ieee transactions on plasma science special issue: …

IEEE TRANSACTIONS ON PLASMA SCIENCESPECIAL ISSUE:SPACE AND COSMIC PLASMA 1

Somerecentdevelopmentsin understandingauroralelectronaccelerationprocesses

A. OlssonandP. Janhunen,

Abstract— Discrete auroral arcs and the associatedinverted-V type electron precipitation are accompanied by potentialstructur es, upgoing ion beams and plasma density depletionsin the acceleration region altitude and above. However, exactlyhow these phenomena depend on altitude in various Kp andsolar illumination conditions as well as at various magneticlocal time (MLT) and invariant latitude (ILA T) is not so wellknown. We review the main resultsof our recentlarge statisticalstudies of the altitude dependenceof these parameters usingPolar data. We also include similar statistical results on someadditional parameters that appear to have major importancein inverted-V auroral acceleration. These parameters includemiddle-energy ( � 50-500eV) electron anisotropies, certain typesof electrostatic wave bursts as well as ion shell distrib utions.One of the most interesting results of the statistical studies isthat many of the parameters mentioned changetheir behaviourrather abruptly at around 4

��radial distance. We draw the

conclusionthat the regionfor auroral energisation processesseemto take place within a closedpotential structur e below � 3-4

� �.

These conclusions lead to a new scenario for auroral plasmaphysicsand the energy flow in the auroral acceleration process.The observational resultsare in agreementwith simulations thatwe also summarise in this paper.

Index Terms— auroral electron acceleration,auroral potentialstructur e, ionosphere-magnetosphere coupling.

I . INTRODUCTION

A N auroral arc is a central element of ionosphere-magnetospherecoupling. Thus, in order to understand

ionosphere-magnetospherecoupling more profoundly oneshouldknow the physicalmechanismsresponsiblefor auroralarcs.

Already 50 yearsago [1] theoreticallypostulatedthe ex-istenceof parallel electric fields which could accelerateau-roral electronsdownward and ions upward from Earth.Soonthereafter, one could observe from rocket and satellitemea-surementsconvergentelectricfields [27], [35] simultaneouslywith downward acceleratedelectronsandupward acceleratedions. Data from low-orbiting satellitesshowed an inverted-Vlike spectraof acceleratedelectrons.[10], [11]. The low andmid-altitude satellite observations (1000-14000km) resultedin a modelof the accelerationregion asa U-shapedpotentialgeometry(seeFigure1) [8] wherethe “bottom” of the U cor-respondsto theregion of theparallelelectricfield. Theclosureof the potential structureand the generationmechanismhasusuallybeenassumedto take placein theoppositehemisphereor far out in the magnetotail.The physical mechanismsfor

Manuscriptreceived March1, 2003;revisedSOMETIME, 2003.This workwassupportedby theSwedishResearchCouncilandtheAcademyof Finland.

A. Olssonis with the SwedishInstituteof SpacePhysics.P. Janhunenand H.E.J. Koskinen are with the Finnish Meteorological

Institute.

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Fig. 1. Low-altitude satellitedatasuggestthat the bottom of the potentialstructureis U-shaped.How the potentialcontinuesto higher altitudesis notconstrainedby thesesatellites,however.

generatingtheupwardparallelelectricfield hasbeenproposedwith theoriesof doublelayers[5], [26], anomalousresistivity[32] or magneticmirror force [2]. To be able to confirm orrefute someof the proposedtheories,a full understandingofthe completegeometryof the potentialis needed.

To get someinformationaboutthe closureof the U-shapedpotentialstructurethe electricfield datafor 65 auroralpassesat � 4 �� altitudewasstudiedwith the Polarsatellite[14]. Alack of convergentelectric fields at high altitude was found.To explain this, the idea of a negative O-shapedpotentialwell was introduced,i.e a potentialmodel wherethe bottompart in all respectis the sameas the traditional accelerationregion of theU-shapedmodel,but which closesalreadybelow� 4 �� . In the closedpotential model there is thus anotherregion of downwardparallelelectricfields at a higheraltitude(presumably2-4 � � ). Thenetresultof particlesgoingthroughsucha closedpotentialstructurewould be zeroacceleration.

However, in [16] a correlation between waves andanisotropy of electrons was observed simultaneouslywithsatellite crossings over various arcs and the energisationmechanismof electronswas suggestedto be related to thebroadbandwave bursts.Accelerationof electronsdueto wavesalone (no potential drop), possibly lower-hybrid waves, hasbeenproposedbefore[6], [3].

In a testparticlesimulation[15] theideaof a self-consistent“Cooperative Model” (CM) wasdeveloped,i.e a modelof theaccelerationprocesswhereboth the closedpotentialstructurein cooperationwith wave-particle interaction takes place.

2 IEEE TRANSACTIONS ON PLASMA SCIENCESPECIAL ISSUE:SPACE AND COSMIC PLASMA

The plasmawaves would push the electronsup an electronpotential hill at � 3-4 � � and the upperpart of the potentialstructurework asanenergisationregion for theparticleswhilethebottompartactsasthe traditionalaccelerationregion. Thewaves are thus the engineand the potential structureis thewheelsand gear. The CM is discussedin detail in [17], [18]and in subsectionVI.A of this paper.

To revisit thequestionaboutclosurealtitudeof thepotentialstructure in a more thorough way, a larger datasetand acompletealtitudecoveragewould be needed.In order to alsodiscussthe physical mechanismsmaintaining the potentialstructureand the energy flow, information about the otherarc-associatedparametersis needed,for exampleobservationsof the altitude profiles of ion beams,cavities and waves.The Polar satellite has the advantageof covering the largealtitude range (5000-32000km) without gaps and thus wecan study the altitude dependenceof various arc-associatedparametersduring several years(1996-2001).It is importantto also study the altitude dependencein different magneticlocal time (MLT) sectorsas well as different Kp conditions,invariant latitudes(ILAT), seasonand solar cycle conditions.We will in this paperreview our latest resultsregarding thealtitudeprofile of the arc-associatedparameters.In this paperwe will alsosummariseresultsconcerningthe altitudeprofileof other relatedparameters,i.e. the middle-energy anisotropyof electronsand ion shell distributions.We believe that thesemight also play a significant role in the auroral scenario.The observational results are section V (Simulations) thencomparedwith a a hybrid simulationmodel for a stablearc.In a two-dimensionalelectrostaticsimulation(subsectionV.B)it is alsoshown that ion shelldistributionscancontainenoughfree energy for waves that could power electronenergisationin the auroralprocess.

