Charge Transport Analysis for Lightning

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CHARGETRANSPORTANALYSISFORLIGHTNINGByWEIFENGADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2013

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c2013WeiFeng 2

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Tomyparents,Iloveyouallthetime 3

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ACKNOWLEDGMENTS IwouldliketothankWilliamW.Hager.Hehasbeenamentorandfriend.Hisguidancemadethisworkdone.IwouldliketothankmydissertationcommitteeofYumeiChen,SergeiS.Pilyugin,VladimirRakovandLeiZhangfortheirsupportovertheyears.Iwouldliketothankthosewhocollecteddataandsharedthemwithus.IwouldliketothankalloftheprofessorswhogavemehelpduringmyPh.D.research. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 LISTOFSYMBOLS .................................... 9 ABSTRACT ......................................... 10 CHAPTER 1INTRODUCTION ................................... 11 2CHARGEREARRANGEMENTFORLIGHTNING ................ 18 2.1Instruments ................................... 18 2.2DescriptionofData ............................... 18 2.3ChannelReconstruction ............................ 21 2.4ElectricFieldAnalysisforPointCharge ................... 25 2.5MathematicalSolverforV0 .......................... 27 2.6ChargeTransportAnalysis .......................... 29 2.6.1DipoleChargeTransportModel .................... 29 2.6.2UniformChargeTransportModel ................... 30 2.6.3SmoothChargeTransportModel ................... 31 3CHARGEREARRANGEMENTBYSPRITESOVERAMESOSCALECONVECTIVESYSTEM ....................................... 42 3.1Overview .................................... 42 3.2DescriptionofDataSet ............................ 44 3.3ElectromagneticFieldDataandLightIntensity ............... 46 3.4MathematicalModel .............................. 48 3.4.1SinglePerfectlyConductingPlaneCase ............... 48 3.4.2DoublePerfectlyConductingPlanesCase .............. 49 3.4.3SphericalCaseApproximation .................... 53 3.5ChargeTransportAnalysis .......................... 56 3.5.1Assumptions .............................. 57 3.5.2AnalysisfortheHumpofSprite2 ................... 58 4CONCLUSIONS ................................... 63 APPENDIX ACONVENTIONS ................................... 65 5

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BDESCRIPTIONOFTHECALCULATIONOFV0 .................. 66 REFERENCES ....................................... 67 BIOGRAPHICALSKETCH ................................ 72 6

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LISTOFTABLES Table page 2-1Dipolechargetransportmodel ........................... 29 2-2Uniformchargetransportmodel .......................... 31 2-3Smoothchargetransportmodel(l=0) ...................... 35 2-4Smoothchargetransportmodel(l=)]TJ /F5 11.955 Tf 9.3 0 Td[(0.06fori2neg)att2 ........... 37 3-1Asummaryofthevideorecords(left)andcloselycorrelatedNLDNstrokerecords(right)forthesprites ..................................... 45 3-2Chargetransportforthesprite ........................... 62 7

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LISTOFFIGURES Figure page 2-1PlanviewsofLMAsourcesfortheICash .................... 19 2-2EastviewandhistogramofLMAsourcesfortheICash ............. 20 2-3ElectricelddatafortheICash .......................... 22 2-4DifferentviewsofreconstructedchannelsfortheICash ............ 24 2-5ComparisonbetweenmodeledandmeasuredelectriceldsforICash .... 38 2-6LinearchargedensityofICashatthreedifferenttimes ............. 39 2-7ChargemovementinnegativeregionforICashat23:05:42.023243UT .... 40 2-8ChargemovementinnegativeregionforICashat23:05:42.1UT ....... 40 2-9ChargemovementinnegativeregionforICashat23:05:42.3UT ....... 41 3-1NationalWeatherServiceStationKFDXNEXRADlevelIIIcompositeradarreectivityforSprites ................................. 43 3-2Aphotoofsprite2 .................................. 46 3-3Measureddataforsprite2 .............................. 47 3-4AverticalantennaofheightHaboveaperfectlyconductingplane ....... 49 3-5Averticalantennasuitedbetweentwoperfectlyconductingplanes ....... 50 3-6Imagedipolesequences ............................... 51 3-7Sphericalcaseforspriteresearch ......................... 54 3-8Plotoftheelectriceldasafunctionofdistance ................. 55 3-9Plotoftheelectriceldasafunctionofdistancefordifferentaltitudes ..... 57 3-10Theelectriceldwhichcorrelateswiththelightforsprite2 ............ 59 3-11Thespritecurrentatthetopofthespritechannelasafunctionoftime ..... 60 3-12Comparisonbetweenthemeasuredelectriceldandthemodeledelectriceldforsprite2 .................................... 61 3-13Threecomponentsoftheelectriceldforsprite2 ................. 62 8

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LISTOFSYMBOLSANDABBREVIATIONS Thelistshownbelowgivesabriefdescriptionofsymbolsandabbreviationsusedinthisdissertation: r2 Laplaciankk EuclideannormE ElectricFieldGPS GlobalPositioningSystemreceiversLEFA LangmuirElectricFieldArrayLMA NewMexicoInstituteofMiningandTechnologyLightningMappingArrayMCS MesoscaleConvectiveSystemNLDN NationalLightningDetectionNetworkUT UniversalCoordinatedTimeULF UltraLowFrequencyVLF VeryLowFrequency 9

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyCHARGETRANSPORTANALYSISFORLIGHTNINGByWeiFengDecember2013Chair:WilliamW.HagerMajor:Mathematics ChargerearrangementisanalyzedforanintracloudashonAugust24th2007.Theanalysisemployeddatafromaballoonborneelectriceldsensor,orEsonde,datafromanexactlyEsondeonamountain,anddatafromtheNewMexicoInstituteofMiningandTechnologyLightningMappingArray(LMA).Anewsmoothchargetransportmodelisusedtoanalyzethechargemovement,demonstratinghighconsistencywithmeasureddata.Basedontheresult,26%ofthenegativechargewastransportedtotheendofthechannel;36%wasdepositedalongthechannelinthepositiveregion;8%wasdepositednearthestartofthechannelinthepositiveregion;and30%wasdepositedinanotherpositiveregionseveralkilometersbeneaththemainchannel.Chargerearrangementbyspritesisalsoanalyzedforamesoscaleconvectivesystem(MCS)onJuly15th2010.TheelectricelddatawererecordedbyLangmuirElectricFieldArray(LEFA)andthemagneticelddatawererecordedbythecharge-momentnetworknearDukeUniversity.AhighspeedvideosysteminLangmuirLaboratoryrecordedtelescopicimagesofthesprites.Foroneoutofthetenspritesthatwererecorded,therewasapositivehumpinelectriceldafewmillisecondsafterthe+CGreturnstroke.Theelectriceldhumpistbyaspritecurrentthatpropagatesfromtheionospheretoabout50kminaltitude.Thetotalchargetransport,whichwastheintegralofthecurrenthump,was23.9Cwhenthevelocityofthecurrentpulsewasbetween0.25cand0.55c. 10

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CHAPTER1INTRODUCTION Lightningisahigh-currentdischargeeventthatoccurswithincloudsaswellasbetweencloudsandtheground.Therearethreebasictypesoflightningashbasedonwheretheashoccurs.Intracloud(IC)ash,themostcommontypeofash,occurswithinacloud;cloudtocloud(CC)ashoccursbetweendifferentclouds;andcloudtoground(CG)ash,themostdangerousash,occursbetweencloudsandtheearth'ssurface. Aboltoflightningisconsideredoneofthemostdangerousnaturalphenomena.Everyyear,aboutonehundredorsopeoplearekilledbylightning,letaloneseveralhundredmoreinjuries,intheUnitedStatesalone.Lightningisalsoresponsibleforthedeathsandinjuriestolivestockandotheranimals,thousandsofbrushandforestres,aswellasdamagetobuildings,communicationssystems,airplanes,powerlines,andelectricalsystemsworthofmillionsofdollars.Becauseofthedamageoflightning,itisnecessaryforustostudyitthatwecanforecastitandprotectourselves.Wemayevenbeabletoutilizeitconsideringthehugeamountofenergystoredinthisuniquenaturalphenomena.Sofar,researchonlightninghasbeendevelopedtheoreticallyandexperimentally(fromground-basedelectriceld-changemetersaswellasfromtriggeredinstruments). Sincelightningispartofthestorm,mosttheoreticalworkoflightningisbasedonthemodelsofstorms.Therearetwotypesofstormmodels:convectivedynamicstormmodelsandmicrophysicalcloudelectricationmodels.Convectivedynamicstormmodels(forexample,KlempandWilhelmson[1978])startswiththedynamicboundaryconditionsandsolvesthemomentumandenergyequations.Theimportantcharacteristicofthismodelisassociatingtheinputswiththephasechangesofwater.Microphysicalcloudelectricationmodels(e.g.ZivandLevin[1974]andChiu[1978])includesdetailedmicrophysicalcalculationsofthechargeseparatedduringdynamic 11

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sizechangesoncollision,meltingorvaporizationinagivenelectriceld.Basedonthesetwomodels,Helsdon[1980]presentedatwo-dimensional,slab-symmetric,time-dependentcloudmodeltosimulatedeepconvectionintheatmosphere.Nisbet[1983]developedatwodimensionalcylindricallysymmetriccomputermodelandfoundthattheelectricationprocessesaremuchmoredirectlyrelatedtothetotalcurrentdensityatthesurfaceunderathundercloudthaneithertheconductioncurrentdensityortheelectriceldatthesurface. Sincetwodimensionalmodelisimpracticalforthreedimensional(realworld)problems,Hageretal.[1989]developedathreedimensionalmodel,combiningconvectiveandmicrophysicalthunderstormmodels,fortheevolutionoftheelectriceldinathunderstorm.Theoutputofthemodelistheelectriceldasafunctionoftimeandtheinputsarecurrentsgeneratedbythemovementofchargedparticleswithinclouds.StartingwithMaxwell'sequation: Gauss'slawfortheElectricField:rE= 0Gauss'slawfortheMagneticField:rB=0Faraday'slawofinduction:rE=)]TJ /F8 11.955 Tf 10.49 8.08 Td[(@B @tAmpere'sLawwithConservation:rB=0J+1 c2@E @t, thecurlofthemagneticeldstrengthHisgivenby rH=@E @t+E+J (1) whereisthepermittivity,istheconductivity,Eistheelectriceld,andJisthecurrentdensityassociatedwiththemovementofchargedparticles.Fromequation( 1 ),they 12

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derivedanequationforcomputingelectriceld: @ @t+rZSrG(r,s) (s)+rJ=0 (1) whereG(,)isGreen'sfunction,and istheLaplacianof-theelectricpotential,i.e.E=r.Byintegratingequation( 1 ),wecanobtain andthencomputetheelectriceldE.Bydiscretizingequation( 1 )forauniformmesh,theyobtainedamatrix-vectorofthediscretizedformula: A[n+1)]TJ /F5 11.955 Tf 11.96 0 Td[(n]+tB[n+1+(1)]TJ /F8 11.955 Tf 11.96 0 Td[()n]=tIn (1) whereAandBaresymmetricpositivedenitematrices,tistimestep,andisthevectorwithcomponents.TheerrorofthemodelisO(tm)+O(h2),wheretandharethetemporalandspatialdiscretizationparametersandm=2fortheCrank-Nicholsonschemewhilem=1otherwise.Whenapplyingthismodeltolightning,theyutilizedtheInverseMatrixModicationFormula(Hager[1989])toget: after=before)]TJ /F7 11.955 Tf 11.95 0 Td[(A)]TJ /F4 7.97 Tf 6.59 0 Td[(1W(WTA)]TJ /F4 7.97 Tf 6.59 0 Td[(1W))]TJ /F4 7.97 Tf 6.59 0 Td[(1WTbefore (1) wherebeforeandafterareelectricpotentialbeforeandafterdischarge. Astheprogressoftheoreticalworkforlightning,alotofground-basedmeasurementshavebeendevelopedforlightningresearch.Thedataweusedinourresearchareprovidedbyseverallightingdetectionsystemswhichisintroducedinbrief. TheNationalLightningDetectionNetwork(NLDN)(Cumminsetal.[1998]),themostaccurateandreliablelightningdetectionsystem,hasbeenprovidingdataforlightningresearchsince1987.Over100ground-basedsensingstationsareconstructedinthenetworkwhichmonitorstheCGlightningactivitiesacrosstheUnitedStates.. NewMexicoInstituteofMiningandTechnologyLightningMappingArray(LMA)wasintroducedbyRisonelal.[1999].LMAisathree-dimensionalvery-high-frequency(VHF)time-of-arrival(TOA)totallightninglocationsystem(Thomasetal.[2001,2004]and 13

