A statistical study of the properties of large amplitude whistler waves and their association with few eV to 30 keV electron distributions observed in the magnetosphere by Wind

3.9 MB·English

by  L. B. Wilson III

A statistical study of the properties of large amplitude
whistler waves and their association with few eV to 30
1 keV electron distributions observed in the magnetosphere
1
0
by Wind
2

n
a
J L.B.WilsonIII∗†,C.A.Cattell∗,P.J.Kellogg∗,J.R.Wygant∗,K.Goetz∗,

7 A.Breneman∗,andK.Kersten∗
1

January18,2011
]
h
p
-
e
c Abstract
a
p
Wepresentastatisticalstudyofthecharacteristicsofverylargeamplitudewhistlerwaves
s
. inside the terrestrial magnetosphere using waveform capture data from the Wind spacecraft as
s
c an addition of the study by Kellogg et al. [2010]. We observed 244(65) whistler waves using
i electric(magnetic)fielddatafromtheWindspacecraftfinding∼40%(∼62%)ofthewaveshave
s
y peak-to-peakamplitudesof≥50mV/m(≥0.5nT).Wepresentanexamplewaveformcaptureof
h thelargestmagneticfieldamplitude((cid:38)8nTpeak-to-peak)whistlerwaveeverreportedinthera-
p diationbelts.TheestimatedPoyntingfluxmagnitudeassociatedwiththiswaveis(cid:38)300µW/m2,
[ roughlyfourordersofmagnitudeabovepreviousestimates. SuchlargePoyntingfluxvaluesare

consistentwithrapidenergizationofelectrons. Themajorityofthelargestamplitudewhistlers
1
occurduringmagneticallyactiveperiods(AE>200nT).Thewaveswereobservedtoexhibit
v
abroadrangeofpropagationangleswithrespecttothemagneticfield,0◦ ≤θ <90◦,which
3 kB
0 showednoconsistentvariationwithmagneticlatitude.Theseresultsareinconsistentwiththeidea
3 thatthewhistlersareallgeneratedattheequator,propagatingalongthemagneticfield,andthat
3 theobservedobliquenessisduetopropagationeffects.Wealsoidentifiedthreetypesofelectron
. distributionsobservedsimultaneouslywiththewhistlerwavesincludingbeam-like,beam/flattop,
1
andanisotropicdistributions. Thewhistlersexhibiteddifferentcharacteristicsdependingonthe
0
1 observed electron distributions. For instance, the whistlers observed with anisotropic distribu-
1 tions in the radiation belts had larger ∆f/f than the rest of the whistlers. The majority of the
v: waveformsobservedinourstudyhavef/fce≤0.5andareobservedprimarilyintheradiationbelts
simultaneouslywithanisotropicelectrondistributions.
i
X
r 1 Introduction
a
Whistlerwavesareoneofthemostubiquitouswavemodesinplasmas. Theyhavebeenobservedin
themagnetosphere[BurtisandHelliwell,1969;Cattelletal.,2008;Cullyetal.,2008;Parrotetal.,
2003;Russelletal.,1969;Santol´ıketal.,2003],inthesolarwind[Brenemanetal.,2010;Neubauer
∗DepartmentofPhysicsandAstronomy,UniversityofMinnesota,Minneapolis,MN
†NASAGoddardSpaceFlightCenter,Greenbelt,Maryland,USA.
1
andMusmann,1977],upstreamofinterplanetaryshocks[WilsonIIIetal.,2009],upstreamofplan-
etarybowshocks[Bertuccietal.,2005;Hoppeetal.,1981],andincometaryforeshocks[Tsurutani
et al., 1987]. Whistler waves are a right-hand polarized electromagnetic mode that can propagate
alongthemagneticfieldorathighlyobliqueanglesasaquasi-electrostaticmodeneartheresonance
coneinteractingwithbothionsandelectrons. Whistlerscanbedrivenunstablebyelectrontemper-
ature anisotropies [Hashimoto and Kimura, 1981; Kennel and Petscheck, 1966], electron heat flux
[Garyetal.,1994],andelectronbeams[SauerandSydora,2010;Tokaretal.,1984;Zhangetal.,
1993]. Forsimplicity,weincludechorus,plasmaspherichiss,andobliquewhistler-modesinouruse
ofthetermwhistlerwave. Becausewhistlersinteractstronglywithenergeticparticles[Kenneland
Petscheck, 1966; Lyons et al., 1972], it has been well accepted that they play an important role in
globalradiationbeltdynamics. Thus,whistlerwaveshavebeenatopicofextremeinterestforover
40yearsinmagnetosphericphysics.
Time-averagedspectralintensitieshavebeenusedtoestimatewhistlerwaveamplitudesforover
30years[Gurnettetal.,1976],butaveragingcanunderestimatetheinstantaneouswaveamplitudes.
