Time Dependent Convection Electric Fields and Plasma Injection

VOL. 84, NO. A8 JOURNAL OF GEOPHYSICAL RESEARCH AUGUST 1, 1979

Time DependentConvection Electric Fields and PlasmaInjection

STANLEY M. KAYE AND MARGARET G. KIVELSON

Instituteof Geophysicsand PlanetaryPhysics and Departmentof Earth and SpaceSciences Universityof California, Los Angeles,California 90024

Large-scaleelectric fields associated with stormsor substormsare responsiblefor inward convection and energizationof plasmasheet plasma. Calculations based on steadystate convection theory show that the responseto such electricfields qualitativelyaccounts for many featuresof the injectedparticle distribution,but quantitativeagreement with the theoryhas not yet beenobtained. It is knownthat the predictionscan be improvedby introducingthe conceptof convectionin responseto a time dependent electricfield. On the other hand, time dependentcalculations are sensitiveto the choiceof initial conditions,and mostmodels have failed to incorporatethese conditions in a realisticand self-consistent manner.In thispaper we presenta morecomplete model consisting of realisticinitial conditionsand time dependentconvection to explaina typicalsubstorm-associated electron injection event. We findvery good agreementbetween the observedelectron flux changesand thosepredicted by our model.

INTRODUCTION ciatedparticle phenomena. One data setwhich has been ana- Transport of collisionlessplasma sheetplasma to the inner lyzedby severalauthors is that of Williamset al. [1974],who observed electron flux enhancements near L = 5 in the dusk to magnetosphereresults from large-scaleelectric fields associated with storms or substorms.As was shown by previous midnightsector on Explorer45 shortlyafter substormonset. workers [Chen, 1970; Kivelson and Southwood,1975; Stern, They tried to explaintheir observationsby invokingplasma convectionin the electricand magneticfield model developed 1975; Cowley and Ashour-Abdalla,1976a, b], the maximum earthward penetration of the adiabatically conv½ctingtail by Mcllwain [1972] to describeparticle injection at geosta- plasmais delimitedin the steadystate by spatialboundaries tionary orbit. Williams et al. found that the magnitudeof determinedby the magnitudeof the convectionelectric field M cllwain'ssteady state electric field wasinsufficient to inject electronsto the low L valuestraversed by Explorer 45. The and the constantsof the particle motion. For magnetospheric Williams et al. paper stressedthat the dispersionof arrival plasmasthe relevantconstants are the two adiabaticinvariants times differedfor high- and low-energyelectrons. Following # and J, where # is the magneticmoment and J the bounce substormonset the delay of the initial flux increaseat the integral.The convectionboundaries, often called'Alfv6n lay- satelliteincreased with energyfor low-energyelectrons (<8 ers' [Wolf 1970], separatespatial regionsin which qualita- tivelydifferent particle drift orbitsare present.For electronsof keV) anddecreased with energyfor high-energyelectrons. To account for this behavior, the authors postulatedan ad hoc prescribed# and J, an Alfv6n layer separatesdrift orbitswhich radial electric field in addition to the Mcllwain field in the are closed about the earth from those which are open to the magnetopauseand tail and thusseparates regions of enhanced evening sector. plasma sheet electron fluxes (antiearthwardof the Alfv6n Subsequently,Kivelson and Southwood[1975] assumeda layer) from regionsof reducedplasma sheet electron fluxes dipolemagnetic field, corotationelectric field, and uniform (earthwardof the Alfv6n layer). For protonsthe natureof drift cross-magnetosphericelectric field and invokedsteady state adiabaticconvection theory to explain the energyand time paths in differentspatial regionsseparated by Alfv6n layers can be more complex [Chen, 1970]. dependentelectron flux increasesin thesame data. Assuming Becausesatellite particle detectorsmeasure particle