We present in situ observations consistent with the ballooning mode in the vicinity of the magnetic equator at XGSM = −10 to −13 RE prior to substorm-associated dipolarization onsets. The ballooning instability is expected to have a wavevector along the Y direction and to give variation to the curvature of the ambient magnetic field lines. The magnetic field fluctuations appearing in the Bx component are transported by the ambient plasma drift in the Y direction. A discrete frequency band would be identified in time series data if the mode has a discrete wavelength. The ballooning mode of this property was identified at the magnetic equator a few min before dipolarization onsets only when the plasma β was large (20 to 70). Using low-energy ion velocity data, we show that the mode has almost zero frequency in the plasma rest frame so that ωsc ∼ ky · vy, where ωsc is the frequency in the spacecraft frame, and ky and vy are the wavenumber and the ambient plasma flow in the Y direction, respectively. This enables us to estimate the wavelengths of the ballooning mode, which were found to be of the order of the ion Larmor radius.
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 Recent theoretical models predict that the ballooning instability is a possible trigger of the substorm expansion phase [e.g., Cheng, 2004]. The idea of the ballooning instability in the near-Earth tail as a substorm trigger was first proposed by Roux et al. . Since then, many theoretical and observational studies have been conducted to see whether or not the ballooning instability plays a crucial role in the near-Earth tail during the late growth phase of substorms [e.g., Bhattacharjee et al., 1998].
 Despite its potential, even the presence of the ballooning mode wave in the Earth's magnetotail has not been well established. First, from a theoretical point of view, the stability analysis is difficult. The stability analysis of the ballooning instability should take into account the equilibrium field shape of the magnetotail [Lee and Min, 1996], boundary conditions of the ionosphere [Miura, 2007], plasma compression [Zhu et al., 2004], and various kinetic effects [Cheng and Lui, 1998]. Second, the near-Earth tail magnetic equator has been scarcely sampled by previous spacecraft since the semi-major axis of spacecraft orbit has been usually set off from the equatorial plane.
 The observed magnetotail structure in the late substorm growth phase was found to satisfy the ballooning unstable conditions which were calculated according to the ideal MHD theory [Pu et al., 1997]. Both the ideal and the Hall MHD theories predict that the instability threshold is given by β > 1. The severe stabilization by kinetic effects, however, increases the critical β to ∼50 [Cheng and Lui, 1998]. They reported an event in which low-frequency magnetic field fluctuations were observed in the high-β plasma sheet just prior to a dipolarization/current disruption onset. The low-frequency fluctuations appeared just after the β value reached above 50, and the period of fluctuations was 50 to 75 s.
 The low-frequency magnetic field fluctuations have been so far the only basis for the presence of the ballooning instability. Detailed analyses that would truly pin down the mode of the low-frequency waves have been desired. In this study, we present observations consistent with the ballooning mode in the near-Earth magnetotail. First, we demonstrate how the ballooning instability deforms the magnetic field lines and produce the observed magnetic field fluctuations. Then we examine a set of six substorm events that satisfy severe selection criteria. All were obtained when the Geotail spacecraft was located continuously in the close vicinity of the magnetic equator. The ballooning mode was identified in four events in which the plasma β was high (20 to 70 on average). Finally, by using the ion moment data as well, we show that the frequency of the mode in the spacecraft frame is mostly due to the Doppler shift. With the knowledge of the plasma flow velocity, we were able to estimate the wavelength.
2. Identification Method
 The ballooning mode signatures are identified by the following characteristics of observed magnetic field fluctuations at the magnetic equator. The ballooning instability would deform the ambient magnetic field curvatures as illustrated in Figure 1. Some field lines become more stretched (shortened) than the average. This results in a striation in Bx along the Y direction in which larger (smaller) values of Bx are obtained with more stretched (shortened) field lines in case of the northern hemisphere, and vice versa in the southern hemisphere. As these field lines are transported by the ambient plasma drift in the Y direction, the magnetic field variations δBx are observed, while δBy is small [Roux et al., 1991]. If a single wavelength mode dominates, a time series analysis of δBx will detect a discrete low-frequency band structure. The above qualitative arguments yield an identification method based on single spacecraft measurements. The properties to be searched are as follows. (1) The spacecraft is near the magnetic equator of the midnight sector. (2) The magnitudes of the fluctuations are ∣δBx∣ > ∣δBz∣ > ∣δBy∣ [Chen and Hasegawa, 1991]. (3) The frequency band of ∣δBx∣ fluctuation is discrete [Elphinstone et al., 1995].