I I . INSTRUMENTS AND DATASETS

We use 3-5 years of Polar data to study the altitudedependenceof the occurrencefrequency of auroral potentialstructures,upgoingion beams,densitydepletions,electrostaticwave bursts,anisotropy of electronsand ion shells.

The potentialstructuresarefoundby locatinglocal minimain the plasma potential which is found by integrating themeasuredelectric field from the EFI instrument[13] alongthe spacecraftorbit. The datasetof the electric fields duringtheyears1996-2001coversthealtituderange5000-30000km.The samedatasetis used for studying the waves as well.Density depletionsare studiedby thresholdingthe Polar/EFIspacecraftpotentialduring the years1996-2000(5000-30000km altitude).

Ion beamsare detectedfrom Polar/TIMAS [37] duringthe years1996-1998and the data cover the altitude ranges5000-10000,20000-30000km. To get a complete altitudecoverageDE-1/EICS [36], data during the years1981-1991arealsoincluded.DE-1/EICScoversthealtitudes8000-23000km during the 10 years.The ion shell distributions are alsomeasuredwith Polar/TIMAS.

The anisotropy of electrons is studied with the Po-lar/HYDRA instrument [34] during the years 1996-2000(5000-30000km altitude).

The magneticfield experiment(MFE) is neededto decom-poseE-field into parallelandperpendicularcomponents[33]

I I I . STATISTICAL RESULTS OF ARC-ASSOCIATED

PARAMETERS

This sectionis a condensedexposition of recentstatisticalstudies.A readernot too interestedin the detailsmay safelyskip to sectionIV.

As mentionedabove, to obtain observational informationabout the physicsof auroral accelerationprocesses,altitudeprofilesof the arc-associatedphysicalquantitiesareof utmostimportance.However, lacking satellite missions especiallydesignedfor producingmagneticconjunctions,instantaneousaltitude profiles are impossibleto obtain. Nevertheless,suchprofiles can be obtainedstatistically, and we have done thisrecentlyfor mostof the parametersthat arerelatedto auroralprocesses.We have also studiedhow the altitude profiles ofthe arc-associatedparametersdependon Kp, ILAT, MLT, thesolar illumination condition of the ionosphericfootpoint andin the ion beamcasealso the solar cycle dependence.Thispresentationis a review and summaryof the output of suchstatisticalstudies.For mostparameterswe will here,however,only show theseasonalandKp effecton thealtitudeprofile.Tobe able to easilycompareresultsfor the differentparameterswe chooseto list them in each subsection.In section IV(Summaryof observations)we take the resultstogether.

A. Potential minima and their associatedeffective electricfields

In [20] we studyhow the altitudedependenceof the meanenergy, the potential minima and the effective electric fieldvary with different MLT, ILAT, Kp and seasonalconditions.The mapped-down effective ionosphericelectric field is de-fined as the depth of the potential minimum divided by themapped-down half-width of the structurein the ionosphericplane.The ionosphereis thususedasa referencealtitudeandthe artificial effective electric fields are introducedjust as away to beableto compareelectricfieldsmeasuredat differentaltitudes.

In Figure 2, we show results for all data collected inthe nightside MLT sectorsfor all Kp values during sunlitand darknessconditions.Panel (a) is the numberof orbitalcrossingsin each �� radial bin. Panel(b) shows thedepth(kV) of potential minima which are less than 0.6� wide inILAT (correspondingto 60 km in the ionosphere)anddeeperthan0.5 kV. Panel(c) shows the effective ionosphericelectricfield �� mV/m. Panel (d) is the occurrencefrequencyof �� mV/m per orbital crossing,which is obtainedbydividing thenumberof datapointsin panelc in eachradialbinby the correspondingnumberof orbital crossingsfrom panela. The error barscorrespondto �� relative errorswhere �is the numberof datapointsexceedingthe threshold.Panel(e)is the occurrencefrequency of �� mV/m.

We summarisethe resultsfrom Figure 2 as well as fromFiguresin [20]:

1) The potentialminima (panelb Figure 2) found at lowaltitude are on the averagemore numerousand deeperthan thosefound at high altitude.

OLSSONET AL.: AURORAL ELECTRON ACCELERATION 3

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Fig. 2. All nightsidepotentialminima deeperthan 0.5 kV and the correspondingeffective ionosphericelectric fields !#" . The ILAT rangeis 65-74: (a)numberof orbital crossingsin eachradial bin, (b) depth in kilovolts and radial distanceof eachpotentialminimum, (c) effective ionosphericelectric fieldassociatedwith eachpotentialminimum with lower limit 100 mV/m, (d) occurrencefrequency of !#" beinglarger than100 mV/m (numberof points in panelc divided by panela), (e) occurrencefrequency of !#" being larger than500 mV/m.

2) The effective electric fields (panel c Figure 2) havehighervaluesat low altitudes.

3) There is a dip in the occurrencefrequency of theeffective electric fields around3.75 �� radial distance(paneld ande Figure2).

4) The high occurrencefrequency of electric fields in the4-6 � � radialdistancemainly comesfrom themidnightMLT (Figure9 [20]). Thesehigh altitudeelectricfieldsmight be substormrelated.Thus in the otherMLT sec-tors the electric fields almostexclusively occursbelowabout3.75 �� .

5) There is a peak in occurrencefrequency of potentialminima at about2.75 �� , however a seasonalandKpdependenceis alsofoundwhich affectsthepeakaltitudeof the potentialminima somewhat. For caseswhen theionosphericfootpoint is sunlit the maximumoccurrencefrequency is around �%$'&(�*) �� while for casesofdarknessconditions there is a downward shift of 0.5� � (paneld ande Figure2).

6) Most potentialminima occur in the dusk and midnightMLT sectorsand their occurrencefrequency increaseswith increasingKp index (Figure9 and10 [20]).

7) When all nightside,Kp valuesand ionosphericillumi-nation conditionsare put togetherto form the baselinecase,themaximumandminimumoccurrencefrequencyof potentialminimaarefoundat �+$,&-� &� and3.75 ��(Figure6, [20]).

8) The mentionedlocal minimum in the occurrencefre-quency of potentialminimaat about �.$,/��0)� is visible

separatelyin all MLT sectors,all Kp andbothsunlit anddarknessionospheres.

