Properties of small-amplitude electron phase-space holes observed by Polar

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1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 110,, doi: /2005ja011095, 2005 Properties of small-amplitude electron phase-space holes observed by Polar J. R. Franz 1 and P. M. Kintner School of Electrical and Computer Engineering, Cornell University, Ithaca, New York, USA J. S. Pickett and L.-J. Chen Department of Physics and Astronomy, University of Iowa, Iowa City, Iowa, USA Received 21 February 2005; revised 20 April 2005; accepted 5 May 2005; published 9 September [1] We present Polar Plasma Wave Instrument (PWI) measurements of electrostatic solitary waves in the high-altitude polar magnetosphere. These waves are electrostatic pulses that move parallel (and antiparallel) to the geomagnetic field and are similar to waves detected in many regions of the magnetosphere by other spacecraft. The PWI instantaneous dynamic range was 72 db with an added 30 db obtained by changing gain states. This large dynamic range enables the study of amplitude-size relations up to a maximum electric field of 44 mv/m in the lowest gain state as well as enabling the investigation of small-amplitude waves (<0.1 mv/m). The Polar PWI data indicate that these small-amplitude solitary waves have typical scale sizes the order of the Debye length, velocities the order of the electron thermal speed, and electrostatic potentials that are small compared with the electron thermal energy per charge (f k B T e /e). Statistical distributions of the wave properties are presented, and the properties are compared with theoretical predictions of electron phase-space holes and electron-acoustic solitons. BGK-type analysis of electron holes predicts a relationship between the minimum allowed scale size and the amplitude and velocity. The observed solitary waves are consistent with these predictions. Citation: Franz, J. R., P. M. Kintner, J. S. Pickett, and L.-J. Chen (2005), Properties of small-amplitude electron phase-space holes observed by Polar, J. Geophys. Res., 110,, doi: /2005ja Introduction [2] In recent years it has become evident that small-scale, coherent electric field structures exist in many regions of space. Many researchers believe that some of these structures are electron phase-space holes, a type of nonlinear electrostatic waves. The purpose of this article is to present the properties of coherent electric field structures detected by the Polar Plasma Wave Instrument (PWI) in the highaltitude polar magnetosphere and to compare the observations with theoretical predictions of electron phase-space holes. In previous publications we presented the first interferometric analysis of the structures in space [Franz et al., 1998] and examined the evidence related to the perpendicular (to B) scale sizes of the structures [Franz et al., 2000]. Here we present a more thorough study of the Polar PWI observations, concentrating on the statistical properties of the structures. [3] The paper is organized as follows. We first present a review of the previous work on solitary waves in space and 1 Now at Lincoln Laboratory, Massachusetts Institute of Technology, Lexington, Massachusetts, USA. Copyright 2005 by the American Geophysical Union /05/2005JA011095$09.00 electron phase space holes in section 2. This review is somewhat lengthy as there are no review papers on the topic. The instrumentation and data used in our study are then described in section 3. In section 4 we present several examples of the observations, while in section 5 we present the statistical properties of the structures. The observations are then compared with theories in section 6 and compared with other experimental studies in section 7. Finally, the major results of this study are summarized in section Solitary Waves and Electron Phase-Space Holes 2.1. Electrostatic Solitary Waves in Space [4] Early wave observations in the Earth s magnetosphere revealed that broadband electrostatic waves occur in many different regions. The waves were first detected in the plasma sheet boundary layer in the geomagnetic tail [Gurnett et al., 1976] and were labeled Broadband Electrostatic Noise (BEN). These waves were characterized by bursty broadband features in electric field spectra. Similar waves were subsequently observed in the auroral zone [Gurnett and Frank, 1977] and the cusp [Gurnett and Frank, 1978]. [5] The first high-resolution waveform measurements of BEN were presented by Matsumoto et al. [1994], 1of14

2 who used waveform measurements from the Geotail spacecraft at 70 R E in the geomagnetic tail. The data showed that BEN consisted of a series of coherent structures, which Matsumoto et al. [1994] labeled Electrostatic Solitary Waves (ESW). The ESW create a bipolar signature in electric field measurements with typical amplitudes of 100 mv/m and pulse widths of 2 5 ms in the time series. The spectra of the waveforms were broadband and accounted for the majority of the BEN. However, a portion of the low-frequency power was not accounted for by the solitary waves. A comparison of the orthogonal electric field measurements indicated that the ESW were one-dimensional and that the transit time of the structures was much smaller than the Geotail spin period (3 s). [6] A companion paper by Omura et al. [1994] used onedimensional (1-D) electrostatic particle codes to show that electron beams are able to produce solitary waves reminiscent of the ESW. The solitary waves were found to coalesce and appeared to be stable. Matsumoto et al. [1994] and Omura et al. [1994] hypothesized that the ESW were BGK mode [Bernstein et al., 1957] electron phase-space holes. Electron phase-space holes, hereafter denoted by EH, contain electron depletions that are localized in both position and velocity. [7] Waves similar to the ESW were subsequently detected in the magnetosheath by the Geotail spacecraft [Kojima et al., 1997] and by the Cluster spacecraft [Pickett et al., 2003]. Other spacecraft have also detected similar waves in other terrestrial regions of space, as well as around other planets. Mottez et al. [1997] report Galileo observations of solitary electrostatic waves in the geomagnetic tail associated with counterstreaming electron beams. Coherent structures have been observed in the auroral acceleration region by Polar [Mozer et al., 1997; Bounds