3. Event Selection
 We examined Geotail data for February 1995 to December 2005. In order to search the data in the vicinity of the magnetic equator of the near-Earth magnetotail prior to substorm onsets, we visually scanned all 2-hour summary plots that were automatically selected when the 2-hour time-series data include the data satisfying the following conditions for at least 12 s: Geotail was located at XGSM = −8 to −12 RE, ∣YGSM∣ < 10 RE, and ∣ZGSM∣ < 5 RE; < 10 nT. We found 14 candidate substorm events, in which Geotail remained near the magnetic equator continuously during the period of interest. Then short-lived (less than 10 min) dipolarization events were excluded. Also, the availability of global aurora images was checked. This led us to finally select 6 substorm-associated dipolarizations. In this data set, we did not exclude possible pseudobreakups and small expansions, because there are no generally accepted definitions of substorm/small expansion/pseudobreakup. However, the selected dipolarizations were related to global reconfiguration process. The events selected here are excellent in the following sense: (1) ∣Bx∣ < 10 nT before the dipolarization onset continuously for more than 15 min; (2) the low-energy ion velocity data are obtained without any indication of an unmeasured component, so that the moment data are reliable; (3) global auroral images during these events are available to provide the substorm onset timings and locations.
Table 1 gives a summary of the events. The timings and the locations of the auroral breakup onsets were determined from Polar ultraviolet imager data [Torr et al., 1995] with an accuracy of 0.5 hr in magnetic local time. The dipolarization onset is defined as the increase of the Bz component. The spacecraft locations were mapped to the ionosphere according to the T96 model [Tsyganenko, 1995] to estimate the local-time difference with respect to the auroral breakup onset region. For 14 August 1996 event, the Polar UVI data were not available, so that we referred to a Polar VIS data plot shown by Frank et al.  and also the high-latitude ground magnetic field data. Out of the 6 substorm-associated dipolarizations (events a–f, which denote date listed from the top to the bottom of Table 1, respectively), the ballooning mode signatures were identified in 4 events a, c, e, and f at XGSM = −10.1, −10.9, −12.3, and −10.2 RE, respectively. Here the plasma β in the table is the ion βi (=ion pressure/magnetic pressure), since the electron contribution would be small (∼20% at most). One can see that βi was large in the events that had the ballooning mode signatures.
Table 1. Summary of Substorm-Associated Dipolarization Events
Onset time of the dipolarization determined by Geotail (t = 0).
Onset time of the auroral breakup.
Magnetic local time of the auroral breakup onset.
Δ MLT = (MLT of Geotail)–(MLT of auroral breakup).
Three minutes averaged from t = −4 to −1 min.
Time of the appearance of the ballooning mode signature.
04:03 ± 1
04:04 ± 2
(−1, 2, 4)
11:08 ± 1
11:06 ± 1
(−5, −3, 9)
09:26 ± 1
(−4, 4, 3)
02:39 ± 1
02:35 ± 1
(0, −2, 20)
09:25 ± 1
09:24 ± 1
(−1, 0, 4)
17:45 ± 1
17:46 ± 1
(2, −3, 3)
 The wavelet analysis of the magnetic field was applied to the 3 s time resolution magnetic field data around the dipolarization onset. Figure 2 is an example of wavelet scalograms and the original time series data in GSM coordinates. The scalograms show the ballooning mode signature of the Bx fluctuations in a discrete frequency range of 0.01 to 0.02 Hz 2 min prior to the dipolarization onset at 0403 UT (Figure 2, second plot). Meanwhile By fluctuations in the same frequency range did not appear (Figure 2, fourth plot). Unlike the windowed Fourier transform, the wavelet transform is sensitive to when a fluctuation starts. For 60 s period fluctuations, the time uncertainty is never larger than 60 s. The wavelet scalograms for all 4 events (not shown here but provided as auxiliary material) showed the same ballooning mode signatures, which appeared at least 1 to 3 min prior to the dipolarization onsets.
 Here we show that these δBx fluctuations had almost zero frequency in the plasma rest frame (ω ∼ 0, where ω is the frequency in the plasma rest frame). According to the linear MHD theory, we have ωδB⊥/B0 = k∥δv⊥, where B0 and k∥ are the ambient magnetic field and the parallel wave vector, respectively, and δB⊥ and δv⊥ are the fluctuations of the magnetic field and the ion velocity perpendicular to the ambient magnetic field, respectively [see Matsuoka et al., 2000; Saito et al., 2008]. If the fluctuations have almost zero frequency, δB⊥x fluctuates with very small δv⊥x according to its value of ω.