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Colemanetal.[2003]).Inatypicaldeployment,tentofteenVHFreceiverstationsarearrangedoveranareaabout90kmindiameter.GPStimingisusedtoaccuratelymeasurethearrivaltimeofimpulsiveradiationeventswithabout40nsaccuracy.Eachstation,receivingandprocessingthelightningsignalsindependently,isabletodetermineupto12,500strongestVHFradiationsourcespersecond.TheLMAmeasurestheTOAofalightningdischargeandlocatesthesourceoftheradiationtogenerateathree-dimensionalmapoflightning. LangmuirLaboratoryforAtmosphericResearch,withfundsfromNationalScienceFoundation,wasbuiltintheMagdalenaMountainsofcentralNewMexicoin1963.Itisabasefortheresearchofcloudprocessesthatproducelightning,hail,andrain.LangmuirElectricFieldArray(LEFA)(Hageretal.[2012])islocatednortheastofLangmuirLaboratoryinaregionofsize25kmby15km,includingnineslowantennastations.Eachstationmeasurestheelectriceldonthreechannelsinparallel:sensitivechannel(withsensitivityfrom30mV/mto1kV/m),mediumchannel(withsensitivityfrom1V/mto30kV/m)andinsensitivechannel(withsensitivityfrom25V/mto750kV/m).LEFAtimesareGPSbased,withfrequency50kHzandaccuracyof20s. Formuchresearch,lightningmustbeobservedascloseaspossibletoensuretheaccuracyofmeasurement.Rocket-triggeredlightning([RakovandUman2003]),animportanttoolforclose-upinvestigation,providesamorecontrolledandpredictablemethodforlightningresearch.Recently,theballoon-borneEsondewasintroducedbySonnenfeldetal.[2006]forcloseobservationoflightning.ItusesGPS-timingandcontainsanarrayoffourelectrodes(slowantennas),whichiscapableofdetectingthundercloudelectriceldvectorwithafrequencybetween1Hzand5000Hz. InChapter2,wewillfocusonthechargetransportanalysisforanICashincentralNewMexicoonAugust24th,2007.WewillstudythechargerearrangementfortheinitialchargetransportofanIC. 14

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Formanyyears,severalchargemodelshavebeenproposedforthunderstorms.Wilson[1916,1920,1925]proposeddipolemodelsforthechargestructure.SimpsonandScrase[1937]andSimpsonandRobinson[1941]latercameupwithtriplemodels.Luetal.[2011]presentedatime-dependentmultidipolemodelforcomputingthechargedistribution. Hageretal.[2007,2010]achievedgoodtbyestimatingCGandICwithdynamicmonopolemodelanddynamicdipolemodelrespectively.Hageretal.[2007]developednewtechniquestoaccuratelylterthebackgroundelectriceldchangesandpresentedpulsegraphtoobtainagraphicalapproximationtothelightningchannel.OneCGashandtwoICasheswereanalyzedbyusingdatafromLMAandaballoonEsonde.Hageretal.[2010]furtherenhancedthetechniquesdevelopedinHageretal.[2007]anddevelopednewtechniquetocomputetheelectriceldwhentheinstrumentrotateseithersteadilyorrapidly.ThestormcharginganddischargingprocesswerestudiedbyusingdatafromLMA,aballoonEsondeandnextGenerationWeatherRadar.ItisshownthatICashestransportedalmost6timesasmuchchargeasCGashesdid. InChapter2,wewillpresentanewsmoothdistributedchargemodelforanalyzingthechargetransportinanIC.ThesubstantialliteratureinsupportofthedistributedchargemodelincludingworkbyFew[1970],Krehbiel[1981],LiuandKrehbiel[1985],Proctor[1981,1997],Weberetal.[1982]andLuetal.[2011]. Datafromground-basedmeasurementshavebeenusedfortheoreticalanalysisofthundercloudchargedistributionsincetheearly20thcentury(JacobsonandKrider[1976],Koshak[1999],KoshakandKrider[1989,1994],Krehbieletal.[1979],Krehbiel[1981,1986],Murphyetal.[1996],Wilson[1916,1930],WorkmanandHolzer[1939]).Wilson[1916,1930]rststudiedtheelectricdischargesinlightningashesbyusingtheelectriceldmeasurements.KoshakandKrider[1994]andKoshak[1999]incorporatedconstraints,suchasconservationofthecharge,intheestimationprocess.More 15

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recently,Sonnenfeldetal.[2006]andHageretal.[2007,2010]usedballoon-basedmeasurementsofthechangeintheelectriceldinchargeretrievalanalysis. OuranalysisinChapter2isbasedondatareportedbyWinnetal.[2011].TheelectricelddataweremeasuredbytwoEsondes:aballoonEsonde(ightEsonde)andagroundEsonde.ThethreecomponentsoftheelectriceldchangemeasuredbyightEsondeandtheverticalelectriceldmeasuredbygroundEsondearethebasisforourchargetransportanalysis.DuetothelocationofthegroundEsonde,themountaingeometryistakenintoaccountinouranalysis.Winnetal.[2011]foundthattheleaderchannelclosetotheightEsondehasaline-chargedensityof-0.36mC/m. Hageretal.[2007,2010]incorporatedthemodelwithconstraints,includingthechargelocations,conservationofcharge,separatedchargecentersandlocationconstrainedbyLMAdata,toeliminatedegreesoffreedom.OurnewsmoothchargetransportmodelconstrainsthetransporttooccurwithindomainofLMAsourcesinasmoothway.Thissmoothnessinthechargetransportparallelsthesmoothnessinchargedensityseeninballoonsoundingsofthunderclouds(MarshallandRust[1991,1993]),StolzenburgandMarshall[1994],Stolzenburgetlal.[1994,1998a,1998b,1998c,2001,2002]).Thesmoothnessconstraintsallowsustotthevastlyunder-determinedsystemofequationsrelatingchargetoelectriceldinaphysicallyplausibleway. InChapter3,wewillstudychargerearrangementinassociationwithsprites.Spritesarelarge-scaleelectricaldischargingeventthatoftentriggeredbylarge,positivecloud-to-groundlightning(Boccippioetal.[1995]).Theyextendhighabovethundercloudsfromabout50kmto90km.ResearchonspritesstartedbyFranzetal.[1990]in1989.Theoriesfortheinitiationofspritesincludetheconventionalbreakdowntheory(Paskoetal.[1997])andarunawaybreakdownmodel(e.g.Belletal.,[1995]).Ithasbeenpredictedthatthe+CGprecedingspriteiscapableoftransferringlargeamountsofcharge.Estimateshaverangedfrom410Cto1500Cfordaytime 16

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sprites[Stanleyetal.2000]tobetween50Cand500Cfornighttimesprites(CummerandInan[1997]).CummerandInan[2000]analyzedthecurrentandchargetransportinspritesbyextractingalowfrequencysfericwaveformfromdistantmagneticeldmeasurements.Thenadeconvolutionmethodwasusedtoestimatethecurrentandchargemomentassociatedwithsprite.InChapter3,wewillfocusonwherechargeismovedbythespriteitselfusingdatafromtheLEFAforspritesassociatedwithlightningstrokeslocatedwithin500kmofLEFA. InChapter4,wepresentourconclusions. 17

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CHAPTER2CHARGEREARRANGEMENTFORLIGHTNING 2.1Instruments Winnetal.[2011]launchedaballoonborneEsonde(ightEsonde)fromtheballoonhangernearLangmuirLaboratoryatabout22:53:51UTAugust24th2007.TheballoontraveledprimarilytoeastandwasretrievedatthewestofInter-state25andtheRioGrandeRiver.Atabout23:05:42UT,theightEsonde,atanaltitudeofabout9100m,passedwithin181mofanLMAsource(oftheICash)detectedbyLMA.ThreecomponentsoftheelectriceldchangenearthelightningchannelwererecordedbytheightEsonde.ThegroundEsonde,whichwasidenticaltotheightEsondeandlocatedonamountainatanaltitudeof3226mnearLangmuirLaboratory,measuredtheverticalelectriceldchangeontheground. 2.2DescriptionofData 719LMAsourceswererecordedfortheICash.Figure 2)]TJ /F5 11.955 Tf 11.96 0 Td[(1 and 2)]TJ /F5 11.955 Tf 11.95 0 Td[(2 (Figure2andFigure3inHageretal.[2013])showdifferentviewsofLMAdatafortheICash.WecalledthisICashFlash83141b.Here,83141meanstheICashoccurred83141secondspastmidnight;bmeansthisICashisoneofthetwoashesoccurredatthesametime.TherstLMAdatawasrecordedat23:05:41.917228UTandthelastat23:05:42.263568UT.TheLMAsourcesrstpropagatedeastward-theseLMAsourcesarereferredaschannel1.ThentheLMAsourcesreturnedtotheinitialpointandpropagatednorthward-theseLMAsourcesarereferredaschannel2(seeFigure 2)]TJ /F5 11.955 Tf 11.95 0 Td[(2 A).Channel1andchannel2constructpositiveregionoftheICashbecausechargewithinthisportionarepositivelycharged.Correspondingly,theLMAsourcesbelowthechannelconstructnegativeregionoftheICash.ItcanbeseenfromFigure 2)]TJ /F5 11.955 Tf 11.95 0 Td[(2 Athatthepositiveregionextendsfrom8to12kmaltitudeandthenegativechargeregionextendsfrom6to8kmaltitude. 18

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A B Figure2-1. PlanviewsofLMAsourcesfortheICash.A)PlanviewoftheLMAsources,depictedasdots,intheLangmuirLaboratorycoordinatesystem.Thecolorofthedotsisbasedonthetimesincethestartoftheash;therstLMAsourcesarecoloreddarkbluewhilethelastaredarkred.Theballoonappearsasalargeblackdotwitharededge.B)ThesameviewoftheLMAsourcesbutwiththedotcolorbasedonthealtitudeofthesource.ThelowaltitudeLMAsourcesinthenegativeregionarebluewhilethehighersourcesinthepositiveregionaregreen,yellow,andred. 19

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A B Figure2-2. EastviewandhistogramofLMAsourcesfortheICash.A)AviewoftheLMAsourcesfromthewestlookingeast.Channel1appearsontheleftside,whilechannel2appearsontheright.ThepartoftheplotlabeledNegativeindicatestheregionwheremostofthenegativechargewaslocated.B)ThehistogramcountsthenumberofLMAsourcesin200mthickhorizontalslicesthroughthecloud. 20

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Figure 2)]TJ /F5 11.955 Tf 11.95 0 Td[(3 (Figure6inHageretal.[2013])showstheelectricelddatameasuredbytheightEsondeandthegroundEsonde.TheelectricelddatashowninthegurewereobtainedbyusingthealgorithmpresentedinHageretal.[2010].Hageretal.[2010]developedtwopolynomialinterpolationmethodtorecoverythemeasuredelectriceld:oneistocomputetheelectriceldchangeattheEsondewhentheinstrumentrotatesslowlyandsteadily;theothercanonlycomputetheverticalelectriceldchangeandthehorizontalelectriceldwhentherotationoftheinstrumentislargeenough(30ormoredegreespersecond).Sincewecanonlymeasuretheelectriceldchangeinthezdirection,Ezisinitializedtozeroatthestartoftheash.Duringthe300msoftheash,theightEsonderotated24.Thiswassufcientrotationtorecoverboththexandycomponentsoftheelectriceld.Hence,theinitialxandycomponentsoftheelectriceldarenonzero,andcorrespondtotheirtruevalues.ThedashedverticallinespresentthetimethattherstLMAsourceoftheICash,thetime(t0)thattheLMAsourcesreachedtheendofchannel1,andthetimethattheLMAsourcesreachedtheendofchannel2.Theinstantoftimet1occurredaftertheLMAsourcesreachedtheendofchannel2.Andtheinstanttimeoft2occurredafterthecompletionoftheICash.Atbothtimet1andt2,theelectriceldwasrelativelystable.NotethatK1,K2andK3inFigure 2)]TJ /F5 11.955 Tf 11.95 0 Td[(3 representthreeK-changesduringthisICash,referringHageretal.[2013]fordetailedstudyofK-changesforthisIC-ash. 2.3ChannelReconstruction SinceoursmoothchargetransportmodelconstraintsthetransporttooccurwithinthedomainofLMAsourcesinasmoothway,weneedtoreconstructasmoothlightningchannelbasedonthelocationoftheLMAsources.Ourgoalistoconstructarelativelysmoothone-dimensionalcurveinthree-dimensionalspacethatinsomesensepassesthroughthemiddleoftheLMAdata.Thealgorithmisthefollowing: 1.ExtractalltheLMAsourcesthatweuseforreconstructingthechannel.ForthisICash,weextractedalltheLMAsourcescorrespondingtochannel1andchannel2, 21

