Comparison of Three Techniques of Determining the Resonant Frequency of Geomagnetic Pulsations

C. T. Russell1, P. J. Chi1, V. Angelopoulos2, W. Goedecke3, F. K. Chun4, G. Le1, M. B. Moldwin5 and E. G. Reeves6


1 Institute of Geophysics and Planetary Physics and Earth and Space Sciences Department, University of California Los Angeles

2 Space Sciences Laboratory, University of California Berkeley

3 Colorado School of Mines, Boulder, CO

4 US Air Force Academy, Colorado Springs, CO

5 Florida Institute of Technology, Melbourne, FL

6 Los Alamos National Laboratory, Los Alamos, NM



Three techniques for determining the frequency of ULF pulsations that form standing waves in the magnetosphere are compared: dynamic spectra of the ratio of the H and D-components of the magnetic field at a single station; dynamic spectra of the ratios of the H-components at neighboring stations along a magnetic meridian; and dynamic spectra of the phase difference between H-components at neighboring along a meridian station. The H/D ratio at a single station appears to detect magnetospheric standing waves but over a broad latitude band. The ratio of the H-components detects resonances local to the stations and has some advantages over the more popular phase-gradient technique that finds resonances between station pairs. The stations used need not be strictly confined to a single magnetic meridian. We find resonance signatures using stations up to 1300 km in east-west separation from the magnetic meridian of the other stations in the chain. Near L=2 the resonant waves do not have harmonic structure. The observed wave period, when assumed to be the fundamental of the standing Alfven wave, gives densities that agree with in situ observation at earlier epochs. The diurnal variation and latitudinal variation of these parameters also agree with expectations.


While many spacecraft pass through the plasmasphere on their way to apogee, few spend much time within this region. Hence many of the studies of this region depend on ground-based observations. Originally VLF whistlers were used. Their dispersed frequency signature could be used to determine the L-value of the whistler's path through the magnetosphere and the equatorial electron density along that path [see e.g. Helliwell, 1965]. This method of determining the electron density does not provide a continuous measure of the equatorial density because whistlers are not always present. They depend on the existence of lightning in the appropriate region and a duct to keep the whistler aligned along the field. Suitable ducts may not exist at all L-values for which measurements are desired. Furthermore, recording of VLF signals requires a substantial bandwidth, if done continuously at multiple stations.

An alternative approach is to identify the resonant frequency of Pc 3-4 ULF waves. Such waves are present almost continually in the dayside magnetosphere and can be monitored continually for modest data bandwidth. The current paradigm for the source of these waves is as follows [Greenstadt and Russell, 1994]. Waves are generated in the solar wind by ion beams traveling upstream from the bow shock. These waves are convected through the bow shock and against the magnetopause. There they compress the boundary and the compressional disturbance inside the magnetosphere travels across magnetic field lines as a fast mode wave. When the fast mode wave encounters a field line whose standing wave period equals that of the fast mode wave, coupling to transverse oscillations (in the east-west or toroidal direction) can occur. Although the radial perturbation of the waves and the east-west perturbation might be expected at first glance to contribute to the H and D-components respectively, the high Hall conductivity of the dayside ionosphere is thought to rotate the perturbation by 90o so that the toroidal resonance is seen in the H-component and the fast mode largely in the D-component [Hughes and Southwood, 1976].

This paradigm in turn suggests the following strategies for determining the resonant wave periods. At a single station the H-component oscillation should be strongest relative to the D-component oscillations at the resonant frequencies. A dynamic spectrum of the ratio of the wave amplitudes in H versus frequency and time should then give a maximum or ridge that indicates the diurnal variation of the resonance [Vellante, et al., 1993]. If one has a pair of stations in the north-south direction one can calculate the ratio of the H-component amplitudes at each. The resonances above the stations in the numerator should result in ridges and above the station in the denominator should result in troughs [Baransky et al., 1985, 1989; Best et al., 1986]. Finally, one can also compare the phases of the H-component oscillations at two stations separated in the north-south direction [Baransky et al., 1989]. The phase of oscillating field lines on either side of a resonance should differ significantly being largest for the resonance centered between the stations and approaching 180 degrees for infinite separation. The technique has been most extensively exploited by Waters et al. [1991; 1994] who pioneered the use of the dynamic spectrum to display the results of the phase - gradient technique. In this paper we examine these three techniques using data obtained by the newly installed IGPP-LANL array and discuss the relative limitations and advantages of the three techniques, and illustrate that the technique returns physically reasonable results.