Typical whistler time-averaged amplitudes are ∼0.5 mV/m for the electric field [Meredith et al.,
2001]and∼0.01-0.1nTforthemagneticfield[Horneetal.,2003,2005]. Theunderestimatedwave
amplitudes may mask important wave-particle interactions which require larger amplitude waves.
RecentobservationsbySantol´ıketal.[2003],Cattelletal.[2008],Cullyetal.[2008],andLeContel
etal.[2009]showedthatprevioustimeaveragedspectralintensitymeasurementsmayoftenunder-
estimate the wave amplitude of terrestrial whistler waves by more than two orders of magnitude.
TestparticlesimulationshavefoundthatelectronscanbeacceleratedtoMeVenergiesinrelatively
shorttimesbytheseverylargeamplitudewhistlerwaves[Bortniketal.,2008;Cattelletal.,2008;
Omuraetal.,2007]. Theseobservationshaveraisednewquestionsregardingtheenergizationand
lifetimeofradiationbeltparticles.
During an eight year period, the Wind spacecraft went through a number of petal orbits into
the terrestrial magnetosphere. We report on whistler wave statistics for 13 of those petal orbits,
buildingonthestudybyKelloggetal.[2010],whousedanautomatedsearchalgorithmtofindlarge
amplitudewhistlers.AsearchbyeyeyieldedsignificantlymorewhistlerwavesthanreportedinKel-
loggetal.[2010]. Thisisthefirststudytoexaminethestatisticsoftherelationshipbetweenwave
propagationangleandmagneticlatitudefortheseverylargeamplitudewhistlers,findingnocorre-
lationbetweenpropagationangleandmagneticlatitude. Thisisthefirstobservationofveryintense
whistlerPoyntingflux. Wealsopresentthefirststatisticalstudyofthecharacteristicsoftheelectron
distributionsovertheenergyrangeofafeweVto30keVassociatedwithlargeamplitudewhistler
wavesinsidetheterrestrialmagnetosphereusingwaveformcapturedatasimultaneouslywiththree
differenttypesofelectrondistributions. Thepaperisorganizedasfollows:Section2introducesand
outlines the data sets and analysis techniques, Section 3 describes the observations, and Section 4
discussestheresultsandconclusionsofourstudy.
2 Data Sets and Analysis
WaveformcaptureswereobtainedfromtheWind/WAVESinstrument[Bougeretetal.,1995],using
the time domain sampler (TDS) receiver, which provides a waveform capture (herein called TDS
sample) of 2048 points with timespans ranging from ∼17 ms to ∼1000 ms, depending on sample
rate. TDS samples are utilized from the fast (TDSF) and slow (TDSS) TDS receivers. The TDS
receivershavealowfrequencycutoffofroughly3.3HzforTDSSand130HzforTDSF.TheTDS
bufferstoresandevaluateswaveformsbasedupontheiramplitude,keepingonlythelargestevents.
TDSFsamplesarecomposedoftwoelectricfieldvectorsintheXY-GSEplane,definedasE .TDSS
j
2
samplesarecomposedoffourvectors,eitherthreeelectric(E )andonemagnetic(B )orthreemag-
j j
neticandoneelectricfieldcomponents. TheTDSSsamplesarerotatedintomagneticfield-aligned
coordinates (FACs), determined with magnetic field measurements from the Wind magnetic field
instrument(MFI)[Leppingetal.,1995],wherethesubscript(cid:107)(⊥)refertoparallel(perpendicular)to
the magnetic field. We define the wave amplitude as |E | for electric fields and |B | for magnetic
w w
fields. TheWind/WAVESinstrumentalsocontainsanonboardtime-averagedspectralintensityin-
strument,thethermalnoisereceiver(TNR),usedtoanalyzelowamplitudesignalsanddeterminethe
localplasmadensityusingtheplasmalinewhenpossible. NotethatweonlyusetheTNRreceiver
fordeterminationofthelocalplasmaline,notforwhistlerwaveidentification.
The wave vector, k, and propagation angle with respect to the magnetic field, θ , were de-
kB
termined using Minimum Variance Analysis (MVA) [Khrabrov and Sonnerup, 1998] on bandpass
filteredTDSSsampleswiththreemagneticfieldcomponents. Thefrequencyrangesforeachband-
passfilter,determinedfromspectralanalysis,werechosenindependentlyforeachTDSsample. We
requiredintermediatetominimumeigenvaluesofthespectralmatrix,λ /λ ,tosatisfythecondi-
int min
tion≥10.0iflessthan50fieldvectorswereusedintheanalysis[KhrabrovandSonnerup,1998].