energy that the flux increasein a givenlow-energy channel occurred when the outbound satellite passedthrough the SSDB and and pitch angle,not # and J, it is convenientto identifya type of boundarysimilar to an Alfv6n layer but definedin termsof entereda regionaccessible to plasmasheet electrons of that energy,they calculated the points of intersectionof thesatellite energyand pitch angleinstead of # and J [Kivelsonand South- wood, 1975]. We introduce the nomenclature'steady state trajectorywith the SSDBsfor a uniformconvection electric field. Althoughthey were successfulin explainingthe energy demarcation boundary' (SSDB) for this related boundary which is determinedby the convectionelectric field magnitude dependenceof arrival times,they obtainedonly qualitative and specifiedfor a particleenergy W and pitch angle a. The agreementwith the timing of theobserved flux increases. They madeno attemptto interpretthe subsequentflux decreases. SSDB is the same as the boundary called LA(qb)by Kivelson and Southwood[1975] and called the critical boundary by Using the sameconvection model as Kivelsonand South- Kivelsonet al. [1979]. Now, the SSDB is not a particle drift wood[1975], Cowley and ,4shour-,4bdalla[1976a] interpreted the Williams et al. data in a slightlydifferent way. For each orbit, sinceenergy is not a conservedquantity, but it does energychannel of thedetector they calculated the electric field delimit the spatial region inside of which an electron whose necessaryto placethe correspondingSSDB at the satellite local energyand pitch angleare the prescribedW and a is on a closed orbit. Outside the SSDB an identical electron will be on positionof the observedflux increase in thatenergy channel. an openorbit and thusmay have a sourcein the plasmasheet. They found that differentconvection electric fields were re- Later in thispaper we extendthe definitionof the demarcation quiredto accountfor increasesin differentenergy channels. boundaryto includetime dependenteffects. Theyconcluded that the convectionelectric field had eithera Steadystate convection theory has been used with differing strongspatial or a strongtime dependence. degreesof successby many authorsto explain substorm-asso- Furtherapplications of convectiontheory were presented in a companionpaper by Cowleyand ,4shour-,4bdalla[1976b], Copyright¸ 1979by the AmericanGeophysical Union. who turned their attention to the storm time proton noseevent

Paper number 9A0533. 4183 0148-0227/79/009A-0533501.00 4184 KAYE AND KIVELSON: TIME DEPENDENT CONVECTION observednear duskby Smithand Hoffman [1974] on Explorer To illustratethe use' of ourmodel, we reinterpret the sub- 45. They arguedthat the observeddistribution was a natural storm electron observationsof Williams et al. [1974], the same steadystate spatial feature produced by adiabaticconvection, data set analyzed by Kit)elsonand Southwood[1975] and Cow- but their model predicteda nose penetrationmuch further ley and Ashour-Abdalla[1976a]. For reasonableinitial and earthward than was observed.Cowley [1976] later broughtthe final convectionelectric fields, whoseamplitudes are the only predictedand observednose positions into agreementby in- free parameters,we can model the times and positionsof the voking proton lossat the strongdiffusion rate over the entire plasmapauseand of flux enhancementsand decreasesin the trajectory. four different energy channelsof the Explorer 45 particle de- The above interpretationsof particleinjection all relied on tector in terms of crossingsof the TDDB. For a simple con- use of steadystate convectiontheory. Implicit in the steady vectionfield model we obtain very good agreementwith obser- state model is the assumptionthat the plasma particleswill vations. have sufficienttime to reach their steadystate configuration, We find that in our time dependentpicture of substorm- an assumptionwhich cannot be valid consideringthe time associatedconvection, the initial particle boundaries are as scaleof