et al., 1999] and FAST [Ergun et al., 1998a, 1998b, 1998c, 1999] and in the highaltitude polar magnetosphere by Polar [Cattell et al., 1998a, 1998b; Franz et al., 1998; Tsurutani et al., 1998; Pickett et al., 1999a, 1999b]. Solitary waves have been observed in the free solar wind by Wind [Mangeney et al., 1999], in short, large-amplitude magnetic structures (SLAMS) by Cluster [Behlke et al., 2004], at the bow shock by Wind [Bale et al., 1998], and at the magnetopause by Polar [Drake et al., 2003] and Cluster [Cattell et al., 2003]. Finally, using Cluster waveform data, Pickett et al. [2004b] conducted a survey of the amplitudes and time durations of the bipolar and tripolar solitary waves observed throughout the 4 19 R E Cluster orbit from the solar wind to the near-earth plasma sheet. Most of these researchers have interpreted the bipolar solitary waves as electron phase-space holes, although some of them, particularly in the auroral acceleration region, have been interpreted as ion phase-space holes. [8] The primary results of space observations are listed below. Kojima et al. [1999a] first summarized the Geotail observations of solitary waves. Since then, substantial progress has been achieved using the Cluster, FAST, and Polar data sets. Consequently, a new review is appropriate before presenting our observations. [9] 1. The solitary waves are positive potential pulses moving along magnetic field lines [Franz et al., 1998; Ergun et al., 1998a, 1998b]. [10] 2. The electrostatic potentials of the solitary wave structures are reasonably modeled by a Gaussian [Krasovsky et al., 1997; Ergun et al., 1998c], fðz; tþ ¼ f 0 exp! ðz vtþ : ð1þ 2L 2 k [11] 3. Solitary waves have been correlated with electron beams in the auroral acceleration region [Ergun et al., 1998a, 1998b, 1998c; Bounds et al., 1999] and with diffused electron beams in the geomagnetic tail [Matsumoto et al., 1999; Omura et al., 1999a]. [12] 4. There is a trend for larger amplitudes of both the bipolar and tripolar solitary waves as the strength of the background magnetic field increases [Pickett et al., 2004b]. [13] 5. Typical parallel electric field amplitudes, E k, differ in different regions of space. E k is typically hundreds of mv/m in the geomagnetic tail [Matsumoto et al., 1994], the order of 100 mv/m near the bow shock [Matsumoto et al., 1997; Bale et al., 1998], hundreds of mv/m in the auroral acceleration region [Mozer et al., 1997; Ergun et al., 1998a], and 0.1 to hundreds of mv/m in the high-altitude polar magnetosphere [Cattell et al., 1998a; Franz et al., 1998; Cattell et al., 1999]. [14] 6. Inferred parallel scale sizes (Gaussian half-lengths, L k ) are 1.8 l D (average) in the auroral acceleration region [Ergun et al., 1998c], 2.5 l D (average) in the high-altitude polar magnetosphere [Franz et al., 2000], l D in the geomagnetic tail [Kojima et al., 1999b], and l D in the bow shock transition region [Bale et al., 1998]. [15] 7. While the perpendicular scales of the structures have not been measured directly, ratios of parallel and perpendicular electric field measurements have provided estimates of the oblateness of the structures. The ratio of the perpendicular and parallel scale sizes (L? /L k ) for structures in the auroral acceleration region is about 1.4 [Ergun et al., 1999], whereas in the geomagnetic tail (R = R E ) the majority of the structures have E? /E k 0.2 [Omura et al., 1999b], roughly implying L? /L k 5. Finally, structures in the high-altitude polar magnetosphere have been shown to scale as L? /L k (1 + r e 2 /l d 2 ) 1/2 [Franz et al., 2000]. [16] 8. Solitary waves in the auroral acceleration region show absorption spectra at ion cyclotron frequencies are correlated with heated ions and are sometimes spaced at the ion gyroperiod [Ergun et al., 1998a, 1998b, 1998c]. Thus ion dynamics are important for the waves in the auroral acceleration region. Experimental studies of the waves in other regions of space have not observed such strong interactions with ions, so the physics of waves in the auroral zone may be somewhat different than in many other regions of space. [17] 9. The solitary waves measured in the auroral acceleration region are consistent with models of electron phasespace holes [Muschietti et al., 1999a, 1999b; Chen and Parks, 2002a, 2002b; Chen et al., 2004]. The solitary waves in other regions of space have not been quantitatively compared with theoretical predictions regarding electron phase-space holes. [18] 10. Magnetic components have been measured in the auroral acceleration region [Mozer et al., 1997; Ergun et al., 2of14

3 1998c], as well as in the high-altitude polar magnetosphere [Tsurutani et al., 1998]. Ergun et al. [1998c] found that the magnetic fluctuations in the FAST data were consistent with the Lorentz field, indicating that the waves are electrostatic in their frame of reference Electron Phase-Space Holes [19] Electron phase-space holes (EH), sometimes called electron phase-space vortices, were first observed in early simulations of electron beam instabilities [Roberts and Berk, 1967; Morse and Nielson, 1969]. These simulations showed that electron beams generated sinusoidal perturbations that eventually developed holes or vortices in phase space. Roberts and Berk [1967] deduced that the phasespace hole is the most probable fluctuation in a system with a waterbag phase space density. (A waterbag phase-space density has only two possible values: 0 and 1.) [20] The first experimental studies of electron holes were performed in a Q-machine [Saeki et al., 1979; Lynov et al., 1979]. A pulse generator was used to inject an electron beam into a plasma column with w e W e. Positive potential solitary waves with parallel scale sizes (L k )of10 50l D and velocities near the electron thermal velocity were generated. Furthermore, the larger-amplitude structures had larger scale sizes than the smaller-amplitude structures, indicating that the structures were not classical solitons. Saeki et al. [1979] and Lynov et al. [1979] interpreted the structures in terms of electron phase space holes and performed one-dimensional numerical simulations, which agreed with the experimental data. [21] The laboratory experiments spawned theoretical interest in the electron holes. Several groups derived BGK equilibrium models of the EH [Schamel, 1979; Turikov, 1984; Lynov et al., 1985]. In