broadband electrostatic wave observations in the auroral region on

Broadband electrostatic wave observations in the

auroral region on Polar and comparisons with theory

C. L. Grabbe1,2 and J. D. Menietti1

Received 6 January 2006; revised 27 April 2006; accepted 5 June 2006; published 26 October 2006.

[1] Broadband electrostatic wave (BEN) data gathered from Polar for passes throughthe aurora are analyzed and compared with theoretical predictions for the origin ofmagnetized solitary waves in BEN. Two passes exhibiting solitary waves were chosen forcomparison, and times were selected that show numerous such waves. Examination of thedata shows that even for these cases the nonsolitary form of BEN occurs much morecommonly than the solitary form that is being particularly examined here. The first passwas for the period of 1430–1550 UT on 7 April 1996 at distances near 7RE, and thesecond pass was for the period of 0600–0800 UTon 10 May 1996 at distances near 8.8 RE.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 indeveloped theory to give predictions for the minimum and maximum electric field forpropagation ofBGKsolitarywaves,which have been the focus of theory for the nonlinear formof BEN for over a decade. This predicted minimum is compared to observations for solitarywaves observed in these spacecraft time intervals along with parameters of the plasmaenvironment at the time of observation, identified from the HYDRAdata for these passes. Theactual electric field is found to lie below the predictedminimum for all cases.Our conclusion isthat these solitary waves are clearly not BGK waves in their observed form and maypossibly have been producedbymeans other than electron trapping processes forBGKsolitarywaves. 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, A10226, doi:10.1029/2006JA011602.

1. Introduction

[2] Intense broadband electrostatic waves (traditionallyreferred to as BEN) in the plasma sheet boundary layer(PSBL) of the magnetotail, originally discovered in the1970s [Gurnett et al., 1976] were shown to be correlatedwith the occurrence of ion beams in the PSBL from ISEE-1observations [Eastman et al., 1984, 1985]. Ion beam modelswere developed and widely investigated as capable ofproducing the broad spectrum that is observed [Grabbeand Eastman, 1984; Grabbe, 1985a, 1985b, 1987; Akimotoand Omidi, 1986; Schriver and Ashour-Abdalla, 1987,1989, 1990; Nishikawa et al., 1987, 1988]. A variation onthe Grabbe-Eastman (GE) model was proposed and debated[Dusenberry and Lyons, 1985; Dusenberry, 1986, 1987,1988; Omidi and Akimoto, 1988]. The consistency of ionbeam models with signatures in the wave data on ISEE-1were examined by Grabbe [1989]. All of these workssupported the view that multiple streaming instabilities orsources were necessary to account for the origin of BEN.Separate models involving trapped particle modes were also

proposed for narrow-band electrostatic noise observed inthe 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 waveformcapture receiver on Geotail in 1994, and a new model forBEN was proposed involving Bernstein-Greene-Kruskal(BGK, or trapped particle) modes, as the observationsprovided the first clear evidence for waves showing non-linear effects in BEN [Kojima et al., 1994;Matsumoto et al.,1994]. Several models and simulations were analyzedcentered around these trapped particle modes [Omura etal., 1996; Krasovsky et al., 1997] and compared withsubsequent observations of similar spiky turbulence in datafrom 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 tomake an unmagnetized model good for a weakly magne-tized plasma. Simulation of these models with the inclusionof the magnetic field in particle trapping was analyzed byMiyake et al [1998]. This study showed that the magnitudeof the magnetic field critically affects the formation ofBGK-type electrostatic solitary waves and can prevent theirformation.

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1Department of Physics and Astronomy, University of Iowa, Iowa City,Iowa, USA.

2SeaLane Consulting, Iowa City, Iowa, USA.