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A B Figure2-3. ElectricelddatafortheICash.A)TheelectriceldattheightEsondestartfrom23:05:42UT.TherstdashedverticallineshowstheinstantoftimewhentherstLMAsourcewasrecorded;Theseconddashedverticallineshowsthetimet0thattheLMAsourcesreachedtheendofchannel1.t1occursaftertheLMAreachedtheendofchannel.t2occursafterthecompletionoftheash.B)TheelectriceldatthegroundEsondeonthegroundnearLangmuirLaboratory. 22

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i.e.thoseLMAsourcesoccurredbetweenthetimeoftherstLMAsourcesandthetimeoftheLMAsourcesreachedtheendofchannel2above8kminaltitude. 2.Initializethealgorithmwiththelocationoftheinitialpointandtheinitialunitvectorpointingalongthechannel.WechosetherstLMAsourceastheinitialpointofthechannel,denotedasp1.WecalculatedtheunitvectorspointingfromtheinitialpointtothersttenLMApoints,andchosetheonethatpointedalongthechannelandpassedthroughthemiddleofthersttenLMAsources,denotedthisunitvectorasd1. 3.Choosetheappropriateparametersforthechannelreconstruction.Inordertoreconstructthechannel,weneedtochoosethelengthofthechannelsegments,denotedasl,andtheregionthatcontainsthoseLMAsourcesthatweusedforcalculatingthechannelsegment.Herewechoselengthofthechannelsegmentl=100m,andaspherewithradiusr=750mthatcontainsLMAsourcesforcalculating. 4.Assumingthekthpointpkandthecorrespondingdirectiondkofthereconstructedchannelareknown,weusedthefollowingalgorithmtochoosethe(k+1)stpointpk+1andthecorrespondingdirectiondk+1.WeextractedalltheLMApointsintheintersectionofthesphereofradiusrcenteredatpkwiththehalf-spacepassingthroughpkwithinwardnormaldk.AssignedthesetofthesepointasH.Foranydirectiondstartedfrompk,wetooktheunitvectordk+1astheminimizerthatminimizesthesumofthedistancesofpointsinHtothedirection.Andpk+1=pk+ldk+1.Astheprocesswent,weremovedthosepointsinpreviousspheresbutoutsideofthecurrentsphere.Eventually,weranoutoftheLMApointsandgotthecompletereconstructedchannel. 153channelpointspk,i.e.152channelsegments,wereobtainedforthechanneloftheICash.Figure 2)]TJ /F5 11.955 Tf 11.95 0 Td[(4 (Hageretal.[2013]Figure7)plotteddifferentviewsofthereconstructedchannel. Ourchoicesforthesphereradiusrandthelengthlofthechannelsegmentswerebasedonthefollowingconsiderations:SincetheLMApointsarespreadoutspatiallyaroundthechannel,theradiusofthespherershouldbeseveraltimesthespatial 23

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A B Figure2-4. DifferentviewsofreconstructedchannelsfortheICash.A),B)Differentviewsofreconstructedchannels.EmptycirclesareLMAsourcesandlledcirclesarereconstructedchannelpoint,i.e.pks. 24

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spread.Becausethespatialspreadisintherangeorderof100to200m,r=750mislargeenoughtogetpastthespreadanddeterminethechanneldirection.Thelengthlofthechannelsegmentswaschosentoapproximatethelengthofthechannelsegmentsassociatedwiththesteppedleader.l=100misareasonablevaluebecauselightningbuildsitschannelinsegmentsthatvaryfrom10to200m(RakovandUman[2003]). 2.4ElectricFieldAnalysisforPointCharge Forourash,theanalysisoftheelectriceldfromapointchargeiscomplicatedbythemountainonwhichthegroundEsondeissituated.AsnotedbyHageretal.[2012],themountaingeometrycanstrengthentheelectriceldbyafactorontheorderof2.Hence,inordertocorrectlyinterprettheelectricelddatafromthetwoEsondes,oneonthemountainandtheotherintheair,itisimportanttotakeintoaccountthegeometryofthemountain. TheelectriceldofchargedparticleswithinashescanbecalculatedbysolvingthePoisson'sequation0r2V=)]TJ /F8 11.955 Tf 9.3 0 Td[(on. whereVistheelectricpotential,0isthevacuumpermittivity,ischargedensityandisthedomainoftheproblem.Theboundaryofalwaysdenoted@.Theelectriceldisthenegativegradientofpotential:E=rV. Poisson'sequationdoesnothaveauniquesolution.InordertoobtaintheuniquenessofV,Vmustsatisfyconditionsontheboundaryof,@.Inourcase,theboundaryconditionsaresimplythehomogeneousboundaryconditions:V=0onthesurfaceoftheearth,attheionosphereand100kmawayfromthechargepoint.Thereasonofourchoiceoftheconditionisthefollowing:[AdlermanandWilliams,1996]presentedthatthepotentialontheverticalsidesofisfaireldif@isfarfromthestorm,andthepotentialabovethestormisionosphericpotentialwhencomputingtheelectricpotentialassociatedwiththestorm.However,weareonlyinterestedinthe 25

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potentialchangeassociatedwithachargepointintheatmosphere,i.e.weneedtosolvePoisson'sequationwithreplacedbyadeltafunctioncorrespondingtoapointcharge.Hence,Theboundaryconditionsarereducedtothehomogeneousboundaryconditions.Notethatourboundaryconditionsareapproximationsbecausetheearthandtheionospherearenotperfectconductors,andthedistantpotentialisnotzero,butdecayslikethereciprocalofdistance.Nonetheless,theerrorsduetoourapproximationstothetrueboundaryconditionsarerelativelysmallcomparedtotheuncertaintyinthedata. InordertoimplementthehomogeneousboundaryconditiononthesurfaceoftheEarth,weusedthe1/3arc-secondelevationdatafromU.S.GeologicalSurvey.Thisdataroughlycorrespondstoelevationsona10mby10mgridoverthesurfaceoftheUnitedStates.ThedistancefromthesurfaceoftheEarthtothecenteroftheEarthwas6378135m(theapproximateradiusoftheEarth)plustheelevationgivenbytheU.S.GeologicalSurvey. Inordertomodelthechargetransport,weplaceddeltafunction(chargepoint)atthemidpointsofthe100msegments,obtainedfromsection2.3,onthelightningchannelsandmodeledtheelectriceldgeneratedbythesechargepointstomatchthemeasuredtheelectricelddata.IfwenumericallysolvePoisson'sequationwithadeltafunction,largeerroroftenoccursduetothesingularityinthepotentialatthelocationofthedeltafunction.Inordertocomputethesolutionwithgreateraccuracy,wedividedVintotwoparts:V=Vf+V0.HereVfisthefundamentalsolutionforLaplace'sequationontheexteriorofasphere Vf(P)=1 401 kP)]TJ /F3 11.955 Tf 11.96 0 Td[(P0k)]TJ /F3 11.955 Tf 47.07 8.09 Td[(R=kP0k kP)]TJ /F5 11.955 Tf 11.95 0 Td[((R2=kP0k2)P0k whereRistheradiusoftheEarth,P0isthelocationofthechargepointandPisthelocationofthepointwherethepotentialismeasured.V0isaharmonicfunctionthat 26

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satisesr2V0=0,andV0=)]TJ /F3 11.955 Tf 9.3 0 Td[(Vfon@.ThustheelectriceldisexpressedasE=)]TJ /F5 11.955 Tf 9.3 0 Td[((rVf+rV0),whereVfwascomputedanalyticallyandV0wascomputednumerically. 2.5MathematicalSolverforV0 Inthissection,weshowthediscretizationprocessforcalculatingtheharmonicfunctionV0insection2.4.Sincetheelectricelddata,measuredbythegroundEsonde,isonthesurfaceoftheearth,weneedanemeshnearthegroundEsondewhilearoughmeshissufcientfarfromthegroundEsondewherethepotentialissmall.ThissuggeststhatweshoulduseasphericalcoordinatesystemwithoriginatthecenteroftheearthandzaxispointsthroughthegroundEsonde. Letusconsiderasphericalcoordinate(,,)correspondstoarectangularcoordinatesystem(x,y,z),whereisthedistancetothecenteroftheearth,istheangleofrotationin(x,y)plane(=0correspondstoxaxis,= 2correspondstoyaxis),andistheangleofrotationawayfromthezaxis(=0correspondstozaxis).Thesphericalcoordinateandtherectangularcoordinateofapointarerelatedasfollows:x=sincos,y=sinsin,z=cos Wesplitthewholedomainintomeshes:i:1iI,j;1jJ,k:1kK. WecanapproximatethesolutionV0atthecentroidofeachvolumeelementinthemesh.Atypicalvolumeelement(Sijk)hastheformf(,,):ii+1,jj+1,kk+1g. 27

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Thecentroid(ri,tj,pk)ofthevolumeelement(Sijk)isgivenbyri=i+i+1 2,tj=j+j+1 2,pk=k+k+1 2. Next,weintegrater2V0=0overthevolumeelementSijkandapplythedivergencetheoremtoobtain:Z@SijkrV0dS=0. Thegradientinsphericalcoordinatesisr=^@ @+^1 sin@ @+^1 @ @ Thenthevolumeelementapproachat(ri,tj,pk)hastheapproximation:)]TJ /F4 7.97 Tf 13.89 5.26 Td[(3Xl=1Cl@+l)]TJ /F4 7.97 Tf 18.37 14.94 Td[(3Xl=1Dl@)]TJ /F10 7.97 Tf -.65 -8.27 Td[(lVijk=0 where@)]TJ /F10 7.97 Tf -.65 -8.28 Td[(l=@+l)]TJ /F4 7.97 Tf 6.59 0 Td[(1,@+1Vijk=Vi+1,jk)]TJ /F3 11.955 Tf 11.96 0 Td[(Vijk ri+1)]TJ /F3 11.955 Tf 11.96 0 Td[(ri,@+2Vijk=Vi,j+1,k)]TJ /F3 11.955 Tf 11.96 0 Td[(Vijk tj+1)]TJ /F3 11.955 Tf 11.95 0 Td[(tj,@+3Vijk=Vij,k+1)]TJ /F3 11.955 Tf 11.96 0 Td[(Vijk pk+1)]TJ /F3 11.955 Tf 11.96 0 Td[(pk, andC1=2(j+1)]TJ /F8 11.955 Tf 11.95 0 Td[(j)2i+1sinpksink+1)]TJ /F8 11.955 Tf 11.96 0 Td[(k 2D1=2(j+1)]TJ /F8 11.955 Tf 11.95 0 Td[(j)2isinpksink+1)]TJ /F8 11.955 Tf 11.95 0 Td[(k 2C2=1 sinpk(i+1)]TJ /F8 11.955 Tf 11.96 0 Td[(i)(k+1)]TJ /F8 11.955 Tf 11.95 0 Td[(k)=D2C3=sink+1(i+1)]TJ /F8 11.955 Tf 11.96 0 Td[(i)(j+1)]TJ /F8 11.955 Tf 11.96 0 Td[(j)D3=sink(i+1)]TJ /F8 11.955 Tf 11.96 0 Td[(i)(j+1)]TJ /F8 11.955 Tf 11.95 0 Td[(j) 28

