CRRES electric field study of the radial mode structure of Pi2 pulsations

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(relative to H) was clustered both at $180° and $0 for events occurring near the plasmapause. This observation still is consistent with the cavity mode according ...

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 108, NO. A5, 1210, doi:10.1029/2002JA009761, 2003

CRRES electric field study of the radial mode structure of Pi2 pulsations Kazue Takahashi,1 Dong-Hun Lee,2 Masahito Nose´,3 Roger R. Anderson,4 and W. Jeffrey Hughes5 Received 1 November 2002; revised 31 January 2003; accepted 25 February 2003; published 23 May 2003.

[1] The radial mode structure of Pi2 pulsations in the inner magnetosphere (L < 7) and

its relation to the plasmapause are studied using data acquired by the Combined Release and Radiation Effects Satellite (CRRES) between August 1990 and September 1991. Low-latitude Pi2 pulsations detected on the ground at Kakioka (L = 1.25) are used as the reference signal to determine the relative amplitude and phase of the electric field oscillations detected at CRRES. The plasmapause is identified using electron density inferred from the plasma wave spectra observed on CRRES. Pi2 events at CRRES are defined to be 10-min intervals of high coherence between oscillations in the Kakioka horizontal northward magnetic field (H) and CRRES duskto-dawn electric field (Ej) components within the Pi2 band (6–25 mHz). The Ej component represents the poloidal oscillation of the geomagnetic field lines for satellite local times near midnight. Fifty-five high-coherence Ej-H Pi2 events occurred when both CRRES and Kakioka were within 3 hours of magnetic midnight. For these events CRRES was on L shells ranging from 2 to 6.5 and was either in the plasmasphere or in the close vicinity of the plasmapause, providing evidence for the plasmaspheric origin of low-latitude Pi2 pulsations. The amplitude of Ej varied significantly but there is an indication of a maximum near L = 4. The phase of Ej (relative to Kakioka H) remained near 90 at all distances. These properties are consistent with the radial structure of the fundamental cavity mode oscillations confined in the plasmasphere. For some events observed at L > 3.5 it was also possible to determine the amplitude and phase of the compressional component Bz at CRRES. In contrast to Ej, the phase of Bz (relative to H) was clustered both at 180 and 0 for events occurring near the plasmapause. This observation still is consistent with the cavity mode according to a numerical simulation using a dipole magnetic field and a realistic plasmapause plasma density structure, which indicates that the node of Bz is located near the plasmapause. Depending on the satellite position relative to the node, the phase can be either –180 or 0. A negative correlation is found between the Pi2 frequency and the distance of the plasmapause, which is additional support for the cavity mode origin of low-latitude Pi2 INDEX TERMS: 2752 Magnetospheric Physics: MHD waves and instabilities; 2768 pulsations. Magnetospheric Physics: Plasmasphere; 2411 Ionosphere: Electric fields (2712); 2788 Magnetospheric Physics: Storms and substorms; KEYWORDS: Pi2 pulsations, plasmasphere, cavity mode, CRRES, electric field Citation: Takahashi, K., D.-H. Lee, M. Nose´, R. R. Anderson, and W. J. Hughes, CRRES electric field study of the radial mode structure of Pi2 pulsations, J. Geophys. Res., 108(A5), 1210, doi:10.1029/2002JA009761, 2003.

1

Applied Physics Laboratory, Johns Hopkins University, Laurel, Maryland, USA. 2 Department of Astronomy and Space Science, Kyung Hee University, Kyunggi, Korea. 3 Data Analysis Center for Geomagnetism and Space Magnetism, Graduate School of Science, Kyoto University, Kyoto, Japan. 4 Department of Physics and Astronomy, University of Iowa, Iowa City, Iowa, USA. 5 Department of Astronomy, Boston University, Boston, Massachusetts, USA. Copyright 2003 by the American Geophysical Union. 0148-0227/03/2002JA009761$09.00

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1. Introduction [2] Pi2 pulsations are damped oscillations of the geomagnetic field with periods from 40 to 150 s. Because they commonly occur in association with the expansion phase onset or with intensifications of substorms and usually last only 5 to 10 min, they are considered to represent a basic magnetohydrodynamic (MHD) response of the inner magnetosphere to a source mechanism(s) that operates for a brief period. Pi2 pulsations on the ground are observed from the auroral altitudes to the magnetic equator but the spectral content of the pulsations changes with latitudes. Accord-

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TAKAHASHI ET AL.: RADIAL MODE STRUCTURE OF PLASMASPHERIC PI2

Figure 1. Schematic illustration of the radial structure of the fundamental cavity mode oscillation (adapted from Takahashi et al. [2001]). The inner and outer boundaries of the cavity are located at Ls and Lpp, respectively. (a) The amplitude of the azimuthal component Ej of the electric field. (b) The amplitude of the compressional component Bz of the magnetic field. The node of this component is located at Ln. (c) The phase of Ej and Bz. The phase of Bz is taken to be zero at the inner boundary.

ingly, there is consensus that there are different source or response mechanisms for high and low latitudes [Olson, 1999]. In this study we will focus on the generation mechanism of Pi2 pulsations observed on the ground at low-latitude stations, where the term ‘‘low-latitude’’ refers to magnetic L shell approximately lower than 2. An outstanding feature of low-latitude Pi2 pulsations is that they are observed at all local times [Sutcliffe and Yumoto, 1989], which implies that MHD fast mode is an essential ingredient for the phenomenon because this mode propagates across the ambient magnetic field. Two mechanisms have been proposed for low-latitude Pi2 pulsations: plasmaspheric cavity mode and forcing by periodic bursty bulk flows. The latter will be referred to as driven Pi2. The primary goal of the present study is to determine whether either of these mechanisms provides a satisfactory explanation for the combined observations from space and the ground. [3] The cavity mode model was first suggested by Saito and Matsushita [1968] to explain the observation of a common Pi2 frequency over a wide range of latitude. Figure 1 illustrates the radial structure of the cavity mode, adapted from Takahashi et al. [2001]. For simplicity we assume that the mode is excited in a box magnetosphere with the upper and lower boundaries representing a perfectly reflecting ionosphere. The box is filled with cold plasma of constant density and has a uniform magnetic field directed parallel to the z axis. The inner boundary Ls and outer boundary Lpp represent the Earth’s surface and the plasmapause, respectively, and both are assumed to be a rigid boundary; that is, there is no field line displacement or electric field on these boundaries. The field components relevant to the cavity mode are the electric field azimuthal component Ej, positive eastward, and the magnetic field compressional component Bz, positive northward. In the dipole field geometry these components represent the axi-

