NASA Technical Reports Server (NTRS) 20110010215: Recent Developments in the Radiation Belt Environment Model

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JournalofAtmosphericandSolar-TerrestrialPhysics](]]]])]]]–]]]
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Journal of Atmospheric and Solar-Terrestrial Physics
journalhomepage:www.elsevier.com/locate/jastp
Recent developments in the radiation belt environment model
M.-C. Foka,n, A. Glocera, Q. Zhenga,b, R.B. Hornec, N.P. Meredithc, J.M. Albertd, T. Nagaie
aGeospacePhysicsLaboratory,NASAGoddardSpaceFlightCenter,Greenbelt,MD,USA
bDepartmentofAstronomy,UniversityofMaryland,CollegePark,MD,USA
cPhysicalScienceDivision,BritishAntarcticSurvey,Cambridge,UK
dAirForceResearchLaboratory,Hanscom,MA,USA
eDepartmentofEarthandPlanetarySciences,TokyoInstituteofTechnology,Tokyo,Japan
a r t i c l e i n f o a b s t r a c t
Articlehistory: Thefluxesofenergeticparticlesintheradiationbeltsarefoundtobestronglycontrolledbythesolarwind
Received1April2010 conditions.Inordertounderstandandpredicttheradiationparticleintensities,wehavedevelopeda
Receivedinrevisedform physics-basedRadiationBeltEnvironment(RBE)modelthatconsiderstheinfluencesfromthesolarwind,
28September2010 ringcurrentandplasmasphere.Recently,animprovedcalculationofwave-particleinteractionshasbeen
Accepted30September2010
incorporated.Inparticular,themodelnowincludescrossdiffusioninenergyandpitch-angle.Wefindthat
theexclusionof cross diffusioncould causesignificant overestimationof electronfluxenhancement
Keywords: duringstormrecovery.TheRBEmodelisalsoconnectedtoMHDfieldssothattheresponseoftheradiation
Radiationbelts beltstofastvariationsintheglobalmagnetospherecanbestudied.Weareabletoreproducetherapidflux
Stormandsubstorm
increaseduringasubstormdipolarizationon4September2008.Thetimingismuchshorterthanthetime
Wave-particleinteractions
scaleofwaveassociatedacceleration.
Spaceweather
PublishedbyElsevierLtd.
1. Introduction wasfoundrelatedtothesolarwindspeed(PaulikasandBlake,1979;
Reevesetal.,2003)andthedirectionoftheIMFBzduringthestorm
TheEarth’sradiationbeltsconsistofenergeticelectron((cid:2)100keV recoveryphase(Ilesetal.,2002).Ontheotherhand,Ukhorskiyand
toseveralMeV)andions((cid:2)100keVtoseveralhundredMeV)trapped Sitnov(2008)suggestedthattheouterbeltcanresponddifferentlyto
inthe magnetosphere roughlyfrom1.2oLo8. Theenergeticelec- similarsolarwinddriving.Thereisabroadrangeofprocessesthat
tronsresidein2distinctregions:theinnerbeltandtheouterbelt, shapetheradiationbelts.Someofthemarenonlinearmechanismsof
whichareusuallyseparatedbytheslotregion(1.8oLo3)ofdepleted localparticleacceleration,suchas,interactionswithwhistlermode
particlepopulations.Pitch-anglediffusionlossofelectronsbyinter- choruswaves(Summersetal.,2004;OmuraandSummers,2006),and
actingwithwhistlermodeplasmaspherichissisbelievedtobethe relativistic electrons drift-resonance with large amplitude, narrow
causeoftheslotregion(Lyonsetal.,1972;Albert,1994;Meredithetal., bandwidthPc5waves(Degelingetal.,2008).
2007).Theinnerbeltisrelativelystablewhiletheouterbeltishighly Theintensificationoftheradiationbeltshassignificantspace
variablewithgeomagneticactivity.Thefluxesofenergeticelectronsin weatherconsequences.Moderateenergy((cid:2)10–100keV)electrons
theouterbeltdecreaseduringthemainphaseofamagneticstormdue can cause surface charging effects and relativistic ((cid:2)0.1–5MeV)
toadiabaticeffect(DesslerandKarplus,1961;KimandChan,1997). electrons can cause deep-dielectric charging on space systems
Additionalnon-adiabaticprocessesalsocontributetothefluxdecrease (Baker,2001).Therefore,understandingthephysicalprocessesthat
inthestormmainphase(Greenetal.,2004;Ukhorskiyetal.,2006). arecontrollingthedevelopmentoftheradiationbeltsduringactive
Duringtherecoveryphasethefluxofenergeticelectronscanchange periods and being able to predict their variability have both
dramaticallyaswell.Whileapproximatelyhalfofallmoderateand scientificandpracticalsignificance.
intensestormscauseanetincreaseinthefluxofenergeticelectronsby Thereexistmultiplesourcesofradiationbeltparticles.Radial
afactorof2ormore,approximatelyaquarterofthesestormsresultin diffusion has traditionally been considered to be the leading
anetdecreaseinthefluxesbymorethanafactorof2(Reevesetal., transportandenergizationmechanismintheinnermagnetosphere
2003).Thisvariabilityiscausedbyanimbalancebetweenacceleration, (Schulz and Lanzerotti, 1974). However, it has recently been
transport,andlossprocessesallofwhichbecomeenhancedduring suggested that electrons can be accelerated efficiently by
geomagneticstorms(Horne,2002;Thorneetal.,2005;Horneetal., resonant wave-particle interactions with whistler mode chorus
2006;Summersetal.,2007).Theratioofpost-stormtopre-stormflux waves (Horne and Thorne, 1998; Summers et al., 1998) and fast
magnetosonicwaves(Horneetal.,2007).
