Broadband electrostatic wave observations in the auroral region on Polar and comparisons with theory

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1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 111,, doi: /2006ja011602, 2006 Broadband electrostatic wave observations in the auroral region on Polar and comparisons with theory C. L. Grabbe 1,2 and J. D. Menietti 1 Received 6 January 2006; revised 27 April 2006; accepted 5 June 2006; published 26 October [1] Broadband electrostatic wave (BEN) data gathered from Polar for passes through the aurora are analyzed and compared with theoretical predictions for the origin of magnetized solitary waves in BEN. Two passes exhibiting solitary waves were chosen for comparison, and times were selected that show numerous such waves. Examination of the data shows that even for these cases the nonsolitary form of BEN occurs much more commonly than the solitary form that is being particularly examined here. The first pass was for the period of UT on 7 April 1996 at distances near 7R E, and the second pass was for the period of UT on 10 May 1996 at distances near 8.8 R E. Four short intervals were chosen from the passes that show interesting solitary waves, each at slightly different radial distance, and the plasma parameters measured are used in developed theory to give predictions for the minimum and maximum electric field for propagation of BGK solitary waves, which have been the focus of theory for the nonlinear form of BEN for over a decade. This predicted minimum is compared to observations for solitary waves observed in these spacecraft time intervals along with parameters of the plasma environment at the time of observation, identified from the HYDRA data for these passes. The actual electric field is found to lie below the predicted minimum for all cases. Our conclusion is that these solitary waves are clearly not BGK waves in their observed form and may possibly have been produced by means other than electron trapping processes for BGK solitary waves. Feasible alternative theories for the solitary waves are discussed. Citation: Grabbe, C. L., and J. D. Menietti (2006), Broadband electrostatic wave observations in the auroral region on Polar and comparisons with theory, J. Geophys. Res., 111,, doi: /2006ja Introduction [2] Intense broadband electrostatic waves (traditionally referred to as BEN) in the plasma sheet boundary layer (PSBL) of the magnetotail, originally discovered in the 1970s [Gurnett et al., 1976] were shown to be correlated with the occurrence of ion beams in the PSBL from ISEE-1 observations [Eastman et al., 1984, 1985]. Ion beam models were developed and widely investigated as capable of producing the broad spectrum that is observed [Grabbe and Eastman, 1984; Grabbe, 1985a, 1985b, 1987; Akimoto and Omidi, 1986; Schriver and Ashour-Abdalla, 1987, 1989, 1990; Nishikawa et al., 1987, 1988]. A variation on the Grabbe-Eastman (GE) model was proposed and debated [Dusenberry and Lyons, 1985; Dusenberry, 1986, 1987, 1988; Omidi and Akimoto, 1988]. The consistency of ion beam models with signatures in the wave data on ISEE-1 were examined by Grabbe [1989]. All of these works supported the view that multiple streaming instabilities or sources were necessary to account for the origin of BEN. Separate models involving trapped particle modes were also 1 Department of Physics and Astronomy, University of Iowa, Iowa City, Iowa, USA. 2 SeaLane Consulting, Iowa City, Iowa, USA. Copyright 2006 by the American Geophysical Union /06/2006JA proposed for narrow-band electrostatic noise observed in the more distant magnetotail [Coriniti and Ashour-Abdalla, 1989; Coriniti et al., 1993]. [3] Spiky pulses a few milliseconds in length on BEN in the PSBL were reported as observed by the waveform capture receiver on Geotail in 1994, and a new model for BEN was proposed involving Bernstein-Greene-Kruskal (BGK, or trapped particle) modes, as the observations provided the first clear evidence for waves showing nonlinear effects in BEN [Kojima et al., 1994; Matsumoto et al., 1994]. Several models and simulations were analyzed centered around these trapped particle modes [Omura et al., 1996; Krasovsky et al., 1997] and compared with subsequent observations of similar spiky turbulence in data from Geotail [Matsumoto et al., 1998, 1999; Omura et al., 1999]. [4] These BGK models examined generally ignored the influence of the magnetic field. However, as Robinson [1988] pointed out, there are rather stringent conditions to make an unmagnetized model good for a weakly magnetized plasma. Simulation of these models with the inclusion of the magnetic field in particle trapping was analyzed by Miyake et al [1998]. This study showed that the magnitude of the magnetic field critically affects the formation of BGK-type electrostatic solitary waves and can prevent their formation. 1of12