Full4π steradianparticledistributionsforelectronsandionswereobtainedfromtheWind/3DP
EESAandPESAparticledetectorsforlowenergies(<30keV)andthesolidstatetelescopes(SSTs)
forhighenergy(∼30-500keV)electrons(SSTFoil)and(∼70-6000keV)protons(SSTOpen)[Lin
et al., 1995]. Electron distribution functions, from a few eV to 30 keV, were examined for pitch-
angle anisotropies as a possible free energy source. Higher energy particle distributions using the
3DPsolidstatetelescope(SST)detectorswereusedtoidentifytheperiodswhenWindenteredthe
radiationbelts. However,sincethedetectorswerebuiltforsolarwinduse,theyoftensaturateinthe
radiationbelts. Thus,theycouldnotbeusedtolookforpotentialfreeenergysourcesorforcorre-
lationsbetweenfluxenhancements/lossesandstrongwaves. WeusetheMFIdata,combinedwith
PESAdata, todefinetheregionwecalltheoutermagnetosphereastheregionbetweentheterres-
trialmagnetopauseandtheradiationbelts. Themagnetopausewasdefinedbythesharpgradientin
flowspeed,density,B-GSM,andmagneticfieldturbulence. Wedefinedtheradiationbeltregionby
z
thesharpincrease(decrease)inomni-directionalfluxofthehighenergyelectrons(>100keV)and
protons(>1MeV)asWindapproached(departedfrom)perigeeinitspetalorbits. Toallowforthe
possibilityofdifferentsourcesoffreeenergyand/ordamping, outermagnetosphericandradiation
beltwhistlerswereexaminedseparately.
Wealsoexaminedelectrontemperatureanisotropies(T /T (cid:54)=1)forboththeEesaLow(j=c)
⊥j (cid:107)j
andEesaHigh(j=h)separatelytocomparetothethresholdcriteriaforwhistleranisotropyinsta-
bility[KennelandPetscheck,1966]. ThetwoEESAdetectorshaveenergyranges((cid:46)30keV)well
below the typical >100 keV radiation belt particles. Thus, using these detectors limits our ability
to accurately describe the full particle distributions in the radiation belts. However, we will show
that many of the observed whistlers are in resonance with electrons below 30 keV. Recall that we
mentioned that the SST detectors often saturate in the radiation belts, thus we are limited to (cid:46)30
keVelectrondistributionsinthisstudy.
3 Observations
Figure 1 shows an example of a Wind perigee pass from 1998-11-13/13:00:00 UT to 1998-11-
14/02:00:00UT.ThetoppanelshowsoneminuteaveragedspectraldatafromtheTNRreceiverin
√
µV/ Hz. Overplotted is the upper hybrid frequency (f ) line. Due to the large amounts of noise
uh
intheTNRdata,thedensityforthef linewasestimatedfromPESAHighdensitymeasurements.
uh
3
However,thePESAmomentsgiveanunderestimateofdensityatlowL-shells,soourdensityesti-
matesarelikelytoolow. ItcanbeseenthatthewhistlersgenerallyoccurbetweenL∼3-5(locations
markedbyverticallinesinallpanels),intheradiationbelts. Alsonotethatthelowestenergybins
oftheSSTdetectorssaturatebetweenroughly16:00and20:00UTon1998-11-13. Thisisthemost
obvious in the SST Foil panel for the 40 keV line which drops in flux just after 16:00 UT. This
saturationlimitedourabilitytousetheSSTdetectorswhenexaminingthewhistlerwaves.
Figure2plotstwowhistlerwavesobservedneartheequatorialplane,postmidnightnearL∼4.
ThesewhistlerswereobtainedfromtheTDSSinstrument(withthreeelectricandonemagneticfield
sampledat7.5kHz); thewaveformsshownwerefiltered(frequencyrangeshownintoppanels)to
removesuperposedlowandhighfrequencysignals. TheL-shell,MLT,GSMlatitude,lowerhybrid
√
frequency(f = f f ),andf foreachcorrespondingwhistlerarelabeledinthefigure. Notethat
lh ce ci ce
the8nTpeak-to-peakamplitudewereporthereisonlyonecomponentofthemagneticfield. The
total magnetic amplitude for both waveforms shown in Figure 2 is larger. The polarization of the
electricfieldsisellipticalforbothexamples. The1998-11-13exampleismuchmoreoblique(θ ∼
kB
73◦)andelliptical(λ /λ ∼10.7)thanthe2000-04-10example(θ ∼39◦andλ /λ ∼1.9).
max min kB max min
Omuraetal.[2007]gavearelationshipforthemaximumchangeinkineticenergygivenby:
5.6×104 δB
(∆KE) ≈ [MeV] (1)
max L2(cid:112)1+ξo2 Bo
where L is the L-shell, ξ 2 = ω (Ω - ω)/ω 2, Ω is the equitorial cyclotron frequency, and
o EQ pe EQ
δB/B istheratioofwaveamplitudetobackgroundmagneticfield. ForthewavesshowninFigure
o
2, (∆KE) ∼ 61 MeV(28 MeV) for the wave on 1998-11-13(2000-04-10). The examples show
max
that: (1) large amplitude whistler waves in the radiation belts are bursty; (2) electric fields are in
excessoftwoordersofmagnitudeabovepreviousestimates;(3)thefirstobservationsofthesevery
large amplitude whistler waves with search coil magnetic fields shows amplitudes exceeding two
ordersofmagnitudeabovepreviousestimates;and(4)thewavesarecapableofproducingelectrons
withenergiesgreateraMeV.