Anumberofkineticmodelshavebeenestablishedtosimulate
nCorrespondingauthor.Tel.:[phone]. the radiation belt dynamics and to provide interpretation for
E-mailaddress: [email protected](M.-C.Fok). observablefeatures.Inakineticmodel,theequationfortheparticle
1364-6826/$-seefrontmatterPublishedbyElsevierLtd.
doi:10.1016/j.jastp.2010.09.033
Pleasecitethisarticleas:Fok,M.-C.,etal.,Recentdevelopmentsintheradiationbeltenvironmentmodel.JournalofAtmosphericand
Solar-TerrestrialPhysics(2010),doi:10.1016/j.jastp.2010.09.033
2 M.-C.Foketal./JournalofAtmosphericandSolar-TerrestrialPhysics](]]]])]]]–]]]
distribution function is solved analytically or numerically. One enhancementduringthedipolarizationevent.Inthefollowing,a
simpleapproachisbasedonastandardradialdiffusionequation briefdescriptionof theRBEmodelis given.Wethenoutlinethe
withdiffusioncoefficientsdrivenbythesolarwindconditionsor algorithm of the ADI method and present the simulation results
geomagneticactivity(Lietal.,2001;Albertetal.,2001).Varotsou withcrossdiffusionincluded.FinallywepresenttheRBEsimula-
etal.(2005)andHorneetal.(2006)combinedradialdiffusionwith tions of a substorm dipolarization and compare the calculations
acceleration and loss due to whistler mode chorus waves and withAkebonoelectrondata.
confirmedthatwaveaccelerationbywhistlermodechorusisan
importantaccelerationmechanismintheouterradiationbelt.Bya
similarapproachbutincludingcrossdiffusiontermsduetochorus 2. Radiationbeltenvironment(RBE)model
waves,Albertetal.(2009)simulatedthe9October1990magnetic
storm. They found both chorus wave acceleration and radial The RBE model is a kinetic model that solves the bounce-
diffusion were required to account for the observed increase in averagedBoltzmannequationtoobtainthedistributionfunctionof
relativisticelectronfluxduringtherecoveryphase.Formodelsthat energeticelectrons(Foketal.,2008)
coverawiderangeofenergy,driftmotionmustbeconsideredsince
convectionisanimportanttransportmechanismforlower-energy @f þ/l_ S@f þ/f_ S @f
@t i @l i @f
(o50keV) particles (Bourdarie et al., 1997; Zheng et al., 2003; i i
Miyoshietal.,2006;Foketal.,2008).Usingtherelativistickinetic 1 @ (cid:4) (cid:2) @f @f(cid:3)(cid:5)
modelwithdipolemagneticfield,Miyoshietal.(2006)simulated ¼G@ao G Daoao@aoþDaoE@E
thedynamicsofenergeticelectronsduringtheOctober2001storm. 1 @ (cid:4) (cid:2) @f @f (cid:3)(cid:5) (cid:2) f (cid:3)
Theyreproducedtheobservedlocaltimefluxasymmetryforhot þG@E G DEE@EþDEao@ao (cid:3) 0:5tb loss cone ð1Þ
electrons (30keV). They also concluded that only convective
transport and radial diffusion cannot explain the enhancement
where
ofrelativisticelectronsseenduringstormrecovery.
Aconvection-diffusionmodel,namelytheRadiationBeltEnvir- G¼TðaoÞsin2aoðEþEoÞpffiEffiffiðffiffiEffiffiffiþffiffiffiffi2ffiffiffiEffiffioffiffiÞffiffi
onment (RBE) model, has been developed to understand the
radiation belt dynamics in order to predict the flux variation f¼(t,l,f,M,K),istheaveragedistributionfunctiononthefield
I I
during active times (Fok et al., 2001, 2005, 2008; Zheng et al., linebetweenmirrorpoints.l andf arethemagneticlatitudeand
i i
2003). The RBE model employs time-varying, realistic magnetic
local time, respectively, at the ionospheric foot point of the
field so that radial diffusion effects due to slow magnetic
geomagnetic field line. M is the relativistic magnetic moment
p
fluctuations and inductive radial transport owing to fast andK¼J= ffi8ffiffimffiffiffiffiffiffiffiMffiffiffiffi,whereJisthesecondadiabaticinvariant.The
o
magnetic reconfiguration can be modeled. The RBE model was
motionoftheparticlesisdescribedbytheirdriftsacrossfieldlines,
usedtosimulateasubstorminjectionduringadipolarizationofthe
which are labeled by their ionospheric foot points. The inner
magnetic field (Fok et al., 2001). Observable features during boundary of l is at 11.81, corresponding to L¼1.06. The outer
i
substorms, such as dispersionless injection and drift echoes, are edgeofthemodelisboundedbyfieldlineswithl notgreaterthan
i
successfully reproduced. Electron flux enhancements during 70.21andanequatorialcrossingat10Earthradius(R ),whichever
E
magnetic storms were also studied using the RBE model (Zheng
iscloser.TheMrangeischosentofullyrepresenttheenergiesof
etal.,2003;Foketal.,2005,2008).Theyfoundthatenergizationby
electrons from 10keV to 6MeV. The K range is chosen to cover
theinductiveelectricfieldandbywhistlermodewavesiscrucial
the loss cone so that particle precipitations can be estimated as
forthefluxincreaseduringmagneticstorms.Asimplifiedversionof well.E istheelectronrestenergy,a istheequatorialpitch-angle,
o o
theRBEmodeliscurrentlyrunninginreal-timetoprovideradiation andT(a )isgivenby
o
belt now-casting updated every 15min. The geosynchronous
fluxes at longitudes of GOES-11 and 13 are extracted from the Tða Þ¼ 1 Z sm ds ð2Þ
RBE real-time run and are plotted together with real-time GOES o Ro 0 cosa
electron (40.6MeV) data. The model-data comparison is
where R is the radial distance in R of the field line equatorial
o E
continuallypostedathttp://mcf.gsfc.nasa.gov/RB_nowcast/.