2 [5] BGK models were reexamined by Grabbe [1998, 2000a, 2000b] with the magnetic field included and other more realistic parameters used, considering not only the BGK modes proposed from the Geotail data, but also a magnetized kinetic model for the generation of part or all of the wave spectrum. BEN wave data from ISEE-1 and ISEE-3 observed in the midmagnetotail and distant magnetotail, which show evidence of large angles of propagation with respect to the magnetic field for frequencies near and below f ce [Grabbe and Eastman, 1984; Coriniti et al., 1990], were discussed as evidence of non-bgk modes. An alternative model was proposed for the midmagnetotail and distant magnetotail, which has both nonlinear BGKtype waves producing the highest frequencies that are closely field aligned, and standard beam instabilities driving the bulk of the broadband spectrum of frequencies below that, which are propagating obliquely to the magnetic field. [6] The full magnetized kinetic plasma model with instabilities driven by electron and ion beams for realistic kappa distributions was examined for parameters typical of the midmagnetotail (typically at R 15 R E ), and found to produce substantial growth rates, up to about 10 20% of the plasma frequency f pe. The ISEE-1 and ISEE-3 data discussed support this model with generation of these waves by beam instabilities, but not by trapped particle modes, so these studies imply instability processes for standard linear modes play an important role for a large part of the BEN observed, up to approximately the electron cyclotron frequency f ce. However, trapping effects can be important at frequencies well above f ce, where the trapped particle models examined in simulation studies (developed for magnetotail plasmas closer to the Earth) are approximately valid. These highest frequencies are observed only in a narrower angular range with respect to the magnetic field for distance beyond about 10 R E (lower magnetic field), and are consistent with BGK model predictions. This model predicts that, at further radial distances out into the magnetotail, BGK-type solitary waves should only be present in the source region, but not in BEN that has propagated well outside the source region. [7] Franz et al. [1998] described Polar observation of solitary wave structure in the high-altitude polar magnetosphere. Cattell et al. [1999] described observations on Polar of solitary waves in the high-altitude cusp. The structures exhibited bipolar electric fields. Observations from Polar of BEN obtained poleward of and within the near-earth extension of the plasma sheet boundary layer (PSBL) were analyzed by Grabbe and Menietti [2002]. The wave data examined for BEN poleward of the PSBL (in the plasma mantle) exhibited essentially little or no evidence of solitary waves. However, the wave data for crossing into the plasma sheet boundary layer source region shows both a very turbulent region immediately at the crossing and the appearance of solitary waves a very short time later. The low-frequency portion of the observed electric field wave data fits the theory of standard beam instabilities, but the higher-frequency portion running up to about f pe exhibits nonlinear characteristics, with the magnetic field apparently playing an important role in that nonlinearity. [8] Grabbe [2002] made a theoretical analysis of nonlinear electrostatic waves using coupled plasma equations for a magnetized plasma, yielding a BGK-like equation generalized for trapped particle distributions that produce these structures in the guiding center approximation for a magnetized plasma. This was extended by Grabbe [2005], who included finding and analyzing the solution of this magnetized BGK-like equation. In these studies a conditional requirement was derived for electron trapping, and the amount of trapping shows a marked decrease as the angle of the electric field relative to the background magnetic field increases, generally ceasing at a critical finite angle. The criterion for trapping found by Grabbe [2002] was sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 b cos > 1 þ þ 2 tw Dmax Dmax tw Dmax where is the angle with respect to the magnetic field, Dmax = E/cB (note E/B is the maximum E B drift velocity), b = jv b j/c, tw = Dw! ce /2c, with E and D w the BGK mode electric field and width, v b the electron beam velocity, and! ce the cyclotron frequency Be/m e. The results were applied to broadband electrostatic waves (BEN) in the magnetotail, and the analysis showed upper limits existed on the angular range with respects to the background magnetic field at which these BGK solitary waves can exist. It was found that trapping can occur over larger angles in the near-earth case because of the large magnetic field there, accounting for oblique solitary waves. However, trapped particle modes are confined to virtual alignment (within about ) with the magnetic field at distances out at 10R E and above in the magnetotail, and cannot exist outside those small-angle ranges. This strongly supports the conclusion made by Grabbe [2000a, 2000b] that in cases further out into the magnetotail that solitary waves are highly field aligned. [9] Catell et al. [2005] reported observations