To illustrate the occurrence rate of these large amplitude whistlers we will discuss two Wind
perigeepassesindetail. Duringtheperigeepassesof1998-11-13and2000-04-10,Windobserved
some of the largest whistler events ever recorded in the magnetosphere. On 1998-11-13 between
18:15:08.170UTand18:21:17.884UT,orroughly6minutes,Windobserved11TDSFand3TDSS
samples. All 14 TDS samples had |E | ≥ 100 mV/m and all 3 TDSS samples had |B | ≥ 4 nT.
w w
Thelargenumberofwhistlersobservedduringsuchashortdurationandspatialregionisconsistent
with STEREO observations [Cattell et al., 2008]. On 2000-04-10 between 02:45:39.736 UT and
04:32:44.807UT,orroughly1.75hours, thespacecraftmovedthrough∼3.48hr(∼52.2◦)ofMLT
and∼0.55L-shell. Duringthisinterval,Windobserved34TDSsamples,27TDSFand7TDSS.Of
those34samples,31had|E |≥80mV/mandall7TDSSsampleshad|B |≥1nT.
w w
SinceweonlyhavefourfieldcomponentsforeachTDSSsample,wecannotfullydescribethe
wavePoyntingflux.However,wecancalculatepartoftwocomponentstoestimatethemagnitudeof
thePoyntingfluxforthesewaves. The1998-11-13(2000-04-10)whistlerinFigure2hasPoynting
flux magnitude of (cid:38)300 µW/m2((cid:38)30 µW/m2), which is roughly four(three) orders of magnitude
largerthantheestimatesfoundbySantol´ıketal.[2010].Toestimatethepossibleimpactofthelarge
Poyntingfluxes,weperformthecalculationofSantol´ıketal.[2010]assumingthesameestimatesof
(cid:38)1MeVelectronfluxes,backgrounddensities,andfield-alignedcolumnarea(ofafluxtube). We
findthatitwouldtakeroughly5ms(50ms)forthewhistlersseeninFigure2todeposittheneces-
sary energy density (∼10−4 J/m2) to accelerate plasma sheet electrons to 1 MeV, assuming 100%
efficiency, in the outer radiation belt. If we now assume a 1% efficiency, we find a time scale of
0.5seconds(5.5seconds)necessaryforthesewhistlerstoproducethesameeffect. Theseestimates
4
are five to six orders of magnitude shorter than the typical estimates. However, as one can see in
Figure2,thesewhistlersareveryburstyandtheiramplitudesarenotsustainedforlongerthan10’s
of milliseconds. We should also note that the larger amplitude waves we observe will likely not
interactwithelectronsinthesamequasi-linearfashionasdescribedbySantol´ıketal.[2010].
Figure 3 shows the relationship between peak whistler wave amplitudes and one minute AE-
Indexforthewhistlers. Theplotsshowaslighttrendofincreasingpeakamplitudeswithincreasing
AE,consistentwithincreasedsubstormactivityprovidingmorefreeenergyforwhistlergrowth. A
cursoryexaminationofLANLgeosynchronouslowenergy(30-300keV)electrondatashowslarge
injectionsonthenightsidejustpriortothetwowhistlersobservedinFigure2, consistentwiththe
AE-indexanalysis.
Figure4illustratesthedifferenttypesofelectrondistributionfunctionsobservedattimesclose
to and/or concurrently with the whistlers. Each plot represents the parallel and perpendicular cuts
of the electron distribution function seen as the solid red and dashed blue lines, respectively. The
firstpanel,TypeA,showsbeam-likeand/orflattopfeaturesintheparallelcut,calledabeam/flattop
electrondistributionfunction. Thesecondpanel,TypeB,doesnotshowaflattopfeature,butdoes
showbeam-likefeatures. Finally,thethirdpanel,TypeC,isanexampleofanelectrondistribution
functionthathasT /T >1.0for∼400eVto≥30keV,calledananisotropicelectrondistribution
⊥ (cid:107)
function. TheanisotropyseeninTypeCextendedwellbeyond1keV,whilethecharacteristicfea-
turesofTypesAandBoccurredprimarilybelow1keV.WewillrefertowhistlersseennearTypes
A and B as beam/flattop-related whistlers and whistlers seen near Type C as anisotropy-related
whistlers. Notethatduetothedifferencesinsamplingtimes(∼3secondsforelectrondistribution
functionsversus∼17-250msforwhistlers)theremaybeshort-livedfeaturesintheelectrondistri-
bution functions that are not observed. Table 1 shows that most of the whistlers were observed in
theradiationbeltsandwereassociatedwithanisotropicelectrondistributionfunctions.