particular, Turikov [1984] and Lynov et al. [1985] were able to deduce that largeramplitude EH should be wider than smaller-amplitude EH. Dupree [1982] argued that a phase-space hole is the maximum entropy (and hence the most probable) fluctuation in a plasma. Dupree [1982] also showed that one large hole has a larger entropy than two small holes, and thus the holes are expected to have a tendency to coalesce. [22] The maximum entropy property also suggests that the phase-space holes are expected to be stable. Schamel [1982] analytically showed that his one-dimensional BGK model for small-amplitude electron holes was linearly stable to one-dimensional perturbations. On the other hand, Collantes and Turikov [1988] used a similar procedure (although to higher order) to investigate one-dimensional EH with ef 0 k B T e and L k l D. Such EH were found to be unstable, although EH in other parameter regimes (e.g., L k l D ) were not investigated. Additionally, both onedimensional [Ghizzo et al., 1988; Omura et al., 1996] and magnetized two-dimensional [Miyake et al., 1998; Goldman et al., 1999; Oppenheim et al., 1999; Muschietti et al., 1999c] numerical simulations have found that the EH can be stable for hundreds and even thousands of plasma periods. In multidimensional (two- and three-dimensional) unmagnetized simulations, the electron holes decay within a few plasma periods [Morse and Nielson, 1969]. [23] Recent measurements in space have fueled a new growth in theoretical investigations of EH. Several new analytical studies of EH BGK equilibria have been performed. Krasovsky et al. [1997] performed a general analysis for an approximate BGK equilibrium model of an EH. Included in this model was a Gaussian-shaped electrostatic potential in a Maxwellian plasma. The theory predicts a minimum allowed scale size (parallel to B) as a function of the velocity and amplitude of the EH. Muschietti et al. [1999a, 1999b, 1999c] derived an exact one-dimensional BGK equilibrium solution for a Gaussian-shaped electrostatic potential in a flat-topped electron distribution and found that the FAST data are in agreement with the minimum scale size predicted by the analysis. The analytical solution was also found to be stable when used as an initial condition in a two-dimensional particle simulation. Krasovsky et al. [1999] analytically investigated the interaction of two small holes. They argued that when two holes collide, the trapped electrons no longer undergo adiabatic motion. Combined with phase mixing, this causes the collision process to be effectively irreversible, resulting in the tendency of electron holes to coalesce. [24] Chen and Parks [2002a, 2002b] and Chen et al. [2005] have examined EH as nonlinear solutions of the Vlasov-Poisson equations and extended the BGK formulation [Bernstein et al., 1957] to three spatial dimensions, assuming a Gaussian solitary potential. Noting that the trapped phase space density cannot take on negative values, they developed a relationship between the EH width and amplitude as an inequality. The inequality forms a bounding line or surface which is the minimum width for a given value of the EH amplitude and speed. From this inequality they predict that the observed distributions of widths should occupy the extended region permitted by the inequality, especially EH with small electric field amplitudes. [25] Additionally, there have been several new simulation studies of electron beam instabilities and EH. Omura et al. [1996] used one-dimensional particle simulations to study a variety of electron beam instabilities and found that a bistream instability with hot ions and the bump-on-tail instability are both able to generate solitary waves. [26] More recently, there have been several two-dimensional simulation studies of electron beam instabilities in magnetized plasmas. Miyake et al. [1998] performed twodimensional magnetized simulations of electron beam instabilities, ignoring ion dynamics, and concluded that solitary waves should form for the three cases they investigated: W e /w p = 1.0, 0.4, 0.2. Mottez et al. [1997] investigated the bistream instability with hot ions (similar to the instability considered by Omura et al. [1996]) in a two-dimensional magnetized plasma with W e /w p = 0.2. They found that solitary waves were generated in a two-step process, resulting in fast (v 1.5 times the electron thermal velocity), small-amplitude waves that were stable. Two studies [Goldman et al., 1999; Oppenheim et al., 1999] investigated electron beam instabilities in a two-dimensional magnetized plasma with mobile ions. Both studies investigated plasmas with W e /w p = 5 and found that after many (1000 or more) plasma periods, the two-dimensional shapes of the solitary waves were approximately circular. Muschietti et al. [1999a, 1999c] inserted a one-dimensional analytical BGK mode into a two-dimensional magnetized plasma simulation with W e /w p = 5. The BGK solution was found to be stable, as long as the relative velocity between 3of14

4 Figure 1. A schematic of the Polar electric field antennas used for this study. See color version of this figure at back of this issue. the wave and the ions was greater than the ion acoustic speed. [27] Finally, Oppenheim et al. [2001] performed threedimensional PIC simulations to conclude that simulation in 3-D produced more stable EH than two-dimensional simulations. 3. Instrumentation and Data [28] The Polar spacecraft has a polar orbit (90 inclination) with a perigee of 1.8 R E and an apogee of 9 R E. The data presented here were measured during passes through the northern hemisphere with R between 4 and 9 R E. Polar features comprehensive instrumentation for in situ measurements of fields and particles, as well as several cameras for imaging the aurora. Descriptions of the various Polar instruments may be found in the work of Russell [1995]. [29] The Polar PWI High Frequency Waveform Receiver (HFWR), used as the primary receiver for the results discussed herein, employs a 25 khz digital bandpass filter with band edges at approximately 50 Hz on the low end and 25 khz on the high end. The HFWR has an instantaneous dynamic range of 72 db, adjustable up to 102 db by programmable gain amplifiers. The waveforms are captured from up to six channels (three electric, three magnetic) simultaneously as snapshots, with the sampling rate within each snapshot