Copyright 2006 by the American Geophysical Union.0148-0227/06/2006JA011602$09.00

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[5] BGK models were reexamined by Grabbe [1998,2000a, 2000b] with the magnetic field included and othermore realistic parameters used, considering not only theBGK modes proposed from the Geotail data, but also amagnetized kinetic model for the generation of part or all ofthe wave spectrum. BEN wave data from ISEE-1 andISEE-3 observed in the midmagnetotail and distant magne-totail, which show evidence of large angles of propagationwith respect to the magnetic field for frequencies near andbelow fce [Grabbe and Eastman, 1984; Coriniti et al.,1990], were discussed as evidence of non-BGK modes.An alternative model was proposed for the midmagnetotailand distant magnetotail, which has both nonlinear BGK-type waves producing the highest frequencies that areclosely field aligned, and standard beam instabilities drivingthe bulk of the broadband spectrum of frequencies belowthat, which are propagating obliquely to the magnetic field.[6] The full magnetized kinetic plasma model with insta-

bilities driven by electron and ion beams for realistic kappadistributions was examined for parameters typical of themidmagnetotail (typically at R � 15 RE), and found toproduce substantial growth rates, up to about 10–20% ofthe plasma frequency fpe. The ISEE-1 and ISEE-3 datadiscussed support this model with generation of these wavesby beam instabilities, but not by trapped particle modes, sothese studies imply instability processes for standard linearmodes play an important role for a large part of the BENobserved, up to approximately the electron cyclotron fre-quency fce. However, trapping effects can be important atfrequencies well above fce, where the trapped particlemodels examined in simulation studies (developed formagnetotail plasmas closer to the Earth) are approximatelyvalid. These highest frequencies are observed only in anarrower angular range with respect to the magnetic fieldfor distance beyond about 10 RE (lower magnetic field), andare consistent with BGK model predictions. This modelpredicts that, at further radial distances out into the magne-totail, BGK-type solitary waves should only be present inthe source region, but not in BEN that has propagated welloutside the source region.[7] Franz et al. [1998] described Polar observation of

solitary wave structure in the high-altitude polar magneto-sphere. Cattell et al. [1999] described observations on Polarof solitary waves in the high-altitude cusp. The structuresexhibited bipolar electric fields. Observations from Polar ofBEN obtained poleward of and within the near-Earthextension of the plasma sheet boundary layer (PSBL) wereanalyzed by Grabbe and Menietti [2002]. The wave dataexamined for BEN poleward of the PSBL (in the plasmamantle) exhibited essentially little or no evidence of solitarywaves. However, the wave data for crossing into theplasma sheet boundary layer source region shows both avery turbulent region immediately at the crossing and theappearance of solitary waves a very short time later. Thelow-frequency portion of the observed electric field wavedata fits the theory of standard beam instabilities, but thehigher-frequency portion running up to about fpe exhibitsnonlinear characteristics, with the magnetic field apparentlyplaying an important role in that nonlinearity.[8] Grabbe [2002] made a theoretical analysis of nonlin-

ear electrostatic waves using coupled plasma equations for amagnetized plasma, yielding a BGK-like equation general-

ized for trapped particle distributions that produce thesestructures in the guiding center approximation for amagnetized plasma. This was extended by Grabbe[2005], who included finding and analyzing the solutionof this magnetized BGK-like equation. In these studies aconditional requirement was derived for electron trapping,and the amount of trapping shows a marked decrease asthe angle of the electric field relative to the backgroundmagnetic field increases, generally ceasing at a criticalfinite angle. The criterion for trapping found by Grabbe[2002] was

cos � >

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ �b

�Dmax

� �2

þ �tw�Dmax

� �2s

� �tw�Dmax

� �ð1Þ

where � is the angle with respect to the magnetic field,�Dmax = E/cB (note E/B is the maximum E � B driftvelocity), �b = jvbj/c, �tw = Dw!ce/2c, with E and D w theBGK mode electric field and width, vb the electron beamvelocity, and !ce the cyclotron frequency Be/me. The resultswere applied to broadband electrostatic waves (BEN) in themagnetotail, and the analysis showed upper limits existedon the angular range with respects to the backgroundmagnetic field at which these BGK solitary waves can exist.It was found that trapping can occur over larger angles inthe near-Earth case because of the large magnetic fieldthere, accounting for oblique solitary waves. However,trapped particle modes are confined to virtual alignment(within about 15�–20�) with the magnetic field at distancesout at 10RE and above in the magnetotail, and cannot existoutside those small-angle ranges. This strongly supports theconclusion made by Grabbe [2000a, 2000b] that in casesfurther out into the magnetotail that solitary waves arehighly field aligned.[9] Catell et al. [2005] reported observations on Cluster