B. Upgoing ion beams

In [21] we study the occurrencefrequency of upgoingionbeamsas a function of altitude in different MLT sectorsas well as invariant latitudes(ILAT), Kp, seasonsand solarcycle conditions.To get the samealtitude coverage(5000-30000km)as for the other parameters,we use two differention instrumentsTIMAS and EICS from two satellitesPolarandDE-1.

In Figure 3, we show resultsfor all data collectedin thenightsideMLT sectors18-06for conditionswherethesatelliteionosphericfootpoint is sunlit (left column) and in darkness(right column).Thetop panelof Figure3 shows all ion beamsbetween0.5 and10 keV asa functionof radialdistance� andbeampeakenergy. In the middle panelswe show the numberof hoursspentby theinstrumentin eachradialbin. By dividingthe numberof points in eachbin by the numberof samplescoming from the bin we get the occurrencefrequency of theion beamsin the bottompanels.

We summariseinformation from Figure 3 as well as fromFiguresin [21].

1) At low altitude ( �21�&(�0 �� ) and high Kp, ion beamsare mainly an evening sector phenomenon,while athigh altitude ( �3�54�� ) and low Kp they are morea midnight sectorphenomenon(Figure 4 in [21]). Thissuggeststhat some high altitude ion beams in themidnight sectordo not come from the ionospherebut


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Fig. 3. Polar/TIMAS andDE-1/EICSion beamsin all the nightsideMLT sectorsas a function of radial distance6 for sunlit (left column) and darknessconditions(right column).The ILAT rangeis 65-74.Top panel:Ion beamevents(one event correspondsto 12 s sample)as function of radial distanceandpeakenergy. Middle panel:Hours spentby the instrumentin eachradial bin. Bottom panel:Occurrencefrequency of ion beams(numberof points in eachbin divided by the numberof samplescoming from the bin). Small Kp ( 798 ) shown by blue andfilled circles, large Kp ( :98 ) by red and triangles.Blackline is both Kp’s put together.

originate as wave-particle interactionsenergising coldplasmaat someintermediatealtitude.

2) In the evening sector, a peak is also present in theinvariant energy and particle fluxes carriedby upwardion beams,where“invariant” meansthatbothquantitiesareprojectedto the ionosphericplane(Figure10 [21]).

3) ConcerningILAT, the peak is most visible in the 68-71 ILAT bin, probablybecausethat correspondsto theaverageauroraloval latitude(Figure6 in [21]).

4) The solar cycle doesnot have much influenceon theion beamoccurrencefrequency except through the Kpparameter(Figure5 in [21]).

5) In the eveningsector, a simpleMonte Carlo simulationdemonstratesthat an O-shapedpotentialstructurewithsomewave energisationcan explain the altitudeprofileof the ion beam occurrencefrequency, energy flux,particleflux andmeanenergy (Figure10 in [21]).

C. Densitydepletions

Density depletions(auroral cavities) are studied in [23]by thresholdingthe Polar/EFIspacecraftpotentialduring theyears1996-2000(5000-30000km altitude). The altitude de-pendenceof the cavities is studiedduring varying conditionsof MLT, ILAT, Kp and season.In this review we only showthe Kp andseasonalconditions.

The density depletionsare shown with three thresholds-11, -18, -25 V, the correspondingdensitiesare 0.3, 0.1, 0.06cm;=< .

In Figure4 we show the occurrencefrequenciesof auroralcavities for the caseswhere the ionosphericfootpoint is in

darkness(left subplot)or sun-illuminated(right subplot).AllnightsideMLT (18-06)sectorsareput togetherand the ILATrange is 65-74. The first panel shows the orbital coverageand the following panelsshow the occurrencefrequency forthe auroral cavities when three different spacecraftpotentialthresholdsare used (-11 V, -18 V and -25 V) representingshallow, medium-deepand deepcavities, respectively. SmallKp index values( 9> Kp >?& ) are shown by blue line withfilled circles,largeKp index values(Kp �@& ) by red line withopen triangles,and all Kp’s put togetherby black lines. Welist the resultsbelow.

1) Thereis a peakin cavity occurrencefrequency at about3.25 �� for darknessconditions and the region ofcavities endsat about4.25 �� . In sunlit conditionsthepeak in occurrencefrequenciesis at 3.75 � � and theregion of cavities extendsup to 5.25 � � (Figure4).

2) The peak in occurrencefrequenciesof auroralcavitiesis clearestfor deepestcavities, correspondingto plasmadensitiessmallerthan � 0.6 cm;=< (Figure4).

3) The Kp index doesnot have a clear effect on the peakaltitude(Figure4).

4) For low ILAT, cavities occurmainly only for large Kp,probablybecausethe auroraloval extendsequatorwardduring disturbedconditions.For other ILAT, however,cavities aremorecommonfor low Kp thanfor high Kp(Figure5 and6 [23]).

5) Cavities show an auroral zone dependency, which in-dicatesthat most of them are associatedwith auroralprocesses(Figure3 [23]).

6) In the midnight sectorthereare cavities also in the 4-6 � � radial distancerangefor both low and high Kp.


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Fig. 4. Occurrencefrequency of nightsideauroral cavities for radial distancesbetween1.5 to 6 6 � . Left subplot show the occurrencefrequenciesforconditionswhenthe ionosphericfootpoint is in darknessandthe right subplotfor sun-illuminatedconditions.Top panels:the orbital coveragein hours.Panels2-4: the occurrencefrequency for spacecraftpotentialthresholds-11 V, -18 V and -25 V. All Kp’s put togetheris shown by a black line. Low Kp ( 7A8 ) isshown by blue lines with filled dotsandhigh Kp ( :B8 ) by red lines andopentriangles.Error barsarealsoshown.

They alsooccurin themorningsector, but only for highKp. This holds for ILAT range 68-74. Thesecavitiesmay be relatedto substorms(Figure5 and6 [23]).

D. Electrostaticwavebursts

In [31] we study the occurrencefrequency of the 1-10 Hzelectric wave componentin the auroral zone (65-74 ILAT)using observationsfrom the Polar/EFI instrument.We inves-tigate the altitude dependenceof the occurrencefrequencyduring varying Kp, MLT, ILAT and seasonalconditions.Weespeciallydiscusswaves with significantparallel component( �DCB1E-� 4��GF and ��FIHJ� mV/m). In this review we onlyshow the seasonalandKp dependence(seeFigure5).