being khz and the time between successive samples being 14 ms. The snapshots are obtained every 9.2 s and their length varies, depending on how many channels have been chosen. When the HFWR is in interferometry mode (the case for all of the waveform data analyzed and included in this paper), usually two or three channels (all electric) were selected, meaning that the snapshot length was 1.34 s or 0.89 s, respectively. The antennas for the three channels are represented in Figure 1. The first antenna is the U and consists of two spheres separated by 130 m. The remaining two antennas, V+ and V, are each composed of the spacecraft and a sphere 50.6 m from the spacecraft, with one sphere biased positive (V+) and one biased negative (V ). Both the U and the V antennas, which are perpendicular to each other, are contained within the spin plane. These two separated, collinear electric field antennas are used to estimate velocities and scale sizes of plasma waves only during those times when the HFWR is in interferometry mode. That is, time delays or phase shifts of the measurements from the V+ and V channels provide estimates of velocities and scale sizes [Franz et al., 1998; LaBelle and Kintner, 1989]. The orientation of the antennas relative to the local magnetic field will be used in determining the electric field of the solitary waves parallel and perpendicular to the magnetic field. [30] A schematic of the spin-plane electric field antennas is presented in Figure 1. The larger circle in the middle of the figure represents the spacecraft, while the smaller circles labeled U+, U, V+, and V represent the 8 cm spheres on the long wire booms. Note that the figure is not to scale. All of the waveform data presented in this paper are from the interferometry mode of the PWI. In this mode the potential differences between the spacecraft and the V+ and V spheres are measured, as indicated by Figure 1. These two measurements will be labeled Ev+ and Ev in subsequent figures. These two collinear channels allow us to estimate the time delay between the two channels when an electrostatic structure propagates along the antennas. [31] One important issue for our study is the saturation voltages of the waveform receivers. The waveform receivers included an automatic gain control (AGC) that selected the gain for a given snapshot, based upon the amplitude levels in the previous snapshot. Within a given snapshot the gain level is fixed. The waveform receivers had three available gain levels, 0 db, 15 db, and 30 db, with respective peak clipping voltages of V, V, and V. Thus the 50.6 m interferometer channels had peak clipping electric field amplitudes of 44.0 mv/m, 7.9 mv/m, and 1.7 mv/m for the 0 db, 15 db, and 30 db gain states, respectively (Table 1). [32] Since the bipolar pulses are relatively small in their parallel dimension, the antennas are expected to distort the waveforms, primarily by broadening and attenuating the waveforms. However, this effect is small as long as 2L k > d k, where L k is the Gaussian half-width of the electrostatic potential and d k is the projected antenna length along the direction of the magnetic field [Franz, 2000]. [33] The HFWR contains an automatic gain system implemented in software. There are three gain states, namely 0, 15, and 30, with changes made in 15 db increments. Only one gain change can be made prior to any given snapshot based on the analytical results of the previous snapshot. Whatever gain level has been automat- Table 1. Peak Clipping Levels of the Plasma Wave Instrument Waveform Receivers Gain, db Voltage, V Ev+ and Ev, mv/m Eu, mv/m of14

5 Figure 2. Shown are 1.5 hours of Polar Plasma Wave Instrument (PWI) Sweep-Frequency Receiver (SFR) electric and magnetic field spectra. During this time period, Polar was in the postnoon cusp, 78 degrees invariant latitude, 1400 magnetic local time, and 7 8 R E. See color version of this figure at back of this issue. ically set prior to a snapshot will thus be applicable for the entire upcoming snapshot. The algorithm for determining whether a gain change will be made is as follows. A total of 1024 raw data samples, equally spaced in time, from each snapshot are evaluated. The RMS power is calculated using these 1024 samples. The high trip point is defined so that there is a factor of 3 in headroom for spikes above the RMS value. The low trip point is taken somewhat arbitrarily at 20 db below the high trip point. The instrument will either remove 15 db of gain if the high trip point is reached or add 15 db of gain if the low trip point is reached, provided that the instrument is not at the lowest gain stain (0 db) in the case of tripping the high point or at the highest gain state (30 db) in the case of tripping the low point. Because of the way in which the automatic gain system is implemented in the HFWR, several of the solitary waves in any given snapshot may be clipped, even though the receiver will probably not be in saturation. The 0 db state of the receiver is reached very infrequently, particularly so in the regions in which solitary waves are observed. Regardless, the HFWR is able to resolve solitary waves over a wide range of amplitudes (at least 3 orders of magnitude) from a low of about 30 mv/m up to about 45 mv/m (peak) with solitary wave time durations spanning a large range from around 100 ms to around 10 ms. This capability allows us to perform a scientifically meaningful survey of the solitary waves observed in Polar s orbit and to determine the characteristics of those solitary waves using the interferometry mode of the HFWR. 4. Examples [34] During our study of the EH we have analyzed many examples of the waves. In this section we present data representing typical observations. [35] Figure 2 presents 1.5 hours of PWI Sweep-Frequency Receiver (SFR) data for a representative pass through the postnoon cusp. The top panel contains the electric field wave spectrogram, while the bottom panel contains the magnetic field wave spectrogram. In both panels, the solid lines represent the electron cyclotron frequency. During the pass, Polar was between 7 and 8 R E, 1400 MLT, and at 78 degrees invariant latitude. At least three different phenomena were found during this time period. First, above 100 khz, Auroral Kilometric Radiation (AKR) is seen in both the electric and magnetic field channels. Second, an emission just above