symmetric poloidal mode [Kivelson et al., 1984]. Because the radial boundaries are rigid, Ej has a node on the boundaries. The lowest order (fundamental) mode has half a wavelength between Ls and Lpp with the antinode located at the midpoint of Ls and Lpp as shown in Figure 1a. The Bz component has an opposite amplitude structure, with the antinode located on the radial boundaries and the node in between as seen in Figure 1b. The vertical dashed line labeled Ln indicates the location of the node of Bz, which coincides with the antinode of the electric field. [4] Figure 1c shows the radial phase profile of Ej and Bz. In this figure the range of the phase is taken to be from 270 to 90. We discuss the mode structure of Pi2 pulsations in space using ground Pi2 signals as the reference. Since the Bz component of the fast mode waves maps to the H (horizontal, northward) component on the ground without phase shift [Allan et al., 1996; Takahashi et al., 1999a], we define the phase of oscillations in space relative to H. With this definition, the phase of Bz is zero between Ls and Lnand 180 from Ln to Lpp. In contrast, the phase of Ej is 90 throughout the box. As a consequence, Ej lags Bz by 90 from Ls to Ln whereas Ej leads Bz by 90 from Ln to Lpp. These quadrature relationships between Ej and Bz mean that the time-averaged radial Poynting flux is zero. [5] The cavity mode shown Figure 1 is simplistic. The mode structure in the real magnetosphere should differ due to the dipole field geometry, energy leakage through the boundaries, and limited azimuthal extent of the wave source region. Numerical models and simulations have been presented incorporating some or all of these features [Allan and McDiarmid, 1989; Lin et al., 1991; Lee, 1996; Pekrides et al., 1997; Fujita et al., 2002]. In the discussion section we will show the amplitude structures of Ej and Bz obtained from a numerical simulation. Nonetheless, Figure 1 is a useful guide for what to look for in observational data. [6] The driven Pi2 mechanism was proposed by Kepko and Kivelson [1999] and Kepko et al. [2001]. In this mechanism, low-latitude Pi2 pulsations result from propagation of fast mode pulses that are generated as periodic bursty bulk flows (BBF) brake in the near-Earth magnetotail. The occurrence of BBFs is well documented [Baumjohann et al., 1990; Angelopoulos et al., 1994; Shiokawa et al., 1998], but their direct connection to Pi2 was only recently considered. The mechanism for the periodicity of the flow is not specified in the driven Pi2 model. However, because the oscillation is intrinsic to the source region, it is obvious that no fast mode resonance in the plasmasphere is required to produce the oscillations. Another important property of the driven Pi2 mechanism is that the pulsations are not confined to the plasmasphere. Instead, the same Pi2 oscillations should be observed in the flow-generation region, in the flow-braking region, in the plasmasphere, and on the ground. Shiokawa et al. [1997] estimated that the braking of BBFs occurs at geocentric distances of 10 to 20 RE, well beyond the average plasmapause and the Combined Release and Radiation Effects Satellite (CRRES) apogee. [7] There have been several attempts to reveal the Pi2 mode structure by analyzing data acquired by spacecraft. Takahashi et al. [1992, 1995] studied the magnetic field data from the Active Magnetospheric Particle Tracer Explorers Charge Composition Explorer (AMPTE/CCE)

TAKAHASHI ET AL.: RADIAL MODE STRUCTURE OF PLASMASPHERIC PI2

spacecraft that covered L = 2 to L = 9 and found that Bz has a radial structure consistent with Figure 1. Takahashi et al. [1999a, 2001] reported CRRES case studies of equatorial Pi2 electric and magnetic fields and their relation to the plasmapause. Osaki et al. [1998] (Akebono spacecraft) and Keiling et al. [2001] (Polar spacecraft) reported case studies of plasmaspheric Pi2 pulsations observed away from the magnetic equator. These spacecraft observations were generally in support of the plasmaspheric cavity mode but also provided indication of strong damping of the mode. With only several magnetospheric Pi2 pulsations reported along with plasma density measured in situ, our understanding of the Pi2 mode structure with reference to the plasmapause is quite limited. Note that the most extensive statistical analysis of magnetospheric Pi2 pulsations reported to date [Takahashi et al., 1995] lacked electric field and plasma density measurements. [8] In the present study we use the CRRES electric field and plasma density data as the main source to investigate the radial mode structure of Pi2 pulsations. The electric field data are sensitive to Pi2 pulsations down to L  2, enabling us to statistically construct the mode structure. Similar to the approach employed in our previous studies [Takahashi et al., 1995; Takahashi et al., 2001] we use data from the Kakioka magnetometer to identify low-latitude Pi2 pulsations and to normalize the amplitude and phase of ULF oscillations seen at CRRES. [9] The remainder of this paper is organized as follows. In section 2 we describe the experiments. In section 3 we describe the data analysis technique. In section 4 we analyze satellite data. In section 5 we compare observations and models. Section 6 presents the conclusions.

2. Experiments [10] This section provides a description of the CRRES and Kakioka experiments. The CRRES spacecraft was launched on 25 July 1990 and was operated until 12 October 1991. The spacecraft had an elliptical orbit with initial perigee altitude of 350 km, apogee at the geocentric distance of 6.3 RE, an inclination of 18, and a period of 10 hours. The spacecraft was spin stabilized with the spin axis pointing to the Sun within 15 and a spin period of approximately 30 s. In this study we use the electric field measured by the cylindrical sensors of the electric field experiment [Wygant et al., 1992], vector magnetic field measured by the fluxgate magnetometer [Singer et al., 1992], and electron density Ne derived from spectra measured by the plasma wave experiment [Anderson et al., 1992]. [11] The electric field was measured only in the satellite spin plane. The reference axes in this plane are ey, pointing duskward, and ez pointing toward the ecliptic north. The third component, directed along the spin axis (ex), can be obtained by assuming that there is no electric field E along the ambient magnetic field B, that is, E  B = 0. However, this technique is reliable when the angle between the satellite spin plane and B is larger than 20. Since this condition was satisfied only for a small fraction of the time and Ey  Ej near midnight, we do not rotate the electric field into local mean-field-aligned coordinates but instead use the Ey component (with its sign reversed) as a proxy for Ej. The use of only one component is not a serious