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2.6ChargeTransportAnalysis Inthissection,wewillstudythechargetransportassociatedwiththeelectricelddatainFigure 2)]TJ /F5 11.955 Tf 11.95 0 Td[(3 .Dipolechargetransportmodelanduniformchargetransportmodelareanalyzedbeforepresentingournewsmoothchargetransportmodel.Wefocusonthechargetransportfromthebeginningoftheashupuntileither23:05:42.1UT(t1)or23:05:42.3UT(t2).Atbothofthesetimes,theelectriceldisrelativelystationary. 2.6.1DipoleChargeTransportModel Hageretal.[2007,2010]showedthatthechargetransportinanICwasoftencloselyapproximatedbyadipole.ForthisIC,negativechargeistransportedfromthenegativeregionanddepositedneartheendsofthechannels(positiveregion).Considerthefollowingassumption:achargeQ1isdepositedatalocation700mfromtheendofchannel1,anotherchargeQ2isdepositedatalocation700mfromtheendofchannel2,atotalchargeQistransportedfromthenegativeregiontopositiveregionandthetotalchargeQisequallyassignedtoeachLMAsourceinnegativeregion.Byconventionofcharge,jQ1+Q2j=Q.ChargeQ1andQ2arechosentogivethebestleastsquaresttothemeasuredelectricelddata.Theresultofmeasured,modeleddata,Q1andQ2isshowninTable 2)]TJ /F5 11.955 Tf 11.96 0 Td[(1 (Hageretal.[2013]Table1). Table2-1. Dipolechargetransportmodel t1=23:05:42.1UTt2=23:05:42.3UT MeasuredModeledMeasuredModeled EGz(kV/m)-6.85.5-7.25.0EFx(kV/m)20.4-7.029.9-8.1EFy(kV/m)35.022.641.125.6EFz(kV/m)-9.4-11.3-17.1-18.7Q1(C)31.228.1Q2(C)-31.1-27.9 29

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InTable 2)]TJ /F5 11.955 Tf 11.95 0 Td[(1 ,EGzistheverticalelectriceldatthegroundEsonde.EFx,EFyandEFzarethreecomponentoftheelectriceldattheightEsonde.Itcanbeenseenthatdipolemodelfailstotthemeasureddatainthefollowingway:thesignbetweenmeasuredandmodeleddataofEGzandEFxisopposite;themodeledQ1ispositivewhileitshouldbenegative.ThereasonwhydipolemodeldidnotworkwasthattheightEsondewasnearthelightningchannel,andthenearbychargeonthechannelhadasignicantimpactonitschargemeasurement.Unlessweputchargealongthechannel,wearenotabletomatchtheelectriceldattheightEsonde. 2.6.2UniformChargeTransportModel Thefailureofdipolemodelinspiresusthatweshoulduseamodelwithchargeplacedalongthechannel.Theuniformlydistributedchargetransportmodelisonesimplemodelofthistype.Assumethatanequalamountofchargeq1isplacedatthecentroidofeachsegmentofchannel1,andanotherequalamountofchargeq2isplacedatthecentroidofeachsegmentofchannel2.Thetotalamountofchargeonchannel1andchannel2areQ1andQ2respectively.Weplaceanequalamountofchargeq)]TJ /F1 11.955 Tf -441.06 -22.12 Td[(ineachLMAsourceinthenegativeregion.Thetotalamountofchargeq)]TJ /F1 11.955 Tf 7.09 1.79 Td[(,transportedfromthenegativeregiontothepositiveregion,Q=jQ1+Q2jbyconventionofcharge.TheresultofbestleastsquarestbetweenthemodeledandmeasuredelectricelddataisshowninTable 2)]TJ /F5 11.955 Tf 11.96 0 Td[(2 (Hageretal.[2013]Table2). ItcanbeenseenfromTable 2)]TJ /F5 11.955 Tf 11.95 0 Td[(2 thattheuniformlydistributedchargemodelisabetterttothemeasureddatacomparedtodipolemodel.However,themodeledQ2ispositivewhileitshouldbenegative.Thettothegroundelectriceldisinerrorbyafactorof2att1andbyafactorof4att2.Thus,theuniformlydistributedchargemodelalsofailstoreproducethemeasuredelectriceld,evenforthebestpossiblechoicesofq1,q2,andq.Thisindicatesthatthechargeisnotuniformlydistributedalongthechannel,andthatgreaterexibilityinthechargedepositionisneededinordertoreproducethemeasuredelectriceld. 30

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Table2-2. Uniformchargetransportmodel t1=23:05:42.1UTt2=23:05:42.3UT MeasuredModeledMeasuredModeled EGz(kV/m)-6.8-3.4-7.2-1.8EFx(kV/m)20.424.129.931.7EFy(kV/m)35.033.141.140.9EFz(kV/m)-9.4-6.8-17.1-14.4Q1(C)-7.7-10.3Q2(C)0.17.0 2.6.3SmoothChargeTransportModel Inthissection,wepresentanewSmoothChargeTransportModelbasedonuniformchargetransportmodel.152channelsegmentsforthechanneloftheICashwereobtainedfromsection2.3.Sincethechargeoneachsegmentisunknown,152degreesoffreedomcouldbeusedtotthemeasuredelectriceldatanyinstantoftime.However,thereareonlyfourdataconstraintscorrespondingtothethreecomponentsoftheelectriceldchangeattheightEsondeandtheverticalelectricaleldchangeonthegroundEsonde.Thedegreesoffreedomneedtobereducedtoavoidoverttingthedata,i.e.wecantthemeasureddataverypreciselyusingchargedistributionsthatlosstheactualphysicalsignicance.Intheprevioussubsections,weremoveddegreesoffreedombyassumingthechargedistributionhadveryspecialforms.Inthedipolet,weassumedthatthechargeonallthechannelsegmentswaszeroexceptforthechargeonasegmentneartheendofeachchannel.Intheuniformt,weassumethatthechargeoneachchannelsegmentwasthesame.Eitherofthesemodelsremovednearlyallthedegreesoffreedom,buttherewasasignicantdiscrepancybetweentheelectriceldsproducedbythemodelandthemeasuredelds.Wenowdevelopacompletelydifferent 31

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waytoremovethedegreesoffreedom,whilepreservingtheexibilitythatisneededtomatchthemeasuredelectricelds. Inoursmoothchargetransportmodel,wetrytotthemeasuredelectriceldusingachargedistributionthatisasmooth,slowlyvaryingfunctionofdistancealongthechannel.Thesmoothnessrequirementremovesmanydegreesoffreedomandleadstoatwithphysicalsignicance.Atanyinstantoftime,assumingthatanamountofchargeqiisplacedatthecentroidofithchannelsegment,thesmoothnessisachievedbykeepingjqi+1)]TJ /F3 11.955 Tf 11.15 0 Td[(qijsmallbutnotzero.Intheuniformchargetransportmodel,wemakethischangezero,butfoundthatwewerenotabletoreproducetheobservedelectriceldswiththisrequirement.Wewillnowuseapenaltyapproachtokeepthisdifferencesmall,butnotzero.Asaresult,wearebetterabletottheobservedelectriceld.Inadditiontokeepingthechargechangesmallbetweenadjacentchannelsegments,wealsoincorporatethefollowingconstraints: (C1)Thetotalchargetransportisrestrainedsmalltomatchthephysicalsignicance. (C2)Inthenegativeregion,eachLMAsourceistreatedasachargelocationwithauniformchargeamplitude.ItcanbeenseenfromFigure 2)]TJ /F5 11.955 Tf 11.96 0 Td[(1 thattheLMAsourcesishighlybranchedinthenegativeregion.SincethereareasmallnumberofLMAsourcesassociatedwitheachbranch,wesimplytreateachLMAsourceinthenegativeregionasthelocationofasmallchannelsegment. (C3)Inthepositiveregion,aconstraintqi+1qiistakenintoaccountneartheendsofthechannelsduetodecreaseofthemodeledchargedensityalongthechannelpresentedbyMazurandRuhnke[1998].Withoutthisconstraint,thechargedensitycoulddroptononphysicalvalueof0severalkilometersfromtheendofthechannel.Whenthisconstraintisimposedontheentirechannel(insteadofjustattheendsofthechannel),weobtainedterrorsbetween2%and5%,whicharequitereasonable;however,therecoveredchargedensityexhibitedsomenonphysicalcharacteristics.Forexample,thechargedensitycanbecomecompletelyconstantonthechannelwithout 32

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anygrowthatallneartheendofthechannelwhereend-of-channelstreamersshouldeffectivelyyieldanincreasedchargedensity. Theleastsquaresproblemthatembodiesallourconstraintsisthefollowing: minimize3(EGmodeled)]TJ /F1 11.955 Tf 11.95 0 Td[(EGmeas)2+kEFmodeled)]TJ /F1 11.955 Tf 11.95 0 Td[(EFmeask2+p+Xiq2i+Xi2ch1pi(qi+1)]TJ /F3 11.955 Tf 11.96 0 Td[(qi)2+Xi2ch2pi(qi+1)]TJ /F3 11.955 Tf 11.95 0 Td[(qi)2+p)]TJ /F9 11.955 Tf 14.57 13.15 Td[(Xi2neg(qi)]TJ /F5 11.955 Tf 15.45 0 Td[(q)]TJ /F5 11.955 Tf 7.09 1.79 Td[()2subjecttoXi2ch1qi+Xi2ch2qi+Xi2negqi=0,qi0,i2ch1[ch2,qili,i2neg,qi+1qiforinearendsofchannels Here,EFisthevectorelectriceldattheightEsonde,EGistheverticalelectriceldatthegroundEsonde,andk.kistheEuclideannorm.Whenwecomparethemodeledandmeasuredelectriceld,weneedtogiveequalimportancetothegroundandtheballoonborneEsondes.Hence,weweightedtheleastsquaresdifferencebetweenthemodeledandmeasuredelectriceldatthegroundbyafactorof3.i2ch1referstothechargeplacedatthemidpointofsegmentsofchannel1andq)]TJ /F1 11.955 Tf 10.41 1.8 Td[(denotesthemeanvalueofchargeinnegativeregion.TheconstraintXi2ch1qi+Xi2ch2qi+Xi2negqi=0 isconservationofcharge. Sinceweonlydepositnegativechargeinpositiveregion,qi0fori2ch1[ch2.Thechargemovementinthenegativeregioniscomplicated,andwewillanalyzeitlaterinthissection.Letusrstassumethatnegativechargeisextractedfromthenegativeregionanddepositedinthepositiveregion,i.e.li=0andqi0fori2neg. Thetermpi(qi+1)]TJ /F3 11.955 Tf 12.82 0 Td[(qi)2onlyvanisheswhenqi+1=qi.Hence,thisterm,whichcreatesthesmoothnessinourmodel,keepsthechangeqi+1)]TJ /F3 11.955 Tf 12.56 0 Td[(qiinchargebetweenadjacentchannelsegmentssmall.Increasingpiforcesqiandqi+1toapproacheach 33

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other.SincetheelectriceldmeasuredbytheightEsondeisverysensitivetothechargeplacedonthenearbychannel,wewantqiconstantalongchannelsegmentsneartheEsonde.ThereissimplynotenoughinformationinthefourvaluesoftheelectriceldtodeterminemorethantheconstantchargedensityalongthechannelneartheightEsonde.ThisrequirementisenforcedbytakingpireallybigwhenqiisneartheEsonde. Theterm(qi)]TJ /F5 11.955 Tf 12.37 0 Td[(q)]TJ /F5 11.955 Tf 7.09 1.8 Td[()2onlyvanisheswhenqi=q)]TJ /F1 11.955 Tf 7.08 1.8 Td[(.SincewehavelimitedinformationconcerningtheoriginofthenegativechargeotherthantheLMAsourcebelow8kmaltitude,wepenalizethedeviationofthechargeremovalateachLMAsourcelocationfromthemean,i.e.chargeremovaliskeptrelativelyconstantateachLMAsourcelocationinthenegativeregion. Fromexperimentresult,thepenaltyparameterswerechosenasfollows: (P1)pi=1012whenqiisnearightEsonde (P2)pi=105whenqiisfarfromightEsonde (P3)p+=1012forchargedeposition (P4)p)]TJ /F5 11.955 Tf 10.41 1.8 Td[(=1012fordeviationofchargefrommeaninnegativeregion Here,weapplythelarge(P1)penaltyalongthe1.4kmofchannelsegmentstothewestandnorthoftheEsondeandalongthe4.9kmofchannelsegmentstotheeastandsourthoftheEsonde.Theoptimizationisrelativelyinsensitivetothepenaltyparameteraslongaswekeep(P1)muchlargerthantheotherpenalties.Notethat(P1)forchannelsegmentsneartheballoonmakesqicurveisessentiallyconstant. InTable 2)]TJ /F5 11.955 Tf 11.95 0 Td[(3 (Hageretal.[2013]Table3),wecomparethemeasuredelectriceldtothemodeledelectriceldattimet1andt2.Q1isthetotalchargeonchannel1andQ2isthetotalchargeonchannel2.Fromtheresult,itcanbeenseenthatthetbetweenmodeledandmeasuredelectriceldisalmostperfectatt1.However,therelativeerrorinthetincreasedfrom1%to7%attheinstantoftimet2.NotethattherelativeerroristhenormofthemodeledEminusthemeasuredEoverthenormofthemeasuredE. 34