Figure 5 summarizes the whistler wave location and wave vector direction statistics. The fig-
ure shows the twelve Wind orbits in the top four panels projected onto the XY-GSM plane with
color-codedlinesandthedateforeachlineislabeledinthecorrespondingpanel(e.g. redlinefor
1998-11-13petalorbitintopleftpanel). Theplotsofthepetalorbitsillustratethelimitedcoverage
ofthemagnetospherebyWind. Overplottedoneachorbitarecolor-codedsymbols,∗’s,wherethe
colorsgreen,orange,andpurplerespectivelycorrespondtothefollowingelectrondistributionfunc-
tions:(1)outermagnetosphericbeam/flattop,(2)outermagnetosphericanisotropic,and(3)radiation
beltanisotropicelectrondistributionfunctions. Forreference,thecumulativenumberofeachtype
ofwhistlerseenineachpanelarelabeledforcorrespondingcolor-coded∗’s(e.g. 5whistlersseen
withanisotropicelectrondistributionfunctionsintheoutermagnetosphereintopleftpanel).
The bottom part of Figure 5 shows two sets of histograms of GSM-Latitude of Wind, λ ,
GSM
on the left and θ , the whistler wave vector angle, on the right. The histogram for λ includes
kB GSM
239TDSFandTDSSsamplesandshowsthecorrespondingrangeoflatitudeduringthefourpetal
orbits. Sinceθ couldonlybedeterminedforTDSSeventswiththreemagneticfieldcomponents
kB
andwerequiredthatλ /λ ≥10.0besatisfied,wedeterminedθ foronly45whistlers. Roughly
int min kB
95%(65%) of the whistlers were observed within 2 R (10◦) of the XY-GSM plane largely due to
E
the Wind spacecraft’s orbits. All of the outer magnetospheric whistlers for both electron distribu-
tionfunctiontypesareobservedbetween4.5≤L≤12andallradiationbeltwhistlersareobserved
between 3 ≤ L ≤ 7.5. Also, 92% of the radiation belt whistlers and all the beam/flattop-related
whistlersareobservedwithin±6hoursofmidnight,whileonly44%oftheanisotropicoutermag-
netospheric whistlers occurred in this range. Most whistlers observed in the outer magnetosphere
hadwavevectorsclosetoparallel(∼60%hadθ ≤30◦),whilethewhistlersintheradiationbelts
kB
hadawiderrangeofwavevectors(0◦≤θ <90◦).
kB
Table1showsthestatisticsofalltheidentifiedwhistlerwavesamplesinsideL=15. Twohun-
5
dredfifty-sevenwhistlerswereobservedinsidethemagnetosphere;ofthese,244whistlersoccurred
between3<L<12. Thetoppartofthetableseparatesthewhistlersintothreecolumnsbyregion
and into two rows defined by the type of electron distribution function. The middle part of Table
1 shows the statistics for the whistler wave amplitudes. The values in each column represent the
fractional number of events with values satisfying the criteria for each column with at least one
electric (top row) or magnetic (bottom row) field component. There were 244 whistlers observed
withatleastoneelectricfieldcomponentand65withatleastonemagneticfieldcomponent. Below
theseratiosaretheL-shellrangesoverwhichthewhistlerswiththecorrespondingamplitudeswere
observed. Note that a significant fraction, ∼40%(∼62%), of the electric(magnetic) field whistlers
have≥50mV/m(≥0.5nT)amplitudes.
The bottom part of Table 1 shows the wave amplitudes versus AE index statistics as the mean
± the standard deviation of the mean. The electric(magnetic) field whistlers have large wave am-
plitudes ((cid:38)110 mV/m or (cid:38)1.6 nT) for moderately strong AE (200-400 nT), suggesting substorm
injections may provide some of the free energy for the observed whistlers. The mean amplitudes
appeartodecreaseinthebottompartofTable1duetothelimitednumberofsamplesforAE(cid:38)600
nT,asseenbythetrendinFigure3. Notethatfor10≤AE≤200nT(seeFigure3), theaverage
waveamplitudesare∼0.40±0.04nT(∼31.54±1.39mV/m),muchsmallerthanforAE>200nT.
Although, much of the literature focuses on the relationship between whistlers and higher en-
ergyelectrons[Lietal.,2010],comparisontoelectrondistributionswithenergies≤30keVindicate
thatthepropertiesofthewavesvarywiththeshapeofthedistributionsinthisenergyrange. Also,
previousworkhasshownthataCerenkovresonancecanexcitewhistlerwaves[Kelloggetal.,2010;
Kumagaietal.,1980;Singh,1972;Starodubtsevetal.,1999],whichhasmuchlowerenergiesthan
the typical cyclotron resonance invoked. Note that the effect of Landau damping by electrons in
thisenergyrangeonpropagationanglesofthewhistlershasalsobeenstudied[Bortniketal.,2006].