crossing.Theintegrationisalongthefieldlinefromtheequatorto
Recentdevelopments,discussedinthispaper,expandonpast themirrorpoint.t istheparticlebounceperiod.
b
workinanumberofways.InpreviousRBEsimulationsofwave-
The left hand side of (1) represents the drifts of the particle
particleinteractions,onlypureenergyandpurepitch-diffusionare
population and the terms on the right hand side of (1) refer to
considered(Foketal.,2005,2008).Sincecrossdiffusionmayhavea
diffusion and loss. The calculation of the bounce-averaged drift
comparableeffecttopurediffusions(AlbertandYoung,2005),the velocities across field lines, /l_ S and /f_ S, were described in
RBEmodelhasbeenextendedtoincludecrossdiffusioninenergy i i
detailinFokandMoore(1997).Thesedriftsincludegradientand
and pitch-angle. We choose to use the Alternating Direction
curvaturedrift,E(cid:4)Bdriftfromconvection,andcorotationelectric
Implicit (ADI) method (Xiao et al., 2009) to solve the cross
fields. In this ‘‘drift’’ approach, the RBE model can simulate the
diffusiontermsintheRBEequation.Werevisitthegeomagnetic
effects of radial diffusion only if the electric and magnetic field
storm on 23–27 October 2002 and quantitatively assess the
fluctuationsareproperlyandconsistentlyrepresented.
influence of cross diffusion in electron flux enhancement during
Theeffectsoftheinductiveelectricfieldduetoatime-varying
the recovery phase. Another development in the RBE model is
magneticfieldarealsotakenintoaccountimplicitlyinthemodel
connectingthemodelwithaglobalmagnetohydrodynamic(MHD)
(Foketal.,2005).Wehaveassumedthatfieldlinesarerootedatthe
model.Gloceretal.(2009)coupledtheRBEmodelwiththeBlock-
ionosphere, so that the inductive electric field there is zero.
Adaptive-Tree-Solarwind-Roe-Upwind-Scheme (BATSRUS) MHD
However, the shapes of field lines at higher altitudes vary as a
model to simulate radiation belt development during steady
functionoftimeaccordingtothemagneticfieldmodel.Iffieldlines
solarwindconditionsandduringarealstormevent.Theyfound
areperfectconductors,thefieldlinemotionathighaltitudes,e.g.,
thecoupledcodewasabletocreaterapidinwardtransportonthe
at the equator, will generate an induction electric field (E ) of
ind
timescaleoftensofminutes.Inthiswork,weusetheRBE-BATSRUS
theform,
codetosimulateasubstorminjectionduringthe3–5September
2008 storm. We examine the timing and location of electron Eind¼(cid:3)vo(cid:4)Bo ð3Þ
Pleasecitethisarticleas:Fok,M.-C.,etal.,Recentdevelopmentsintheradiationbeltenvironmentmodel.JournalofAtmosphericand
Solar-TerrestrialPhysics(2010),doi:10.1016/j.jastp.2010.09.033
M.-C.Foketal./JournalofAtmosphericandSolar-TerrestrialPhysics](]]]])]]]–]]] 3
electricfieldbyWeimer2000model(Weimer,2001).Magneticand
electric fields are updated every 5min according to the
instantaneous solar wind condition and Dst index. The bounce-
averaged diffusion coefficients are given by the Pitch-Angle and
EnergyDiffusionofIonsandElectrons(PADIE)code(Glauertand
Horne, 2005). Only resonance with lower-band whistler mode
chorus(0.1fceofo0.5fce)isconsidered. Thepresenceof chorus
waves is confined between–151 and 151 magnetic latitude. The
diffusioncoefficientsarecalculatedasafunctionofLshell,energy,
pitch-angle and f /f , the ratio of plasma frequency to the
pe ce
cyclotron frequency. To calculate f /f , which depends on the
pe ce
plasma density (n), we have embedded inside the RBE model
Fig.1. Anillustrationtoshowthegenerationoftheinductiveelectricfield(Eind) thecoreplasmameodelofOberandGallagher(Oberetal.,1997).
duringfield-linere-configuration.