the electron cyclotron frequency is present in the electric field spectrogram. This emission is indicative of electron Bernstein waves, which are often detected in Polar orbits through the cusp as discussed by Menietti et al. [2001] and Pickett et al. [2001]. The third phenomenon in Figure 2 is the bursty, large-amplitude wave power found primarily in the electric field data. This is an example of the BEN discussed in section 2. Here we see that most of the power is below 300 Hz, with bursts of power extending above the electron cyclotron frequency in the electric field measurements. The plasma frequency in this region of space is around khz, as determined by density measurements from the Hydra instrument. [36] One snapshot of PWI waveform data from 0914:22 is presented in Figure 3. The top panel contains data from the long (130 m) U antenna, while the second and third panels contain data from the V+ and V channels. The bottom panel presents the angles between the antennas and the magnetic fields; the top trace represents the orientation of the V antennas while the bottom trace represents the orientation of the U antenna. Thus the V antennas are within 45 of being parallel to B during the snapshot, and the U antenna is within 40 of perpendicular to B. The spiky 5of14

6 Figure 3. One snapshot of Polar PWI waveform data. See color version of this figure at back of this issue. Figure 5. Cross-correlation analysis of a single solitary wave. From Franz et al. [1998]. Figure 4. Shown are 20 ms of three-channel data when the U antenna is perpendicular to B and the V antenna is 30 degrees away from parallel to B. From Franz et al. [1998]. signatures in the time series are the solitary waves, seen to be clipped within the V channels near the center of the snapshot. A 20 ms portion of the data is plotted in Figure 4. During this short time period, the U antenna was almost perpendicular to B and the V antennas were within 30 degrees of being parallel to B. Here we see the standard signature of the solitary waves: the perpendicular electric fields are unipolar, while the parallel electric fields are bipolar. In this particular case, the peak values of E? are much smaller than the corresponding peak values of E k. This is typical in the cusp, where f pe > f ce. [37] As described by Franz et al. [1998], the two V antennas may be used to estimate the wave velocities. We use the cross-correlation method to estimate the velocities [see LaBelle and Kintner, 1989]. In Figure 5 we present a cross-correlation analysis of a single electron hole. The U antenna waveform is in the top panel, and the V channel waveforms are shown in the middle two panels. The opposite polarities of the signatures in the V+ and V channels are simply the result of the opposite polarities of the antennas. The cross-correlation of the two V time series is plotted in the bottom panel. The extreme value of the cross-correlation is near unity, indicating that the signals in the V+ and V channels are almost identical except for a time delay. In this case we see that the time delay is about 84 ms. Since the V antennas are 28 from being perpendicular to B, we estimate a velocity of 530 km/s. 6of14

7 during the snapshot, and the U antenna is within 40 of perpendicular to B. The spiky signatures in the time series are the solitary waves. In this example the waves cause clipping in all three channels, particularly the V+ and V channels, causing gain to be removed from these two channels in the follow-on snapshot to eliminate the clipping. Although several of the solitary waves observed in Figure 7 are clipped, most of them are not, and interferometry analysis can be applied to these nonclipped solitary waves. During this time period the Polar spacecraft was in the nightside plasma sheet at about 5 R E, 0100 MLT, and 68 degrees invariant latitude. The electron temperature was about 2 kev, and the electron plasma and cyclotron frequencies were f pe 4.5 khz and f ce 10 khz. Finding f pe < f ce in the plasma sheet and plasma sheet boundary layer is common. [40] One interesting feature in Figure 7 is the fact that the signatures in the perpendicular field measurements all tend to have the same polarity. The sign of the perpendicular electric field pulse associated with solitary waves depends on which side of the spacecraft the EH structure passes. Since all or most of the solitary waves have the same polarity in the perpendicular direction, all or most of the solitary waves pass by the same side of the spacecraft instead of being randomly ordered in space. This suggests that the structures are somewhat ordered in the direction perpendicular to B, as depicted in a schematic in Figure 8 where Figure 6. Shown are 30 ms of three-channel data obtained in the nighttime plasma sheet at 5 R E, 0100 magnetic local time, and 68 degrees invariant latitude when the antennas are oblique to B. Note the two different polarizations of solitary waves, corresponding to waves traveling in opposite directions. The spatial distance between the two peaks, 2L k (L k is the Gaussian half width), is then 67 m. For comparison, the Debye length (estimated with Hydra data) is about 23 m in this region of space. [38] It is common to observe solitary waves propagating both parallel and antiparallel to B in the same region of space. An example of this phenomenon as observed in the cusp is provided in Figure 6, where we have plotted 30 ms of data from a time when all of the antennas were oblique to B. Two different polarities of solitary waves are clearly seen in the data. We performed a cross-correlation analysis of the different polarizations [Franz, 2000] and found that the two polarizations simply correspond to the same types of structures moving in opposite directions. Again, these counterstreaming solitary waves are a common occurrence in the Polar data set. [39] One snapshot of three-channel wave data is presented in Figure 7. The top panel contains data from the long (130 m) U antenna (normal voltage mode), while the second and third panels contain data from the V+ and V channels (interferometry mode). The bottom panel presents the angles between the antennas and the magnetic fields; the top trace represents the orientation of the V antennas, while the bottom trace represents the orientation of the U antenna. Thus the V antennas are within 40 of being parallel to B Figure 7. Another snapshot of Polar PWI waveform data. See color version of this figure at back of this issue. 7of14