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limitation as long as the radial propagation of fast mode waves is concerned. To study Pi2 pulsations we use spin averages of the electric field data originally sampled at the rate of 32 s1 [Maynard et al., 1996]. [12] The CRRES magnetometer consisted of triaxial fluxgate sensors mounted on a 6.1 m boom. It had two dynamic ranges ±45,000 nT (low gain) and ±850 nT (high gain). The sampling rate was 2 or 16 vector samples/s. A 12-bit A/D converter was used resulting in the resolution of 22 nT in the low gain range and 0.43 nT in the high gain range. Only data acquired in the high gain range are useful for Pi2 pulsations, which limits the satellite radial distance to greater than approximately 3.5 RE. [13] Data from the plasma wave experiment on CRRES are used to estimate the electron number density Ne based on spectral features in the electric field oscillations. The experiment included an electric field sweep frequency receiver in the frequency range from 100 Hz to 400 kHz. The electron density was calculated from narrow-band emission at the upper hybrid resonance frequency and from the cutoff at the plasma frequency of the continuum radiation [LeDocq et al., 1994]. For most of the time interval studied, the electron density was determined at 8-s resolution. [14] The Kakioka magnetometer [Yanagihara et al., 1973] is located at the geographic longitude of 140.2E and at L = 1.25. The primary reason for using the Kakioka data is the timing accuracy of the data acquisition system. The data are provided in 1-s vectors but we use only the horizontal northward component H because Pi2 signals are usually strongest in this component. [15] In order to enable coherence analysis of CRRES and Kakioka Pi2 events, we generated data files with a common time tag. This process started from the spin-averaged CRRES electric and magnetic field data that were provided by the CRRES science team. The spin averages were generated at every half a spin resulting in a time resolution of approximately 15 s (the spin period varied). To match the CREES data, the Kakioka 1-s data were averaged in a 30-s moving time window and sampled at exactly 15-s intervals. The CRRES and Kakioka data were merged by interpolating the CRRES data by spline fitting and then resampling the data at the time steps of the Kakioka data. Since the 30-s window translates to a nominal Nyquist frequency of 17 mHz (60-s period), we cannot discuss pulsations having periods shorter than 60 s. However, previous statistics indicated that a large fraction of Pi2 pulsations has a period longer than 60 s [Saito, 1969], and we believe that the upper limit of the detectable frequency in the CRRES data does not place any significant limitation on the statistics presented here.

3. Time Series Analysis Technique [16] To find Pi2 events and obtain statistical properties of the CRRES electric field data we developed a moving time window analysis program and applied it to the entire Kakioka-CRRES data files. The program takes a 10-min time segment (40 data points at 15-s time resolution) at a step and generates a cross-spectral matrix SHE( f ) from the Ej and H components, where f is frequency. The time window is stepped forward in 2.5-min increments. From SEH we computed the band-integrated parameters, including the Ej amplitude hAEi, the H amplitude hAHi, the Ej-H

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TAKAHASHI ET AL.: RADIAL MODE STRUCTURE OF PLASMASPHERIC PI2

Figure 2. Moving window analysis of merged CRRESKakioka data for CRRES orbit 243. The solid line in each panel indicates the Pi2 event shown in Figure 5. (a) Location of CRRES orbit in L-MLT coordinates from 1990 day 306 (2 November) 1540 UT to day 307 0130 UT. (b) The L profile of the electron number density for the outbound leg of the orbit. (c) The amplitude of Ej and H integrated in the Pi2 band. (d) The coherence of Ej and H integrated in the Pi2 band. coherence hCEHi, and the Ej-H cross phase hqEHi. These parameters were saved in a file for each CRRES orbit. For the calculation of the spectral matrix we used the autoregressive time series technique. Appendix A provides a brief description of the technique.

4. Data [17] In this section we present the details of data analysis. Examples of Pi2 times series and their spectral properties as well as statistical properties of low-latitude Pi2 pulsations are described. 4.1. Selection of Pi2 Events [18] Figure 2 illustrates processed data for CRRES orbit 243. Figure 2a is the spacecraft orbit in L-MLT coordinates

calculated using a centered dipole. Figure 2b shows the radial profile of electron density for the outbound leg. On this leg the spacecraft was moving near midnight and crossed the plasmapause at L  4.9. Figure 2c shows the band-integrated amplitudes hAEi for CRRES electric field and hAHi for ground magnetic field. The former was highly variable, ranging from 0.02 mV/m to 0.5 mV/m. The latter remained in a somewhat smaller dynamic range from 0.03 nT to 0.3 nT. Both hAEi and hAHi show 10- to 20min enhancements, each of which could be due to a Pi2 burst. However, in Figure 2d we find that not all bursts give rise to an elevated value of band-integrated space-to-ground coherence hCEHi. Periods of sustained high coherence occurred only twice on the orbit, first at 1615 UT and second at 0010 UT. The former, marked by a heavy horizontal bar, was associated with a genuine Pi2 event that will be shown in section 4.3. The latter was observed when CRRES was at 0700 MLT and Kakioka at 0900 MLT. This event appears to have been caused by a dayside compressional Pc4 wave similar to the wave reported by Kim and Takahashi [1999]. [19] Figure 2d shows that throughout the orbit there were spiky enhancements in hCEHi. Most of these were apparently caused by statistical fluctuations. Upon examination of the time series plots corresponding to the spikes we usually find that there is little similarity in Ej and H. We therefore searched for a practical approach to select time intervals that give us meaningful information on the propagation of Pi2 pulsations. After trial and error on several orbits we have defined a high-coherence event to be the occurrence of hCEHi > 0.7 over three or more consecutive 2.5-min time steps. We applied this condition to all available CRRES-Kakioka files and made a list of intervals of high hCEHi. [20] In the next step of data analysis we examined nonoverlapping 10-min segments that contained high-hCEHi time steps. The Ej and H time series in these segments were plotted for a sign of any data anomaly, and when the quality of data was satisfactory we calculated the spectral parameters including the autospectra of the electric field SEE( f ) and the Ej-to-H coherence CEH( f ). High-coherence Pi2 events were then selected by applying the following criteria. [21] 1. The magnetic local time of Kakioka was between 2100 and 0300 to eliminate pulsations of dayside origin. [22] 2. A spectral peak in SEE exists between 5 and 20 mHz. If multiple peaks exist, the strongest peak is selected. The frequency for the peak is denoted fPi2. [23] 3. The coherence between Ej and H must exceed 0.9 at f = fPi2. [24] A total of 67 events were identified by the above procedure. Figure 3 shows the location of CRRES at the time center of the selected 10-min segments. No local time condition was imposed on CRRES. Although the spacecraft spent a substantial amount of time on the dayside, most high-coherence events occurred on the nightside with only a few events found on the dayside, near dawn. This implies that either the electric field amplitude of Pi2 is small on the dayside or that dayside Pi2 is masked by Pc4 waves (45 – 150 s) propagating Earthward from the dayside magnetopause. Since our coherence is defined from the spectral matrix integrated over the entire Pi2 band, any oscillation

TAKAHASHI ET AL.: RADIAL MODE STRUCTURE OF PLASMASPHERIC PI2

Figure 3. CRRES location in L-MLT coordinates for the selected high-coherence Ej-H events. Kakioka local time for the event search was limited to 2100 –0300 MLT, but no local time condition was imposed on CRRES. See the text for the event selection criteria.