The PADIE diffusion coefficients are scaled with a chorus wave
intensityof104pT2.Toobtaintheactualdiffusioncoefficients,we
wherev andB arethefieldlinevelocityandmagneticfieldatthe
o o
estimatethechorusintensityatagivenlocationandtimeduring
equator.Fig.1isanillustrationonfieldlinemappingchangingwith
thestormusingthesurveyofCRRESplasmawavedataforlower-
time and the generation of E . Under the frozen-in condition,
ind
band chorus presented by Meredith et al. (2001, 2003). For our
particles initially moving along a particular field line at t will
1
applicationthewavedatawerebinnedinLshell,magneticlocal
continue to share the same field line at t when the field line
becomesmorestretched,asshowninFig.12.Particlesaredrifting timeand3levelsofmagneticactivity(Kpo2;2rKpo4;KpZ4).
InFoketal.(2008),weonlyconsideredpureenergyandpure
outward and decelerated in this case, and the first and second
pitch-angle diffusion. Recently, we have implemented cross
adiabaticinvariantsareconservedifthefieldlinemotionisslow.In
diffusion in the RBE model. The cross diffusion terms can be
theRBEmodel,electronsareenergizedorde-energizedduringthe
solvedusing variousmethods or numerical schemes (Albert and
course of varying themagnetic configuration.Theelectron pitch
Young,2005;Taoetal.,2008;Xiaoetal.,2009;Shpritsetal.,2009).
angleswillchangeaswell.
We choose the Alternating Direction Implicit (ADI) method,
The first two terms on the right hand side of (1) represent
whichwefoundisstable,efficientandeasytoimplement.Inthe
particlediffusioninenergyandpitch-anglefrominteractionswith
ADIscheme,thecrossdiffusiontermsin(1)arerewrittenas(Xiao
plasma waves. When solving these two terms, we first map the
particle phase space density from (M,K) to (E,a ) coordinates, etal.,2009)
o
dpiesrtfroirbmutiodniffbuasciokntointheE(Man,Kd) caooo,rdanindatethse(nFokmeatpalt.h,e199u6p)d.aTthede @@ft ¼ G1@a@ (cid:2)GDaoE@@Ef(cid:3)þG1@@E(cid:2)GDEao@@af (cid:3)
o o
diffusion terms are followed by losses due to the loss cone, the 1@ðGD~Þ@f 1@ðGD~Þ @f @2f
boundaryofwhichisassumedtocorrespondtomirrorheightof ¼G @a @eþG @e @a þ2D~ @a @e
o o o
120km.Particlesinthelossconeareassumedtohavealifetimeof ¼^I fþ^I fþ^I f ð5Þ
onehalfbounceperiod(0.5t )(Lyons,1973). 1 2 3
b
Eq. (1) includes multiple processes of different timescales.
We use the method of fractional step or operator splitting to where
dsteecpom(Fpokoseettahl.e,1e9q9u3a)t.ioTnosaonldveso(1lv),ewoenlhyaovenetotesrpmeciaftyathferaeclteicotnraicl e¼E=Eo,D~ ¼eDEao=E¼eDaoE=E
and magnetic fields, the initial distribution, and the particle
Thecrossdiffusiontermsarethusexpressedasderivativeine
distribution on the nightside boundary, which is set at 10 R or
E (^I ), derivative in a (^I ) plus mix derivative in e and a (^I ). To
the last closed field line. The effect of radial diffusion is 1 o 2 o 3
advancefintime,atimestepisdividedintotwosub-stepsas
incorporatedviathesetime-varyingelectricandmagneticfields.
TheNASAtrappedradiationmodel(AE8MAX)(Vette,1991;Fung, Dth i
fnþ1=2¼fnþ ^I fnþ1=2þ^I fnþ^I fn ð6aÞ
1996) is used for the initial condition in the entire RBE spatial 2 1 2 3
domain.Thedistributionatthenightsideboundaryisassumedto
Dth i
b(Eea)kmaopdpealefudnbcytiolinnewairthredlaetnisointys w(Niptsh)athneduchpsatrraecatmerisstoilcarenweringdy fnþ1¼fnþ1=2þ 2 ^I1fnþ1=2þ^I2fnþ1þ^I3fnþ1=2 ð6bÞ
ps
conditions(Zhengetal.,2003) where
p
N ðtÞ¼½0:02N ðt(cid:3)2hÞþ0:316(cid:5) ffiaffiffimffiffiffiffiuffiffiffi fn¼fðtÞatt¼nDt
ps sw
E ðtÞ¼0:016V ðt(cid:3)2hÞ(cid:3)2:4 ð4Þ
ps sw
Inthefirstsub-step,^I issolvedimplicitlyand^I explicitly.In
1 2
whereNpsisincm(cid:3)3,Nswisthesolarwinddensityinthesameunit, turn,inthesecondsub-step,^I2issolvedimplicitand^I1explicitly.^I3
amuistheatomicmassunitoftheelectron,E isinkeV,andV is isalwayssolvedexplicitly.Thefinitedifferencerepresentationof
ps sw
thesolarwindvelocityinkm/s.Notethatweassumea2htimelag (6)isgivenintheAppendix.