8 Figure 8. A schematic illustrating the perpendicular ordering observed in the data. most of the solitary waves are ordered to one side of the spacecraft. This ordering is not always present in the data but occurs often enough to be noteworthy. 5. Statistical Properties 5.1. Data Selection and Analysis [41] In this section we present the statistical properties of the structures detected by the PWI. For our analysis, we used a simple algorithm to detect the structures in the time series. Our primary criterion was to prevent false alarms, and hence the algorithm failed to detect a large number (roughly 80%) of the structures, including those that were clipped. For each detected structure, we estimated the parallel electric field amplitude, the width in the time series, the time delay estimate, the cross-correlation value, the polarity of the structure, and the perpendicular electric field amplitude (only when in the three-channel data mode). The resulting data were then used to further discard structures whose optimal cross-correlation magnitudes were not at least 0.7. [42] There are several biases and uncertainties in the resulting processed data. First, the gain on the waveform receiver was often such that field amplitudes larger than about 1.7 mv/m (peak) were not obtained due to waveform clipping. The detection algorithm discards those structures whose waveforms are clipped. A second bias involves estimating the velocities of the structures. Many structures were detected having estimated time delays of zero, and we refer to them as zero-lag structures. These structures have velocities greater than 50.6 m/14 ms = 3614 km/s. This lower bound on the velocity also allows us to calculate a lower bound on the parallel scale sizes of these structures. For comparison, the zero-lag structures observed by Cluster have velocities greater than 2500 m/s [Cattell et al., 2003]. A third bias occurs from the finite antenna length effects, discussed in section 3. Given the antenna response to Gaussian potentials and the uncertainties in the velocity estimates, structures for which we estimate L k < 38 m most likely have been somewhat broadened [Franz, 2000]. Finally, we note that the velocity estimates often have large uncertainties of 25% or more, especially for those structures with high velocities. These uncertainties in the velocities yield correspondingly high uncertainties in size and amplitude estimates. [43] For this study we analyzed Polar PWI data from 19 different days. The particular time periods were chosen from the available interferometry data for which Polar was not in the plasmasphere. Those time periods for which BEN is seen in the SFR data were favored over other regions Structures in the Cusp [44] A total of 2836 structures were detected in the cusp. Of those structures, 769 had time delay estimates that were too small to be measured with our cross-correlation technique. [45] A histogram of the velocities of the structures, normalized to the electron thermal velocity, is plotted in Figure 9. Here we have only included the 2067 structures with measurable velocities. The main piece of information in the figure is that most of the structures have velocities slower than the electron thermal speed. Only 11.6% of the 2067 structures had V > V e. However, we need to consider the bias from the 769 structures for which we do not have velocity estimates. Careful analysis of the data leads us to conclude that between 669 and 769 of those structures had V > V e. Therefore 35.3% to 38.7% of the structures detected in the cusp had V > V e. Note that typical velocities are about 1000 km/s. [46] A histogram of the parallel scale sizes of the detected structures, normalized to the Debye length, is plotted in Figure 10. Here we see that 77.5% of the structures have scale sizes between 1 and 5 Debye lengths, while less than 9% of the structures have scale sizes smaller than the Debye Figure 9. cusp. Histogram of structure velocities observed in the 8of14

9 Figure 10. Histogram of structure parallel scale sizes observed in the cusp. length. Physically, the structures have typical values of L k between 30 and 100 m. Of course, we have experimental biases: the antenna effects are important for 370 of the detected structures, and 769 structures have estimated time delays of zero lags. Lower bounds for the values of L k /l D may be calculated from the temporal widths of the structures along with v > 3600 km/s. The majority (607) of the 769 zero-lag structures must have L k >5l D, while only 40 of the structures have lower bounds that are less than one Debye length. Thus the main effect of the 769 structures with zero lag is to increase the large scalesize tail of the distribution. For the 370 structures where antenna effects are important (L k estimates less than 38 m), 353 are estimated to have l D < L k <3l D and 16 had L k < l D. Thus the distribution of L k /l D should be shifted even more to the left. Of course, it is possible for all 370 of the structures to have scale sizes less than the Debye length. Therefore at most there are 577 (20.3% of the 2836) structures with L k < l D. [47] Figure 11 presents scatterplots of the various properties of the structures versus each other. In the top panel the normalized velocity is plotted versus the normalized scale Figure 11. Structure trends in the cusp. 9of14

