporal ambiguities, the analysis of the DE-1 event given above does suggest that the large-amplitude electric fields, sometimes interpreted as electrostatic field structures, may be more closely connected with field line resonance phenomena than previously thought. The slab geometry simulation model reported by SL1,2 has now been extended to dipole magnetic geometry. Future results from the dipole model, in conjunction with a comprehensive statistical data analysis of the observed wave electric and magnetic fields in such events, and specification of the ambient plasma conditions and associated particle acceleration features over a range altitudes, would further help in clarifying the relationship between field line resonance phenomena, large-amplitude, auroral-zone electric fields and small-scale auroral structure.


The deconvolved DE 1 data was provided by D. Weimer. M. Boehm provided useful comments on the manuscript. The work was sponsored by NASA Grants NAG5-2252, NAG5-1098, and NAGW-3989, and by Dartmouth College, Thayer School of Engineering. WL acknowledges the hospitality of MPE, Garching where a portion of the research was performed.


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