on Cluster of electron holes near the outer edge of the plasma sheet when there were narrow (in pitch angle) electron beams present but not when the beams were broad in pitch angle. This is expected from the theoretical predictions of Grabbe [2002] because farther out in magnetotail, electron holes can only exist propagating along B, formed from beams with energies focused along B. They also reported that the velocity and scale sizes of e holes are consistent with Drake et al. [2003] model for reconnection. [10] The purpose of this study is to make comparisons of some BEN data showing solitary waves from Polar with predictions of the theory on BGK modes with a magnetic field present, as well as other aspects of theory for the generation of those waves. These observations are confined to the auroral region and magnetotail close to the Earth, and the angular limitations discussed by Grabbe [2002] may not be significant here. Thus the focus will be on comparing the electric fields of the observed waves with the electric field predictions from BGK theory. [11] In section 2 we describe the instrumentation on Polar and its capability of presenting three-dimensional views, which are used to show the angle of propagation with respect to the magnetic field. In section 3, details are described on the plasma wave observations on two passes of Polar through the auroral zone. In section 4, comparisons ð1þ 2of12

3 Figure 1. Frequency versus time spectrogram over 80 min for observations on 7 April 1996 for the period of UT, with the electric field intensity color coded from the sweep frequency receiver (SFR) on board the Plasma Wave Instrument (PWI). are made of those observations with theoretical predictions. Section 5 summarizes the results and conclusions. 2. Instrumentation [12] Polar is the first satellite to have 3 orthogonal electric antennas (E u, E v, and E z ), 3 triaxial magnetic search coils, and a magnetic loop antenna, as well as an advanced plasma wave instrument [Gurnett et al., 1995]. This combination can potentially provide the polarization and direction of arrival of a signal without any prior assumptions. [13] The Plasma Wave Instrument (PWI) on the Polar spacecraft is designed to provide measurements of plasma waves in the Earth s polar regions over the frequency range from 0.1 Hz to 800 khz. Five receiver systems are used to process the data: a wideband receiver, a high-frequency waveform receiver (HFWR), a low-frequency waveform receiver (LFWR), two multichannel analyzers, and a pair of sweep frequency receivers (SFR). For the high-frequency emissions of interest here, the SFR is of special interest. The SFR has a frequency range from 26 Hz to 808 khz in 5 frequency bands. The frequency resolution is about 3% at the higher frequencies. In the log mode a full frequency spectrum can be obtained every 33 s. From 12.5 khz to 808 khz, of interest in this study, a full frequency spectrum can be obtained every 2.4 s. The wideband receiver (WBR) provides high-resolution waveform data and is programmable, allowing the selection of 11, 22, or 90 khz bandwidths with a lower band edge (base frequency) at 0, 125, 250, and 500 khz. In the 90 khz bandwidth mode the sampling rate is 249 khz. The LFWR measures electric and magnetic field waveforms in the frequency range of 0.1 Hz to 25 Hz at a 100 Hz sampling rate. The duty cycle of this receiver is typically to take a 2.5 s snapshot of data every 25 s. The HFWR measures waveform data over the frequency range of 20 Hz to 25 khz, but also operates with a 2 khz or 16 khz filter. The sampling rate is khz in the 25 khz mode. The receiver obtains a snapshot of data every 128 s, which is 456 ms in both the 16 khz mode and the 2 khz mode. [14] The Electron and Ion Hot Plasma Instrument (HYDRA) [Scudder et al., 1995] is an experimental threedimensional hot plasma instrument for the Polar spacecraft. It consists of a suite of particle analyzers that sample the velocity space of electron and ions from 100 ev to 35 kev in three dimensions, with a routine time resolution of 1.5 s. The satellite has been designed specifically to study accelerated plasmas, such as in the cusp and auroral regions. 3. Observations [15] We selected Polar auroral region passes for the northern hemisphere for presentation. The data were observed over distances ranging from R = 6.58 R E to R = 8.76 R E out, where the plasma frequency f pe f ce (the electron cyclotron frequency). In a later paper we plan to study a pass for the southern hemisphere, which is much closer to the Earth (out around R 2R E ), where f ce f pe. [16] Because the PWI instrument did not operate in a continuous data stream mode for the high-frequency waveform receivers, it is not possible to obtain an absolute value of occurrence probability of solitary wave structures for the passes observed. However, we can comment on the occurrence of such structures during the observations. For the 3of12