that exists in only Ej or H can dominate the spectral matrix and lower the value of the coherence. That is, our technique identifies high-coherence events only when there are nearly identical oscillations at both locations. [25] In the statistics presented in the following sections we used only those events that occurred when both CRRES and Kakioka were within 3 hours of magnetic local midnight. Of the 67 events shown in Figure 3, 55 (82%) satisfied this condition. There are two reasons for this choice on the local time. First, we used the dawn-to-dusk component Ey of the electric field as a proxy to the azimuthal component Ej to detect the poloidal oscillations. When the spacecraft was near dawn or dusk, Ey was closer to the radial component than the azimuthal component. In this case Ey is more indicative of the toroidal oscillations than the poloidal oscillations. We need to avoid the toroidal oscillations since they have L-dependent frequencies [Takahashi et al., 1996] and could mask poloidal oscillations that might be simultaneously present. Second, we need to eliminate the effect of longitudinal phase delay of Pi2 pulsations. Yumoto [1986] found that Pi2 pulsations observed on the ground between L = 1.15 and L = 1.83 propagate away from midnight at a longitudinal phase velocity of 5/s. For a Pi2 pulsation of 60-s (120-s) period, this translates to a phase delay of 50 (25). Accordingly, for events observed more than 3 hours away from midnight it could be difficult to interpret the ground-satellite phase delay in terms of radial propagation delay. [26] Although we did not impose conditions on the magnetic latitude of CRRES, the selected 55 events all occurred within 13 of the magnetic equator. The CRRES orbits covered the magnetic latitude range of ±30 and when the condition was satisfied that both CRRES and Kakioka were within 3 hours of midnight, CRRES had a 40%

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probability of being more than 13 away from the equator. Thus our events are definitely biased toward low magnetic latitude. A possible reason for the bias is the variation of the Pi2 amplitude with latitude. A wave mode with an amplitude maximum of the electric field at the equator can account for the latitudinal occurrence property. The fundamental cavity mode has this property. [27] In closing this subsection it is appropriate to comment on whether our Pi2 samples are consistent with those selected in other studies. We selected Pi2 events based solely on high ground-satellite coherence. In many other studies the coherence was not a selection criterion. Most commonly, low-latitude Pi2 pulsations were identified based on the suddenness of their onset. In addition, a Pi2 pulsation is often required to accompany signatures of substorms, such as midlatitude positive bay and enhancement of the AE index to qualify as an event [e.g., Takahashi et al., 1995]. [28] To get insight into the consistency among studies, we compared our list of high-coherence Pi2 events with the list of Pi2 events generated by applying the wavelet technique [Nose´ et al., 1998] to the Kakioka H component. Of our 55 high-coherence events that occurred when both CRRES and Kakioka were in 2100 – 0300 MLT, 39 (71%) occurred within 15 min of a wavelet Pi2 event. A possible reason that the match is not perfect is the amplitude threshold for the wavelet event. In the wavelet method only those events with peak-to-peak amplitude of 0.6 nT or greater were selected, whereas our method did not impose any amplitude threshold on Pi2 pulsations. [29] We also examined the unfiltered Kakioka H component for substorm positive bay signatures. A positive change was observed for 36 (65%) events, and for the rest the change was either very small or negative. Thus not all of our Pi2 pulsations were associated with the onset of a welldefined isolated substorm. Approximately one third appears to have occurred in the declining phase of the positive bay. It is well known that Pi2 pulsations occur during pseudo onsets, and it is very likely that a fraction of our events corresponds to these minor geomagnetic activities. Whether Pi2 generation mechanism differs among expansion phase onset, substorm intensification, and pseudo onset is an interesting question. However, we do not attempt to distinguish these. 4.2. Radial Occurrence Probability of High-Coherence Events [30] One objective of the present study is to search for observational evidence of driven Pi2 pulsations. If a driver wave propagates Earthward from the source that is more than 10 RE away, the wave amplitude will increase with distance from Earth. In addition, identical oscillations will be detected at CRRES and on the ground regardless of CRRES’s position relative to the plasmapause. This consideration motivated us to examine the L dependence of the general pulsation amplitude and the occurrence probability of high-coherence Ej-H events. [31] Figure 4 shows the radial profile of Ne, hAEi and the rate of detection of high-coherence Pi2 events. To be consistent with other statistical results shown in this paper we used data that were acquired when both CRRES and Kakioka were in 2100 –0300 MLT. For the statistics in

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TAKAHASHI ET AL.: RADIAL MODE STRUCTURE OF PLASMASPHERIC PI2

[34] The detection rate of high-coherence events (Figure 4c) decreases with L. The rate is 2% at L = 2 and goes down to 0.4% at L = 6. The rate is 0 for L > 7. From a comparison of Figures 4a and 4c we conclude that the high-coherence events are almost exclusively a plasmaspheric phenomenon and that there is no evidence of Pi2 pulsation propagating from large distances. We arrive at the same conclusion after inspecting the plasma density data for the high-coherence events. We believe that the occurrence rate of 2% is not unreasonable. Assume that a substorm occurs every 6 hours and generates a Pi2 pulsation that lasts 10 min. The fraction of time that Pi2 is present is 10 min divided by 360 min, which is 3%. In the following subsections we describe two examples of high-coherence events observed at different CRRES radial distances.

Figure 4. Radial profile of (a) electron density, (b) bandintegrated amplitude of Ej oscillations, and (c) detection rate of high-coherence Pi2 pulsations from merged Ej and H data. The statistics are all compiled from data acquired when both CRRES and Kakioka were near midnight in the MLT range between 2100 and 0300. Figures 4a and 4b, all data, regardless of the presence of Pi2 pulsations, were sorted into 0.1 L shell bins and the median and quartile values were calculated in each bin. For Figure 4c the statistics were done using 1.0 L bins, and the detection rate is defined to be the number of 10-min segments with hCEHi > 0.7 divided by the total number of 10min segments. [32] The Ne profile is smooth, starting at 1.8  103 cm3 at L = 2 and decreasing to 2.7  100 cm3 at L = 7. The variability of Ne, measured by the separation between the upper and lower quartiles, is greatest from L = 4 to L = 5, suggesting that the plasmapause was most often located there. It should be noted that the density gradients seen on individual orbits are often very sharp but occur at different L shells and that detached plasma or plasma tail is common. All contribute to the smearing of the density profile. [33] The radial profile of the electric field amplitude is less smooth but still shows a clear general trend. Nearest the Earth (L  1.5) the median amplitude is 0.1 mV/m, and it drops to 0.05 mV/m at L = 2. The amplitude decreases to a minimum of about 0.03 mV/m at L  4 and increases monotonically to over 0.1 mV/m at L > 7. We believe that the near-Earth enhancement is an artifact that is caused by somewhat degraded data acquired very near the spacecraft perigee. By contrast, the outward amplitude increase that starts at L  4 is real. Comparing Figures 4a and 4b we conclude that the electric field amplitude is larger in the plasmatrough than in the plasmasphere by a factor of 2 to 4.