In Figure 5 we investigatethe occurrencefrequency ofparallel-dominatedwavesdecomposedinto sunlitanddarknessconditionsrespectively. In the first panelthe orbital coveragein hours is shown and in the bottom panel the occurrencefrequency for the parallel dominatedwaves is seenfor theradial distances1.5-6 �� . Redlines correspondto conditionswhereKp �K& and blue lines representL> Kp >K& . Blacklines correspondto caseswhenall Kp’s are put together. Welist the resultsseenin Figure5.

1) The preferred radial distance of parallel-dominatedwaves is 3.25 �� when the ionosphericfootpoint ofthe satelliteis in sunlight,but movesto 2.75 �� whenit is in darkness(Figure5).

2) The occurrencefrequency of parallel-dominatedwavesis usually larger for high Kp than for low Kp index(Figure5).

3) Constantoccurrencefrequency contoursof the parallel-dominatedwavesqualitatively follow theaverageauroral

oval (in the ILAT-MLT plane),(Figure7, [31]).4) The occurrencefrequency of perpendicular-dominated

wavesdecreasessmoothlywith altitude(Figure4, rightsubplot,[31]).

E. Anisotropy

Often the electrondistribution in auroralfield linesconsistsof at leasttwo quasi-Maxwellianpopulations,eachwith pos-sibly differentparallelandperpendiculartemperaturesMNC andMOF . Especiallycaseswherethe hot populationis isotropicbutthe cold populationshas MNC9�PMOF asymmetryare commonandarestudiedin [22].

The anisotropy of electrons is studied with the Po-lar/HYDRA instrumentduring the years 1996-2000(5000-30000 km altitude). The altitude dependenceof especiallythe middle-energy electrons(100-1000keV) is studiedwithvarying MLT, ILAT andKp.

In Figures6 and7 show thebasicstatisticalresultsof M�C��MOF middle-energy (100-1000eV) electronanisotropies.Themajor findingsareas follows.

1) Themiddle-energy anisotropies(100-1000eV) obey theMLT-ILAT dependenceof the averageauroraloval. Thedistributionsof thelow andhigh-energy anisotropiesaremore irregular (Figure6).

2) The altitudeprofile of the anisotropiesis smoothasonewould expect sinceelectronshave high mobility alongthe magneticfield (Figure7).

3) Thereis oftenanisotropicelectrondistribution at higherenergies simultaneouslywith the anisotropy (Figure 2and4 [22]).


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Fig. 5. The occurrencefrequency of the parallel dominatedwaves during conditionswhen the satellitefootpoint is sunlit (left subplot)and conditionsofdarkness(right subplot).All nightsideMLT sectorsare put togetherand the ILAT rangeis 65-74.On the horizontalaxis the radial distancebetween1.5-66 � is shown. Top panel:the orbital coveragein hours.Bottom panel:the occurrencefrequency for the paralleldominatedwaves( !RQNSUTWV XY![Z and ![ZA\^]mV/m). Blue line with filled circlescorrespondsto low Kp ( T_7 Kp 7B8 , red line with trianglesis for high Kp (Kp :B8 ) andwhenall Kp’s areput togetherit is representedwith a black line.

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Fig. 6. Strengthof `�QA:+`aZ electron anisotropiesin different energyranges:10-100eV (top), 100-1000eV (secondpanel),1-10keV (third panel)andorbital coverage(bottom)asa function of MLT andILAT. Especiallythemiddle-energy anisotropies(100-1000eV, secondpanel)follow thepatternofthe averageauroraloval.

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4) The anisotropiesare often up/down symmetric(Figure2 and4 [22]).

5) The middle-energy anisotropiesappearat lower ILATin all MLT sectorswhen the Kp index increasesas isexpectedfor an auroral oval relatedprocess(Figure 8[22]).

F. Ion shells

By ion shellwe meana 3-D sphericalalmosthollow shell invelocity space.Suchdistributionscanbe formedby a varietyof mechanisms,including various time of flight effects andenergy-dependentperpendicularion drifts, andthey potentiallycontaina lot of free energy. An exampleof a half-consumedion shell is given in Figure 8, wherealso the simultaneouslyoccurring electronmiddle-energy anisotropy is shown (rightsubplot).Our interpretationis that below the spacecraft,partof the free energy hasgoneto waves that have energisedtheelectrons,so that only the downgoingpart is a pureion shell.

Statistically, in Polardataion shelldistributionsoccureithercloseto the polar cap boundaryor at low latitudes(Figure 9left subplot).The low latitudeonesare the mostcommon.Inthe radial direction, there is somepreferencefor ion shellsto occur at 3-4 � � radial distance,although quantitativeconclusionscannotbe drawn becauseof incompleteorbitalcoverage(Figure9, right subplot).

Ion shells occur less frequently in the auroral zone thanaround it (Figure 9 left subplot, white area in middle andbottom panels).Thereare two possibleexplanations:(1) themechanismsproducingtheshelldoesnot operatein theauroralzone,(2) the mechanismsoperatealso there,but the auroralzoneshellsarequickly consumedby wave-particleinteractionsand thus erodedaway. To find out which is the more correctexplanation one would need ion shell distribution data athigheraltitude.

IV. SUMMARY OF OBSERVATIONS

We review the latest results on the altitude dependenceof auroral potential structures,upgoing ion beams,densitydepletions,electronmiddle-energy anisotropies(� 50-500eV),electrostaticwave bursts and ion shell distributions. Theseparametersare all probably related to the auroral electronaccelerationprocessandthusfinding out their altitudedepen-dencegives valuableinformation about auroral acceleration.We have also discussedhow the altitude profiles of thementionedparametersdependon Kp, ILAT, MLT andthesolarillumination conditionof the ionosphericfootpoint.

Below we list the major resultsfrom the studies.

1) In all parameters(potential structures,upgoing ionbeams,cavities and waves with a large parallel com-ponent) there is peak in occurrencefrequency at � 3�� radial distance.

2) Peakamplitudeandaltitudedependonsolarilluminationcondition (season),Kp index, ILAT and MLT, in thesameway for potential structures,ion beams,cavitiesandwaves.

3) Increasein occurrenceof potentialstructuresand cavi-tiesabove ��$,4�� takesplacein midnightMLT sector

(22-02). This is likely due to some midnight-specifichigh altitude processwhich is different from ordinaryauroralarcs;it may be relatedto substorms.