betweentheplasmasheetconditionandsolarwindconditionat Havingpreviouslysimulatedtheenergeticelectronfluxesduring
thedaysidemagnetopause(Borovskyetal.,1998). the23–27October2002event(Foketal.,2008),werevisitthisevent
withcrossdiffusiontermsincludedinthegoverningequation.Fig.2
showstheL-timediagramsofequatorialelectronfluxesinenergy
3. CrossdiffusionintheRBEmodel ranges of 20–70keV (left panels) and 0.6–1.8MeV (right panels);
radialtransportduetotime-varyingmagneticandelectricfieldsare
We have previously simulatedthe evolution of radiation belt includedinallthreerows.ThetoppanelsofFig.2displayresultsin
electronsduringthe23–27October2002geomagneticstormusing which wave diffusion is not included in the calculations. The Dst
theRBEmodel(Foketal.,2008).Themagneticfieldisspecifiedby index(blackcurve)isoverlaidontheplots.Theringcurrentelectrons
theTsyganenko2004 (T04)model(Tsyganenko etal., 2003)and (Fig.2a)andMeVelectrons(Fig.2b)behaveverydifferentlyinthe
Pleasecitethisarticleas:Fok,M.-C.,etal.,Recentdevelopmentsintheradiationbeltenvironmentmodel.JournalofAtmosphericand
Solar-TerrestrialPhysics(2010),doi:10.1016/j.jastp.2010.09.033
4 M.-C.Foketal./JournalofAtmosphericandSolar-TerrestrialPhysics](]]]])]]]–]]]
Fig.2. Simulatedelectronfluxeson23–27October2002.Leftpanels:20–70keV.Rightpanels:0.6–18MeV.Toppanelsarefluxeswithoutwave-particleinteractions.Middle
panelsarefluxeswithenergyandpitch-anglediffusion.Bottompanelsarefluxeswithcrossdiffusionincluded.TheblackcurvesinthetoppanelsareDst.
storm.Duringthemainphase,ringcurrentelectronsfromhigherL Foketal.(2008)foundthatthefluxincreaseisaresultofelectron
shellsdriftearthwardandfilltheentireouterbelt.Thefluxesremain injectionandearthwardtransportduringthecourseofthestorm.
highbecausenowaveassociatedlossmechanismisincludedinthis Next we examine the effects of wave-particle interactions on
calculation. For MeV electrons (Fig. 2b), magnetic field effects theringcurrentandradiationbeltelectrons.Themiddlepanelsof
dominate over convection. Noticeable flux dropout is seen in the Fig. 2show theRBEelectron fluxeswithpureenergy and pitch-
heartoftheouterbeltduringthemainphasewhenringcurrentis angle diffusion from interacting with whistler mode chorus.
intensified.ThisisthewellknownDsteffect(DesslerandKarplus, Calculationresultswithcrossdiffusionareplottedinthebottom
1961;KimandChan,1997).Themagneticfieldproducedbythering panels.Atthebeginningofthestorm,waveactivityisweak.There
currentinflatesthemainfield.Thedriftshellsofenergeticparticlesin is no obvious difference between runs with and without chorus
theradiationbeltexpandoutwardcorrespondingly(conservationof waves.Duringthemainandearlyrecoveryphases,waveactivityis
the third adiabatic invariant) and particles decelerate. This de- strong.Forringcurrentelectrons(leftpanels),theeffectsofpitch-
energization causes an adiabatic decrease in particle fluxes since angle diffusion are stronger than those of energy diffusion.
theenergyspectrumslopeisnegativeintheradiationbeltenergy Asignificantamountofelectronsarediffusedintothelosscone.
range.Theexpansionofdriftshellsalsoproducespermanentlossat A flux hole is formed around L¼4 during the storm recovery. In
high L’s when particles encounter the magnetopause (Ukhorskiy contrast, with the consideration of chorus waves, MeV electron
et al., 2006). During the recovery phase, MeV electron fluxes fluxesgraduallyincreaseduringtherecoveryphase(rightpanels,
recoveraswell.Theintensitiesarehigherthanthepre-stormlevel. Fig. 2). The injection of lower-energy electrons forms a seed
Pleasecitethisarticleas:Fok,M.-C.,etal.,Recentdevelopmentsintheradiationbeltenvironmentmodel.JournalofAtmosphericand
Solar-TerrestrialPhysics(2010),doi:10.1016/j.jastp.2010.09.033
M.-C.Foketal./JournalofAtmosphericandSolar-TerrestrialPhysics](]]]])]]]–]]] 5
population for energy diffusion. Particles are accelerated by the Cross diffusion tends to moderate the impacts from pure pitch-
choruswavesandMeVelectronsslowlydiffusetolargerLshellsat angleandenergydiffusion.Forringcurrentelectrons,withcross
laterecovery. diffusionincluded(Fig.2e),pitch-anglediffusionisweakenedand
Whencomparingthefluxesinmiddlepanelswiththoseinthe the overall fluxes are higher than those without cross diffusion
bottompanelsofFig.2,wefindnoticeableeffectsofcrossdiffusion. (Fig. 2c). Similarly in the case of MeV electrons, cross diffusion
reduces the energy gain by energy diffusion and thus the flux
enhancementintherecoveryphase.Fig.3illustratesquantitatively
theeffectofcrossdiffusioninradiationbeltelectrons.Theratioof
MeVelectronfluxwithoutandwithcrossdiffusionisplottedasa
functionofLshellattheendof4-dayssimulation.Inmostpartsof
theinnermagnetosphere,theratioiscloseto1.However,inthe
heartoftheouterbeltaroundL¼4,ignoringcrossdiffusioncould
cause overestimation of electron flux as much as a factor of 5.