4 Figure 2. Electric and magnetic field data in field-aligned coordinates for a 28 ms snapshot starting at 1451: The first three panels of electric field show two significant waveforms. The first is electrostatic electron cyclotron (EEC) waves at high frequency, with f > f ce, waves which are often observed on Polar Northern Hemisphere passes when f pe /f ce > 1 [cf. Menietti et al., 2002]. Superimposed on these waves in the third panel, E k, are the solitary wave (SW) signatures. 16 khz mode of the HFWR used for the passes reported in Figures 1 4, the sampling was discontinuous with 456 ms snapshots every 128 s as noted above. For both passes the HFWR operated at times when intense broadband plasma wave structures were observed by the swept frequency receiver (SFR). There were periods when these broadband emissions occurred simultaneously with the solitary wave structures, but there were also many periods when they did not. [17] Note that the occurrence of BEN without the nonlinear structures is significantly greater than BEN with these nonlinear structures that are the primary focus of this paper. The presence of those standard mode waves is well predicted, e.g., by the analysis of Grabbe [2000a, 2000b]. [18] On 7 April 1996 for the period 1430 to 1550 Polar made a northern hemisphere nightside pass that intercepts the auroral region in the range 6.5 R E < r < 7.5 R E. During this time the plasma frequency (f pe ) and cyclotron frequency (f ce ), most likely are in ratio f pe /f ce > 1, based on the fact that the whistler mode emission is observed to cutoff near f ce.it is known that whistler mode emission has an upper frequency cutoff at either f pe or f ce, whichever is lowest. The particle data from the HYDRA instrument measured electron density during this period in the range of 0.1 < n < 0.4 cm 3, which is consistent with f pe /f ce >1. [19] For the pass of 7 April 1996, between 1441 and 1556 there were seventy-two 28 ms snapshots of data sampled by the HFWR receiver. For this time interval the SFR observed rather intense broadband emission in the electric field antenna almost continuously. Of these, there were 16 snapshots containing clear signatures of solitary wave structures, and eight others containing turbulent and possible examples of SW signatures. 4of12

5 Figure 3. Electric field data starting at 1504: (magnetic wave field data were not reliable at this time). Solitary wave structures are observed in all three electric field components with the largest fields generally, but not always, in E?. Note the absence of EEC waves for this time interval. [20] In Figure 1 we display a frequency versus time spectrogram with the electric field intensity color coded. The data are from the SFR on board PWI. The plot extends over 80 minutes and includes rather intense electrostatic and electromagnetic (magnetic oscillations not shown) for this pass. The white line indicates the local electron cyclotron frequency f ce. The intense emission begins near the poleward edge of the auroral region at about 1450 and extends to about The particle data from the HYDRA instrument on board Polar (not shown) confirm the poleward edge of the auroral precipitation region (plasma sheet boundary) near 1450 and also the equatorward edge of the auroral region near 1545 where more energetic central plasma sheet precipitation begins. [21] For this pass the Polar spacecraft HFWR was in a mode to monitor high-resolution waveforms up to 16 khz. The receiver sampled the data in 456 ms snapshots every 128 s. We have selected for presentation three time intervals where solitary waves are observed. The first time interval is seen in Figure 2, where we display the electric and magnetic field data in a multipanel presentation in field-aligned coordinates for a 28 ms snapshot starting at 1451: The top three panels of electric field show two significant waveforms, including electrostatic electron cyclotron (EEC) waves at high frequency, with f > f ce. These waves are often observed on Polar northern hemisphere passes when f pe /f ce > 1 [cf. Menietti et al., 2002]. Superimposed on these waves in panel 3 (E k ) are the solitary wave (SW) signatures. The correlation of the SWs with EEC waves has been pointed out by Menietti et al [2004] for the magnetopause. Here we note that the SWs are observed only in E k. [22] The next series of data for this pass is shown in Figure 3 for the time interval starting at 1504: Solitary wave structures are observed in all three electric field components with the largest fields generally, but not always, in E?. Note the absence of EEC waves for this time interval. The magnetic field plotted at this time is not meaningful due to a data gap. [23] Finally, for this pass we show a 29 ms snapshot of data starting at 1541:13.896, near the equatorward edge of the auroral region (Figure 4). Here we observe another series of solitary wave structures quite similar to those of Figure 3. Note that the monopolar structures observed in E? are typically larger than the corresponding E k (the scales on each axis is different). Near 1551 EEC waves are again observed (not shown) and solitary wave structures are also present