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Table2-3. Smoothchargetransportmodel(l=0) t1=23:05:42.1UTt2=23:05:42.3UT MeasuredModeledMeasuredModeled EGz(kV/m)-6.8-6.7-7.2-6.9EFx(kV/m)20.420.429.930.0EFy(kV/m)35.035.041.142.3EFz(kV/m)-9.4-9.1-17.1-13.6Q1(C)-15.0-13.1Q2(C)-0.2-1.8 Thereseemstobeverylittlechargetransportonchannel2becauseQ2ismuchsmallerthanQ1.Infact,thechargedepositQ2onchannel2wasonchannelsegmentsveryclosetochannel1,i.e.thestartofthechannel2.Hence,thechargetransportanalysisimpliesthattherewasalmostnochargetransportalongchannel2,and-1.8CgiveninTable 2)]TJ /F5 11.955 Tf 11.95 0 Td[(3 att2actuallyrepresentschargedepositedonchannel1.Inourmodel,wecanforceallthechargetobedepositedonchannel1bysettingpi=1012wheni2ch2. FromTable 2)]TJ /F5 11.955 Tf 11.96 0 Td[(3 ,thetattimet2wasnotasgoodasthetattimet1.ThemainmismatchoccursatEFz.Wehavefoundthattheerrorbetweenthemodeledandthemeasuredeldscanberemovediftheconstraintqi0fori2negisrelaxedtoqiliforanegativenumber.Hageretal.[2013]presentedthatli=)]TJ /F5 11.955 Tf 9.3 0 Td[(0.06Cissufcient.Hence,chargetransportanalysisimpliesthatchargemovementinnegativeregioniscomplicated:duringtheinitialchargetransport,negativechargeisextractedfromthenegativeregionanddepositedinthepositiveregion,i.e.,li=0andqi0fori2neg;aftertheinitialchargetransport,thechargeisrearrangedinthenegativeregion,i.e.qi>0wherethenegativechargeisremovedandqi<0wherenegativechargeisdeposited. 35

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Theelectriceldchangeprovidesstrongevidenceofthechargerearrangementtakingplacebetweent1andt2intheregionlabeledNegativeinFigure 2)]TJ /F5 11.955 Tf 11.96 0 Td[(3 .Duringthistimeinterval,mostofthechangeintheelectriceldattheightEsondeoccurredinthexandzcomponentswiththezcomponentofEdecreasingandthexcomponentincreasing.ThedecreaseinEZcorrespondstonegativechargebeingplacedbeneaththeEsonde.Incontrast,negativechargeplacedattheendofchannel1increasesEzsincetheendofchannel1is2kmabovetheballoon.IfnegativechargeisplacedbeneaththeEsonde,thenweneedtorelaxtheconstraintqi0fori2negsincethisconstraintonlyallowsustoputpositivechargeintheregionbeneaththeEsonde. Wenowrepeatthechargetransportanalysiswithpi=1012fori2ch2andli=)]TJ /F5 11.955 Tf 9.3 0 Td[(0.06Cfori2negattimet2.Wealsoinvestigateelectriceldatanotherinstantoftimet0.ThetbetweenmodeledandmeasuredelectriceldisshowninTable 2)]TJ /F5 11.955 Tf 11.96 0 Td[(4 .Here,jQ)]TJ /F2 11.955 Tf 7.08 1.79 Td[(jisthemagnitudeofthetotalchargetransportinthenegativeregion.Itcanbeseenthattherelativeerrorisnomorethan1%atalltimes.jQ)]TJ /F2 11.955 Tf 7.09 1.79 Td[(j=Q1attimet0andt1.Attimet2,wehave jQ)]TJ /F2 11.955 Tf 7.09 1.79 Td[(j=Xi2negjqij=22.4C, whichisthesumofthechargetransportQ1=)]TJ /F5 11.955 Tf 9.3 0 Td[(15.6Cfromthenegativeregiontothepositiveregion,and-6.8Cthatdepositedtoadifferentlocationbeneath8kmaltitude. Itisnotedthatthechargetransportanalysisdidnotshowmeasurablechargetransportalongchannel2.Toexamineifthisobservationisstatisticallysignicant,wenowstudythesensitivityoftheelectriceldmeasurementstochargedepositiononchannel2.If1Cnegativechargeisplacedatthemidpointofchannel2,thenthechangeinelectriceldis4E=(69,859,98)V/mattheightEsondeand 4Ez=424V/matthegroundEsonde.ThesefournumbersshouldbeaddedtothefourmodeledeldsinTable 2-4 toobtainthetotalelectriceldcorresponding 36

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Table2-4. Smoothchargetransportmodel(l=)]TJ /F5 11.955 Tf 9.29 0 Td[(0.06fori2neg)att2 t0=23:05:42.023UTt1=23:05:42.1UTt2=23:05:42.3UT MeasuredModeledMeasuredModeledMeasuredModeled EGz(kV/m)-4.2-4.2-6.8-6.7-7.2-7.0EFx(kV/m)16.816.820.420.429.930.0EFy(kV/m)26.926.935.035.041.141.3EFz(kV/m)-7.4-7.6-9.4-9.1-17.1-16.4Q1(C)-9.7-15.4-15.6jQ)]TJ /F2 11.955 Tf 7.09 1.79 Td[(j(C)9.715.422.4 toanadditional1Cnegativechargeonchannel2alongwiththechargedepositiongeneratingTable 2-4 .Noticethatwhenthisnegativechargeisplacedonchannel2,allfourcomponentsofthemodeledelectriceldatt1moveawayfromthemeasuredelectricelds.Inparticular,if10,5,or2Cnegativechargewereplacedatthemidpointofchannel2,thenthecorrespondingrelativeerrorsinthemodeledelectriceldatthegroundEsondeare63%,33%,and14%respectively.Sincethetotalchargetransportwas22.4C,weconcludethatrelativelysmallamountsofchargeplacedonchannel2areeasilydetectedandhaveasignicantinputintheanalysis.Sinceanychargethatweputonchannel2seemstomovethemodeledeldsawayfromthemeasureddata,thedatadoesnotsupportmeasurablechargetransportonchannel2 Themodeledandmeasuredelectriceldfortheentiretime[t0,t2]isshowninFigure 2)]TJ /F5 11.955 Tf 11.95 0 Td[(5 (Hageretal.[2013]Figure8).Thelargesterrorisattimet2.TherelativeerroratthegroundEsondeandinthezcomponentofightEsondeareabout3%and4%respectively.TheerrorsinthexandycomponentofightEsondearelessthan1%. Thechargetransportalongchannel1asafunctionofdistancealongthechannelisshowninFigure 2)]TJ /F5 11.955 Tf 11.96 0 Td[(6 (Hageretal.[2013]Figure9),wherelinearchargedensityisinmC/m.Theleastsquarestyieldsthechargeatthemidpointofeach100mchannel 37

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A B Figure2-5. Comparisonbetweenmodeledandmeasuredelectricelds.A)ComparisonbetweenmodeledandmeasuredelectriceldsatightEsonde.B)ComparisonbetweenmodeledandmeasuredelectriceldsongroundEsonde. segment;thechargepermeteronthatchannelsegmentisobtainedbydividingby100.Analysisofthechargedensityonchannel1couldbefoundinHageretal.[2013].Neartheballoon,thelinearchargedensitywasabout)]TJ /F5 11.955 Tf 9.3 0 Td[(5.5mC/matt0,)]TJ /F5 11.955 Tf 9.3 0 Td[(0.69mC/matt1and)]TJ /F5 11.955 Tf 9.3 0 Td[(0.90mC/matt2,whicharelargerthan-0.36mC/mreportedbyWinnetal.[2011].Thedifferencemaybecausedbythechannellocation.Winnetal.[2011]used200mforthedistancebetweentheballoonandthelightningchannelwhile304misusedin 38

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ourassumption.Thusmorechargemustbeplacedonthechanneltotthemeasuredelectriceld. Figure2-6. Linearchargedensityonchannel1asafunctionofdistancealongthechannelatthreedifferenttimes. FromFigure 2)]TJ /F5 11.955 Tf 11.95 0 Td[(7 to 2)]TJ /F5 11.955 Tf 11.95 0 Td[(9 (Hageretal.[2013]Figure11),weplotthelocationsintheregionbeneaththe8kmaltitudewherenegativechargewasremovedordeposited.Theareaofeachcircleisproportionaltothechargemagnitude,andthecenterofeachcircleisanLMAsource.Thecircleislledifnegativechargewasremoved;oropenifnegativechargewasdeposited.Figure 2)]TJ /F5 11.955 Tf 11.96 0 Td[(7 andFigure 2)]TJ /F5 11.955 Tf 11.96 0 Td[(8 correspondtotimet0andt1,wherethereareonlyremovedcharge;Figure 2)]TJ /F5 11.955 Tf 11.95 0 Td[(9 correspondstotimet2,wheretherearebothremovedchargeanddepositedcharge. 39

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Figure2-7. Chargemovementinnegativeregionat23:05:42.023243UT.Theareaofeachcircleisproportionaltochargeamplitude.Thelledcirclerepresentwherenegativechargewasremovedandtheopencirclesrepresentwherenegativechargeisdeposited. Figure2-8. Chargemovementinnegativeregionat23:05:42.1UT. 40

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Figure2-9. Chargemovementinnegativeregionat23:05:42.3UT. 41

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CHAPTER3CHARGEREARRANGEMENTBYSPRITESOVERAMESOSCALECONVECTIVESYSTEM 3.1Overview ThestormproducingthespriteswasaMesoscaleConvectiveSystem(MCS-acomplexmediumscaleorganizedthunderstorm)situatedbetweennorthTexasandeastNewMexicoonJuly15th2010.Theterrainunderneaththestormwasrelativelyatwiththegroundelevationroughly1000m.Tenspritesoccurredintwoclustersbetween5:22UTand7:06UT.Therstsevenspritesoccurredbetween5:22UTand5:56UT,roughly1spriteevery5minutes,ontheeasternsideofthestormwithgroundelevationabout1100m.About45minuteslater,threemorespritesappearedtothewestoftheinitialspritesbetween6:41UTand7:06UTwithgroundelevationabout1300m.Figure 3)]TJ /F5 11.955 Tf 11.96 0 Td[(1 (Hageretal.[2012])showsthattheNEXRADlevelIIIcompositeradarreectivityat5:40UTand6:53UTrecordedbytheNationalWeatherServiceStationKFDX.Here,thelocationoftheNLDNlightningevent,either+CGreturnstrokeorcloudash,wassuperimposedoneachplot.TheseNLDNlightningeventswereassociatewithtensprites.Thenumbers(1to10)indicatetheorderinwhichthespritesoccurred,andalsocorrespondtothenumbersinTable 3)]TJ /F5 11.955 Tf 11.96 0 Td[(1 (Hageretal.[2012]Table2).ThelocationofLEFA2whichisusedinouranalysis,andLangmuirLaboratorywherethespriteswererecordedaremarkedonthegure. ThelightdatapresentedinthischapterwererecordedbyavideosysteminLangmuirLaboratory.ThesystemhastwohighspeedPhantom7camerasoperatingat12,500framespersecondandaWatecvideocameraoperatingat30framespersecondandaWatecvideocameraoperatingat30framespersecond.Thetimeresolutionofthehighspeedcameraswas80microseconds.Ourdataareobtainedfromcamera1,whichhadawidereldofview-roughly3.5Aby7.0A. ThechargetransportanalysisisbasedondatafromLEFA.Hageretal.[2012]presentedthedetailofLEFAcalibration.Horizontalmagneticelddatawererecorded 42