Detailedanalysisoftheeffectsofcrossdiffusionwillbereportedin
aseparatestudy.
4. RBEsimulationofaMHDsubstorm
TheSunwasinitsdeepminimuminyears2008and2009;there
wasminimalsolarandgeomagneticactivity.However,recurring
highspeedstreamscouldtriggersubstormsintheEarth’smagneto-
Fig.3. FluxratioasafunctionoftimeofMeVelectronswithoutandwithcross
diffusion.Ratiosarecalculatedatsimulationtimeof4days. sphereeveninsolarminimum(Bakeretal.,1998).On3September
Fig.4. Dst,AU,ALandsolarwindspeed,density,ByandBzon3–6September2008.
Pleasecitethisarticleas:Fok,M.-C.,etal.,Recentdevelopmentsintheradiationbeltenvironmentmodel.JournalofAtmosphericand
Solar-TerrestrialPhysics(2010),doi:10.1016/j.jastp.2010.09.033
6 M.-C.Foketal./JournalofAtmosphericandSolar-TerrestrialPhysics](]]]])]]]–]]]
2008, a high speed stream arrived at the magnetosphere and
triggered a moderate storm with minimum Dst of (cid:3)51nT on
September4,0500UT.Fig.4plotstheDst,AU,ALandsolarwind
speed,density,ByandBzon3(cid:3)5September2008.
TheAkebonosatelliteobservedenhancementsofradiationbelt
electrons during the active period on 3(cid:3)5 September 2008.
AkebonowaslaunchedinFebruary1989bytheInstituteofSpace
and Astronautical Science in Japan (Takagi et al., 1993). In
September 2008, Akebono was in a high inclination, highly
ellipticalorbit with apogee at 5260km altitude, perigeealtitude
at295km,andorbitperiodof2.5h.TheRadiationMonitor(RDM)
measuredelectronfluxesinthreeenergychannels:0.30–0.95MeV,
0.95–2.5MeVand42.5MeV(Takagietal.,1993).Arapidincrease
inhigh-energyelectronfluxwithafactorof80at4oLo5isseen
between 0321 and 0546UT, around the peak of the storm
(minimum Dst). Similar growths are also seen in the other 2
RDM lower-energy channels. The top panel of Fig. 5 shows the
Akebono electron flux of energy 42.5MeV from September 3,
00UT to September 4, 12UT. The Dst index during this time is
overlaidintheplot.TheAkebonodataareaveragedover3orbit
periods.Thetimescaleofthisenhancementisafewhoursandthus
itistooshortforwaveassociatedenergization.ThelargeALvalues
inthistimeperiod(Fig.4)suggestasubstormmayplayaroleinthis
quickincreaseofouterbeltelectrons(Nagaietal.,2006).
WeperformRBEsimulationstounderstandthecauseoftheflux
enhancement seen in the Akebono electron data. In most of our
previousRBEcalculations,weusedempiricalmodelsofmagnetic
and electric fields (Fok et al., 2001, 2008; Zheng et al., 2003).
Recently,wehaveadaptedthemagneticandelectricfieldsoutput
fromthecoupledmodelofBATSRUS-RiceConvectionModel(RCM)
into the RBE model in order to self-consistently simulate the
responses of the radiation belts to solar wind and ring current
variations during storm time (Glocer et al., 2009). The MHD
electromagnetic fields are updated every 10s to drive the drift
motion and radial transport of radiation belt particles. Fig. 5(b)
depictstheRBEelectronfluxescalculatedwithT04magneticfield
modelandWeimerelectricfieldmodel(Tsyganenkoetal.,2003;
Weimer,2001).Fig.5(c)istheRBEfluxcalculatedintheMHDfields
simulatedfromtheBATSRUS-RCMmodel(DeZeeuwetal.,2004).
NotethattheRBEfluxesshowninFigs.5(b)and(c)areequatorial
fluxesandwithtemporalresolutionof1h.Incontrast,theAkebono
measurements are taken along high inclination orbits. However,
thetemporalvariabilityofhighlatitudefluxeswasfoundnearly
identical with the equatorial fluxes (Kanekal et al., 2001, 2005).
Wave-particle interactions are not included in these RBE
Fig. 5. Top panel: L-Time plot of Akebono electron flux (42.5MeV) on 3–4 calculations. In the quiet period on September 3, the two RBE
September2008.TheblackcurveisDst.Middlepanel:correspondingRBEsimulated simulationsgivesimilarfluxintensity.Duringthemainphaseof
fluxwithT04magneticfield.Bottompanel:simulatedfluxwithBATSRUSfields. thestorm,bothRBErunsproducefluxdropoutintheouterbelt,in
Fig.6. BATSRUS-RCMsimulationat(a)05:00UTand(b)05:20UTon4September2008.Magneticfieldlines(whitelines)andpressure(color)areplottedonX–Zplane.