Belowwewilldiscussestimatesofelectron/whistlerparallelresonanceenergiesforarangeofelec-
tronpitchangles. Thetotalelectronenergyisataminimumforelectronswithzeropitch-angle. For
instance,ifaparticlehasapitch-angleof45◦,thetotalparticleenergyistwicetheparallelenergy.
Wewilldiscussthoseconsequencesbelowaswell,butfirstwewilldiscussjustparallelresonance
energies. Although there are some uncertainties in our density estimates, as explained above, we
found that most of the whistlers were observed under conditions where they were resonant with
electrons within the energy range of the EESA detectors. The well known nonrelativistic parallel
resonanceenergy[KennelandEngelmann,1966]ofaparticleisgivenby:
(cid:16) B 2 (cid:17)(cid:16) Ω (cid:17)(cid:104) ω (cid:105)(cid:104) ω (cid:105)2
E = o ce cosθ − m+ (2)
(cid:107)res 2µ n ωcos2θ kB Ω Ω
o e kB ce ce
whereB isthemagnitudeofthemagneticfield,n istheambientplasmadensity,Ω istheelectron
o e ce
cyclotronfrequency,θ isthewavepropagationanglewithrespecttothemagneticfield,andm=
kB
0(Landau),−1(normalcyclotron),or+1(anomalouscyclotron)forthedifferentresonances.
ForthetwoexamplesshowninFigure2thenecessaryparametersneededtocalculateresonant
energies are the following: f ∼ 900-2200 Hz(800-2100 Hz), θ ∼ 73◦(39◦), N ∼ 168 cm−3(71
kB i
cm−3),and|B|∼452nT(168nT)forthewaveformontheleft(right). Giventheaboveparameters,
theLandauresonantenergiesfromthecoldplasmadispersionrelationare∼560-720eV(170-240
eV)andthenormalcyclotronresonantenergiesare∼16.2-96.2keV(0.37-4.02keV)forthewave-
form on the left(right). Of course if our estimates of either the density or propagation angle are
wrongwereceiveslightlydifferentresults. WeknowthatthePESAHighdetectorsaturatesinthe
radiationbeltsandunderestimatesthedensities. Thus,increasingthedensitylowersboththeLan-
dauandnormalcyclotronresonanceenergiesforboththewhistlerat1998-11-13/18:20:59.590UT
andtheoneat2000-04-10/03:10:43.077UT.Loweringthepropagationanglehasthesameeffect,in
6
thecoldplasmalimitwithallotherparametersinEquation2beingconstant,sinceE isroughly
(cid:107)res
proportionalto1/cosθ .
kB
Particleswithanonzeropitch-anglehaveahighertotalenergythangivenbyEquation2. Thisis
importantbecausethedetectorsmeasurethetotalenergyofaparticleinaspecificangularbin,not
justtheparallelenergy.Thus,iftheparticleshavelargepitch-anglesandE (cid:46)30keV,itispossible
(cid:107)res
thatthedetectorwouldnotmeasuretheparticle. Thisbecomesanissueathighpitch-angles((cid:38)45◦)
which encompasses a significant fraction of the total distribution when anisotropic. However, the
wavesweobservearehighlyoblique(asseenbelow)andthushavelargelongitudinalelectricfields.
TheconsequenceisthattheLandauresonanceenergiesmaybemoreimportantthanthecyclotron
resonanceenergies[Kelloggetal.,2010;KennelandPetscheck,1966;Kumagaietal.,1980;Staro-
dubtsevetal.,1999]. Thus,wearguethatthedistributionsproducedbytheEESAdetectorscanbe
relevanttotheobservedwhistlerwaves.
RecentsimulationworkbySauerandSydora[2010]foundthatobliquewhistlerwavescanbe
excitedbyelectron beamsifthebeamspeed, V , isatleast greaterthantwicetheelectronAlfve´n
√ b
speed,V =B / µ n m ,whereB istheambientmagneticfieldmagnitude, µ thepermeability
Ae o o o e o o
of free space, n the background plasma number density, and m the electron mass. However, the
o e
beams rarelyexceed 6000km/s andthe correspondingV is typically (cid:38) 50,000 km/s. Thus, itis
Ae
unlikelythattheobservedbeam/flattop-relatedwhistlersaredrivenunstablebythemechanismpro-
posed by Sauer and Sydora [2010]. We also note that the electron distributions used in this study
have energies almost entirely below the energies expected by Sauer and Sydora [2010]. Thus, a
beammayexistabove50,000km/sbutweareunabletoobserveitintheEESAdistributions.