a substorm.At substormonset there is a major change crucial to the calculation as the incorporation of the time in the magnitudeand configurationof the earth'selectromag- dependenceitself, sincethe position of a particle within the netic fields. The tail field strengthtypically increasesby •1 first tensof minutesafter onsetdepends greatly on its location kV/RE, and the SSDB moves >•1 RE earthward [Kit)elsonet al., at onset. We suggestthat any time dependent convection 1979]. Particlesnear L = 5 (B = 150 7) need •100 min to model must incorporaterealistic initial conditionsconsistent adjustto the changedconditions. On the other hand, particle with quiet time observationsto give good quantitativeresults. phenomenasuch as those observedby Williamset al. [1974] ANALYSIS OF DATA are sometimes seen • 10 min after substorm onset, and on such a short time scale the use of steady state theory cannot be The particle observationsreexamined in this study are the justified. low-energy(0 to 7.1 keV) electronflux increasesand decreases A time dependentmodel of plasmasheet response to the seenaboard Explorer 45 shortlyfollowing a substormonset on convection electric field was presentedby M. Ejiri, R. A. December 12, 1971 [Williams et al., 1974]. As the Explorer 45 Hoffman, and P. H. Smith (unpublishedmanuscript, 1978), particledetectors measure particles with energiesdown to only who attemptedto explainthe observedposition (L • 4.5) and 1.5 keV, the presenceor absenceof the thermal (•0 keV) energy(W • 15 keV) of the protonsinjected inside the plasma- component was inferred by the measurementfrom the on- pause (proton nose event) during a magneticstorm. They board dc electric field probe [Maynard and Cauffman, 1973]. assumedthat all their particlesstarted from a singleboundary The substormbegan at •2200 UT, at which time the satellite which was azimuthally symmetric in the range +60 ø about was outbound at L = 5.3 and near 2040 LT. As time pro- midnight at L = 10. Substorm onset was simulatedby a gressed, the satellite passed outward through the plasma- stepwiseenhancement of the convectionelectric field. With pause,as determinedby saturationof the on-boarddc electric their ad hoc particleinitial conditionthey predictedthat the field probe, and then 90ø pitch angle electron flux increases noseshould penetrate to L = 4.5 with a time delay from onset were successivelyobserved in the energychannels 1.5-2.1, 2.3- much greater than that normally observed. 3.2, 3.5-4.8, and 5.3-7.1 keV. For this range of energiesthe observedenergy dispersionwas oppositeto that expectedif MODEL OF SUBSTORM CONVECTION injectionoccurred near midnight, and gradientcurvature drift In this paper we describean adiabaticconvection model in then brought the electronsto the dusk-midnightsector. Later, which we incorporatenot only a time dependentconvection on the inbound portion of the orbit, the low-energyelectron electric field but also physically realistic energy dependent fluxes were seen to decrease,first in the 5.3- to 7.1-keV channel initial conditionsconsistent with quiet time observations.We and subsequentlyin lower energy channels.No change (in- assumethat plasmasheet electrons are presentoutside of the creaseor decrease)of electron flux in the 8.0- to 10.8-keV initial electron boundaries, which are themselvesdetermined energychannel was seenat any point along the relevantpor- from steady state convectiontheory. The initial boundaries tion of the orbit. In Table 1 we list the times of the electron have an energydependence appropriate for a smallconvection flux increasesand decreasesalong with the satellitepositions at field representativeof a steady,quiet magnetosphere[Kivelson thesetimes. The inverseenergy dispersion of the flux increases et al., 1979]. We simulatesubstorm onset by an instantaneous and the opposite dispersionof the flux decreasesare clearly enhancementof the cross-magnetosphericpotential drop; evi- seen. dence for enhanced convection and convection