with these waves. [24] For the second auroral pass chosen, the magnetic latitude and L shells are significantly larger (apparently a transpolar arc). In Figure 5 we display a frequency versus time spectrogram with the electric field intensity color coded for this pass. The data are from the SFR on board PWI. The plot extends for 120 minutes and includes rather intense electrostatic and electromagnetic waves for this pass (magnetic oscillations not shown). The white line indicates the local electron cyclotron frequency. The intense emission begins near 8.76 R E in auroral region at about 0654 and extends to about [25] We have selected for presentation a time interval where solitary waves are observed. This time interval is shown in Figure 6 where we display the electric and magnetic field data in a multipanel presentation in fieldaligned coordinates for a 230 ms snapshot starting at 5of12

6 Figure 4. Electric and magnetic field data for a 29 ms snapshot starting at 1541:13.896, near the equatorward edge of the auroral region for the first pass. 0702: For this pass the Polar spacecraft HFWR was in a mode to monitor high-resolution waveforms at frequencies up to 2.0 khz. The receiver sampled the data in s snapshots every 128 s. The top three panels of electric field show the wave signatures for the electric field both in the direction parallel to the magnetic field, and in the two directions perpendicular to the magnetic field. Superimposed on these waves in all three panels (i.e., with both significant E k and E? components), are three solitary wave (SW) signatures. Notice the data exhibits a large turbulent solitary-like structure with E? > E k, as well as two more typical solitary waveforms with E? E k. 4. Comparison of Theory With Wave Data [26] The theoretical analysis for BGK mode solitary waves for a magnetized plasma presented by Grabbe [2002, 2005] will be compared with the data. As pointed out by Grabbe and Menietti [2002], the waveform of BEN associated with linear instabilities is clearly considerably more prevalent than the nonlinear form that has been the focus of the Geotail studies, although the nonlinear form has more interesting features. The previous section discussed how this was likewise found to be true in these auroral regions. As one goes out into the more distant magnetotail, the nonlinear form is expected to become much rarer, as predicted by the theory of Grabbe [2002]. Thus cases presented in these studies for nearby auroral plasmas is expected to be a region where nonlinear waves are more common than in further distances from the Earth, providing a motivation for this study. [27] The theoretical analysis by Grabbe [2002] predicted how the angular range of nonlinear waves about the magnetic field direction continually narrows in range as the distance from the Earth increases, and at distances beyond 10 R E it is limited to a small cone of less than 20 in size about the magnetic field direction. A figure in that paper showed an absolute maximum of this cone angle for the whole gamut of plasma parameters. However, the data in Figures 1 4 were observed at R 6 9R E where that 6of12

7 Figure 5. Frequency versus time spectrogram over 120 min for observations on 10 May 1996 for the UT with the electric field intensity color-coded from the sweep frequency receiver (SFR) on board the Plasma Wave Instrument (PWI). angle can be large. Because the plasma observations were near the Earth the ratio tw/ Dmax 1 in equation (1), so that equation does not serve as a useful test because of the large error produced in its predicted for even quite small errors in other measured parameters. Instead, we will make other comparisons involving the predicted criterion for the existence of the BGK waves in a magnetized plasma. This criterion of Grabbe [2002, 2005] is ðme =2Þ v2dmax sin2 þ v2z < e Z Ez dl ð2þ where vdmax is the maximum drift velocity (i.e., the drift at 90 with respect to the magnetic field direction) and vz, Ez are the electron velocity and electric field along the magnetic field direction, respectively. While vz in (2) is the individual electron velocity range necessary for trapping, it will taken as the beam velocity vb in the comparison of theory with observations, since that criterion must be satisfied and vb can be measured. [28] When E B is the dominant cross-field drift, jvdmaxj = E/B. Real trapped electron solutions to equation (2) exist in the range Emin < E < Emax where ðemin =cbþ ¼ ðemax =cbþ ¼ tw cos qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 cos2 2 ð1 cos2 Þ tw b ð1 cos2 Þ tw cos þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 cos2 2 ð1 cos2 Þ tw b ð1 cos2 Þ ð3þ ð4þ Here tw = Dw!ce/c and b = jvbj/c, with Dw the effective spatial width of the solitary wave packet (the spacecraft velocity times length of observation time, since the solitary wave velocity is much smaller), and vb the electron beam velocity. Emin is the minimum electric field for the BGK modes to exist, while Emax is the maximum electric field the BGK modes can attain. [29] Note