4 Results and Discussion
Wepresentastatisticalstudyofthepropertiesoflargeamplitudewhistlersandcomparisonofwave
amplitudes to AE-index to further characterize the waves described by Kellogg et al. [2010]. We
havealsopresentedthelargestamplitude((cid:38)8nTpeak-to-peak)whistlerwavemeasuredbyasearch
coilintheradiationbelts. Thehighestoccurrenceprobabilitywaswithin±6hoursofmidnightin
theradiationbeltsbetween3≤L≤6. Inaddition,wedescribethedependenceofwhistlercharac-
teristicsontheshapeoftheelectrondistributionsforenergiesbelow∼30keV.However,oneshould
notethatbecausetheWindmissionwasnotfocusedonthemagnetosphere,thecoverageislimited.
Wealsoshowthatverylargeamplitudewhistlersarecommonintheradiationbelts,consistent
withrecentresults[Cattelletal.,2008;Cullyetal.,2008;Kelloggetal.,2010]usinghightimeres-
olutionfielddetectors. Manypreviousstudiesthatfocusedontime-averagedspectralintensityplots
underestimated the amplitude of these bursty whistlers while the satellites with waveform capture
capacities have only recently made measurements in the radiation belts. In addition to their com-
monoccurrence,simulationshaveshownthatthesewavescouldbecapableofproducingsignificant
changes in the relativistic electron fluxes of the radiation belts [Bortnik et al., 2008; Cattell et al.,
2008].
Themajorityofthewaveformsobservedinourstudyhavef/f ≤0.5, occurredwithinL=6,
ce
simultaneously with anisotropic electron distribution functions below 30 keV, and the majority of
thelargestamplitudewhistlersobservedoccurduringmagneticallyactiveperiods(AE>200nT),
consistentwithrecentobservations[Lietal.,2010]. ThecorrelationwithAEsuggeststhesourceof
freeenergymaybeduetoplasmainjectionsfromthegeomagnetictail. ExaminationofLANLlow
energy (30-300 keV) electron summary data (not shown) showed large injections on the nightside
atgeosynchronousorbitshortlybeforeWindobservedthetwowhistlersinFigure2,supportingthe
ideathatplasmainjectionsmayprovidesomeofthefreeenergyforlargeamplitudewhistlergener-
7
ation. BecausetheSSTdetectorsweredesignedforthesolarwind, wecouldnotidentifyfeatures
inthehighenergy(>70keV)electronsassociatedwiththeobservedwhistlers. Regardlessofwhat
thesourceoffreeenergymaybefortheobservedwaves,weobservedifferencesinwaveproperties
dependingonthetypeofelectrondistributionfunctionstheyareobservedwith.
The whistlers associated with anisotropic electron distribution functions in the radiation belts
exhibited broader frequency peaks than the whistlers associated with beam/flattop distributions in
the outer magnetosphere. For the 45 TDSS samples in which we were able to determine θ , we
kB
found that the waves are often very oblique with θ ’s ranging from 0◦ ≤ θ < 90◦, occasion-
kB kB
ally more oblique than the whistlers reported by Cattell et al. [2008] and Kellogg et al. [2010].
The anisotropy-related whistlers in both regions were more oblique than the beam/flattop-related
whistlersobservedintheoutermagnetosphere. Thedifferencesinpropagationangleandwidthof
frequencypeakmaybeduetodifferencesinthesourcesoffreeenergy. Sometheoriessuggestthat
whistlersaregeneratedwithparallelwavevectorsnearthemagneticequatorandtheybecomemore
oblique as they propagate to higher latitudes. So we examined the relationship between θ and
kB
λ . Wefoundnocorrelationbetweenθ andλ , sothewavesdonotappeartobeproduced
GSM kB GSM
asparallelwaveswhichpropagatetothesatelliteandbecomemoreobliqueastheypropagatedue
todispersiveeffects. Theanisotropy-relatedwhistlersinboththeradiationbeltsandoutermagne-
tosphere are highly oblique, which results in a large longitudinal electric field. When this electric
fieldbecomessufficientlylarge,thetypicalcyclotronresonancescanbecomelessimportantthana
Cerenkovresonance[Kelloggetal.,2010;KennelandPetscheck,1966]. Recentstudieshavefound
thatanisotropicbi-Maxwellianelectrondistributionscanproducewhistlerswithpeakgrowthrates
forθ ∼50◦,suggestingthatobliquewhistlersmaybedrivenbytheanisotropicelectrondistribu-
kB
tionsobserved[HashimotoandKimura,1981;Kelloggetal.,2010;Schriveretal.,2010].