Figure 3 shows hodograms of δB⊥x and δv⊥x for all events. The band-passed (40–150 s) data of the magnetic field and the ion velocity are plotted for the 3 min interval from 4 to 1 min prior to the dipolarization onsets. For all the ballooning mode events (events a, c, e, and f), the absolute values of δv⊥x were small. Hence these δB⊥x fluctuations had almost zero frequency in the plasma rest frame. On the other hand, the observed non-zero frequency in the spacecraft frame ωsc is due to the Doppler shift, i.e., ωsc = k · v, where k and v are the wavevector and the ambient plasma flow vector, respectively. From this relation, the wavelength of the ballooning mode wave can be estimated, since Geotail measures ωsc and v.
Figure 4 summarizes the results of the wavelength estimation. The wavelengths were estimated as λy ∼ Tsc · vy using the observed ωsc and the ion velocity vy, where λy, vy, and Tsc are the wavelength, the ambient flow velocity in the Y direction, and the observed period of the fluctuation in the spacecraft frame, respectively. From the wavelet analysis, the observed frequency was between 0.01 to 0.02 Hz, as previously noted by Lui and Najmi . By taking Tsc = 60 s and the average vy over the 3 min interval prior to each dipolarization onset (t = −4 to −1 min), the wavelengths λy were estimated to be 3345, 693, 5471, and 2797 km for events a, c, e, and f, respectively. Figure 4 shows that the estimated wavelengths are of the order of the ion Larmor radius. Here the errors come from a possible variability of Tsc = 50–75 s. The location where the largest wavelength (i.e., the largest westward ion drift velocity) was observed is the closest to the auroral breakup onset region in MLT, and vice versa.
5. Summary and Discussion
 The present study provides solid evidence of the ballooning mode in the equatorial region of the near-Earth tail (X = −10 to −13 RE). The method based on single spacecraft measurements for identifying the drift ballooning mode is presented. The dominant magnetic field perturbations were found in Bx, whereas the perturbations in By and Bz (0.01–0.02 Hz) remained small. The ballooning mode wave appeared 1 to 3 min prior to both the dipolarization onsets and the auroral breakups. The locations where the ballooning mode was identified were within a few hours from the auroral breakup onset region in MLT. The frequency in the spacecraft frame was discrete between 0.01 to 0.02 Hz. Using the ion data, we showed that ωsc ∼ k · v. Hence the wavelengths of the ballooning mode were estimated for the first time.
 As seen in Figure 4, the observed wavelengths are of the order of the ion Larmor radius and are qualitatively explained in terms of the linear theory with kinetic effects. The ballooning instability prefers smaller wavelengths but is stabilized when the wavelength is smaller than the ion Larmor radius [Lehnert, 1961]. For a larger perpendicular wavenumber (k⊥), the instability is stabilized, when k⊥/k∥ is larger than a certain value [e.g., Hasegawa, 1975]. That the ballooning instability was not detected in the two cases where the plasma β was smaller (<20) is also consistent with the theoretical argument including kinetic effects [Cheng and Lui, 1998] and/or plasma compression effect [Zhu et al., 2004].
 The observed wavelengths λy were approximately equal to the ion Larmor radius, ∼ 3000 km. The wavelength of 3000 km in the magnetotail (X ∼ −10 RE) corresponds to 300 km in the ionosphere, assuming an azimuthal compression ratio of 10. The azimuthally structured aurora having a wavelength of 100 km has been observed before auroral breakup or possibly pseudobreakup [Donovan et al., 2007]. From global imager observations, the azimuthal structure have also been reported by Elphinstone et al. , although they found that the wavelength ranged from 132 to 600 km with a sharp cutoff near 130 km.
 In the events without the ballooning mode, the plasma β was smaller. For the smaller plasma β, the ion Larmor radius is small, and the expected wavelength of the ballooning instability becomes smaller accordingly. While we cannot deny the possibility that our identification method misses the ballooning mode wave when the plasma β is low, we believe that, with a wavelength which is negligibly small compared to the ambient spatial scale, the ballooning instability cannot play a key role in the global reconfiguration process. On the other hand, although we showed the presence of the 3000 km-wavelength ballooning mode in the Earth's magnetotail prior to the substorm-associated dipolarization onsets, clarifying its role in the trigger of a substorm requires more studies.
 The Geotail MGF magnetic field data were provided by S. Kokubun and T. Nagai. We thank G. K. Parks for providing the Polar UVI data. The Wind and ACE plasma and magnetic field data were provided through CDAWeb at NASA. The ground magnetic field data were provided by CSSDP. The Dst data were obtained from WDC, Kyoto. We thank K. Shiokawa and C. Z. Cheng for fruitful discussions and valuable suggestions.