electric fields at As was done in the previousanalyses of Kivelsonand South- substormonset may be found in the work of Pytte et al. [1978] wood [1975] and Cowley and Ashour-Abdalla[1976a], we as- and Mozer [1971], respectively.We then calculatethe position sume that the observedflux changesoccurred when the satel- (radial distanceversus azimuth as a functionof time) of the lite passedthrough demarcationboundaries, but we use the inner edge of the region in which plasma sheetelectrons of TDDBs described above. given W and a can be found for the assumedinitial conditions We assumethat the total magnetosphericelectric field is and enhancedfield. The calculationprovides new demarcation derivablefrom a time dependentgeneralization of the poten- boundarieswhich are not steadystate but rather convecting. tial in the equatorial plane given by Volland [1973]: Thereforewe supplementour definitionof the SSDB with an •(kV) = C,(t)L: sin 0 + C:/L (1) additional definition of the 'time dependent demarcation boundary'(TDDB), a boundarywhich followsthe inner edge where C•(t) is proportional to the total potential drop across of convectingplasma sheet electrons of a designatedenergy. A the magnetospherein kilovolts and is time dependent,½ is satellitewill encounterfreshly injectedplasma sheetelectrons azimuth measuredeastwards from noon, and C: = 91 kV is the of energy W when it movesout through the TDDB for that corotationpotential. We adopt the L: dependenceof the con- energy. vection potential based on observationaland theoreticalevi- KAYE AND KIVELSON: TIME DEPENDENT CONVECTION 4185 dence that the convection electric field is partially shielded TABLE 1. Explorer 45 Electron Flux Observations, December from the inner magnetosphere[Heppner, 1972; Volland, 1973; 12-13, 1971, Orbit 86 Jaggi and Wolf 1973; Southwoo& 1977]. Later we will show Energy,keV Time, UT L Time, LT 4•,deg that the use of a shielded potential is not critical to our conclusions and that a uniform electric field in which the first Outbound(Flux Increase) term on the right-hand sideof (1) is proportional to L can be 0 2221 5.5 2048 132 1.5-2.1 2230 5.5 2052 133 used with equal success.For our calculation we assumethat 2.3-3.2 2253 5.5 2116 139 the earth's magneticfield is dipolar and that the field lines are 3.5-4.8 2314 5.5 2128 142 equipotentials. 5.3-7.1 2339 5.3 2152 148 We usethe potential of (1) to model both quiet presubstorm conditions and substorm conditions, imagining the total po- Inbound(Flux Decrease) 5.3-7.1 0025 4.7 2240 160 tential drop [C•(t)] to be small and time independentprior to 3.5-4.8 0029 4.6 2248 162 substormonset. For sucha steadystate the SSDBs are taken 2.3-3.2 0032 4.5 2256 164 to define the inner edgeof the plasma sheetelectron distribu- 1.5-2.1 0039 4.3 2304 166 tion, and this provides us with the neededinitial conditions; 0 0039 4.3 2304 166 i.e., we supposethe plasma sheetelectrons of all energiesare Explorer 45 electron flux observations taken from Williams et al. presentat all tail locationsbeyond the SSDBs. For an initial [1974]. The universaltime (UT) of the flux increasesand decreases convectionpotential drop of ß = 60 kV we have plotted in is given along with the position of Explorer 45 at the time of the Figure I the SSDBs for 90ø pitch angle electronsof 0-, 2.3-, observedflux change.The angle 4• is measuredeastward from noon. and 5.3-keV energies.Kp beforethe substormat 2200 UT was Substorm onset was at 2200 UT. 3-, and the 60-kV potential drop is an upper limit estimateto what might be expectedfor suchKp [Kivelson,1976]. It is seen magnetosphericpotential. For a good fit the predicted radial that the boundaries are elongated so that they are farthest from the earth at dusk and closest to the earth at dawn and positionshould be closeto the radial positionof Explorer45 at the azimuth •bt.The resultsof our calculationare presentedin that higher energy boundarieslie progressivelyfarther from Table 2. In this table we have again