4.3. CRRES Pi2 Event at L  2 [35] Figure 5 shows a high-coherence Pi2 pulsation detected when CRRES was near perigee on its outbound leg of orbit 243. This event gave a high value of hCEHi in Figure 2d and is chosen here to demonstrate that the CRRES electric field experiment was sensitive to Pi2 waves even very deep in the plasmasphere. The Pi2 events reported in previous CRRES studies were all detected near apogee (L  6) [Takahashi et al., 1999b, 2001]. Figure 5a shows a 10-min segment of the Ej and H time series that were detrended by removing a second-order polynomial fitted to the original time series. At the center of the interval, CRRES was located at L = 2.3, MLAT = 3.4 and MLT = 23.5 h, where MLAT is the magnetic latitude measured from the dipole equator. Kakioka was located at 1.7 h MLT, 2.2 h east of CRRES. The root-mean-square amplitude of Ej is 0.2 mV/m, substantially larger than the median value 0.04 mV/m of hAEi at L = 2.3 (see Figure 4b). The rootmean-square amplitude of H is 0.3 nT. The two components exhibit identical Pi2 pulsations with a period of 110 s. However, H leads Ej by approximately a quarter of the wave period. The event is inferred to be associated with the expansion phase onset of a substorm because the unfiltered H component increased by 10 nT from 1610 to 1640 UT. [36] Figures 5b to 5d illustrate the spectral properties of the Pi2 event. The power spectral density functions of the electric field (SEE) and the magnetic field (SHH) are virtually identical and have a pronounced peak at 9.6 mHz (Figure 5b). The vertical dashed line indicates this frequency, denoted fPi2. At fPi2 the coherence between Ej and H is nearly perfect (Figure 5c), and the phase of Ej relative to H, denoted qEH, is close to 90 (Figure 5d). The phase is consistent with the fundamental cavity mode excited in the plasmasphere (see Figure 1). Unfortunately, the CRRES magnetometer data were too noisy to detect Pi2 pulsations at L < 4 so we are unable to determine the relationship between Bz and H for this event as an additional check of the mode structure. 4.4. CRRES Pi2 Event at L  4 [37] There are a total of 16 high-coherence Ej-H events for which Bz data were clean enough for time series analysis. Figure 6 is an example of an event with clean Bz. Figure 6 (upper) shows the location of CRRES (large dot on the orbit trace) and Kakioka (open circle) for this event, and Figure 6 (lower) shows the perturbations of Ej, Bz and H about their 150-s running averages. CRRES was located

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spectral density at fPi2 versus L as shown in Figure 7a. Somewhat to our disappointment, we find a considerable variability of the power density. At a given L, SEE varies up to two orders of magnitude, which translates to an amplitude range of one order of magnitude if we assume a common bandwidth for all events. However, this is not surprising when we realize that Pi2 amplitude varies from event to event and that we did not place any amplitude threshold in event selection. [39] In our second attempt, shown in Figure 7b, we normalized SEE by the spectral intensity SHH of ground Pi2 pulsation. As was the case with our previous study of the AMPTE/CCE magnetic field [Takahashi et al., 1995], this approach helps to reduce the degree of data scattering. Now we find that the normalized Ej intensity has a relatively flat L dependence with a suggestion of a maximum near L = 4. The maximum could be the indication of an electric field antinode of the cavity mode illustrated in Figure 1a. We emphasize that there is no indication of the normalized electric field intensity increasing toward larger L. For waves directly propagating from a distant source one would expect the amplitude to be larger closer to the source. In this sense the result does not favor directly driven Pi2. By contrast, strong support of the cavity mode type oscillation is found in the qEH versus L plot shown in Figure 7c. The relative phase stays near 90 for all L values from 2 to 6, consistent with the model phase diagram shown in Figure 1c.

Figure 5. A Pi2 pulsation simultaneously detected at CRRES and Kakioka. (a) Time series plots of Ej and H. (b) Power spectra of the field components shown in Figure 5a. (c) The coherence between Ej and H. (d) The cross phase between Ej and H. at L = 4.1, MLAT = 7.0, and MLT = 21.74 h. Kakioka was located at 2.38 h MLT. The Pi2 pulsation started at 1726 UT and exhibited nearly identical oscillations in all three components. A positive bay is evident in the unfiltered Kakioka H component but its magnitude is modest at 5 nT. CRRES was moving outward and encountered the plasmapause at L  6. Using the spectral analysis described in Appendix A, we find a spectral peak of Ej at 8.6 mHz and that at this frequency the phase of Bzrelative to Ej, qBE, is 98. The phase of Bz relative to H, qBH, is 29. In reference to the phase and amplitude diagrams shown in Figure 1, we find that these phase properties are roughly consistent with the cavity mode observed Earthward of the node Ln. The model values are qBE = 90 and qBH = 0, respectively. 4.5. Radial Profile of EJ [38] Based on the 55 high-coherence events occurring at various L values, we can gain insight into the radial mode structure of Pi2 pulsations. We first plotted the electric field

Figure 6. A Pi2 pulsation seen in Ej and Bz at CRRES and in H at Kakioka. The upper illustration shows the MLTL plot of CRRES orbit 455 and the location of CRRES (heavy line) and Kakioka (open circle) at the time of the Pi2 pulsation.

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TAKAHASHI ET AL.: RADIAL MODE STRUCTURE OF PLASMASPHERIC PI2

Figure 8. Radial profile of the relationship between Ej and Bz. The plotted parameters were evaluated at fPi2. (a) The Bz to Ej power ratio. (b) The phase of Bz relative to Ej.

Figure 7. Radial profile of high-coherence Pi2 pulsations detected when both CRRES and Kakioka were between 2100 and 0300 MLT. The number of data points is 55. All spectral parameters were evaluated at fPi2, the frequency of the spectral peak for Ej. (a) The power spectral density of Ej. (b) The ratio of the power density of Ej to the power density of H. (c) The cross phase between Ej and H.