of electron holes near the outer edge of the plasma sheetwhen there were ‘‘narrow’’ (in pitch angle) electron beamspresent 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 canonly exist propagating along B, formed from beams withenergies focused along B. They also reported that thevelocity and scale sizes of e holes are consistent with Drakeet al. [2003] model for reconnection.[10] The purpose of this study is to make comparisons of

some BEN data showing solitary waves from Polar withpredictions of the theory on BGK modes with a magneticfield present, as well as other aspects of theory for thegeneration of those waves. These observations are confinedto the auroral region and magnetotail close to the Earth, andthe angular limitations discussed by Grabbe [2002] may notbe significant here. Thus the focus will be on comparing theelectric fields of the observed waves with the electric fieldpredictions 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 withrespect to the magnetic field. In section 3, details aredescribed on the plasma wave observations on two passesof Polar through the auroral zone. In section 4, comparisons

A10226 GRABBE AND MENIETTI: BEN POLAR DATA COMPARED WITH THEORY

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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 electricantennas (Eu, Ev, and Ez), 3 triaxial magnetic search coils,and a magnetic loop antenna, as well as an advanced plasmawave instrument [Gurnett et al., 1995]. This combinationcan potentially provide the polarization and direction ofarrival of a signal without any prior assumptions.[13] The Plasma Wave Instrument (PWI) on the Polar

spacecraft is designed to provide measurements of plasmawaves in the Earth’s polar regions over the frequency rangefrom 0.1 Hz to 800 kHz. Five receiver systems are used toprocess the data: a wideband receiver, a high-frequencywaveform receiver (HFWR), a low-frequency waveformreceiver (LFWR), two multichannel analyzers, and a pairof sweep frequency receivers (SFR). For the high-frequencyemissions of interest here, the SFR is of special interest. TheSFR has a frequency range from 26 Hz to 808 kHz in5 frequency bands. The frequency resolution is about 3% atthe higher frequencies. In the log mode a full frequencyspectrum can be obtained every 33 s. From 12.5 kHz to808 kHz, of interest in this study, a full frequency spectrumcan be obtained every 2.4 s. The wideband receiver (WBR)provides high-resolution waveform data and is programma-ble, allowing the selection of 11, 22, or 90 kHz bandwidthswith a lower band edge (base frequency) at 0, 125, 250, and500 kHz. In the 90 kHz bandwidth mode the sampling rateis 249 kHz. The LFWR measures electric and magneticfield waveforms in the frequency range of 0.1 Hz to 25 Hz

at a 100 Hz sampling rate. The duty cycle of this receiver istypically to take a 2.5 s snapshot of data every 25 s. TheHFWR measures waveform data over the frequency rangeof 20 Hz to 25 kHz, but also operates with a 2 kHz or16 kHz filter. The sampling rate is 71.43 kHz in the 25 kHzmode. The receiver obtains a snapshot of data every 128 s,which is 456 ms in both the 16 kHz mode and the 2 kHzmode.[14] The Electron and Ion Hot Plasma Instrument

(HYDRA) [Scudder et al., 1995] is an experimental three-dimensional hot plasma instrument for the Polar spacecraft.It consists of a suite of particle analyzers that sample thevelocity space of electron and ions from 100 eV to 35 keV inthree dimensions, with a routine time resolution of 1.5 s. Thesatellite has been designed specifically to study acceleratedplasmas, such as in the cusp and auroral regions.

3. Observations

[15] We selected Polar auroral region passes forthe northern hemisphere for presentation. The data wereobserved over distances ranging from R = 6.58 RE to R =8.76 RE out, where the plasma frequency fpe � fce (theelectron cyclotron frequency). In a later paper we plan tostudy a pass for the southern hemisphere, which is muchcloser to the Earth (out around R � 2RE), where fce � fpe.[16] Because the PWI instrument did not operate in a

continuous data stream mode for the high-frequency wave-form receivers, it is not possible to obtain an absolute valueof occurrence probability of solitary wave structures for thepasses observed. However, we can comment on the occur-rence of such structures during the observations. For the

Figure 1. Frequency versus time spectrogram over 80 min for observations on 7 April 1996 for theperiod of 1430–1600 UT, with the electric field intensity color coded from the sweep frequency receiver(SFR) on board the Plasma Wave Instrument (PWI).