Thoughweobservedistinctdifferencesinthewhistlercharacteristicsfordifferentelectrondis-
tribution functions, we cannot definitively state that the observed features are the source of free
energyforthesewaves. Kelloggetal.[2010]presentedawarmplasmainstabilityanalysisforone
anisotropic electron distribution function measured by EESA Low near the whistler observed in
Figure 4 on the right finding the distribution to be unstable. The analysis showed that maximum
growthoccurredatrealfrequenciescorrespondingtotheobservedwavefrequency,butthegrowth
rate(0.08s)andpropagationangle(∼10◦)weretoolow.TheinstabilitywasfoundtobeaCerenkov
resonance[KennelandPetscheck,1966]near150eV,notthetypicalcyclotronresonance,duetothe
highlyobliquenatureofthewaves. OwingtothelongdurationnecessaryfortheEESALowinstru-
menttocollectafull4π steradiandistribution,onecanassumethattheinstantaneousdistributions
arelessisotropic. Thus, Kelloggetal.[2010]performedasecondinstabilityanalysiswithamore
anisotropic hot electron component finding the growth rate (1.5 ms) and propagation angle (up to
∼60◦)tobeapproximatelythesameasthemeasuredvalues(theymeasuredθ ∼61◦). Although,
kB
therealfrequencyproducedbytheanalysisdidnotmatchtheobservedfrequency. Theydefinitively
showedthatthedistributionwasunstabletowhistlerwavesthroughaCerenkovresonance. Mean-
ing,theparallelLandauresonantenergiesestimatedbyEquation2fallwellwithintheenergyrange
oftheEESAdetectors.
5 Conclusions
Tosummarize,weobserved244(65)whistlersinsideL=15usingelectric(magnetic)fielddatafrom
theWindspacecraft;∼40%(∼62%)ofthewaveshavepeak-to-peakamplitudesof≥50mV/m(≥0.5
nT).Themajorityofthewhistlershadf/f ≤0.5andwereobservedwithin±6hoursofmidnight
ce
between3≤L≤6. Thewaveshadabroadrangeofpropagationangles,0◦ ≤θ <90◦,andθ
kB kB
8
wasnotcorrelatedwithλ . Thus,assuminganequatorialsource,thewavepropagationanglesare
GSM
notaconsequenceofpropagationeffects.
OnewhistlerobservedbytheWindsearchcoilhadawaveamplitudeovertwoordersofmag-
nitudeabovepreviousobservations,largeenoughtosaturatethedetector. Also,weonlymeasured
onecomponentofthemagneticfieldmagnitudeforthiswave. Thus,thewaveamplitudeof∼8nT
peak-to-peakandourPoyntingfluxestimateof∼300µW/m2areunderestimatesofthetruevalues.
Aquasi-linearcalculationofthetimescalesnecessaryforawavewiththisamplitudetoaccelerate
plasmasheetelectronsto∼1MeV,assuming1%efficiency,yieldsatimescaleofroughly0.5sec-
onds,muchshorterthanpreviousestimatesbasedonsmallamplitudewhistlerswhichwereonthe
orderofdays. UsingtheestimatesofmaximumchangeinkineticenergyfromOmuraetal.[2007],
wefoundthe8nTwhistlertobecapableofproducing∼61MeVelectronsthroughrelativisticturn-
ingacceleration.
Although we were not able to determine a source of free energy for the observed waves there
were physical differences in the whistlers associated with different electron distributions. In addi-
tiontotheanisotropy-relatedwhistlershavinglarger∆f/fthantherestofthewhistlers,weobserved
differencesinwavecharacteristicswiththefollowingelectronparameters: T /T ,T /T ,
⊥,EL (cid:107),EL ⊥,EH (cid:107),EH
T ,andT (whereEL=EESALowandEH=EESAHigh). Thewhistlersobservedintheradi-
EL EH
ationbeltssimultaneouslywithanisotropicelectrondistributionshadlarger∆f/fthantherestofthe
whistlersobserved. WealsoexaminedtheLANLlowenergy(30-300keV)electronsummarydata
andfoundthatourlargesteventswereassociatedwithlargesubstorminjections,consistentwiththe
increase in wave amplitudes with increasing AE-index. Our study adds to the mounting evidence
thatverylargeamplitudewhistlerwavesareanimportantphenomenainradiationbeltdynamics.
6 acknowledgments
WethankR.Lin(3DP),K.Ogilvie(SWE),R.Lepping(MFI),andE.Dors(LANLdata)fortheuse
ofdatafromtheirinstruments. WewouldalsoliketothankM.Pulupa,S.D.Bale,andP.Schroeder
fortechnicalhelpwiththe3DPsoftwareandanalysis. WethankL.Wangforhelpincalibrationof
theSSTFoildata. TheauthorsthankWorldDataCenterforGeomagnetism, Kyoto, forproviding
theAEindex. ThisresearchwassupportedbyNESSFgrantNNX07AU72H,grantNNX07AI05G,
the Dr. Leonard Burlaga/Arctowski Medal Fellowship, and a contract from APL for the develop-
mentofRBSP/EFW.
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