4) Increasein occurrencefrequency of ion beamsabove�b$c4a� � takes place probably due to wave heating.This is a processwhich is peculiar to ion beamsandthus it hasno counterpartin the otherparameters.

5) There is indirect event-basedand simulation evidencethat Bernsteinwavesdriven unstableby ion shell distri-butionsenergise the electronsin the parallel directions,thus forming MNCd�eMfF electronanisotropies.

V. SIMULATIONS

We use2-D electrostatichybrid and kinetic simulationstomodel the developmentof auroral potential structures.Thehybrid code shows how a potential structure arises fromwave-particle interactions,and the kinetic simulation showsa possibility how such waves may be formed: free energyassociatedwith a hot ion shell distribution.

A. Electrostatichybrid simulation

The2-D hybridsimulationcodeof [19] hasparticleionsandquasineutralfluid electronsobeying theBoltzmannresponse.Itemploys a new type of ion pusher(“monopolesolver”) whichmakesit possibleto usetimestepsas long as g �h��ikj[C insteadof the usual g �h��iljmF , where g � is the maximumion velocityand ikj[Con F is thegrid spacingin theparallelandperpendiculardirections,respectively. Thesimulationbox is alignedwith thedipolemagneticfield andis 20 km wide in theionosphereand38000km long alongthemagneticfield. Theenergy sourceisprovidedby wave-particleinteractionswith the electrons.Thewaves are assumedto be localisedin ILAT and thus definethe boundariesof the auroralarc that forms.

Figure10 shows the ionosphericion densityandthe poten-tial in thehybridsimulationafterthewave-particleinteractionshave beenon for 10 minutes.A densitycavity, upgoing ionbeamanda closedpotentialstructurehave formed.

We summarisethe main resultsfrom thehybrid simulation:1) With wave-particleinteractionsturnedon,thesimulation

producesa self-consistentO-shapedpotential structure(Figure 10, right subplot) with an associateddensitycavity (Figure 10, left subplot).If a shearflow topsideboundarycondition is usedwithout wave-particleinter-actions,a U-shapedpotentialis produced.

2) Strongconvergentperpendicularelectricfields arecon-fined in a relatively narrow altitude range(about5000-11000km, Figure7 [19]).

3) Upward electric fields of about 1 mV/m exist at thebottomof thepotentialstructure,while at higheraltitudethereareweaker downward fields (Figure8 [19]).

4) When the driver is turned off, the potential structuredisappearsin the electrontime scale ( �%� s) and thedensitydepletionin theion timescale( � 10min) (Figure9 [19]).

5) The arc becomesstrongerif the anisotropy is increased.The potential structurebottom altitude becomeslowerif the ionosphericsourceplasmadensity is decreased(Tables1 and2 [19]).


−1.0×106 0.0×100 1.0×106 1.8×106

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TOTION 19970411, 05:46:54 .. 05:47:6

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ELE 19970411, 05:46:54 .. 05:47:6

vPerp (m/s)

vPar

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/s)

1/(cm2 s sr keV2)

1e+03

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Fig. 8. Left subplot: Ion shell distribution in Polar/TIMAS data(greencircular area).Upgoing low energy ion shell is also seen(yellow and red). Rightsubplot:Simultaneouselectrondistribution, wheremiddle-energy `pQN:q`aZ anisotropy is seen.

65

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er q

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PSBL shell events, MLT=18−06

Fig. 9. Left subplot:Ion shellsas function of MLT andILAT. Orbital coverage(top), occurrencefrequency of ion shellsobtainedby thresholding(middle)andtheir 95-percentile(bottom).Right subplot:Ion shellsasfunctionof radialdistance.Detectedion shell freeenergy densities(top), orbital coverage(secondpanel),occurrencefrequency obtainedby thresholding(third panel)andupperquartile (bottom).

B. 2-D electrostaticPIC simulation

The 2-D electrostatickinetic simulation is a conventionalexplicit particle-in-cellcodewith realisticelectronmass[24].Thebox sizeparallelandperpendicularto themagneticfield is800and10 km, respectively (Figure11). Initially thereis freeenergy in theplasmain theform of a hot ion shelldistribution.Thefreeenergy drivesunstableion Bernsteinwaves.Themainresultsare:

1) A hot ion shell distribution may contain enoughfreeenergy to power the aurora(Figure2 panelj [24]).

2) Ion shelldistributionsdo occuron auroralfield lines.Wehave found caseswhere the dataare clearly consistent

with the assumptionthat the shell distribution providesthe free energy for wave-inducedparallelelectronener-gisation(Figure2 [24]).

3) The perpendiculare-folding distanceof the ion Bern-stein waves in questionis of the order of the auroralarc width (a few kilometres)or smallerwhenprojectedto the ionosphere(Table1 [24]). The parallel e-foldingdistanceis a few hundredkilometres,which is smallerthan the field-alignedextent of the energisationregion.Thesepropertiesof the Bernsteinwaves are consistentwith the physicalsizeof the region wherethey exist.

4) Electrons are heatedin the parallel direction by theBernsteinwaves at a rate �Ir� eV/s (Figure 13 [24]).


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Fig. 10. Left subplot:density of ionosphericions after auroralarc has formed in the hybrid simulation.Horizontal axis is ILAT and vertical is altitudemeasuredalongdipolar magneticfield. Right subplot:correspondingpotentialstructure,with maximumdepth2.52 kV.

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Fig. 11. Time developmentof Bernsteinwave electricpotentialin thekineticsimulation.Horizontalaxis is alongmagneticfield (box length800 km) andvertical axis is perpendicular(box width 10 km). From top to bottom thetimesare5, 70, 130 and250 ms. The maximumwave amplitudeis 1.5 kV.

Cold ions receive only 10%of theheating(their heatingis perpendicular).

5) An inversecascadeprocessfor the Bernsteinwaves isin operationwhich transfersenergy from short to longwavelengthsby nonlinearmechanisms(Figure11).

VI . THEORY AND A SMALL SIMULATION

The observations reviewed above give rise to importantconsequencesconcerningthe physicsof auroral electronac-celeration.Exactly what theseconsequencesare is not quitestraightforward. We now presentthe model which we have

arrived at studyingthe dataand which we find also theoreti-cally pleasing.Afterwardswediscussto whatextentthemodelis unique, i.e. could our observationsbe explained by othermodels.