10 Figure 12. Histogram of structure velocities observed in the plasma sheet/plasma sheet boundary layer. size. Thus the larger-scale structures tend to have larger velocities. In the second panel the normalized amplitude of the structures is plotted versus the scale size, indicating that larger-amplitude structures tend to have larger scale sizes. Note that this trend is the opposite of that predicted by theories for small-amplitude solitons. Also, the amplitudes of the structures are small. Most of the structures have f 0 < 1 V, corresponding to ef 0 k B T e. The third panel presents the amplitudes versus the velocities, illustrating that largeramplitude structures tend to have larger velocities. However, since all of the correlation coefficients associated with the trends in Figure 11 are smaller than 0.4, we cannot make any stronger inferences about the trends Structures in the Plasma Sheet/Plasma Sheet Boundary Layer [48] As noted above, for the purposes of our study we did not distinguish between the plasma sheet and the plasma sheet boundary layer. We refer to these regions as PS/PSBL. [49] The statistical properties of the structures in the PS/ PSBL are more difficult to determine than those for structures in the cusp. The reason for this is that from the total of 1260 structures detected, almost 40% (496) of the detected structures had velocities so large that the time delay estimate was zero lag (less than 14 ms). Also, 30% of the structures had velocities corresponding to one lag of time delay. These structures with one lag have large uncertainties associated with their velocity estimates and hence large uncertainties in the resulting scale sizes and potential amplitudes. [50] A histogram of the velocities of the structures, normalized to the electron thermal velocity, is presented in Figure 12. Of course, for this histogram we have utilized only the 764 structures for which we have finite time delay estimates. All of the 764 structures have velocities smaller than the electron thermal velocity, which is not surprising since the temperature in the plasma sheet is typically larger than 100 ev, corresponding to a thermal velocity of at least 4200 km/s. The measured thermal velocities were larger than 3614 km/s for all but eight of the zero-lag structures. Since the zero-lag structures have velocities greater than 3614 km/s, it is possible that only eight (out of the 1260) structures have velocities larger than the thermal velocity. On the other hand, since we only have a lower bound for the velocities of the zero-lag structures, it is also possible that all of the zero-lag structures have velocities greater than the thermal velocity. Thus at most 496 (39.4%) and at least eight (0.6%) of the structures detected in the PS/PSBL have velocities larger than the electron thermal velocity. Since ion acoustic solitons can propagate at speeds less than 1.6 of the ion sound speed [Treumann and Baumjohann, 1997] and assuming a generous temperature of 4 kev for pure H + ions, the maximum velocity for ion acoustic solitons is 1400 km/s. Hence almost all of the structures have velocities larger than those predicted by the theory of ion acoustic solitons. [51] A histogram of the parallel scale sizes of the structures, normalized to the Debye length, is presented in Figure 13. The most striking feature is that 575 (75.3% of the 764) structures have scale sizes smaller than the Debye length. Of the 496 zero-lag structures, 231 have minimum scale sizes smaller than the Debye length. Thus at least 575 (45.6%) and at most 806 (64%) of the 1260 detected structures had scale sizes smaller than the Debye length. Note that the structures in the PS/PSBL have typical scale sizes between 50 and 500 m. [52] Figure 14 presents scatter plots of the relationships between the normalized properties of the structures. As with the cusp examples, larger-scale structures tend to have larger velocities and amplitudes. The fact that the largeramplitude structures tend to have larger scale sizes is contrary to predictions of theories for small-amplitude solitons [Treumann and Baumjohann, 1997]. Again, as in the cusp, the structures have small amplitudes with ef 0 k B T e (most have f 0 < 5 V). The correlation coefficient between the lengths and velocities is 0.6, which is moderate. However, since almost half of the structures used in the plot have time delays of a single lag, the correlation could be artificially enhanced. Thus as we found in the cusp, all of these parameters are weakly correlated and these plots indicate nothing more than general trends. 6. Interpretation [53] In this section we compare the observed properties of the coherent structures with existing theories of solitary Figure 13. Histogram of structure parallel scale sizes observed in the plasma sheet/plasma sheet boundary layer. 10 of 14

11 Figure 14. Structure trends in the plasma sheet/plasma sheet boundary layer. waves. For reference, the results of the statistical analysis are summarized briefly in Table 2. There are basically two types of structures that could produce the bipolar signature in the parallel electric field observations: solitons and phasespace holes. [54] Let us first consider solitons. Three types of electrostatic solitons propagate parallel to the geomagnetic field: ion acoustic, electron acoustic, and Langmuir. Langmuir solitons are envelope solitons and therefore have much internal structure, so they would not produce a bipolar pulse. Ion acoustic solitons are positive (compressive) or negative (compressive) potential structures that propagate with velocities such that c s < v <1.6c s [see, e.g., Treumann and Baumjohann, 1997]. The vast majority of the waves discussed in this paper have v 1.6c s and therefore cannot be ion acoustic solitons. Electron acoustic solitons [Dubouloz et al., 1991] are negative (compressive) or positive (rarefactive) potential pulses that propagate with velocities larger than the electron acoustic speed. For a given potential amplitude of solitons, there is only one allowed size, and for small amplitudes, the width decreases with increasing amplitudes. Therefore electron acoustic solitons are also inconsistent with the observations. [55] We therefore conclude that phase-space hole interpretation of the coherent structures is the only explanation that is consistent with the observations. Since the frequencies associated with the structures are much larger than typical ion frequencies and the potentials associated with the structures are positive, the structures are not ion phase- Table 2. Properties of Coherent Structures in the Cusp and Plasma Sheet Plasma Sheet/Plasma Cusp Sheet Boundary Layer Total number Zero lag 27% (769) 40% (496) V < V e 61% 65% 60% 99% L k > l D 80% 89% 35% 55% ef/k B T e 1 100% 100% je? /E k j 80% have 80% have je? /E k j < 0.45 je? /E k j < of 14