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16 kHz mode of the HFWR used for the passes reported inFigures 1–4, the sampling was discontinuous with 456 mssnapshots every 128 s as noted above. For both passes theHFWR operated at times when intense broadband plasmawave structures were observed by the swept frequencyreceiver (SFR). There were periods when these broadbandemissions occurred simultaneously with the solitary wavestructures, but there were also many periods when they didnot.[17] Note that the occurrence of BEN without the non-

linear structures is significantly greater than BEN with thesenonlinear structures that are the primary focus of this paper.The presence of those standard mode waves is well pre-dicted, 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 interceptsthe auroral region in the range 6.5 RE < r < 7.5 RE. During

this time the plasma frequency (fpe) and cyclotron frequency(fce), most likely are in ratio fpe/fce > 1, based on the fact thatthe whistler mode emission is observed to cutoff near fce. Itis known that whistler mode emission has an upper fre-quency cutoff at either fpe or fce, whichever is lowest. Theparticle data from the HYDRA instrument measured elec-tron density during this period in the range of 0.1 < n <0.4 cm�3, which is consistent with fpe/fce > 1.[19] For the pass of 7 April 1996, between 1441 and 1556

there were seventy-two 28 ms snapshots of data sampled bythe HFWR receiver. For this time interval the SFR observedrather intense broadband emission in the electric fieldantenna almost continuously. Of these, there were 16 snap-shots containing clear signatures of solitary wave structures,and eight others containing turbulent and possible examplesof SW signatures.

Figure 2. Electric and magnetic field data in field-aligned coordinates for a 28 ms snapshot starting at1451:51.468. The first three panels of electric field show two significant waveforms. The first iselectrostatic electron cyclotron (EEC) waves at high frequency, with f > fce, waves which are oftenobserved on Polar Northern Hemisphere passes when fpe/fce > 1 [cf. Menietti et al., 2002]. Superimposedon these waves in the third panel, Ek, are the solitary wave (SW) signatures.


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[20] In Figure 1 we display a frequency versus timespectrogram with the electric field intensity color coded.The data are from the SFR on board PWI. The plot extendsover 80 minutes and includes rather intense electrostatic andelectromagnetic (magnetic oscillations not shown) for thispass. The white line indicates the local electron cyclotronfrequency fce. The intense emission begins near the pole-ward edge of the auroral region at about 1450 and extendsto about 1544. The particle data from the HYDRA instru-ment on board Polar (not shown) confirm the poleward edgeof the auroral precipitation region (plasma sheet boundary)near 1450 and also the equatorward edge of the auroralregion near 1545 where more energetic central plasma sheetprecipitation 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 every128 s. We have selected for presentation three time intervalswhere solitary waves are observed. The first time interval isseen in Figure 2, where we display the electric and magneticfield data in a multipanel presentation in field-alignedcoordinates for a 28 ms snapshot starting at 1451:51.468.The top three panels of electric field show two significantwaveforms, including electrostatic electron cyclotron (EEC)waves at high frequency, with f > fce. These waves areoften observed on Polar northern hemisphere passes whenfpe/fce > 1 [cf. Menietti et al., 2002]. Superimposed onthese waves in panel 3 (Ek) are the solitary wave (SW)signatures. The correlation of the SWs with EEC waves hasbeen pointed out by Menietti et al [2004] for themagnetopause. Here we note that the SWs are observedonly in Ek.