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A B Figure3-1. NationalWeatherServiceStationKFDXNEXRADlevelIIIcompositeradarreectivityonJuly15th2010atA)5:40UTandB)6:53UT.Eachspriteissuperimposedontheplot.A+signcorrespondstoa+CGreturnstokeandtheblackdotcorrespondstoacloudash.Thenumbersindicatetheorderinwhichthespriteoccurred.LEFA2andLangmuirLaboratoryarethewhitedots. 43

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byDukesultralowfrequency(ULF)andverylowfrequency(VLF)instrumentation.ThefrequencyrangefortheULFsystemis1to400Hz,whilethefrequencyrangefortheVLFsystemis50Hzto30kHz.ThesamplingfrequencyoftheULFsystemandVLFsystemsis2.5and100kHzrespectively.Accordingly,thetimingresolutionandaccuracyis400msfortheULFsystemand10msfortheVLFsystem.Thenoiseleveloftheinstrumentsisabout10pTRMS. 3.2DescriptionofDataSet LEFAstations2(LEFA2),LEFA5,andLEFA7wereoperationalduringthestormofJuly15th2010.OnlythesensitivechannelprovidedareliablemeasurementoftheverticalelectriceldchangebecausetheMCSwasfar(between300and500km)fromtheLEFAnetwork.Bycomparingthedatarecords,theamplitudeofthenoisewassmallestforLEFA2.Moreover,LEFA2wasthecloseststationstotheMCS,i.e.theamplitudeoftheelectriceldatLEFA2waslargest.Therefore,ouranalysisisbasedonthedataobtainedfromthesensitivechannelofLEFA2. Datafromthehighspeedcamera1andNLDNfortenspritesisgiveninTable 3)]TJ /F5 11.955 Tf 11.96 0 Td[(1 .Therstcolumndescribesthepredominantstructureofthesprite,eithercarrot,column,orastreamertipsplittingspriteasdiscussedbyMcHargetal.[2010].Sprite9wasnotclassiedsincethecameradidnotcatchthewholesprite.ThecolumnlabeledDelayisthedifferenceofthesourcetimebetweenthespriteanditsparentstrokegivenbyNLDN.ThecolumnlabeledLightSumgivesameasureoftherelativeintensityofthesprite.ThecolumnlabeledDischags.Beforegivesthenumberofdischargesinthe3secondsprecedingtheNLDNeventassociatedwithsprite.Columns9and10givethelat/lonoftheparentCGforeachsprite.ThelastcolumniCMC,denedbyCummerandLyons[2005],isbasedonananalysisoftheULFmagneticelds. 44

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Table3-1. Asummaryofthevideorecords(left)andcloselycorrelatedNLDNstrokerecords(right)forthesprites CameraNLDNDuke SpriteDelayLightCurTimeTriggerDischgs.iCMCTypeTime(UT)(ms)Sum(kA)(s)TypeLatLonDist.(km)Before(Ckm) 1Carrot05:22:01.7138092.12??6801.710146+CG34.3355-102.2054460313482.Carrot05:27:09.6964090.7790.478509.694066+CG34.2303-102.1123467326103.Carrot05:32:57.58216916.342.473357.564323+CG34.3625-102.3073450351184.Carrot05:37:59.5443293.314.815459.539386IC34.7223-101.9781485472075.Carrot05:45:14.49488966.621.562214.426730+CG34.2059-102.192646031??6.Carrot05:50:01.58472913.993.083101.569140IC34.4187-102.0331476181007.Column05:55:54.8116092.742.245254.807285+CG34.3785-102.0794471182268.Column06:41:36.0837691.000.327936.081595+CG34.0208-103.3953349173569.??07:00:31.8463290.37??4031.844769+CG34.0088-103.32473551840810.Split07:06:09.8099290.05??11209.808870+CG33.7988-103.930630120718 45

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Figure 3)]TJ /F5 11.955 Tf 11.95 0 Td[(2 (Hageretal.[2012]Figure4)showsaphotoofthesprite2takenbyWateccamera.Theboxshowstheeldofviewofthehighspeedcamera1. Figure3-2. AphotoofSprite2takenbytheWateccamera.Thespriteextendsbetween50and90kminaltitude.Therectangleshowstheeldofviewofthehighspeedcamera. 3.3ElectromagneticFieldDataandLightIntensity Startfromthissection,wewillfocusonsprite2,aspecialspritewithbroadhumpassociatedwithluminosityandmagneticeld.TheanalysisofotherspritescanbefoundinHageretal.[2012]. Theelectriceldforsprite2isshowninFigure 3)]TJ /F5 11.955 Tf 11.95 0 Td[(3 A(Hageretal.[2012]Figure7).ThezeropointonthetimeaxisrepresentsthetimeofthereturnstrokeasreportedbyNLDN.Theinitialsharppositivestepintheelectriceldisthe+CGreturnstroke.Theelectriceldforsprite2isdifferentfromthatoftheother9spritesinthefollowingaspect:Between4and8msafterthe+CGreturnstroke,thereisasignicanthumpintheelectriceldthatcloselycorrelateswithapeakintheluminousvolumeofthespriterecordedbyhighspeedvideo(Figure 3)]TJ /F5 11.955 Tf 11.95 0 Td[(3 B)andahumpinthemagneticeld(Figure 3)]TJ /F5 11.955 Tf 11.96 0 Td[(3 B).Inourstudy,wehaveexcellentconsistencybetweenallthree 46

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A B C Figure3-3. Measureddataforsprite2.A)Theelectricelddata,B)lightintensity,andC)azimuthalmagneticeldchangeversustimeforsprite2. 47

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distinctinstrumentsappliedtothesamesprite.Notethatonlyoneoutoftenspriteshasthishump. ThehumpwitnessedinallpanelsofFigure 3)]TJ /F5 11.955 Tf 11.95 0 Td[(3 willbeanalyzedinsection3.5.Wewilldemonstratethatmodelingaspriteasadownwardpropagatingverticalcurrentpulsereducingthepositivechargeintheionospherecanproducejustsuchahump.TheexplanationofthesignofthespritehumpcanbefoundinHageretal.[2012]. 3.4MathematicalModel ThemathematicalmodelforanalyzingthespritehumpisbasedontheexactsolutiontoMaxwell'sequationgivenbyUmanetal.[1975]foraverticalantennaonaperfectlyconductinggroundplane. 3.4.1SinglePerfectlyConductingPlaneCase ThegeometryconsideredinUmanetal.[1975]isshowninFigure 3)]TJ /F5 11.955 Tf 11.96 0 Td[(4 (Hageretal.[2012]Figure9).AverticalantennaofheightHisonaperfectlyconductinggroundplane.zaxisisperpendiculartothegroundplane.Thecurrentatanyheightisi(z,t)andi(z,t)=0fort0.Pistheobservationpoint.ThedistancefromPtothebaseoftheantennaisD.Umanetal.[1975]presentedthattheverticalelectricaleldE(t)attimetandobservationpointPonthegroundplanecanbeexpressed E(t)=1 2"0ZH0Zt02)]TJ /F5 11.955 Tf 11.96 0 Td[(3sin2 R(z)3i(z,)]TJ /F3 11.955 Tf 13.15 8.09 Td[(R(z) c)ddz+ZH02)]TJ /F5 11.955 Tf 11.95 0 Td[(3sin2(z) cR(z)2i(z,)]TJ /F3 11.955 Tf 11.95 0 Td[(R(z)=c)dz)]TJ /F9 11.955 Tf 11.29 16.27 Td[(ZH0sin2(z) c2R(z)@i(z,)]TJ /F3 11.955 Tf 11.95 0 Td[(R(z)=c) @tdz (3) wherei(z,t)=i(t)]TJ /F3 11.955 Tf 11.95 0 Td[(z=c),R(z)=p D2+z2andsin(z)=D R(z). Thethreetermsontherightsideofformula( 3 )aretheelectrostaticeld,theinductioneldandtheradiationeld.Formula( 3 )wasderivedbycombiningtheeldsassociatedwithadipolecurrentsourcewiththeeldassociatedwithan 48

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Figure3-4. AverticalantennaofheightHaboveaperfectlyconductinggroundplane. oppositecharged(image)dipolecurrentbeneaththeconductinggroundplane.Wenotethatimagechargetechniquesdevelopedfortheelectrostaticeldassociatedwithaconductingspheredonotextendtotimevaryingelectromagneticelds,sotheplanargeometryassociatedwithformula( 3 )isanimportantrequirement. 3.4.2DoublePerfectlyConductingPlanesCase Formula( 3 )includestheboundaryconditionassociatedwithaperfectlyconductinggroundplane.However,theboundaryconditionattheionosphere,whichisrelativelyhighlyconductive,isnottakenintoaccount.Weapproximatetheionosphereasaperfectlyconductingplane,andconsiderthetwoperfectlyconductingplanescase.SuchassumptioncanbeseeninFigure 3)]TJ /F5 11.955 Tf 11.96 0 Td[(5 (Hageretal.[2012]Figure10).Thetwoplanesareparalleltoeachotherwithz-axisperpendiculartothem.Thegroundconductingplaneisatz=0andtheionosphereconductingplaneatz=H.Theverticalantennatravelsfromz=z0toz=H.SupposethereisasmallcurrentdipoleSoflengthdzataltitudezontheantenna.ThesubterraneanimageofS,associatedwithgroundplane,isS1.ThedistantfromS1togroundplaneisz1=z.Applyingformula( 3 ),the 49

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totalelectriceldofSandS1atobservationpointPisE1(t)=ES1(t)+ES(t)=1 2"0ZHz0Zt02)]TJ /F5 11.955 Tf 11.95 0 Td[(3sin21 R1(z)3i(z,)]TJ /F3 11.955 Tf 13.15 8.08 Td[(R(z) c)ddz+ZHz02)]TJ /F5 11.955 Tf 11.96 -.01 Td[(3sin21(z) cR1(z)2i(z,)]TJ /F3 11.955 Tf 11.95 0 Td[(R(z)=c)dz)]TJ /F9 11.955 Tf 11.29 16.27 Td[(ZHz0sin21(z) c2R1(z)@i(z,)]TJ /F3 11.955 Tf 11.95 0 Td[(R(z)=c) @tdz whereR1(z)=p z21+D2isthedistancefromthedipoletotheobservationpoint,andtheassociateangleis1.sin1=D R1(z). Figure3-5. Averticalantennasuitedbetweentwoperfectlyconductingplanes. TheimageofS1aboutionosphereplaneisI2,andtheimageofI2aboutgroundplaneisS3.ThedistancefromS3togroundplanez3=2H+z.ReferringFigure 3)]TJ /F5 11.955 Tf 11.96 0 Td[(6 (Hageretal.[2012]Figure11)forthesequenceofimages.Wecanuseformula( 3 )tocalculatethetotalelectriceldofS2andI2atP: E3(t)=ES3(t)+EI2(t)=1 2"0ZHz0Zt02)]TJ /F5 11.955 Tf 11.95 0 Td[(3sin23 R3(z)3i(z,)]TJ /F3 11.955 Tf 13.15 8.09 Td[(R(z) c)ddz+ZHz02)]TJ /F5 11.955 Tf 11.96 0 Td[(3sin23(z) cR3(z)2i(z,)]TJ /F3 11.955 Tf 11.95 0 Td[(R(z)=c)dz)]TJ /F9 11.955 Tf 11.29 16.27 Td[(ZHz0sin23(z) c2R3(z)@i(z,)]TJ /F3 11.955 Tf 11.95 0 Td[(R(z)=c) @tdz whereR(z)3=p z23+D2andsin3=D R3(z). 50