A. TheCooperative Model

A central concept in the model is a closed (O-shaped)negative potential structure.The structureis formed by thefollowing steps:

1) Broadbandplasmawaves(maybeion Bernsteinmodes,[24]) get unstableby a presenceof somefree energysource.Thefreeenergy is probablyin thehot ion distri-bution, possiblyin the form of an ion shell distribution[24].

2) The wavesenergiseelectronsin paralleland to a lesserextent ions in perpendiculardirection. In caseof ionBernsteinwaves driven unstableby an ion shell dis-tribution, the particle-in-cell simulation producedtherate of energisation 80 eV/s [24]. The result is ananisotropic( MNCd�eM F ) electrondistribution [16], [22]. Iftheelectronenergisationcomesfrom Landauresonance,the anisotropy may be limited in energy [22].

3) Theelectronanisotropy causestheelectronmirror pointsto movedownwardsincethepitchanglesdecrease.Sincethe flux tube becomesnarrower at lower altitude, thenumberof electronsper volume tendsto increase.Asthe ions do not act similarly, the result is a tendency toform a negative charge cloud whosedensity increasesdownward.

4) Since the electron Debye length is only �s�� m,quasineutralitymust be maintained.Thus the chargecloudneveractuallyforms,but a downwardelectricfieldis setup which restoresquasineutrality. This happenssofast that ions do not have time to react.


tvu wxy0z{z{|{} |�~{y0� � ��0�{�� 0�W��(��

� �� W��W� {¡� {¢ £�¤ ¥§¦0¨ ¤ ©�ª«0¬{�® ¯�°±0²W³�´�µ(¶�·

Fig. 12. Sketchof our proposedmodelfor auroralacceleration.With thehelpof wave-particleinteractions,electronsareenergisedin the paralleldirectionsand are thus able to climb the potential hill in the energisation region.Thereafterthey undergo electrostaticacceleration,which producespeakedenergy spectraat low altitude. The engine is the wave-particle interaction,while the “gears and wheels” (geometryand transferbetweenkinetic andpotentialenergy) is providedby thepotentialstructure.Thepotentialstructureis formed by spacecharges producedby the wave-particleinteractionsandmirror force. The systemis maintainedself-consistentlyas long as there isfree energy for the waves to consume.

5) The ionosphereand magnetosphereare maintainedinapproximatelyzero potential. Thus the potential con-tours must close by forming a closedpotential struc-ture. Another way to see this is to note that parallelelectricfield cannotexist in plasmawhich containscoldelectrons,and at low enoughaltitude cold electronsofionosphericorigin alwaysexist.

Oncethe O-shapedpotentialstructurehasformedwith thehelp of wavesandanisotropies[22], it givesrise to a numberof additionalobservationalconsequences:

1) Strongperpendicularelectric fields are confinedto thealtitudewherethe potentialstructureexists [20].

2) Since the bottom of the structurecontainsan upwardparallel electric field, inverted-V electron spectraareproducedat low altitude [15].

3) Upgoing ion beamsgain energy at the bottom of thepotentialstructureandloseit againin the upperpart ofit [21].

4) O-shapedpotential structureis a density depletion: itrepelselectrons,and ions speedup inside it [23]. Thusthe structureremainsstable[19].

5) 1-10 Hz plasmawaves with significant parallel com-ponentexist at the samealtitude where the topsideofthe potentialstructureprobablyresides[31]. The wavesmight be anotherbyproductof the potential structure,e.g.ion acousticwavesgeneratedby rapidmovementsofthe potentialstructure.The exact natureof the parallel-dominatedwaveshasnot beenconfirmedyet, however.

The energy flows throughthe systemin the following way:

1) Ion shelldistribution givesenergy to plasmawaves[24].2) Plasmawaves energise electronsin parallel direction,

andasa byproductthey alsocontributeto perpendicularion heating[24].

3) When energising the electrons,the waves must workagainstthe downward electric field in the energisationregion (Figure12). This meansthat theenergy extractedfrom the wavesis soontranslatedfrom kinetic to poten-tial energy. In caseof Landauresonance,this may helpkeepthe electronsin the resonantvelocity rangefor alonger than would be the casewithout the downwardelectric field. Thus the presenceof the downward fieldmay increasethe total power that is extractedfrom thewaves [15].

4) When an electron reachesthe centre of the potentialstructure, it experiencesa “free fall” accelerationto-wards the ionosphereexactly as in the traditional U-potential model. This releasesits potential energy andturns it into kinetic energy. The observationalmanifes-tation is the appearanceof inverted-V electron spec-tra at low altitude. The auroral electron beamshavemany secondaryphenomenaassociatedwith thembelowthe accelerationregion (optical auroralarcs,secondarywave-particleinteractions,etc.).

We call the model as a whole the “Cooperative Model”(CM), becausein it, the plasma waves and the potentialstructurebothcontributeto electronaccelerationin anessentialway. Onecharacterisationof the CM is that the plasmawavesare the engineand the potential structureis the “gears andwheels”. The energy is provided by the plasmawaves, butmany of the importantobservationalcharacteristicswould notexist without the potentialstructure.Indeed,it is meaninglessto ask how electron accelerationwould look like withouta potential structure,since the emergenceof the potentialstructureis an inescapableconsequenceof the action of thewaves.

B. Is Cooperative Model unique?

Although we have pointedout above that many aspectsoftheCM arein accordancewith observationsandthatthemodelis also compatiblewith test particle and plasmasimulations,one can ask if the observationscould be explainedby othermodels:

1) The observation that strongperpendicularelectricfieldsare confined below �¸4a� � radial distance(Figure 3[20]) could perhapsalsobe explainedby a wideningofa U-shapedpotentialstructureas in the potentialfingermodelof [28].

2) The fact that auroral density depletionsalso have apreferredaltitude range(Figure 4) could be explainedif there is another layer of upward electric fields athigh altitude [9]. The auroral cavity would then besandwichedbetweentwo layersof upward fields.

3) Theelectronanisotropies[22] couldcorrespondto iono-spheric return current regions (upward beams)insteadof being producedin the magnetosphere[25]. The factthattheanisotropiesareoftenup-down symmetricmightbe due to insufficient instrumenttime resolutionor by


symmetricreturncurrentregionsworking in theoppositehemisphere.

4) The fact that ion beamshave a dip at �¹4a� � radialdistance(Figure3) is hardto explain in the U-potentialmodel, i.e. without downward electricfields.