[22] The next series of data for this pass is shown inFigure 3 for the time interval starting at 1504:44.296.Solitary wave structures are observed in all three electricfield components with the largest fields generally, but notalways, in E?. Note the absence of EEC waves for this timeinterval. The magnetic field plotted at this time is notmeaningful 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 ofthe auroral region (Figure 4). Here we observe anotherseries of solitary wave structures quite similar to those ofFigure 3. Note that the monopolar structures observed in E?are typically larger than the corresponding Ek (the scales oneach axis is different). Near 1551 EEC waves are againobserved (not shown) and solitary wave structures are alsopresent with these waves.[24] For the second auroral pass chosen, the magnetic

latitude and L shells are significantly larger (apparently atranspolar arc). In Figure 5 we display a frequency versustime spectrogram with the electric field intensity colorcoded for this pass. The data are from the SFR on boardPWI. The plot extends for 120 minutes and includes ratherintense electrostatic and electromagnetic waves for this pass(magnetic oscillations not shown). The white line indicatesthe local electron cyclotron frequency. The intense emissionbegins near 8.76 RE in auroral region at about 0654 andextends to about 0800.[25] We have selected for presentation a time interval

where solitary waves are observed. This time interval isshown in Figure 6 where we display the electric andmagnetic field data in a multipanel presentation in field-aligned coordinates for a 230 ms snapshot starting at

Figure 3. Electric field data starting at 1504:44.296 (magnetic wave field data were not reliable at thistime). Solitary wave structures are observed in all three electric field components with the largest fieldsgenerally, but not always, in E?. Note the absence of EEC waves for this time interval.


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0702:11.344. For this pass the Polar spacecraft HFWR wasin a mode to monitor high-resolution waveforms at frequen-cies up to 2.0 kHz. The receiver sampled the data in 0.456 ssnapshots every 128 s. The top three panels of electric fieldshow the wave signatures for the electric field both in thedirection parallel to the magnetic field, and in the twodirections perpendicular to the magnetic field. Superim-posed on these waves in all three panels (i.e., with bothsignificant Ek and E? components), are three solitary wave(SW) signatures. Notice the data exhibits a large turbulentsolitary-like structure with E? > Ek, as well as two moretypical solitary waveforms with E? � Ek.

4. Comparison of Theory With Wave Data

[26] The theoretical analysis for BGK mode solitarywaves for a magnetized plasma presented by Grabbe[2002, 2005] will be compared with the data. As pointedout by Grabbe and Menietti [2002], the waveform of BENassociated with linear instabilities is clearly considerably

more prevalent than the nonlinear form that has been thefocus of the Geotail studies, although the nonlinear form hasmore interesting features. The previous section discussedhow this was likewise found to be true in these auroralregions. As one goes out into the more distant magnetotail,the nonlinear form is expected to become much rarer, aspredicted by the theory of Grabbe [2002]. Thus casespresented in these studies for nearby auroral plasmas isexpected to be a region where nonlinear waves are morecommon than in further distances from the Earth, providinga motivation for this study.[27] The theoretical analysis by Grabbe [2002] predicted

how the angular range of nonlinear waves about themagnetic field direction continually narrows in range asthe distance from the Earth increases, and at distancesbeyond 10 RE it is limited to a small cone of less than20� in size about the magnetic field direction. A figure inthat paper showed an absolute maximum of this cone anglefor the whole gamut of plasma parameters. However, thedata in Figures 1–4 were observed at R � 6–9RE where that

Figure 4. Electric and magnetic field data for a 29 ms snapshot starting at 1541:13.896, near theequatorward edge of the auroral region for the first pass.


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angle can be large. Because the plasma observations werenear the Earth the ratio �tw/�Dmax � 1 in equation (1), sothat equation does not serve as a useful test because of thelarge error produced in its predicted � for even quite smallerrors in other measured parameters. Instead, we will makeother comparisons involving the predicted criterion for theexistence of the BGK waves in a magnetized plasma. Thiscriterion of Grabbe [2002, 2005] is

me=2ð Þ v2Dmax sin2 �þ v2z

� �< e

ZEz � dl ð2Þ

where vDmax is the maximum drift velocity (i.e., the drift at90� with respect to the magnetic field direction) and vz, Ez

are the electron velocity and electric field along the magneticfield direction, respectively. While vz in (2) is the individualelectron velocity range necessary for trapping, it will takenas the beam velocity vb in the comparison of theory withobservations, since that criterion must be satisfied and vb canbe 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