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Figure3-6. Imagedipolesgeneratedbythesourcedipolecurrentataltitudez. Likewise,wehaveI4whichistheimageofS3aboutionosphereplane.TheimageofI4aboutgroundplaneisS5.Thus,wehaveaninnitepairofI2k)]TJ /F4 7.97 Tf 6.58 0 Td[(2andS2k)]TJ /F4 7.97 Tf 6.58 0 Td[(1,whereI2k)]TJ /F4 7.97 Tf 6.58 0 Td[(2istheimageofS2k)]TJ /F4 7.97 Tf 6.59 0 Td[(3aboutionosphereplaneandI2k)]TJ /F4 7.97 Tf 6.59 0 Td[(2istheionosphericimageofS2k)]TJ /F4 7.97 Tf 6.59 0 Td[(1.ThedistancefromS2k)]TJ /F4 7.97 Tf 6.59 0 Td[(1togroundplaneisz2k)]TJ /F4 7.97 Tf 6.59 0 Td[(1=2(k)]TJ /F5 11.955 Tf 12.19 0 Td[(1)H+z,k=2,3....Applyingformula( 3 ),thetotalelectriceldforonepairatPisEz2k)]TJ /F15 5.978 Tf 5.75 0 Td[(1(t)=ES2k)]TJ /F15 5.978 Tf 5.75 0 Td[(1(t)+EI2k)]TJ /F15 5.978 Tf 5.76 0 Td[(2(t)=1 2"0ZHz0Zt02)]TJ /F5 11.955 Tf 11.96 0 Td[(3sin22k)]TJ /F4 7.97 Tf 6.58 0 Td[(1 R2k)]TJ /F4 7.97 Tf 6.58 0 Td[(1(z)3i(z,)]TJ /F3 11.955 Tf 13.15 8.08 Td[(R(z) c)ddz+ZHz02)]TJ /F5 11.955 Tf 11.96 0 Td[(3sin22k)]TJ /F4 7.97 Tf 6.58 0 Td[(1(z) cR2k)]TJ /F4 7.97 Tf 6.59 0 Td[(1(z)2i(z,)]TJ /F3 11.955 Tf 11.95 0 Td[(R(z)=c)dz)]TJ /F9 11.955 Tf 11.3 16.27 Td[(ZHz0sin22k)]TJ /F4 7.97 Tf 6.59 0 Td[(1(z) c2R2k)]TJ /F4 7.97 Tf 6.58 0 Td[(1(z)@i(z,)]TJ /F3 11.955 Tf 11.95 0 Td[(R(z)=c) @tdz 51

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whereR2k)]TJ /F4 7.97 Tf 6.58 0 Td[(1(z)=q z22k)]TJ /F4 7.97 Tf 6.58 0 Td[(1+D2andsin=D R2k)]TJ /F15 5.978 Tf 5.76 0 Td[(1(z). Similarly,theimageofSaboutionosphereplaneisI1.TheimageofI1aboutgroundplaneisS2.Byusingformula( 3 ),thetotalelectriceldofS2andI1isE2(t)=ES2(t)+EI1(t)=1 2"0ZHz0Zt02)]TJ /F5 11.955 Tf 11.95 0 Td[(3sin22 R2(z)3i(z,)]TJ /F3 11.955 Tf 13.15 8.09 Td[(R(z) c)ddz+ZHz02)]TJ /F5 11.955 Tf 11.96 0 Td[(3sin22(z) cR2(z)2i(z,)]TJ /F3 11.955 Tf 11.95 0 Td[(R(z)=c)dz)]TJ /F9 11.955 Tf 11.29 16.27 Td[(ZHz0sin22(z) c2R2(z)@i(z,)]TJ /F3 11.955 Tf 11.95 0 Td[(R(z)=c) @tdz whereR2(z)=p z22+D2,sin2=D R3(z)andz3=2H)]TJ /F3 11.955 Tf 11.96 0 Td[(z. Also,wehaveaninnitepairwisepointsI2k+1andS2k+2,whereI2k+1istheimageofS2kaboutionosphereplaneandS2k+2isthesubterraneanimageofI2k+1.ThedistanceofS2k+2tothegroundisz2k+2=(2k+2)H)]TJ /F3 11.955 Tf 12.01 0 Td[(z,k=1,2....ThetotalelectriceldofI2k+1andS2k+2isE2k+2(t)=ES2k+2(t)+EI2k(t)=1 2"0ZHz0Zt02)]TJ /F5 11.955 Tf 11.95 0 Td[(3sin22k+2 R2k+2(z)3i(z,)]TJ /F3 11.955 Tf 13.15 8.09 Td[(R(z) c)ddz+ZHz02)]TJ /F5 11.955 Tf 11.96 0 Td[(3sin22k+2(z) cR2k+2(z)2i(z,)]TJ /F3 11.955 Tf 11.95 0 Td[(R(z)=c)dz)]TJ /F9 11.955 Tf 11.29 16.27 Td[(ZHz0sin22k+2(z) c2R2k+2(z)@i(z,)]TJ /F3 11.955 Tf 11.96 0 Td[(R(z)=c) @tdz Fromabove,thetotalelectriceldEattheobservationpointPis E(t)=1Xk=1Ek(t) (3) 52

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whereEk(t)=ESk(t)+EIk)]TJ /F15 5.978 Tf 5.76 0 Td[(1(t)=1 2"0ZHz0Zt02)]TJ /F5 11.955 Tf 11.95 0 Td[(3sin2k Rk(z)3i(z,)]TJ /F3 11.955 Tf 13.15 8.08 Td[(R(z) c)ddz+ZHz02)]TJ /F5 11.955 Tf 11.96 -.01 Td[(3sin2k(z) cRk(z)2i(z,)]TJ /F3 11.955 Tf 11.96 0 Td[(R(z)=c)dz)]TJ /F9 11.955 Tf 11.29 16.27 Td[(ZHz0sin2k(z) c2Rk(z)@i(z,)]TJ /F3 11.955 Tf 11.96 0 Td[(R(z)=c) @tdz,Rk(z)=p z2k+D2,sink=D Rk(z)andzk=8><>:kH)]TJ /F3 11.955 Tf 11.95 0 Td[(zifkisevenkH+z)]TJ /F3 11.955 Tf 11.95 0 Td[(Hifkisodd SinceE(t)=O(1 k2),E(t)isconvergent.E(t)satisestheboundaryconditionongroundplanebecauseeachpairaresymmetryaboutgroundplane.IfwerearrangethetermsofE(t)E(t)=(ES1(t)+ES(t))+(ES2(t)+EI1(t))...+(ESk(t)+EIk)]TJ /F15 5.978 Tf 5.75 0 Td[(1(t))+...=(ES(t)+EI1(t))+(ES1(t)+EI2(t))...+(ESk(t)+EIk+1(t))+... Becauseeachnewpairsatisestheboundaryconditiononionosphere(symmetryaboutionosphereplane),sodoestheentiresum.Thus,formula( 3 )satisesboundaryconditiononbothconductinggroundplaneandconductingionosphereplane. 3.4.3SphericalCaseApproximation Inpractice,theinniteseriesinformula( 3 )mustbeterminatedwhenkissufcientlylarge.InthissectionweanalyzeboththeatEarthapproximationusedinsection3.4.2andtheeffectoftruncatingthesuminformula( 3 ).First,wefocusonadoubleconductingsphericalcase:InFigure 3)]TJ /F5 11.955 Tf 11.96 0 Td[(7 (Hageretal.[2012]Figure13),wecomputetheelectriceldattheobservationpointP0duetoapointchargeQatlocationP.ChargepointPwithQchargeishabovethesphereS1.Thesecondconducting 53

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sphereS2surroundsthetheconductingsphereS1.TheimagechargeofPaboutS1isPwithchargeQ=)]TJ /F10 7.97 Tf 12.75 4.7 Td[(QR R+handdistanceR2 R+h,whereRistheradiusofS1.TheelectricofPandPatP0is E0=1 40[Q(P0)]TJ /F3 11.955 Tf 11.96 0 Td[(P) kP0)]TJ /F3 11.955 Tf 11.96 0 Td[(Pk3+Q(P0)]TJ /F5 11.955 Tf 13.24 2.66 Td[(P kP0)]TJ /F5 11.955 Tf 13.24 2.65 Td[(Pk3]. (3) ThedirectionoftheelectriceldisperpendiculartothesurfaceofS1. Figure3-7. Themethodofimagesforsphericalelectriceldresearch. PalsohasanimagechargePaboutS2andthetotalelectriceldofPandPatP0canbecalculatedbyusingformula( 3 ).ByusingthesamemethodwithSection3.4.2,thetotalelectriceldatP0isaninnitesequenceofformula( 3 ). SupposetheEarthisaperfectconductingsphereofradius6,371km,andtheionosphereistheperfectlyconductingsphericalshellofradius6,471km(100kmabovethesurfaceoftheEarth).Figure 3)]TJ /F5 11.955 Tf 11.95 0 Td[(8 (Hageretal.[2012]Figure14)plotstheelectriceldduetoa)]TJ /F5 11.955 Tf 9.3 0 Td[(100Cchargepointatanaltitudeof75km.ThehorizontalaxisisthecircledistancefromtheobservationpointtothechargelocationontheEarthandthe 54

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verticalaxisisthebase10logoftheelectriceld(V/m).Thesolidblackcurve,showingtheelectriceldasafunctionofdistancebyusing80imagepairs,iscorrectto16digits.Thebase10logofelectriceldisnearlyalinearfunctionofdistance.ThisimpliesthatEc10)]TJ /F13 7.97 Tf 6.58 0 Td[(D,whereistheslopeoftheline.Forasingleimagechargeinformula( 3 ),thedecayoftheelectriceldisproportionalto1/D3forP0nearP.The1/D3correspondstothesmallbendatthetopofthecurveinFigure 3)]TJ /F5 11.955 Tf 11.95 0 Td[(8 .ThecurveisessentiallylinearwithDisgreaterthan50km.Thistransitionfrom1/D3to10)]TJ /F13 7.97 Tf 6.59 0 Td[(Disduetotheinnitesequenceofimagechargesproducedbytheionosphere.Thereiscancelationintheseriescausingtheelectricelddecayexponentiallyinsteadof1/D3. Figure3-8. Thebase10logoftheelectriceldasafunctionofdistancefromanelectriceldmeterfora-100Cchargepoint75kmabovethesurfaceoftheEarth.ThesolidcurvecorrespondstoasphericalEarthandionosphere.ThedashedcurvescorrespondtoatEarthandionosphere,andeither50,200,800,or3200imagepairsintheapproximation. Next,supposetheEarthandionosphereareperfectlyconductingplanes.Formula( 3 )canstillbeusedtocalculatetheelectriceldbysubstituteQ=)]TJ /F3 11.955 Tf 9.3 0 Td[(QandPthe 55

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mirrorimageofPintheplane.InFigure 3)]TJ /F5 11.955 Tf 11.96 0 Td[(8 ,Thedashedcurvesareobtainedbyusing50,200,,800,3200imagepairs.FromFigure 3)]TJ /F5 11.955 Tf 11.96 0 Td[(8 ,itcanbeseenthattheelectriceldgottenfromplanemodelisveryclosetotheresultfromsphericalmodelwhenenoughpairsareused. Ifwevarythealtitudeofthecharge,similarplotscanbeobtainedwiththesameslope.Figure 3)]TJ /F5 11.955 Tf 11.95 0 Td[(9 (Hageretal.[2012]Figure15)plotsthebase10logoftheelectriceldforthesame-100Cchargeat3,10,and50kmaltitude.Byttingtheplotswithstraightlines,weobtainthattheverticalelectriceldonthesurfaceoftheEarthisgivenby E=)]TJ /F3 11.955 Tf 9.3 0 Td[(Qc(h)10)]TJ /F13 7.97 Tf 6.59 0 Td[(D,=(.0140930.000002)=km,c(h)=4Xi=0aihi, whereDisthegreatcircledistancefromthesensortothechargeinkilometers,histheheightofthechargeinkilometers,andthecoefcientsaiare: a0=4:84546e)]TJ /F5 11.955 Tf 11.96 0 Td[(3,a1=2:00712e)]TJ /F5 11.955 Tf 11.96 0 Td[(1,a2=2:32032e)]TJ /F5 11.955 Tf 11.96 0 Td[(4,a3=)]TJ /F5 11.955 Tf 9.3 0 Td[(4:53918e)]TJ /F5 11.955 Tf 11.96 0 Td[(5,a4=2:30095e)]TJ /F5 11.955 Tf 11.96 0 Td[(7. 3.5ChargeTransportAnalysis Wenowapplythemodeldevelopedinsection3.4.2tothehumpofsprite2.Accordingtotheconventionalbreakdowntheoryforsprites[Paskoetal.1997],thelightemittedbythespritecorrespondstoregionsoftheatmospherethatwereionizedbytheelectriceld.Totheextentthattheionizedchannelisanequipotential,itwillhaveapolarizationchargeoneitherend.Astheionizationspreadstoloweraltitudes,itlooks 56