SEE (Figure 8a) and the phase of Bz relative to Ej, qBE (Figure 8b), both evaluated at f = fPi2. Unfortunately,the power ratio is highly scattered near L = 6, and there are too few points on other L shells to conclude if there is any indication of the node of Bz. The phase is also scattered near L = 6. However, qBE is close to 90 near L = 4, consistent with the cavity mode model. [42] Figure 9 shows the L dependence of the spectral properties of high-coherence Ej-H. These events are again a subset of the 55 high-coherence Ej-H events, and their spectral parameters were evaluated at fPi2, which is deter-

[40] The normalized intensity (Figure 7b) still exhibits significant variability at a given L, which warrants discussion. A likely reason for the variability is the latitudinal and longitudinal variation of the Pi2 amplitude. Any ULF wave in the magnetosphere has spatially varying amplitude and Pi2 is no exception. In order to determine the Pi2 radial mode structure we would ideally use observations made when Kakioka and CRRES were at exactly the same local time (e.g., 2400 MLT) and CRRES was exactly on the magnetic equator (or at another fixed latitude). Of course, we would not find many events satisfying such a strict condition. In addition, even when the above condition was satisfied, the variability would not be completely reduced. This is so because the plasma distribution and the location of the Pi2 source can also change from event to event, resulting in different mode structures. Given these possible sources for data scatter, it is remarkable that single spacecraft observations covering just one season on the night side still yield a radial mode structure that can be quantitatively compared with numerical models. 4.6. Radial Profile of Bz [41] The model cavity mode structure (Figure 1) indicates that the amplitude and phase relationship between Ej and Bz also provides important information on the Pi2 mode structure. We therefore used the Bz data available for 16 events to examine the L dependence of the Ej-Bz relationship. Figure 8 shows the results: the B to E power ratio SBB/

Figure 9. Radial profile of the relationship between Bz and H. The plotted parameters were evaluated at fPi2. (a) The power spectral density of Bz. (b) The Bz to H power ratio. (c) The phase of Bz relative to H.

TAKAHASHI ET AL.: RADIAL MODE STRUCTURE OF PLASMASPHERIC PI2

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mined from the spectral peak in the electric field. The normalized power SBB/SHH (Figure 9b) does not exhibit a clear L dependence, so we cannot confirm the Bz node illustrated in the cavity mode model (Figure 1b). However, the phase of Bz relative to H, qBH (Figure 9c) shows an indication of phase switch at L  4 from 0 to 180, consistent with the model shown in Figure 1c. Near L = 6, there is a set of events with qBH  0, which does not fit the model, in addition to a group of qBH  180, which fits the model. 4.7. Distance Normalized to Plasmapause [43] With CRRES we have the advantage of having electron density measurements. This allows us to scale the radial distance to the plasmapause in ordering the property of Pi2 pulsations. We examined the electron density for every CRRES orbit on which at least one high-coherence Pi2 event was identified in Ej and H and then determined the plasmapause location. Operationally, the plasmapause was defined to be a density drop of a factor of 10 or greater over an L distance smaller than 0.5. The inner edge of the steep gradient is denoted Lpp. Since the value of Lpp usually differed between the outbound and inbound legs of the same orbit, we chose the leg on which the Pi2 event was found. On some orbits no plasmapause could be identified using the above definition. Two interpretations are possible for these cases: (1) there was no steep density gradient at any distance or (2) the plasmapause was formed beyond the CRRES apogee. Overall, orbits with a ‘‘missing plasmapause’’ were rare, and we were able to determine Lpp for 45 (85%) out of the 55 high-coherence events. For the 45 events we defined a normalized radial distance Lnorm  (L  1)/(Lpp  1). The distance Lnorm = 0 corresponds to the Earth surface and Lnorm = 1 to the plasmapause. [44] Figure 10 shows the Pi2 spectral parameters plotted as a function of Lnorm. We first note that the events are distributed from Lnorm = 0.3 to Lnorm = 1.1. The inability of CRRES to detect Pi2s below about L = 2 (see Figure 7) causes the cutoff at Lnorm = 0.3. On the other hand, the outer limit is explained most easily by the absence of ULF waves that are directly related to low-latitude Pi2 pulsations. As Figure 4a indicates, the average electric field amplitude is large at large L, especially in the region outside the average plasmapause. But unlike at low L, the elevated amplitude is due to oscillations of natural origin. [45] Another important feature of Figure 10 is that most data points associated with Bz (Figures 10b, 10c, and 10e) are located between Lnorm = 0.8 and Lnorm = 1.1, that is, near the plasmapause. In this distance range, qBE values are mostly clustered near 90, which is predicted by the simple cavity mode model, but there are also values near 90, which is the opposite to the model. Also, for some events qBH is close to 0 instead of 180 that is expected for the cavity mode. We will discuss this in more detail using simulation results obtained with realistic magnetic field geometry. [46] As for the electric field intensity SEE/SHH (Figure 10a) and phase qEH (Figure 10d), they cover a wide range of normalized radial distances. As we have already seen in Figure 7, there is no indication of an electric field node within the plasmasphere and the phase stays near 90 throughout. These features are consistent with the cavity mode.

Figure 10. Amplitude and phase of Pi2 pulsations plotted against the distance normalized to the plasmapause distance Lpp. All spectral parameters were evaluated at fPi2. [47] Note that the amplitude behavior near the plasmapause argues against the surface wave origin of Pi2, which we did not include in the introduction but is a model that was considered previously [Sutcliffe, 1975]. The surface wave will have the amplitude peak at the plasmapause, and its amplitude will decrease exponentially away from the plasmapause. In agreement with the Akebono spacecraft observations [Osaki et al., 1998] we conclude that the observed radial variation of the electric field amplitude does not provide support for plasmapause surface waves. However, we do not eliminate the possibility that there is a mode highly localized to the plasmapause. Such a mode would not propagate to Kakioka so it would not give rise to high coherence between Ej and H. 4.8. Pi2 Frequency and Plasmapause Distance [48] The frequency of Pi2 pulsation varies considerably, and the factors controlling the frequency provide informa-

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TAKAHASHI ET AL.: RADIAL MODE STRUCTURE OF PLASMASPHERIC PI2

[52] The radial cavity mode structure shown in Figure 12 was obtained using the MHD code described by Lee and Lysak [1999]. Briefly, the numerical technique incorporates a dipole magnetic field, a radial density profile including the plasmasphere, and energy escape into the nightside magnetotail. The inner boundary of the simulation domain is located at L = 2 corresponding to Ls of Figure 1, and the outer boundary is located at L = 10.5. The ionosphere is taken to be partially reflecting with a reflection coefficient of 0.7. The choice of the reflection coefficient is somewhat arbitrary. Runs with different ionospheric reflection coefficients indicate that the ionospheric boundary condition does not much affect the radial mode structure of the cavity mode. MHD waves are excited by an impulsive disturbance localized to nightside. The ingredient essential for the Figure 11. Dependence of Pi2 frequency fPi2 on the plasmapause distance Lpp. For the samples shown here, a plasmapause and a Pi2 were identified on the same leg of a CRRES orbit. The curve indicates the fundamental cavity mode frequency that is derived assuming simple box geometry and a constant Alfve´n velocity. tion on the Pi2 generation mechanism. Figure 11 shows the relationship between fPi2 and Lpp for the 44 high-coherence Ej-H events for which the plasmapause was identified. There is considerable scatter of data, but a general trend is clear. The frequency decreases as Lpp increases. A linear regression analysis yields a correlation coefficient of 0.68 between fPi2 and Lpp. [49] This type of relationship is expected for the plasmaspheric cavity mode. As the size of the plasmasphere increases, the cavity mode frequency decreases as a result of the increase in the fast mode bounce time between the radial boundaries. We can make a rough estimate of the frequency for the box magnetosphere (Figure 1) using the approach of Saito and Matsushita [1968]. 1 R L The fundamental frequency of the cavity mode is 2RE Lspp dL=VA , where the integral is taken a radial path of a wave packet between the radial boundaries. For a constant Alfve´n velocity the mode frequency is    fPi2 ¼ VA = 2RE Lpp  1 :