Emin=cBð Þ ¼�tw cos ��

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2tw cos

2 �� 2b 1� cos2 �ð Þq

1� cos2 �ð Þ ð3Þ

Emax=cBð Þ ¼�tw cos �þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2tw cos

2 �� 2b 1� cos2 �ð Þq

1� cos2 �ð Þ ð4Þ

Here �tw = Dw!ce/c and �b = jvbj/c, with Dw the effectivespatial width of the solitary wave packet (the spacecraftvelocity times length of observation time, since the solitarywave velocity is much smaller), and vb the electron beamvelocity. Emin is the minimum electric field for the BGKmodes to exist, while Emax is the maximum electric field theBGK modes can attain.[29] Note that (4) implies that Emax exists only because of

the drift across the magnetic field. When no cross-tail driftexists, i.e., when � = 0�, then Emax goes to 1, and there isnot upper bound on E. Correspondingly, at � = 0�, Emin goesto the finite value �b

2/2�tw. Equations (3) and (4) have threegeneral predictions for these nonlinear BGK waves to existin cases like this, where the E � B drift is the dominantcross-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 wavepacket to exist.[33] Table 1 summarizes the effective electron beam

energies from the HYDRA data for the three of the fourperiods of observation. These were determined from theelectron distribution functions at nearby times. There is noHYDRA data for the 1504:44.296 observations on April 7,1996, but data for nearby times showed a bulk electron flowenergy of 800 eV, so a ballpark estimate for an electronbeam energy of 200–400 eV is used.[34] The predicted electric field, extended from (3) to also

include the smallerrB and the polarization drifts (as treatedby Grabbe [2005]), is plotted in Figure 7 along with data

Figure 5. Frequency versus time spectrogram over 120 min for observations on 10 May 1996 for the0600–0800 UT with the electric field intensity color-coded from the sweep frequency receiver (SFR) onboard the Plasma Wave Instrument (PWI).


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points gathered from the observations in Figures 1–6 as afunction of distance in Earth radii, where the Emin predictionshave been calculated for values in the range of observedplasma measurements that minimize it. The results show thatthe solitary waves cannot be BKG modes! In no case do thesolitary waves from Figures 1–6 meet the minimum electricfield 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 observa-tions and the theoretically predicted minimum in Figure 8.This more clearly shows that the electric fields for the wavesdo not meet the minimum requirements for the onset ofBGK waves. The conclusion is that these solitary waves arenot 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 stillmore intense than other background waves. The data inFigures 1–6 suggest this possibility, since there is clear

evidence of the (so-called) solitary waves mixing with otherwaves. These do not appear to be really robust solitarywaves, although the bipolar structure along the magneticfield appears to be generally preserved. To establish thispossibility, one must examine the solitary waves near theirsource 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 byGoldman et al. [1999] from reported observations of soli-

Figure 6. Electric and magnetic field waveform data for a 230 ms snapshot starting at 0702:11.344 on10 May 1996. Solitary wave structures are observed in all three electric field components.

Table 1. Plasma Parameter Range From HYDRA for Snapshotsa

R/RE fce, Hz Eb, eV

7.3 3764 200–4007.13 3911 no data6.58 4462 125–2008.8 2589 150

aData used in the theoretical curves in Figures 7 and 8.


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tary bipolar structures similar to these on the FAST satellite[Ergun et al., 1998] as arising from the nonlinear stage ofthe two-stream instability, a process for which 2-D simu-lations [Oppenheim et al., 1999; Miyake et al., 2000] and3-D simulations [Oppenheim et al., 2001] were subsequentlyrun. Similarly, Singh et al. [2001] and, later, Lu et al. [2005]presented theoretical models in which the solitary wavesobserved in the auroral zone are electron acoustic. Theseelectron acoustic solitary waves were previously investigatedby Dubouloz et al. [1991, 1993] and by Berthomier et al.[1998]. The analysis by Singh et al. [2001] predicts severalfeatures such as soliton dependence on the hot electrondensity, 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 worthfurther 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 beamvelocity has an upper limit in the criterion for the existenceof solitary waves. The solitary waves that were the closest tothe onset criterion for BGK waves were those of Figure 4 atR = 8.8RE, but these are also solitary waves observed at alower beam velocity. Figures 7 and 8 show the minimum Erequired for onset is lower than in the other cases. Thuseven though there is less energy available in the electron