includedthe Explorer 45 the earth. The region insidethe zero-energySSDB, or plasma- radial distance and azimuth at the time of the observed flux pause,is filled with a high-densitycold plasmaon closedorbits increasesand decreasesin the energychannel labeled by Wt. In about the earth, and the cold plasma densitydrops sharplyas addition, we give the delay of the increaseor decreasefollow- the satellitemoves out acrossthe plasmapause[Chappell et al., ing substormonset (tt - to) and the radial positions,Lt øtea,of 1970]. Becausethe fluxesof electronswhich have convectedin the TDDBs at tt - to and •b,for a = 90ø electronsof energy from the tail are excluded from regions earthward of the W•. The differencebetween predicted and observedvalues of L relevant SSDB, it is clear that finite energy electronswhose is entered as AL. sourceis in the magnetotailare excludedfrom regionsinside It is clear from Table 2 that there is very good agreement the plasmapause.This exclusiondoes not apply to positively betweenthe radial positionsof Explorer45 at the timesof the charged particles. For example, protons with nonzero but observedflux changesand thoseof the TDDBs as predictedby _<20-keV energiescan penetrateearthward of the plasmapause our calculation. The maximum difference between observation to form the proton nosestructure [Smith and Hoffman, 1974]. and prediction occursfor the 5.3-keV electron flux decrease, Having selectedthe initial potential drop •o = 60 kV, we but evenin this worst casethe TDDB liesonly •0.2 Re outside then determinedthe SSDBs for 90ø pitch angle electronswith the Explorer 45 orbit. The averageabsolute difference between initial energiesfrom 0 to 10 keV in stepsof I keV. Particles observationand prediction for thiscase is (I zXLI)= 0.1. Fig- were placedalong their respectiveSSDBs every 3 ø in azimuth ure 2 illustratesthe relation of observedand predictedposi- from 120ø to 240ø (60ø either side of midnight) to model a magnetotailsource. To simulatesubstorm onset at time to,we instantaneouslyincreased the cross-magnetosphericpotential Steady State Demarcation Boundaries to •f = 140 k V and tracked individual particles,calculating for Electrons(C•o:90 ø) of FixedEnergy their position and energyevery I min. From thesecomputed Noon ..... • ..... •0 = 60 kV trajectorieswe were able to constructthe innermostlocations .. . .. at which plasma sheetparticles of any given energywould be 0 keV ..' '-. found as a function of time. These locationsprovided a time (plasmapause)-..,.,' dependent demarcation boundary (TDDB) for each energy channel. For the particular data being studied, we chose boundaryenergies Wo = 0 and W, (i = 1-4) equal to 1.5, 2.3, 3.2, and 5.3 keV, correspondingto the minimum energy in eachof the four lowestenergy channels of the electrondetector of Williams et al. .5.3keV•' ,,, Turning to the Williamset al. [1974] observations(Table 1), L=10 ..' we note that the data give an azimuth and radial distanceat which flux changesin the electronsof energy W, are observed Midnight at a time t, - tofollowing substormonset. Here we take toto be Fig. 1. Steadystate demarcation boundaries (SSDB) for electrons the substormonset time at 2200 UT asdetermined by Williams (ao = 90ø) are displayedfor electronenergies of 0, 2.3, and 5.3 keV for et al. from groundmagnetograms. To compareour calcu- a cross-magnetosphericpotential drop of • = 60kV. Asis described lationswith the observations, wepredict the radial position of inthe text, these boundaries separate thespatial region inwhich the electronsof the specifiedenergy are on closedorbits (earthward of the theTDDB for electrons ofenergy W, at the satellite azimuth •bt boundary)from that in which they are on open orbits (antiearthward and at a time tt - to followingthe increaseof the cross- of theboundary). 4186 KAYE AND KIVELSON: TIME DEPENDENTCONVECTION