We have calculated the model Pi2 frequency using VA = 500 km/s [e.g., Takahashi et al., 1992] and included the result in Figure 11. The agreement between the model and observation is good, which gives support to the cavity mode model of low-latitude Pi2 pulsations.

5. Discussion [50] In this section we discuss the ground-satellite observations with existing models: cavity mode Pi2 and driven Pi2. 5.1. Cavity Mode Pi2 [51] The spatial variations of Pi2 pulsations at CRRES summarized in Figure 10 are consistent with the cavity mode in several ways. To interpret the observations we refer to the simple cavity mode diagram of Figure 1 and also introduce a more realistic mode structure obtained by numerical simulation.

Figure 12. Radial structure of the fundamental cavity mode resonance obtained using the simulation code of Lee and Lysak [1999]. The vertical line labeled Ls indicates the inner boundary for the simulation. (a) The equatorial Alfve´n velocity employed in the simulation. The solid vertical line at L = 5 indicates the inner edge of the plasmapause. (b) The amplitude of the azimuthal component of the electric field (Ej) at the equator. (c) The amplitude of the compressional component of the magnetic field (Bz) at the equator. The node (Ln) of this component is located at L = 4.7 and is indicated by a dashed vertical line. (d) The phase of Ej and Bz. The phase is defined to be zero for the Bz component sampled at the inner boundary Ls.

TAKAHASHI ET AL.: RADIAL MODE STRUCTURE OF PLASMASPHERIC PI2

trapping of fast mode energy is the radial variation of the Alfve´n velocity, which is illustrated in Figure 12a. A sharp increase of Alfve´n velocity occurs at L = 5, corresponding to our observationally determined Lpp. The outer edge of the plasmapause is located at L = 6. [53] We obtained the radial amplitude structure of the field perturbations by Fourier transforming the simulated time series sampled at midnight and on the magnetic equator. The radial profile of the amplitude of Ej, the amplitude of Bz, and the phase of these components are shown in Figures 12b, 12c, and 12d, respectively. Note that we used a 10-min segment starting 2 min s after the onset of the Pi2 trigger and that the Fourier component was evaluated at 8 mHz. Lee and Lysak [1999] identified this to be the frequency of the fundamental cavity mode (or virtual resonance according to Lee [1998]). In Figures 12b and 12c we confirm that the radial mode structure is consistent with the fundamental cavity mode in that the electric field has an antinode and the magnetic field has a node within the plasmasphere. The radial phase structure (Figure 12d) also exhibits properties consistent with the cavity mode, including the 180 shift for Bz at L = Ln and the Ej phase that is nearly constant at 90. Near the outer boundary of simulation (L = 10.5) the phase difference between Ej and Bz becomes small, reflecting the open boundary condition used for this boundary. [54] The simulation indicates a few notable differences from the simple cavity mode illustrated in Figure 1. First, the oscillation is not strictly localized to the plasmasphere. The wave energy tapers off with distance from the plasmapause, but there is substantial amplitude at L > Lpp, in particular for the electric field. Second, the node of the magnetic field component, Ln, is located very close to the plasmapause. The difference between Ln and Lpp is only 0.3. Third, the mode structure is far from sinusoidal. The electric field is similar to a skewed Gaussian and the magnetic field is highly asymmetric about the node with much larger amplitude Earthward of the node. [55] Now we compare the observations with the cavity mode model. The first point to be made is that our events are limited to the plasmasphere with only a few exceptions detected outside. This strongly favors the cavity mode type oscillation. We have shown that despite a large volume of CRRES measurements made outside the plasmapause very few high-coherence events were found there. The occurrence of a small number of events outside the plasmapause does not contradict the cavity mode interpretation since the simulation shows that both Ej and Bz extend beyond the plasmasphere. [56] The second and most important point is that the electric field oscillates in phase in the whole plasmasphere. This is exactly what one expects for a fundamental cavity mode as illustrated in Figure 1c. If the Ej Pi2 pulsations observed in the plasmasphere resulted from radially propagating fast mode waves, the phase would change linearly with L. For a quantitative estimate of the propagation phase delay, assume a fast mode speed of 500 km/s [Takahashi et al., 1992] and a wave frequency of 10 mHz. The wave travel time over 4 RE is 51 s, which is approximately one half the wave period. In this case the observers near the Earth and at the plasmapause (located at L  5 RE) should

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see a phase lag of 180. Such a large phase lag was not observed, which excludes simple radial propagation. [57] Third, the mixed phase structure of Bz at CRRES can be explained by the simulation, which indicates that the node of Bz is located very close to the plasmapause. Near the plasmapause there is no observational preference between qBH = 0 or qBH = 180. Possibly, the spacecraft location relative to Ln changed from event to event or even during a single event. In contrast, the Bz component observed at lowest L shells oscillated nearly in phase with H, consistent with the model illustrated in Figure 1c. [58] Finally, the dependence of fPi2 on Lpp (Figure 11) is consistent with the plasmaspheric cavity mode. The frequency is known to depend on the solar wind speed and Kp [Saito, 1969]. The frequency is higher for higher Kp and solar wind. These correlations fit the plasmapause control of fPi2 because the plasmapause distance becomes smaller when Kp and solar wind speed go up. For example, Gallagher et al. [1995] derived an empirical relationship Lpp = (5.6  0.46 Kp)F(MLT), where F is a function of magnetic local time. 5.2. Driven Pi2 [59] The high-coherence events do not provide strong support for the driven model of low-latitude Pi2 pulsations. We scanned all Ej data acquired by CRRES, but high coherence between Ej and H occurred only when CRRES was either in the plasmasphere or in the close vicinity of the plasmapause. If Pi2 directly propagated from the source region located at 10 RE, there would be events showing high ground-satellite coherence even when the satellite was outside the plasmasphere. [60] In Figure 4 we noted a monotonic outward increase of electric field amplitude starting at L  4. Because wave amplitude will in general increase if the observer gets closer to the source region, this suggests that there could be a wave generation region outside the CRRES apogee of L  7. However, the total absence of high-coherence events for L > 7 (Figure 4c) means that the oscillations at L > 7 do not directly drive low-latitude Pi2 pulsations. Note that the observation of large-amplitude oscillations outside the plasmasphere does not contradict the cavity mode. The mode also derives energy from MHD disturbances in the magnetotail, but in the cavity mode scenario the disturbances only need to provide random impulses to the plasmasphere. Once inside, the wave energy is amplified only at frequencies that match the eigenfrequencies of the cavity. Obviously, the coherence will be low between the random impulse and the cavity mode oscillations.