Figure 7. Theoretically predicted minimum electric fields (circles, triangles, squares, and diamonds) forthe plasma conditions in Figures 2–4 and 6 at which electrons can be trapped by electrostatic BGK wavesas a function of distance into the magnetotail (measured in units of Earth radii RE), plotted along with thedata on measured E fields presented in those figures (crosses). The points from the theory use estimatesfor the plasma parameters from the data of Polar. The observations at all different radii show the waveelectric fields are not adequate to drive BGK modes.


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beams for driving the trapped electron BGK waves, theefficiency by which that energy is harnessed in electrontrapping may be greater. To establish this feature in theobservations, one would need to show these waves areindeed BGK remnants originating from a BGK sourceregion in which the onset criterion is satisfied, and furtherdemonstrate similar properties in actual BGK waves neartheir source.

5. Summary and Conclusions

[40] The auroral data, when compared with the theoreti-cally predicted electric field required for the existence ofBGK modes, shows that condition is not satisfied for any of

the solitary wave cases that were examined. These appar-ently nonlinear forms of BEN are distinct from the morecommon forms of BEN which appear consistent withstandard beam instabilities. Thus these solitary waves maywell originate from a different nonlinear process thanelectron trapping.[41] The observations do not rule out the possibility that

BEN of sufficiently large electric fields near the sourceregion which is confined at sufficiently narrow angles withrespect to the magnetic field, can satisfy the criteria forBGK waves. However, well outside these source regions thetrapping process is suppressed, but the solitary waveformsmay still exist because the forms have not yet broken up.

Figure 8. Theoretically predicted minimum electric fields for the plasma conditions in Figures 2–4 and6 at which electrons can be trapped by electrostatic BGK waves as a function of angle � of the electricfield direction with respect to the magnetic field (circles, triangles, squares, and diamonds), plotted alongwith the data on measured E fields presented in those figures (crosses). The points from the theory useestimates for the plasma parameters from the data of Polar. The observations at all different � show thewave electric fields are not adequate to drive BGK modes.


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BEN much more commonly takes on its standard apparentlylinear form, so the trapped electron BGK mode forms ofBEN would be a much rarer form of BEN confined to nearthe source region.[42] BGK solitary waves could still exist out to larger

angles with respect to the magnetic field direction at thesedistances out under 10 RE because fce > fp there (equiva-lently �De > rce), provided that source regions are presentwhere the trapping criterion is satisfied. That is because, tolowest order the wave is experiencing the ‘‘infinite magneticfield’’ approximation in all directions, except at very largeangles with respect to the magnetic field. However, atdistance >10 RE we have fce < fp, or equivalently �De <rce. Thus the larger electron cyclotron radius is preventingthe one-dimensional solitary wave from forming except atangles very close to the magnetic field direction, where thecyclotron effects are no longer significant. The minimumelectric field requirement still must be satisfied for thoseBGK waves to exist, so they would typically be confined toa very narrow angle with respect to the magnetic field rightin the source region (the plasma sheet boundary layer inthose regions).[43] However, the solitary waves are likely produced by a

process other than particle trapping, such as by the nonlin-ear 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 furtherexplored for realistic models so comparisons between thetheory and experiment can adequately supply strong andclear evidence for it. Future developments on models for thenonlinear waves and comparisons with the BEN data inpotential source regions will help answer the question as towhether such alternative processes are the probable sourceof these solitary waves.

[44] Acknowledgments. We would like to thank Jack Scudder’sresearch group for access to HYDRA data to determine input parametersfor the computer programs. This research was supported by the NationalScience Foundation under grant ATM0335583 with the University of Iowaand by NASA through grant NAG511942 with NASA Goddard SpaceFlight Center.[45] Amitava Bhattacharjee thanks David Schriver and S. Singh for

their assistance in evaluating this paper.

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