Thusit seemsin principlepossibleto explain thebehaviourof all mentionedparametersexcept the ion beambehaviourwithout necessarilyinvoking a closedpotentialstructure.Howcompatiblethesealternative explanationswould be with low-altitude observations is another matter. For example, if aU-shapedpotential structurewidens, the correspondinglow-altitude inverted-V regions shouldalso be wide, not narrow,since the electronsundergo potential drop accelerationre-gardlessof the altitude of the potential drop. The multitudeof perpendicularscalesizespresentin the auroral arcs mayleave room for both types of models [30]. If it is so, thenthe O-shapedmodel probablycorrespondsto narrow auroralarcs(few kilometrewidth in the ionosphere).We remarkthata closure of the potential before the opposite hemisphere(which in our terminology is an O-shapedpotential) wasindependentlysuggestedby Hallinan and Stenbaek-Nielsen[12] from studying the degree of conjugacy of auroral arcsin the two hemispheres.

C. How anisotropy producespotentialstructure

We now comebackto item3 in subsectionVI.A (TheCoop-erativeModel)above,i.e. thequestionhow a MNCd�eMfF electronanisotropy producesa downwardelectricfield. Considera bi-Maxwellian equatorialplanedistribution function º�»a¼�g-C�½¾g F�¿ ,º » ¼�g-CW½¾g�F ¿ $ �f»MOFdÀ MNC ÁRÂ&�Ã�Ä <ÆÅ§ÇfÈYÉ(ÊÌËÎÍ Â gaÇ C&�MNC Í Â gaÇF& MOF9Ï (1)

where Â is the electron mass, � » the electrondensity andM�C , MOF the parallelandperpendiculartemperaturesin energyunits,respectively. At otherpointson thefield line, Liouville’ stheorem states that º�¼{g-CÐ½og�F ¿ $sº » ¼�g » C ½og »F ¿ where g » C ,g »Fare the equatorialplane valuesof the particle velocity. Theconnectionbetweeng-C ,g�F and g » C ,g »F canbe workedout fromthe conservation of total energy ��$�¼¾��& ¿ Â ¼�gÑÇC�Ò gÑÇF ¿ ÒÔÓ Õand the magneticmoment Ö+$%¼¾��& ¿ Â gÑÇF � × . The electrondensityat point Ø ( Øq$� correspondsto equatorialplane) isgiven by ��¼ÙØ ¿ $.&�ÃlÚeÛ; Û�Ü g-CmÚÝÛ» Ü g F g F º�¼{g-C�½og F�¿ (2)

which givesafter doing the integrations��¼ÙØ ¿ $ �f» ÈÐÉ(ÊqÞoÍ Ó Õ ¼ÙØ ¿ ��MNC§ßà áàÑâ Þ � Í+ãfäã ß Ò ãfäã (3)

where ×G» is theequatorialplanemagneticfield. Oftena usefulapproximationis ×�»�å, in which casewe obtain��¼ÙØ ¿ $ M?æM F � » ÈYÉ(Ê Þ Í Ó Õ ¼ÙØ ¿ ��MNC ß � (4)

All particlepopulationsobey a similar relation.Thequasineu-trality gives the equation �mç � çè $5�mç � ç� which binds together

differentpopulations.If thereis only oneion andoneelectronpopulation, this equationcan be solved analytically, whichyields é Õ ¼ÙØ ¿ $ M Cè M C�M Cè Ò M C�ëêíì�î Ë M Fè M C�M Cè M F� Ï (5)

in the limit × »ðï . Alfv en and Falthammar[2] considereda similar problem,but asthey useddeltafunctionpopulationsinsteadof Maxwellians,their formulasare not directly com-parableto ours. If ions are isotropic ( M C� $JM F�'ñ Mf� ) andMf�#òóM Cè ½¾M Fè , we obtainé Õ ¼ôØ ¿ $ Í M Cè ê�ì�î Ë M CèM Fè Ï (6)

which givesthemagnitudeof thenegative potentialthatarisesfrom a cigar-shaped( M Cè �@M Fè ) electronanisotropy far awayfrom theequatorialplane.In caseof multiple ion andelectronpopulationswhile still using × »�ï theequationfrom whichÕ ¼ÙØ ¿ mustbe numericallysolved readsõ è+ö è � è ÈYÉ(Ê ¼ é Õ ��M Cè ¿ $ õ � ö � � � ÈYÉ(Ê ¼ Í é Õ ��M C� ¿ (7)

where ö $vM�C÷��MOF is the temperatureanisotropy of eachpopulationand �O� is thepartialdensityof thepopulationin theequatorialplane.When Eq. (7) is solved with an anisotropicpopulationpresent(oneof the ö ’s is largerthanunity), Õ mustbeadjustedsothattheequationis satisfied.Thesolution Õ willbe of the orderof M C of the coldestpopulationwhosepartialdensity � exceeds ¼ ö Í � ¿ � of the anisotropiccomponent.Thus, if cold electronsare present,the potential is small aslong astheanisotropy is below somethresholdwhich dependson the partial densitiesand becomeslarge if the thresholdisexceeded(seealsoEqs.9 and10 in reference[19]).

All the formulas of this subsectionignore the nonlinearfeedbackbetweenthe potential structureand the electrons,sincethey assumethat the electrondistribution is not affectedby the potential.Thereforeonly potentialswhich are of thesameorder as the parallel temperatureof the electronscanbe produced.According to low-altitudesatelliteobservations,however, the potential is often �v/ times larger than thethermalenergy [29], [30]. We shall now dealwith a possibletheoreticalexplanation.