that (4) implies that Emax exists only because of the drift across the magnetic field. When no cross-tail drift exists, i.e., when = 0, then Emax goes to 1, and there is not upper bound on E. Correspondingly, at = 0, Emin goes to the finite value 2b/2 tw. Equations (3) and (4) have three general predictions for these nonlinear BGK waves to exist in cases like this, where the E B drift is the dominant cross-field drift. [30] 1. The E field of the wave has a lower limit. [31] 2. The electron beam velocity vb has an upper limit. [32] 3. The observed width Dw of the wave packet has a lower limit. fcedt 1 has to be large enough for the wave packet to exist. [33] Table 1 summarizes the effective electron beam energies from the HYDRA data for the three of the four periods of observation. These were determined from the electron distribution functions at nearby times. There is no HYDRA data for the 1504: observations on April 7, 1996, but data for nearby times showed a bulk electron flow energy of 800 ev, so a ballpark estimate for an electron beam energy of ev is used. [34] The predicted electric field, extended from (3) to also include the smaller rb and the polarization drifts (as treated by Grabbe [2005]), is plotted in Figure 7 along with data 7 of 12

8 Figure 6. Electric and magnetic field waveform data for a 230 ms snapshot starting at 0702: on 10 May Solitary wave structures are observed in all three electric field components. points gathered from the observations in Figures 1 6 as a function of distance in Earth radii, where the E min predictions have been calculated for values in the range of observed plasma measurements that minimize it. The results show that the solitary waves cannot be BKG modes! In no case do the solitary waves from Figures 1 6 meet the minimum electric field required for the existence of BGK modes. [35] To make the result clearer, we have plotted the electric field versus the wave angle for both the observations and the theoretically predicted minimum in Figure 8. This more clearly shows that the electric fields for the waves do not meet the minimum requirements for the onset of BGK waves. The conclusion is that these solitary waves are not BGK modes. [36] This conclusion implies one of two possibilities. [37] 1. The solitary waves are still remnants of BGK waves that are observed well away from their source region, but still have not broken up very much, since they are still more intense than other background waves. The data in Figures 1 6 suggest this possibility, since there is clear evidence of the (so-called) solitary waves mixing with other waves. These do not appear to be really robust solitary waves, although the bipolar structure along the magnetic field appears to be generally preserved. To establish this possibility, one must examine the solitary waves near their source region and show they meet the BGK criterion. [38] 2. Some other process is producing these solitary waves. One example of such a process was proposed by Goldman et al. [1999] from reported observations of soli- Table 1. Plasma Parameter Range From HYDRA for Snapshots a R/R E f ce,hz E b,ev no data a Data used in the theoretical curves in Figures 7 and 8. 8of12

9 Figure 7. Theoretically predicted minimum electric fields (circles, triangles, squares, and diamonds) for the plasma conditions in Figures 2 4 and 6 at which electrons can be trapped by electrostatic BGK waves as a function of distance into the magnetotail (measured in units of Earth radii R E ), plotted along with the data on measured E fields presented in those figures (crosses). The points from the theory use estimates for the plasma parameters from the data of Polar. The observations at all different radii show the wave electric fields are not adequate to drive BGK modes. tary bipolar structures similar to these on the FAST satellite [Ergun et al., 1998] as arising from the nonlinear stage of the two-stream instability, a process for which 2-D simulations [Oppenheim et al., 1999; Miyake et al., 2000] and 3-D simulations [Oppenheim et al., 2001] were subsequently run. Similarly, Singh et al. [2001] and, later, Lu et al. [2005] presented theoretical models in which the solitary waves observed in the auroral zone are electron acoustic. These electron acoustic solitary waves were previously investigated by Dubouloz et al. [1991, 1993] and by Berthomier et al. [1998]. The analysis by Singh et al. [2001] predicts several features such as soliton dependence on the hot electron density, which bear comparison with the data. Most of these investigations of alternative models used a 1D analysis. Figures 7 and 8 suggest these proposed processes are worth further investigation, specifically in a magnetized plasma (i.e., at least 2-D and anisotropic). [39] Another feature Figures 7 and 8 may shed light on is the unusual conclusion made above that the electron beam velocity has an upper limit in the criterion for the existence of solitary waves. The solitary waves that were the closest to the onset criterion for BGK waves were those of Figure 4 at R = 8.8R E, but these are also solitary waves observed at a lower beam velocity. Figures 7 and 8 show the minimum E required for onset is lower than in the other cases. Thus even though there is less energy available in the electron 9of12