The Los Alamos National Laboratory through the mini grant program of the Institute of Geophysics and Planetary Physics has established a chain of high resolution, high sampling rate fluxgate magnetometers in the western US to study the inner region of the magnetosphere presently from 1.7 = L = 2.2 Re. In this report we examine data from the first three of these stations to be installed the Air Force Academy in Colorado Springs. The Los Alamos National Laboratory and the San Gabriel Dam, California. The locations of these stations are given in Table 1. The first two sites are roughly along the same magnetic meridian and separated by 350 km. The San Gabriel Dam site is located 1300 km to the west of the other two sites as well as being 3.5 degrees south of Los Alamos.

Table 1. Station Locations

The IGPP/LANL stations record at one sample per second continuously with an amplitude resolution of 0.01 nT, and precise GPS timing. The electronics unit is mounted on a single board in a PC that provides power and data storage. The sensors are buried in the ground at distances from 100 to 200' from the electronics unit. The sensors, the analog to digital converter and the feedback resistors are kept at constant temperature. Figure 1 shows dynamic spectra of the signals in H, D, and Z from 0700 UT on October 3 to 0700 UT on October 4, 1998 corresponding to a full day of data starting at approximately local midnight. The data have been averaged to six seconds and overlapped by three seconds and the derivative taken before being Fourier transformed. The dynamic spectra here consist of 512 point individual FFT spectra shifted 128 data points at a time and summed in the frequency direction in groups of five. As can be seen in Figure 1 the spectrum is non-descript. There are no sharp spectral features, but rather the spectrum exhibits a broad enhancement from about 15 mHz to 60 mHz. The major temporal feature is that the power is elevated during the day. The H-component power is generally the strongest followed by D and Z in that order. The spectra at the other two stations are similar but exhibit less power with increasingly lower latitude.

Figure 1.Dynamic spectra of the power spectral density of the three components observed by the IGPP/LANL magnetometer at the Air Force Academy in Colorado Springs, Colorado for 24 hours beginning at midnight local time on October 4, 1998. The vertical scale is logarithmically spaced. The original one-second data were averaged to six-second averages with a three-second overlap. The derivative of the field components were calculated for prewhitening before the dynamic power spectra were constructed. In constructing the dynamic spectra, the fast Fourier spectrum was calculated for 512 data points and shifted by 128 for the next calculation. Spectral estimates have 10 degrees freedom.

The Three Resonance Finding Techniques

As discussed in the introduction three different methods have been proposed to find the resonance frequencies of the field lines above a magnetometer site: comparing H and D powers at a single site; comparing H powers at two adjacent sites; and determining the phase difference between the H components at neighboring sites. We now apply these techniques in turn to this set of data at our three sites. We will examine only the daytime hours in the sections to follow i.e. 1300 UT on 10/3/98 to 0100 UT on 10/4/98.

Single Station H/D Ratio. If our overly simplified picture of the rotation of the magnetospheric signal through 90o by the ionosphere is correct, we might expect that the D-component would respond to the cross L-shell propagating fast mode and the H-component to the toroidal mode. If so, then the ratio of H to D at a single station would show a ridge in the power ratio at the toroidal resonance frequency. The three panels of Figure 2 shows this ratio at each of our three stations. The AFA station shows a broad maximum at a wave period ranging from 30 to 55 seconds as the day progresses. AFA also shows a trough for part of the day at slightly shorter periods from 1700 to 2100 UT. The LAN station shows the same broad maximum as AFA but also exhibits a new ridge changing from 20 seconds to 25 seconds in the course of the day. This ridge covers the region of the trough seen at AFA. The bottom panel shows the dynamic spectral ratio at San Gabriel (SGD) an hour earlier local time for the same universal time. The H/D ratio is weaker at SGD than LAN and AFA but shows some of the structure seen at AFA and LAN in the lower frequency band at about 35s period. In short there appears to be waves with enhanced H/D ratios at all three stations at lower frequencies, and a ridge of enhancement seen only at LAN at higher frequencies. If this technique is sensing standing wave resonances, it is sensitive to a wide band of latitudes not simply those above the station.

Figure 2. Dynamic spectra of the ratios of the power in the north-south to east-west directions at each of the Air Force Academy, Los Alamos and San Gabriel Dam. See Figure 1 for details about the construction of this figure.

Dual Station H Ratio. With our three stations we can create two ratios of adjacent pairs of H-component powers. These two ratios are shown in Figure 3 for the daylight hours. As expected there is a clear trough and a clear ridge running across the top panel. We interpret the ridge as a resonance over the AFA station. It varies from 32 to 35 seconds. The trough varies from 20 seconds to 28 seconds as the day progresses. The crossover line between the two is at 25 seconds at dawn and at 32 seconds at dusk. We interpret this crossover line as the resonant frequency halfway between the stations.