TABLE 2. Comparison Between Explorer 45 Observations and energy dependentplasma sheet boundary penetration. The Predictions of the Present Model final potential drop in the applicationof this model to the Wt, keV Ltøø• ckt,deg tt - to,min L? •a fiL electrondata of Williamset al. [1974]was fixed by the timesof the return of the satelliteto the regionearthward of the plasma Flux Increase sheet.Other observationswhich can be explainedin a similar 0 5.5 132 21 5.3 -0.2 fashionhave recentlyappeared in the literature [Maeda et al., 1.5 5.5 133 30 5.4 -0.1 2.3 5.5 139 53 5.4 -0.1 1978; Kivelsonet al., 1979]. 3.2 5.5 142 74 5.4 -0.1 The critical differencebetween our calculationand the pre- 5.3 5.3 148 99 5.4 0.1 viouswork of Kivelsonand Southwood[1975] and Cowleyand Ashour-Abdalla[1976a, b] is the use of a time dependent Flux Decrease model. Our calculation differs from that of M. Ejiri, R. A. 5.3 4.7 160 145 4.9 0.2 3.2 4.6 162 149 4.7 0.1 Hoffman, and P. H. Smith (unpublishedmanuscript, 1978) 2.3 4.5 164 152 4.5 0.0 because we introduce initial conditions consistent with the 1.5 4.3 166 159 4.4 0.1 assumedmodel. Two differentforms of the convectionpoten- 0 4.3 166 159 4.2 -0.1 tial have been usedin the paperscited. The presentwork and The Explorer 45 observationsare given in the first four columns, the work of M. Ejiri, R. A. Hoffman,and P. H. Smith(unpub- where tt - to is the time after substorm onset at which the flux in- lishedmanuscript, 1978) usea convectionpotential (first term creasesand decreaseswere observed.The predicted radial positions of ( 1)) which variesas L •',while Kivelsonand Southwood [ 1975] of the TDDBs at time tt - to at the satellite azimuth are listed in the and Cowleyand Ashour-Abdalla[1976a, b] usea uniform con- fifth column. The differencebetween observed and predictedradial vection field with a potential that varies as L. However, we distances,/XL, is given in the last column. find that the modelsare insensitiveto the choiceof L depen- dencebecause of the limited spatialrange over which injection tions of the plasmapauseand of flux increasesand decreasesin is observed. the four designatedenergy channels relative to the Explorer45 We tested the sensitivityof calculationsto the form of the trajectory. potential in two different ways. First, we asked whether a steady state convectiontheory could successfullypredict the locations of flux increases and decreases for either form of DISCUSSION OF RESULTS convectionpotential. We found the potentialdrop whichmini- By use of a simple approximationto the earth's electricand mized(I/XL I) for eachof the assumedforms of potential. magneticfields, substorm-associatedelectron injection and en- Using thesevalues, 45 kV for the uniform field and 90 kV for ergy dispersioncan be quantitativelyinterpreted in terms of a the partially shielded field, we calculated the SSDBs for the convectionmodel in which the potential drop acrossthe mag- two differentpotential forms; the resultsare displayedin Fig- netosphereincreases discontinuously at substormonset. The ure 3. (We note that the 45 k V calculated for the uniform field two free parametersof the calculationare the potential drop is lessthan that calculatedby Kivelsonand Southwood[1975]. before and after the start of the substorm,and they are highly They determineda potentialdrop of 55 kV by requiringthat constrained.The initial potential drop gives consistentinitial the Explorer45 trajectorydid not intersectthe 8.0-keVSSDB.) conditionsfor the calculation,and its magnitudeis fixed by the The inaccuracy of the prediction is similar for the two cases, time delay betweensubstorm onset and the times of the initial with (I ALl) - 0.3 for boththe uniformfield and the partially shieldedfield. Recognizingthat for any given potentialform a ----Flux Increoses---- changein the magnitudeof the potentialdrop would causethe 6- •o = 60 kV entire SSDB pattern to shift to higheror lower L, we seefrom •f =140 kV Figure 3 that no singlechoice of a potential could reducethe scatter illustrated or consistentlyaccount for both the flux increases and the flux decreases.