Pleasecitethisarticleas:Fok,M.-C.,etal.,Recentdevelopmentsintheradiationbeltenvironmentmodel.JournalofAtmosphericand
Solar-TerrestrialPhysics(2010),doi:10.1016/j.jastp.2010.09.033
M.-C.Foketal./JournalofAtmosphericandSolar-TerrestrialPhysics](]]]])]]]–]]] 7
responsetotheringcurrentintensificationsimulatedintheT04 etal.,2003)havebeenusedtosimulatethetemporalvariationsof
andBATSRUS-RCMmodels.However,at05–06UTonSeptember4, magnetic field in the RBE model. We found strong electron
asuddenincreaseinelectronfluxisseenat4oLo5intheRBE- energization by the inductive electric field associated with the
BATSRUScalculation,consistentwiththeAkebonodata.Thereisno time-varyingmagneticfield(Zhengetal.,2003;Foketal.,2008).
significantenhancementintheRBE-T04runduringtherecovery However,asshowninSection4,wecannotreproducetherapidflux
phaseofthestorm. enhancement during a substorm dipolarization with the T04
Since wave-particle interactions are not considered in these model. In contrast, the RBE model driven by MHD fields
particularRBEcalculations,theenhancementseeninFig.5(c)must successfully produces the observed increase in electron flux
bearesultofparticletransport.Whenweexaminethemagnetic duringasubstorm.AnotherlimitationofthecombinedRBE-T04-
configuration during the enhancement, we find the MHD model Weimer models is that they cannot fully simulate the effect of
predictsasubstormdipolarizationat(cid:2)05UTonSeptember4.Fig.6 radialdiffusionevenwithT04andWeimermodelinputparameters
showstheBATSRUSfieldlinesinwhiteandpressureincoloronthe updated in time. Both the Tsyganenko and Weimer models are
X–Z plane before (left) and after (right) the substorm onset on empiricalandnotconsistentwitheachother.Thisproblemcanbe
September4.At05:00UT,thefieldlinesonthenightsidearegreatly resolvedby,again,applyingMHDfieldsintheRBEmodel.Recently
stretched. 20min later, the dipolarization takes place in the tail. workbyHuangetal.(2010)hasshownthattheULFwaves(mHz
Field lines are convecting earthward and have more dipole-like range) predicted by theLyon–Fedder–Mobarry MHDmodel well
shape. Electrons which are gyrating along these collapsing field representtheULFwavedataobservedbytheGOESsatellites.
linescanbeacceleratedsignificantlyonatimescaleofminutes(Fok In summary, we have reported recent developments and
etal.,2001;Gloceretal.,2009),muchfasterthanthetimescalefor improvementsinourradiationbeltmodel.Wehaveimplemented
energizationbywhistlermodechoruswaves,whichistypicallyof pitch-angle-energycrossdiffusioninourwavediffusioncalcula-
theorderof1–2days(SummersandMa,2000;Horneetal.,2005). tion.Wehavesimulatedasubstormdipolarizationeventwiththe
TheT04model,whichisdrivenbyDstandsolarwindparameters, RBE model embedded in MHD magnetic and electric fields. The
does not contain clear substorm signatures. Empirical magnetic findingsfromthismodeldevelopmentworkinclude
field models of this kind cannot directly simulate substorm
reconfiguration unless special tricks are applied (Delcourt et al., (1) Crossdiffusionmoderatestheeffectsofpurepitch-angleand
1990,1997;Pulkkinenetal.,1991;Foketal.,2001).Formoderate pure energy diffusion. Exclusion of cross diffusion would
stormssuchasthisoneon3–5September2008,convectionisweak significantlyoverestimatethefluxenhancementsofrelativistic
andthedominant energization and transportmechanism issub- electronsduringstormrecovery.Inoursimulationofthestorm
stormreconfigurationandtheresultingdipolarizationelectricfield. on 23–27 October 2002, at the heart of the outer belt, the
The rapid enhancements of radiation belt fluxes during modest overestimationcanbeashighasafactorof5.
stormscannotbeexplainedwithouttheconsiderationofsubstorm (2) The strong inductive field during substorm dipolarization
effects. produces rapid increase in energetic electron flux on a time
scale of an hour, much shorter than that from wave
acceleration.
5. Discussionandconclusions
Onemustbecautioustointerpretthetimingsignatureseenin
theAkebonodata.Thetemporalresolutionislimitedbytheorbit
Acknowledgment
periods (2.5h). In our separate study of the storm in September
2008, we will identify storm and substorm signatures using
continuous high resolution data, such as measurements from This research was supported by NASA Science Mission Direc-
NOAAandGOESsatellites. torate,HeliophysicsDivision,LivingWithaStarTargetedResearch
So far we have considered only low-latitude whistler mode and Technology Program, under Work Breakdown Structures:
choruswaves.However,studyhasshownthatchoruswavesathigh 936723.02.01.06.78and936723.02.01.01.27.