Figure 3. Dynamic spectra of the ratios of the north-south component of the magnetic field at adjacent pairs of stations. Comments of Figure 1 apply.

If we examine the ratio of the H-component at LAN to that at SGP we obtain the bottom panel. Here we see only a ridge signaling that there is a resonance over LAN (where the ridge was in the upper panel) but the absence of any trough suggest that there is no resonance over SDG.

Phase Gradient in the H-Component. The more common technique to find resonant frequencies is the phase gradient technique. Again we can do this for two pairs of stations as shown in Figure 4. The AFA-LAN phase difference maximizes at a wave period that changes from 22 seconds to 32 seconds in the course of the day. The LAN-SGD phase difference maximizes at a wave period that changes from 16 seconds to 25 seconds in the course of the day. These periods are consistent with the periods deduced above from the dual station H-ratio technique when account is taken for the fact that the H-ratio technique has ridges and troughs at the resonances over the stations used and the phase difference is a maximum for the resonance between two stations.

Figure 4. Dynamic spectra of the phase differences between the components of the north-south component of the magnetic field at adjacent stations. Comments of Figure 1 apply.

Discussion and Conclusions

Of the three techniques for finding the standing wave resonance frequency, the power ratio method using the H-component appears to be the most informative by providing the resonant period over both stations. Moreover, this technique does not require the precise timing needed to perform the phase gradient technique. The IGPP/LANL array does have such accurate timing, much more than needed in fact, and can confirm that the same resonant frequency is returned by both the dual station H-ratio and the phase gradient methods. In contrast the single station H/D ratio technique is not successful. It does see the same signals as other techniques but it does not in general return narrow features in the ratioed spectra. The technique seems to be sensitive to a broad latitude band. It shows that resonant oscillations are present but does not define precisely where those resonances are occurring.

We can use the periods found using the dual station H-ratio and phase-gradient techniques to deduce the mass density above AFA, LAN and a latitude midway between SGD and LAN. Table 2 lists the resonant periods at each location at both dawn and dusk and the inferred mass density at the equator under the assumption that the observed pulsations are oscillating at the fundamental, second and third harmonic frequencies. (n=1, 2, 3 respectively) [Schulz, 1996]. We have assumed an r-3 (m=3) dependence for the density but the rate of falloff of the density does not much affect our derived equatorial densities. The densities for n=1 are most similar to the densities observed in this region by the OGO-5 thermal ion spectrometer [Chappell et al., 1971]. Thus it appears that we are observing the fundamental frequency. This is also consistent with observations on four consecutive passes of CCE that recorded only the fundamental mode in the inner magnetosphere [Takahashi et al., 1990] near L=2.

The density increases from dawn to dusk as expected due to the upward flux of ions from the sunlit ionosphere. This result is consistent with the earlier work of Waters et al. [1994]. The diurnal change in density is close to a factor of 2 at L=1.9 and about 20% at L=2.2. This change in density is consistent with similar upward fluxes of ions at the two locations flowing into different volumes of the flux tubes. The flux tube volume is proportional to the fourth power of the L value so we expect the change in density at the two locations to differ by a factor of 1.8, a value very similar to that observed.

Table 2.

*Resonance point halfway between SGD and LAN

In summary the IGPP/LANL array can detect field line resonances routinely by two techniques: dual station H-component ratios and dual station H-component phase gradients. We strongly recommend the former technique as it provides more information and is simpler to implement. Near L=2 the pulsations appear to be occurring at the fundamental of the field line resonance for the limited sample of data that we have examined. The deduced mass densities are similar to those observed on the very infrequent observational passes through this region with spacecraft. It is clear that the mass density of the inner plasmasphere is being accurately measured by the IGPP/LANL observatory chain which we intend to maintain as permanent observatories of the plasmaspheric density.


The development of the magnetometers for the IGPP/LANL array was made possible by many different agencies including grants from SCOSTEP, Calspace, NSF, in addition to the major grant from IGPP/LANL. We are particularly indebted to the efforts of four members of the UCLA engineering team: D. Pierce, D. Dearborn, W. Greer and J. D. Means who worked long hours to build and perfect the instrument. Special thanks for the support also goes to Joe Allen, Dave Sibeck, Bob Clauer and Galen Gisler for their support and assistance. We are also grateful for numerous discussions of these results with Colin Waters and Brian Fraser. Arrays of similar instruments are now being installed along the eastern seaboard by M. Moldwin and in China by Y. F. Gao.


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