õ Having decided that the steady state calculation was not significantlyimproved by useof an L •'dependent potential, we askedwhether the time dependentcalculation could be carried L out with equal successfor either form of the potential.Accord- • •Exp!orer 45 ingly, we reran our time dependentcalculation using a uniform

4- field and, although we do not present our resultshere, ob- tained resultsas good as thoseshown in Figure 2 usingvalues q0 = 35 kV and qf = 75 kV. Consequently,we believethat it is the time dependenceand the self-consistentinitial conditions which are the crucial features of our calculation, and not the form of the potential. I10 120 130 140 150 160 170 180 190 Although we have presentedthe results of only one com- t 4"ø't t puter run for the shieldedpotential, a numberof suchcompu- •'000LT •'•'00LT •400 LT tationshave beenperformed in whicheither •0 or •f or both Fig. 2. Explorer 45 trajectory with positions of flux increasesin were varied, and we did find the results to be sensitive to energy channels Wx to W4 and flux decreasesin channels W4 to Wx changesin thesepotentials. We varied •0 from 20 to 60 kV and shown as vertical tick marks. Plasmapauseentry and exit are marked • from 100 to 160 kV; these values are typical polar cap as solid dots. The calcualted radial distances to the TDDB for the ith energy channel at the time of an observedflux increaseor decreaseare potential drops observedby Heppner [1972] for quiet and shownby the circlednumbers, and the positionof the W = 0 TDDB at disturbed times, respectively.We found 'good agreement' times of plasmapauseencounter is shown by the circled P. ((I/xLI) _<0.2) for all combinationsof q0 and qt suchthat KAYE AND KIVELSON: TIME DEPENDENT CONVECTION 4187

-----Flux Increoses---- 45 kV < $o < 60 kV and 130kV < St < 160 kV. The resultsof 6• (4:) the differentruns also showedthat the predictedflux increases were far more sensitiveto changesin $o than to changesin ,'h ".:•) whereas the predicted flux decreaseswere more sensitiveto .,,,,,,.4.•,,• -"-FluxOecreoses---- changesin St. Thesedependences may be easilyunderstood by noting that the flux increasesoccurred shortly after substorm onsetwhen the initial distributiondetermined by $o is of great importance. The flux decreaseswere seenafter the distribution had experiencedthe final convectionfield for about 2 hours Explorer 45 and therefore were more directly dependenton the value se- Trojectory lected for •t. These dependencesalso point out why a single 4-- choiceof potential could not consistentlyaccount for both the flux increases and the flux decreases. For our study we did not take to,the substormonset time, to be a free parameter,although it was pointed out by Williamset al. [1974] that there could have been a timing error of +10 min I10 120 I$0 140 150 160 170 180 190 in the substormonset time. We found that this uncertaintyhad t t negligible effect on our calculation, causing the range of $0 •000 LT •00 LT •400 LT valuesyielding good agreementto shift by a few kilovolts. Throughout this paper we have stressedour use of initial Fig. 3. Same format as that of Figure 2; however,here the bestfit SSDBs are shownfor a uniform electricfield (dashedcircles) and for a conditions which are representativeof a steady state unper- partially shieldedelectric field (solid circles). turbed convectingplasma in the magnetosphere.To illustrate the initial conditionsactually usedin our calculation, we took particles with energiescharacteristic of each energy channel [1974] observationsfor any ,:I:,r using the Ejiri et al. initial and tracedthem backwardsin time (conserving# and J) from condition. We proposethat useof initial conditionssimilar to their positionsat t = tt to their initial positionsat t = to, as those we have adopted could reduce the unacceptablylong shown in Figure 4. Note that the positionsof theseparticles at time required for nose penetration in the Ejiri et al. calcu- t = tt were chosento be the predictedpositions of the corre- lation. spondingTDDBs associatedwith the flux increases.The posi- Finally, at this point in the analysiswe must posea philo- tions of the observedplasmapause crossing and subsequent sophicalquestion as to whether it is worthwhile to manipulate flux increasesgiven in Table 1 are indicatedby the solid dots further the form of the convectionpotential or its time varia- along the satellitetrajectory. Two important featuresemerge tion to improve the fit of our predictionsto the observedflux from the calculation. First, as might be expected from the changes.Although