latitude are important in the loss and acceleration of energetic
electrons (Horne and Thorne, 2003). Other wave modes, such as
plasmaspherichissandelectromagneticioncyclotronwaves,also Appendix:FinitedifferencerepresentationoftheADIscheme
playcrucialrolesinthedevelopmentanddecayoftheradiation
belts(Meredithetal.,2007;Lorentzenetal.,2000;Summersand Eq. (6)outlinestheADI schemeinsolvingthecross diffusion
Thorne, 2003). All these wave modes have different sources of equation.Thesecondorderfinitedifferencediscretizationof(6)can
excitation and are found in different regions of the inner bewrittenas
magnetosphere. We plan to gradually include all the important
m1 fnþ1=2þfnþ1=2(cid:3)m1 fnþ1=2
wavemodesintheRBEmodel.Inthatcasewewillbeinabetter k,m k(cid:3)1,m k,m k,mkþ1,m
(cid:7) (cid:8)
position to understand and identify the physical processes that ¼fn þm2 fn (cid:3)fn
k,m k,m k,mþ1 k,m(cid:3)1
controltheobservedvariabilityintheradiationbelts. (cid:7) (cid:8)
þm3 fn þfn (cid:3)fn (cid:3)fn ð7aÞ
Inclusion of the cross diffusion term does make a noticeable k,m kþ1,mþ1 k(cid:3)1,m(cid:3)1 k(cid:3)1,mþ1 kþ1,m(cid:3)1
differenceinelectronfluxintheheartoftheouterbelt.However,
the effect is relatively mild and localized when compared with
otherprocessessuchasparticleinjection,transportandaccelera- m2 fnþ1 þfnþ1(cid:3)m2 fnþ1
k,m k,m(cid:3)1 k,m k,mk,mþ1
tion.Itisdifficulttoshowquantitativelythatwithcrossdiffusion ¼fnþ1=2þm1 (cid:7)fnþ1=2(cid:3)fnþ1=2(cid:8)
willimprovethedatamodelcomparison.Nevertheless,inclusionof k,m k,m kþ1,m k(cid:3)1,m
(cid:7) (cid:8)
crossdiffusionwillgiveabetterestimationofthediffusiveeffect þm3 fnþ1=2 þfnþ1=2 (cid:3)fnþ1=2 (cid:3)fnþ1=2 ð7bÞ
k,m kþ1,mþ1 k(cid:3)1,m(cid:3)1 k(cid:3)1,mþ1 kþ1,m(cid:3)1
frominteractingwithaparticularwavemode. where
Thedynamicsofradiationbeltelectronsisstronglycontrolled
bythemagneticconfigurationanditsfluctuations.Theempirical m1 ¼ Dt Gk,mþ1D~k,mþ1(cid:3)Gk,m(cid:3)1D~k,m(cid:3)1
modelsofTsygenenko(TsyganenkoandStern,1996;Tsyganenko k,m 2 Gk,mðaomþ1(cid:3)aom(cid:3)1Þðekþ1(cid:3)ek(cid:3)1Þ
Pleasecitethisarticleas:Fok,M.-C.,etal.,Recentdevelopmentsintheradiationbeltenvironmentmodel.JournalofAtmosphericand
Solar-TerrestrialPhysics(2010),doi:10.1016/j.jastp.2010.09.033
8 M.-C.Foketal./JournalofAtmosphericandSolar-TerrestrialPhysics](]]]])]]]–]]]
Dt G D~ (cid:3)G D~ Horne,R.B.,Thorne,R.M.,1998.Potentialwavesforrelativisticelectronscattering
m2k,m¼ 2 G kþða1,m kþ(cid:3)1a,m kÞ(cid:3)ðe1,m (cid:3)k(cid:3)e1,mÞ and stochastic acceleration during magnetic storms. Geophys. Res. Lett. 25,
k,m omþ1 om(cid:3)1 kþ1 k(cid:3)1 3011.
m3 ¼ DtD~k,m Horne,R.B.,Thorne,R.M.,2003.Relativisticelectronaccelerationandprecipitation
k,m ða (cid:3)a Þðe (cid:3)e Þ duringresonantinteractionswithwhistler-modechorus.Geophys.Res.Lett.
omþ1 om(cid:3)1 kþ1 k(cid:3)1 30(10),1527.doi:10.1029/2003GL016973.
kiseindexandmisa index. Horne,R.B.,Thorne,R.M.,Glauert,S.A.,Albert,J.M.,Meredith,N.P.,Anderson,R.R.,
0
2005.Timescaleforradiationbeltelectronaccelerationbywhistlermodechorus
Eq. (7) represents two tri-diagonal systems similar to the
waves.J.Geophys.Res.110,A03225.doi:10.1029/2004JA010811.
Crank–Nicolson method. There are well established numerical Horne,R.B.,Meredith,N.P.,Glauert,S.A.,Varotsou,A.,Boscher,D.,Thorne,R.M.,
techniques and stability analysis for this type of problem Shprits,Y.Y.,Anderson,R.R.,2006.Mechanismsfortheaccelerationofradiation
(LeVeque,2002;BurdenandFaires,2004). beltelectrons.In:Tsurutani,B.T. (Ed.),RecurrentMagneticStorms:Corotating
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Horne,R.B.,Thorne,R.M.,Glauert,S.A.,Meredith,N.P.,Pokhotelov,D.,Santolik,O.,
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