D. Simple1-D PIC simulation

Thus far we have simulatedthe global formation of theO-shapedpotential structurewhen a primary charge cloud(more accurately:a tendency to form such a cloud) is putin by hand [19], and on the other hand the wave-particleinteraction betweenion Bernsteinwaves and electronshasbeendirectly simulated[24]. What is still missingis a combi-nation of the two, i.e. a particle-in-cell simulation with solarge simulation box that the whole potential structurefitsin it. While waiting for the computing power necessarytoperform such a calculationto becomeavailable, we presenthereaone-dimensionalelectrostaticparticle-in-cellsimulation.The basic characteristicsof the simulation are as follows.The ion dynamicsis ignored,thus the ions form a stationary


TABLE I

POTENTIAL DROP DEPENDENCE ON STRENGTH OF WAVE-PARTICLE

INTERACTION ø . STRENGTH OF AVERAGE AND MAXIMUM DOWNWARD

ELECTRIC FIELD IS ALSO GIVEN.ø (sù=ú ) Potentialdrop (V) Average! (mV/m) Maximum !0.5 620 0.03 0.081.0 1380 0.07 0.221.5 2100 0.1 0.212.0 2700 0.135 0.32

backgroundchargedensity. Theequationsto besolvedaretheadiabaticelectronequationof motion containing the mirrorforceandtheelectricforce terms,andPoisson’s equation.Im-plicit time integrationis usedto reduceunwantedoscillationsat theplasmafrequency [4]. Thedensityis artificially reduced(electronDebyelength enlarged) to make the simulationrunquickly. 200 grid cells and 300 electronsper cell are usedwith quadraticsplineweighting [4]. The magneticfield hasadipolar ��û�< dependence.The Poissonequationis written asÍGü Ç�ý $ Í �û <ÿþþ û�� û < þ ýþ û�� $ é ¼�� Í � è ¿� » ½ (8)

which we justify by analogywith the sphericalcase(mag-netic monopolecase)where the Laplacianwould be ü Çÿ$¼ ��û Ç ¿ ¼ þ � þ û ¿ ¼�û Ç ¼ þ � þ û ¿o¿ . Notice that argument cannot bemade rigorous as the dipole geometry is symmetric withrespectto only onecoordinate(thelongitudeangle).However,we have run the programbothusingthe monopoleanddipolefields and the resultsare qualitatively very similar. Sincethedipolecaseis easierto comparewith observations,we useit inthe following. The radial distancerangeof the simulationboxis 2-5.1 �� ( &�� m). Theinitial electrontemperatureis 300eV in both parallelandperpendiculardirections.Particlesaremirrored at the bottom boundary, i.e. ionosphericlossesareignored.Whenanelectronreachesthetopboundaryits parallelvelocity is re-randomisedanddrawn from 300eV Maxwelliandistribution. This correspondsto neglecting the effect of theoppositehemisphere,which is justified if a large volume ofthe flux tubeexists above the top boundary.

Thewave-particleinteractionsareaddedto thesimulationasfollows.At eachtimestep,if theparallelenergy of anelectronis in the range100-1000eV, its parallelvelocity is multipliedby ¼¾� Ò i�� ¿ , where i�� is the timestep(30 ms). Here is a parametercontrolling the strengthof the wave-particleinteractions.

Table 1 shows the potential drop dependenceon . Ineach casethe simulation is run for 30 s and the potentialdrop is determinedfrom the time-averagedelectricfield. Thepotential dependsroughly linearly on û so that the time-averageddownward electric field � is roughly constant.Theaverageandmaximum � is alsogivenin Table1. For large ,fluctuationsduring the run becomestronger. The fluctuationsare slowly reduced if the number of particles per cell isincreased.We verified that the resultspresentedin the tabledo not changemuchif 3000particlespercell areusedinsteadof 300.

Although this simulationlacks many importantfeatures,itillustrateshow a downwardelectricfield canform becauseof

wave-particleinteractionsand that dependingon the strengthof thewave-particleinteractions,thestrengthof theassociatedpotentialdrop canmuchexceed300 eV, which is the originalelectronthermalenergy. This nonlineareffect occursbecausethe downward electric field enableselectronsto stay in theresonantvelocity range for a longer time than what wouldbe otherwisepossible,which increasesthe total power drawnfrom thewaves.If onevariestheparameters(e.g.the resonantenergy range),the resultingnumbersdiffer, but the behaviourremainssimilar.

VI I . CONCLUSIONS

Using statisticalanalysisof electricfield, ion, electronandwave datafrom Polarsatellitetogetherwith differenttypesofsimulations,we have shown that the dataandsimulationsareconsistentwith a picture of auroralaccelerationthat we callthe Cooperative Model. The modeldescribesthe shapeof theelectricfield structureandhow energy is transferredin severalstepsfrom the hot ion distribution finally to auroralelectrons.Themodeldoesnot yet explain thehorizontalscaleof auroralarcsor their multiplicity. Explainingtheseobservationalchar-acteristicsmust wait for a moredetailedstudy of the hot iondistribution free energy sourceandhow that is formeddeeperin the magnetosphere.Also the return current region is notyet a self-consistentpart of the model:what is missingis themechanismfor generatingperpendicularelectric currentsinthe magnetosphere.Finally, the role of Alfv en waves is stillunclear. It has beenwidely known for decadesthat Alfv enwaves exist and probablyplay somerole in auroralelectronenergisation, but their relationshipto the potential structureaccelerationhas not yet been studied much. Our guessisthat Alfv enic power is responsiblefor the energy neededindeforming auroral arcs and maybe also for the associatedionosphericJouleheating,but that the electronsforming thearcsare mostly energisedby the processesdescribedin thispaper.

While we have found statistical studies very useful, tofinally judge what is the instantaneousshapeof the potentialstructureone would needmagneticconjunctionsof at leastthreesatellitesin different altitudesbelow, within and abovethe potential structure.The occurrencefrequency of three-satelliteconjunctionsis currentlyalmostzero.To obtainsuchconjunctions,a specificallydesignedmulti-satellitemissionisrequired. Having three or more satellites in circular orbitswhose periods are related to each other as small integerswould be ideal. Furthermore,choosingthe orbit periods tobe integer fractionsof the siderealdaywould allow a ground-basedobservation of eachconjunction,which would increasethe benefitsof sucha missioneven more.

ACKNOWLEDGEMENT

Theauthorswould like to thankW.K. Peterson,F.S.Mozer,C.A. Kletzing and C.T. Russell for providing the PolarTIMAS, EFI, HYDRA andMFE data,respectively. They arealsogratefulto Harri Laakso,Andris VaivadsandMatsAndrefor usefuldiscussions.We thankHannuKoskinenfor readingthe manuscriptandgiving a numberof comments.The work


of AO was supportedby the SwedishResearchCouncil andthat of PJby the Academyof Finland.

REFERENCES

[1] Alfv en,H., On thetheoryof magneticstormsandaurora,Tellus,10, 104,1958.

[2] Alfv en,H. andC.-G.Falthammar, Cosmicalelectrodynamics,Clarendon,Oxford, 1963.

[3] Bingham,R., D.A. Bryant and D.S. Hall, Auroral electronaccelerationby lower-hybrid waves,Ann. Geophys.,6, 159-168,1988.

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