additional free parameterswould yield im- generally eastward drift at low L of electrons,the particles start at azimuthswest of their final positions.They do not start Noon closeto midnight.Second, the particlesstart at radial distances I which are within 2 Rs of their radial position at t = tt. Note that all of the particlesstart insideof L = 7 and that the initial radial distanceis stronglyenergy dependent. M. Ejiri, R. A. Hoffman, and P. H. Smith (unpublished Explorer45 manuscript, 1978) have recentlycompleted some calculations of time dependent proton convection in which they started particlesof all energieson a singleboundary which was azimuthally symmetricabout midnight at L = 10. Our calculations indicate that such initial conditions seem inappropriate and that energy dependent boundaries at small radial distances 0 key•[, / shouldbe assumed.In fact, a study of ATS 5 data just com- W,•', / ,' pleted [Kivelsonet al., 1979] supportsthe view that low-energy ..' (< 15 keV) electronsand protons at synchronousorbit exhibit a steadystate Alfv6n boundary structureduring quiet times. In W4/•- -t--L=5 the Kivelsonet al. [1979] study, the energydispersion seen in the electronsand protons in UCSD energy spectrogramsfor I quiet times is quantitatively shown to be consistentwith the Midnight motion of the satellite through the SSDBs for a weak convection electric field of the sort that we assume for the t < to time period. Becausewe believethat SSDBs exist in the vicinity of synchronousorbit and that details of plasma injection are very sensitiveto initial conditions, we are convincedthat Fig. 4. Explorer 45 trajectoryin the equatorialplane with trajec- time dependentcalculations which assumedistant sourcesof tories of electronswith energiescharacteristic of eachenergy channel. injectedparticles and which neglectto incorporatethe energy The electronswere t•:hced backwards in time from theirpositions at dispersionof initial conditions,such as thoseperformed by M. t = tt to their initial positionsat t = to.The positionsof the particles at t = tt were chosento be the predictedpositions of the correspond- Ejiri, R. A. Hoffman, and P. H. Smith (unpublishedmanu- ing TDDBs associatedwith the flux increases.The solid dots indicate script, 1978), can be misleading.In fact, we found it impossible the positions of the observedplasmapause crossing and subsequent to find quantitative 'good agreement'with the Williams et al. flux increasesas given in Table 1. 4188 KAYE AND KIVELSON: TIME DEPENDENT CONVECTION proved agreement with observations,such improved agree- Jaggi, R. K., and R. A. Wolf, Self-consistentcalculation of the motion ment would not necessarilyimprove our understandingof the of a sheetof ions in the magnetosphere,d. Geophys.Res., 78, 2852, 1973. injection processin this particular event or in qualitatively Kivelson, M. G., Magnetosphericelectric fields and their variation similar events unless we could independentlydetermine the with geomagneticactivity, Rev. Geophys.Space Phys., 14, 189, 1976. added parameters.We do not think that improvedfits would Kivelson, M. G., and D. J. Southwood,Approximations for the study add to the evidencepresented to support the view that the of drift boundariesin the magnetosphere,J. Geophys.Res., 80, 3528, 1975. inclusion of time dependenteffects and self-consistentenergy Kivelson, M. G., S. M. Kaye, and D. J. Southwood,The physicsof dependentinitial boundariesin a simplemodel of convection plasma injection events,submitted to SpaceSci. Rev., 1979. is useful in understandingparticle observations,similar to Maeda, K., N. K. Bewtra, and P. H. Smith, Ring current electron thosepresented in this study,shortly after substormonset. The trajectoriesassociated with ELF emissions,d. Geophys.Res., 83, applicabilityof our approachover a wider rangeof substorm 4339, 1978. Maynard, M. C., and D. P. Cauffman, Double floating probe mea- event types may be testedin the future through simultaneous surementson S•-A, J. Geophys.Res., 78, 4745, 1973. measurementsat two points in space;such measurements are Mcllwain, C. E., Plasma convectionin the vicinity of the geosynch- now available from the Isee 1 and 2 spacecraft. ronousorbit, in Earth's MagnetosphericProcesses, edited by B. M. McCormac, p. 268, D. Reidel, Hingham, Mass., 1972. Acknowledgments. We gratefullyacknowledge comments and sug- Mozer, F. S., The origin and effectsof electric fields during isolated gestionsby S. W. H. Cowley,D. J. Southwood,and R. J. Walker.This magnetosphericsubstorms, J. 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