6. Conclusions [61] In conclusion, we presented a statistical analysis of Pi2 pulsations that give rise to nearly identical waveforms in the magnetosphere at the CRRES spacecraft and at the low-latitude station Kakioka. The dawn-to-dusk electric field measured by CRRES was used as the primary field component for the analysis. The component exhibited Pi2 oscillations on L shells as low as 2. This enabled us to construct the radial mode structure of the pulsations from an ensemble of events observed at various radial distances.

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TAKAHASHI ET AL.: RADIAL MODE STRUCTURE OF PLASMASPHERIC PI2

As was done in a previous AMPTE/CCE magnetic field study [Takahashi et al., 1995], we used Kakioka magnetic field time series as the reference to define the amplitude and phase of the oscillations at CRRES. The AMPTE/CCE study had an advantage of clearer magnetic field data for low L shells, but it lacked the electric field and plasma density measurements. Therefore the present study provided a complementary view of the Pi2 structure in the inner magnetosphere. We have confirmed that the plasmapause is essentially the outer limit of the region in which pulsations are observed with high coherence with the low-latitude Pi2 pulsations. This, combined with the radial phase structure of the electric field, provides strong evidence for cavity mode type oscillations confined in the plasmasphere. In addition, the Pi2 frequency varied with the plasmapause distance as expected for the cavity mode. We note, however, that not all observations fitted the simple cavity mode model. In particular, the magnetic field amplitude and phase showed somewhat confusing L dependence. We pointed out that a recent simulation study indicates that the cavity mode has a magnetic field node very near the plasmapause and that this may explain the behavior of the magnetic field near the plasmapause.

The corresponding band-integrals are hCEH i ¼

jhSEH ij2 hSEE ihSHH i

ðA5Þ

and hqEH i ¼ tan1

RehSEH i ImhSEH i

ðA6Þ

where h i indicate band integral, which in this study is done from 7 mHz to 25 mHz. Also, the band-integrated amplitude can be define for Ej and H as hAE i ¼

pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi hSEE i; hAH i ¼ hSHH i

ðA7Þ

[65] Acknowledgments. The ground magnetometer data were provided by the Kakioka Observatory. Work at JHU/APL was supported by NSF grants ATM-9901102 and ATM-0001687 and NASA grant NAG58964. [66] Lou-Chuang Lee thanks Anthony Walker and Kazuo Shiokawa for their assistance in evaluating this paper.

References

Appendix A [62] We fit an autoregressive (AR) process to the twocomponent observational time series (Ej(t), H(t)). An mdimensional AR time series, denoted Zn, has the form Zn ¼

p X

Ai Zni þ Un

ðA1Þ

i¼1

where Ai (i = 1, . . ., p) is an m  m matrix, p is an integer specifying the order of the AR process, and Un is random impulse. The suffix n indicates time steps. The coefficients Ai and the covariance of Un are estimated by a leastsquares approach, and they are uniquely related to the spectral matrix of the time series. For details of AR spectral analysis technique as applied to ULF waves, see Ioannidis [1975]. [63] Let the spectral matrix of a two-component time series Ej(t) and H(t) be given as 2 SEH ð f Þ ¼ 4

SEE ð f Þ

SEH ð f Þ

3 5

ðA2Þ

SHE ð f Þ SHH ð f Þ

where each element of the matrix is a function of frequency and is defined from 0 to the Nyquist frequency. [64] The coherence and cross phase between Ej and H are given, respectively, as CEH ð f Þ ¼

jSEH j2 SEE SHH

ðA3Þ

and qEH ð f Þ ¼ tan1

ReðSEH Þ ImðSEH Þ

ðA4Þ

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Takahashi, K., B. J. Anderson, and S.-I. Ohtani, Multisatellite study of nightside transient toroidal waves, J. Geophys. Res., 101, 24,815 – 24,825, 1996. Takahashi, K., B. J. Anderson, and K. Yumoto, Upper Atmosphere Research Satellite observation of a Pi2 pulsation, J. Geophys. Res., 104, 25,035 – 25,045, 1999a. Takahashi, K., W. J. Hughes, R. R. Anderson, and S. I. Solovyev, CRRES satellite observations associated with low-latitude Pi2 pulsations, J. Geophys. Res., 104, 17,431 – 17,440, 1999b. Takahashi, K., S. Ohtani, W. J. Hughes, and R. R. Anderson, CRRES observation of Pi2 pulsations: Wave mode inside and outside the plasmasphere, J. Geophys. Res., 106, 15,567 – 15,581, 2001. Wygant, J. R., P. R. Harvey, D. Pankow, F. S. Mozer, N. Maynard, H. Singer, M. Smiddy, W. Sullivan, and P. Anderson, CRRES electric field/Langmuir probe instrument, J. Spacecr., 29, 601 – 604, 1992. Yanagihara, K., M. Kawamura, Y. Sano, and T. Kobuki, New standard magnetic observation system of Kakioka (KASMMER), Geophys. Mag., 36, 217 – 281, 1973. Yumoto, K., Generation and propagation mechanisms of low-latitude magnetic pulsations—A review, J. Geophys., 60, 79 – 105, 1986.



R. R. Anderson, Department of Physics and Astronomy, University of Iowa, Iowa City, IA 52242, USA. ([email protected]) W. J. Hughes, Department of Astronomy, Boston University, 725 Commonwealth Avenue, Boston, MA 02215, USA. ([email protected]) D.-H. Lee, Department of Astronomy and Space Science, Kyung Hee University, Yongin, Kyunggi-Do, 449-701, Republic of Korea. ([email protected] khu.ac.kr) M. Nose´, Data Analysis Center for Geomagnetism and Space Magnetism, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan. ([email protected]) K. Takahashi, Johns Hopkins University Applied Physics Laboratory, 11100 Johns Hopkins Road, Laurel, MD 20723-6099, USA. (kazue. [email protected] jhuapl.edu)

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