10 Figure 8. Theoretically predicted minimum electric fields for the plasma conditions in Figures 2 4 and 6 at which electrons can be trapped by electrostatic BGK waves as a function of angle of the electric field direction with respect to the magnetic field (circles, triangles, squares, and diamonds), plotted along with the data on measured E fields presented in those figures (crosses). The points from the theory use estimates for the plasma parameters from the data of Polar. The observations at all different show the wave electric fields are not adequate to drive BGK modes. beams for driving the trapped electron BGK waves, the efficiency by which that energy is harnessed in electron trapping may be greater. To establish this feature in the observations, one would need to show these waves are indeed BGK remnants originating from a BGK source region in which the onset criterion is satisfied, and further demonstrate similar properties in actual BGK waves near their source. 5. Summary and Conclusions [40] The auroral data, when compared with the theoretically predicted electric field required for the existence of BGK modes, shows that condition is not satisfied for any of the solitary wave cases that were examined. These apparently nonlinear forms of BEN are distinct from the more common forms of BEN which appear consistent with standard beam instabilities. Thus these solitary waves may well originate from a different nonlinear process than electron trapping. [41] The observations do not rule out the possibility that BEN of sufficiently large electric fields near the source region which is confined at sufficiently narrow angles with respect to the magnetic field, can satisfy the criteria for BGK waves. However, well outside these source regions the trapping process is suppressed, but the solitary waveforms may still exist because the forms have not yet broken up. 10 of 12

11 BEN much more commonly takes on its standard apparently linear form, so the trapped electron BGK mode forms of BEN would be a much rarer form of BEN confined to near the source region. [42] BGK solitary waves could still exist out to larger angles with respect to the magnetic field direction at these distances out under 10 R E because f ce > f p there (equivalently De > r ce ), provided that source regions are present where the trapping criterion is satisfied. That is because, to lowest order the wave is experiencing the infinite magnetic field approximation in all directions, except at very large angles with respect to the magnetic field. However, at distance >10 R E we have f ce < f p, or equivalently De < r ce. Thus the larger electron cyclotron radius is preventing the one-dimensional solitary wave from forming except at angles very close to the magnetic field direction, where the cyclotron effects are no longer significant. The minimum electric field requirement still must be satisfied for those BGK waves to exist, so they would typically be confined to a very narrow angle with respect to the magnetic field right in the source region (the plasma sheet boundary layer in those regions). [43] However, the solitary waves are likely produced by a process other than particle trapping, such as by the nonlinear stages of beam instabilities for a magnetized plasma [Goldman et al., 1999; Singh et al., 2001; Oppenheim et al., 2001; Lu et al., 2005]. Those processes should be further explored for realistic models so comparisons between the theory and experiment can adequately supply strong and clear evidence for it. Future developments on models for the nonlinear waves and comparisons with the BEN data in potential source regions will help answer the question as to whether such alternative processes are the probable source of these solitary waves. [44] Acknowledgments. We would like to thank Jack Scudder s research group for access to HYDRA data to determine input parameters for the computer programs. This research was supported by the National Science Foundation under grant ATM with the University of Iowa and by NASA through grant NAG with NASA Goddard Space Flight Center. [45] Amitava Bhattacharjee thanks David Schriver and S. Singh for their assistance in evaluating this paper. References Akimoto, K., and N. Omidi (1986), The generation of broadband electrostatic noise by an ion beam in the magnetotail, Geophys. Res. Lett., 13, 97. Berthomier, M., R. Pottelette, and M. Malingro (1998), Solitary waves and weak double layers in a two-electron temperature auroral plasma, J. Geophys. Res., 103, Cattell, C. A., et al. (1999), Comparisons of Polar satellite observations of solitary wave velocities in the plasma sheet boundary and the high altitude cusp to those in the auroral zone, Geophys. Res. Lett., 26, 425. Catell, C., et al. 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