12 space holes. Thus electron phase-space holes are the only possible explanation based upon existing theories. [56] In comparing the Polar PWI observations of coherent structures to existing theories of phase-space holes, one must keep in mind their principal feature: they are smallamplitude in both potential and electric field. Typically, dn e /n e < 0.01 and 87% of the EH have df(u)/f 0 (u) <0.2 [Franz, 2000]. In addition, there are very few observations where ef/k B T e exceeds 0.1 and most examples are less than This is in contrast to many other investigations of EH, which focus on large-amplitude structures presumably in environments where they are being created. On the other hand, small-amplitude structures are the phenomena most likely to reflect an inequality between amplitude and width, since the trapped electron population must be nonnegative. That is, the allowed range of amplitude and width is a region bounded by a surface, and the precise bounding surfaces are shown in the companion paper [Chen et al., 2005]. Here we note that for a given EH amplitude, there is roughly a one-order magnitude range in width normalized to the Debye length for both the cusp/cleft populations and the plasma sheet populations (middle panels of Figures 11 and 14). The relationship between velocity and width is similar in that as the velocity increases, the lower bound for width also increases [Turikov, 1984; Chen et al., 2005]. This inequality is reflected in the top panels of Figures 11 and 14 where, for a given velocity, there is roughly a one-order magnitude range in width normalized to the Debye length. The range of cusp EH velocities is concentrated around V/V e, so we do not expect a trend (the lower bound increasing with velocity) to be present in Figure 11 but in Figure 14, with a larger range of plasma sheet EH velocities, the trend is apparent. [57] The result that the relationship of the EH width to the EH amplitude is an inequality has also been presented by Muschietti et al. [1999a, 1999b, 1999c] in their equation (7). However, they then conclude that the width of electron holes must increase with increasing amplitude and interpret the FAST data relationship between EH width and EH amplitude as an equality showing that L k /l D (ef/k B T) 1/4, for ef/k B T e from 0.05 to 0.7. We suspect that the FAST data behave as an equality because the selection of data biased the results toward the largest electric fields (>50 mv/m). From Chen et al. [2005, Figure 1] we can see that for electron phase space holes with a fixed potential, the largest electric fields will occur at the smallest scale lengths where the scaling relation between potential and size becomes an equality. 7. Discussion [58] Above we have presented the statistical observations from the Polar PWI HFWR interferometer in the highaltitude polar magnetosphere. The observations disagree with soliton theories but are consistent with theories of electron phase-space holes. In this section we compare our results with other experimental studies. [59] The scale sizes we report for the cusp data are in agreement with observations in other regions of the magnetosphere. On the other hand, our observations of structures in the PS/PSBL generally imply smaller scale sizes than those reported elsewhere in the literature. Partly, this may be due to the bias in our data set: the range of our instrument prevents determining amplitudes of structures larger than about 45 mv/m (peak) with a gain of 0. For a significant number of structures with amplitudes larger than 1.7 mv/m, the most typical gain state would have the smaller gain of 15 or 0 db. In support, note that Figure 6c of Pickett et al. [2004a] clearly shows that the highest percentage of bipolar solitary waves in this same region of space will fall around 1 2 mv/m peak-to-peak or less. They performed their survey of solitary waves in this region using a wideband receiver with very fast gain update changes, so the results were not biased by the gain state of the instrument. [60] Cattell et al. [1999] analyzed solitary waves in the high-altitude polar magnetosphere using the Electric Field Instrument (EFI) on the Polar spacecraft. They analyzed data from the plasma sheet boundary layer and the cusp, utilizing particle data to aid in interpreting the electric field data. Cattell et al. [1999] found that the solitary waves in the plasma sheet boundary layer propagated both up and down the geomagnetic field, just as we report. In the cusp, they found that solitary waves propagated toward the Earth when there was an earthward ion injection and away from the Earth when there was no such injection. We have also found that the structures propagate both parallel and antiparallel to B in the cusp, although we have not carefully analyzed the particle data to determine the correlation between ion injections and the direction of solitary wave propagation. [61] Cattell et al. [2003] used Polar and Cluster data to examine EH in PS/PSBL and in the cusp. They concluded that the EH velocity, not normalized to the electron thermal velocity, is faster in PS/PSBL than the cusp, as expected for the hotter electron temperatures in PS/PSBL. Additionally, they noted a general trend for EH amplitude, not normalized to kt e, increases with increased hole velocity and scale size, again not normalized to electron temperature or Debye length. The larger peak clipping amplitude of the Polar EFI instrument allowed this investigation to detect the very largest events with efkt e 1. [62] As shown by Muschietti et al. [1999a, 1999b], the FAST measurements of solitary waves in the auroral acceleration region are quantitatively consistent with predictions of an EH model. It is interesting to find that both the FAST and the high-altitude Polar data sets are consistent with such theories. The EH models were one-dimensional theories describing only the parallel electron dynamics, yet both data sets indicate that the EH are not one-dimensional structures. Also, while ions clearly play an important role for the FAST EH, possibly determining the perpendicular scale, the parallel scale sizes are consistent with a theory that includes only electrons. This may indicate that parallel dynamics are somewhat decoupled from perpendicular dynamics. [63] However, there are several differences between the FAST measurements and the high-altitude Polar measurements. For FAST, ef 0 /k B T e is tens of percent [Ergun et al., 1999], while for almost all of the Polar observations it is less than 1 percent. This indicates that the EH detected by FAST correspond to deeper holes in phase-space than those detected by Polar. That is, jdf(u)/f (u)j is larger for the FAST EH. Finally, the FAST EH showed absorption spectra at the ion cyclotron frequencies and are spaced the order of ion 12 of 14