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Figure3-9. Thebase10logoftheelectriceldasafunctionofdistancefromanelectriceldmeterfora-100CpointchargeplacedatdifferentaltitudesabovethesurfaceoftheEarth. likeadescendingpositivecurrent.Itisshownthattheelectriceldhumpcorrespondstothisionizationcurrentsincetheamplitudeofthehumpalmostexactlymatchesthelightintensityofthesprite. 3.5.1Assumptions Ourapplicationofthemodeldevelopedinsection3.4.2entailsthefollowingapproximations: 1.Theionosphereistreatedasaperfectconductor.Theconductivityoftheionosphereisabout10)]TJ /F4 7.97 Tf 6.58 0 Td[(7to10)]TJ /F5 11.955 Tf 7.08 -4.34 Td[(4siemens/m,whichismuchlargerthantheconductivityoftheair(3.010)]TJ /F4 7.97 Tf 6.59 0 Td[(15to10)]TJ /F2 11.955 Tf 7.08 -4.34 Td[()]TJ /F5 11.955 Tf 9.3 0 Td[(13siemens/m)nearthesurfaceoftheEarth[Pawaretal.2009;U.S.AirForce,1960]. 2.Thesurfaceoftheearthistreatedasaplane.Thedistancebetweentheequipmentandthespriteisabout442km.DuetothecurvatureoftheEarth,the 57

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equipmentdropsabout15kmbeneaththehorizon,whichhavealine-of-sightviewofthesprite.Moreover,Insection2.2.3,Itisindicatedthattheplanemodelisrelativelyaccuratebyincludingenoughimagescharges,. 3.Thecurrenttransportinthespriteisapproximatedbythatofaninnitelysmallwire.Thediameterofthespriteissmallrelativetothedistancebetweenequipmentandthesprite. 4.Thecurrentmaintainsthesameshapeasitpropagatesatconstantspeedonthewire.Thisisanapproximationthatallowsustocalculateaspeedandacurrentwaveform. 5.Conductioncurrentintheairabovethecloudisneglected.Sincethehumponlylastsforafewmilliseconds,thenegativechargetransportthatcanbeaccomplishedbyconductionisafractionofthenegativechargetransportpossibleonahighlyconductivelightningchannel. 3.5.2AnalysisfortheHumpofSprite2 TheelectricelddataofthehumpisplottedinFigure 3)]TJ /F5 11.955 Tf 11.96 0 Td[(10 (Hageretal.[2012]Figure16).Thelengthoftheintervalisabout3ms.ItbeginswhentheelectriceldreacheszeroinFigure 3)]TJ /F5 11.955 Tf 11.95 0 Td[(3 ,anditcontinuesuntiltheendofthehump,justbeforethelargeoscillationsthatoccurattheendofthespritecurrentpulse. Thevideoofthesprite2showsaninitialbroadglowextendingfrom70to85km.Latertheluminositywaveproceedsdownwardtoabout60km,thensuddenlybloomsinbothdirections,proceedingbackuptotheionosphereanddowntoabout50kminaltitude.[Paskoetal.1997]presentedthattheextentoftheluminositywaveindicatestheextentoftheionizedregion.Inouranalysis,wealsoassumethattheregionilluminatedbythespritecorrespondstotheregionwheretheatmosphereisionized. Byapplyingthemodeltothehump,weplacetheionosphereatanaltitude100kmaboveLEFA.Supposethespriteextendsfromz0=50km(thebottomofthesprite)up 58

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Figure3-10. Theelectriceldwhichcorrelateswiththelightforsprite2. totheionosphere(thetopofthesprite).Thecurrentiisatransmissionline(UmanandMcLain,[1969,1970]):i(z,t)=i(t+z v) wherezisthealtitudeandvisthevelocityofthedownwarddescendingcurrentpulse. Sincewedonotknowthecurrentpropagationvelocity,wetrieddifferentvelocitiestotthemeasuredelectriceld.Thevelocityofspritestreamersisintherange106)]TJ /F5 11.955 Tf 10.84 0 Td[(108m/s(Stanleyetal.[1999];Cummeretal.[2006]),i.e.0.003cto0.3c.Inourcase,theagreementwithmeasurementswasrelativelypoorwhenthevelocitieswentbelow0.1c.Onepossiblereasonisthatthecurrentwaveneedsatleastthismuchspeedtotraversethedistancefromtheionospheretothebottomofthespriteduringthetimeintervaloftheelectriceldhump.Hence,wefocusonvelocitiesofatleast0.1c.Resultsaregivenforthreevelocities0.25c,0.40cand0.55c.Thewidthofthecurrentpulseischosenso 59

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thatthecorrespondingelectriceldpulsedetectedbyLEFAhasthewidthseeninFigure 3)]TJ /F5 11.955 Tf 11.96 0 Td[(10 .Thecurrentpulseisapproximatedbyapiecewiselinearsplinewith20equallyspacedknots.Wecomputethecurrentithatresultsinthebestleastsquaresttothemeasureddataforeachcurrentpulsevelocity. ThebestttingcurrentpulseforthethreevelocitiesareshowninFigure 3)]TJ /F5 11.955 Tf 11.95 0 Td[(11 (Hageretal.[2012]Figure17).Theplotsdemonstratethecurrentattheionosphereasafunctionoftime.Weshift0.40cresultup20kAand0.25cresultup40kAtoavoidoverlapoftheplots.Forthetransmissionlinemodel,thecurrentatanyaltitudebetweenthebottomofthespriteandtheionospherewouldbethesameasthecurrentatthetopofthechannel,butdelayedbythetimeittakestothepulsetotraveltothataltitude. Figure3-11. Thespritecurrentatthetopofthespritechannelasafunctionoftime. InFigure 3)]TJ /F5 11.955 Tf 11.96 0 Td[(12 (Hageretal.[2012]Figure18),wecomparethemodeledelectriceldtothemeasuredelectriceldforthecurrentvelocity0.40c(currentvelocitiesof0.25cand0.55calsotthemeasuredelectriceldwithcomparableaccuracy).Thepeakcurrentforthecurrentpulsetravelingat0.55cis-17.7kA,whichislargerinmagnitudethanthe3.3kA,3.3kA,and1.6kAobtainedbyCummeretal.[1998]forthreedifferentsprites,butlessthanthe25kAobtainedforaspritebyCummer[2003].ThetechniquesusedtorecoveraspritecurrentweredevelopedinCummerandInan[2000],Lietal.[2008],andHuandCummer[2006].Intheirtechniques,acurrent 60

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momentiscomputedandthenfromthelengthofthechannel,acurrentisobtained.Ourtechnologydistinguishesitselffromtheirsbyprovidingthestartandtheendofthechannelalongwhichthecurrentpropagates. Figure3-12. Comparisonbetweenthemeasuredelectriceldandthemodeledelectriceldforthecurrentpulsetravelingat0.40c. InFigure 3)]TJ /F5 11.955 Tf 11.96 0 Td[(13 (Hageretal.[2012]Figure19),wedecomposethemodeledelectriceldintoelectrostaticeld,theinductioneldandtheradiationeldforthecurrentvelocity0.4c.Theradiationeldsdominatestheelectriceldatthebeginning.Astimeprogresses,theinductiontermbecomesmoresignicant.Theradiationeldchangesfastestwhiletheelectrostaticeldgrowsslowest.Theelectrostaticeldisthesmoothestcomponentoftheelectriceld. Byintegratingthecurrent,weobtainthetransportchargeforthethreevelocities(seeTable 3)]TJ /F5 11.955 Tf 11.96 0 Td[(2 ,Hageretal.[2012]Table3).Cummeretal.[1998]reportedthatthechargetransportfor3spritesare5C,6C,and42Cbyassuming50%ofthetotalchargetransportwenttothegroundand50%totheionosphere.Ouranalysisisbasedontheassumptionthatthechargetransportonlyfromtheionosphere. 61

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Figure3-13. TheelectriceldassociatedwithethecurrentpulseofFigure 3)]TJ /F5 11.955 Tf 11.95 0 Td[(11 traveling0.40cisdecomposedintoitsthreecomponents. Table3-2. Chargetransportforthesprite VelocityCharge(C) 0.25c-24.00.40c-23.90.55c-23.9 62

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CHAPTER4CONCLUSIONS AnICashonAugust24th2007wasanalyzedinChapter2.TheightEsondewasabout300mfromtheIClightningchannel,providingauniqueopportunityfortheanalysisoflightningchargetransport.Boththedipolemodelandtheuniformchargetransportmodelfailedtotthedata.Wepresentedanewsmoothchargetransportmodelallowinggreaterexibilityinchoosingthechargeamplitude.Inthepositiveregion,weplacednegativechargeatthemidpointsofsegmentsformingthelightningchannel.Inthenegativeregion,weplacedchargeatthelocationofLMAsources.Penaltieswereusedtocreateasmoothlyvaryingchargedensity.Withthisexibility,weobtainedaverygoodttothemeasuredelectriceldchangeshowninTable 2)]TJ /F5 11.955 Tf 11.96 0 Td[(4 .Thechargetransportanalysisshowedthefollowing:Afterthetransportwascomplete,about-1.8Cand-5.8Cwaslocatedatthebeginningandendofchannel1respectively;-8.0Cwasdistributedalongchannel1withalinearchargedensityofabout-0.9mC/m.Hence,relativetothe-15.6Cdepositedalongchannel1,37%wasdepositedneartheendofthechannel,11%wasdepositednearthebeginningofthechannel,and51%wasdistributedalongthechannel.Inthenegativeregion,outof-22.4Cthatwastransportedfromthenegativeregion,30%ofthechargewasmovedtoaloweraltitudebeneaththelightningchannel,while70%wasdepositedalongchannel1.Therewasnomeasurablechargetransportalongchannel2. AspritewithsignaturehumponJuly15th2010wasanalyzedinChapter3.Weshowedthatthehumpintheelectriceldthatfollowedthe+CGreturnstroke,couldbemodeledbyacurrentpulsethattraveledverticallythroughthevolumeilluminatedbythesprite.Weadaptedatransmissionlinemodelwhichhasbeenroutinelyappliedtolightningreturnstrokesandchangedtheboundaryconditionstoapplyittospritesbyincludingtwoconductingboundaries(ionosphereandground).Usingthetransmission 63

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linemodelandapulsevelocitybetween0.25cand0.55c,thechargetransporttotheionospherewasabout23.9C. 64

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APPENDIXACONVENTIONS PolarityConvention:Thesignconventionfortheelectriceldisthesameasthephysicsconvention,i.e.theeldvectorpointsinthesamedirectionoftheforceexertedonapositivetestcharge.Inourcase,positiveelectriceldpointsawayfromthesurfaceoftheearthforthemeasurementsoftheverticalelectriceld. CoordinateSystem:ThecoordinatessystemusedinChapter2isdenedasfollows:theoriginpointsitsatsealeveldirectlybelowapointnearNWcorneroftheannexbuildingofLangmuirLaboratory;x)]TJ /F1 11.955 Tf 9.3 0 Td[(axispointstotheeast;y)]TJ /F1 11.955 Tf 9.3 0 Td[(axispointsto(true)north,andz)]TJ /F1 11.955 Tf 9.3 0 Td[(axisverticallypointsawayfromthesurfaceoftheEarth. 65

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APPENDIXBDESCRIPTIONOFTHECALCULATIONOFV0 Inthisappendix,wegiveabriefdescriptionofhowtocalculateV0derivedfromsection2.5.Thedomainthatwediscretizedextendedfromthesurfaceoftheearthtotheionosphere(100kmabovethesurfaceoftheearth)andradiallyadistance100kmawayfromLangmuirLaboratoryalongthesurfaceoftheEarth.Weconsideredameshofsize298inthedirection,101inthedirection,and101inthedirection.NeartheLanmuirLaboratory,themeshelementshavesidesbetween20and50m,while100kmaway,thelargestsideofmeshelementsisaround4.5km.Wecomputedthepotentialatroughlythreemillion(298101101)pointsinthedomain.WeusedMETIS(KarypisandKumar[1998])toreorderthelinearsystemtoasymmetric,positivedenitelinearsystem.ThenCHOLMODpackage(DavisandHager[2009])wasusedtosolvethelinearequations(ReferringHageret.al[2013]formoredetail). 66

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BIOGRAPHICALSKETCH WeiFengwasbornininJiamusi,China.ShewasawardedaBachelorofSciencedegreeinappliedmathematicsin2006fromUniversityofScienceandTechnologyofChina.Aftergraduation,WeistartedhergraduatestudyinmathematicsattheUniversityofFlorida,fromwhichshereceivedherPh.D.inmathematicsin2013. 72