Charge Transport Analysis for Lightning

MISSING IMAGE

Material Information

Title:
Charge Transport Analysis for Lightning
Physical Description:
1 online resource (72 p.)
Language:
english
Creator:
Feng, Wei
Publisher:
University of Florida
Place of Publication:
Gainesville, Fla.
Publication Date:

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Mathematics
Committee Chair:
HAGER,WILLIAM WARD
Committee Co-Chair:
PILYUGIN,SERGEI S
Committee Members:
CHEN,YUNMEI
ZHANG,LEI
RAKOV,VLADIMIR ALEK SANDROVICH

Subjects

Subjects / Keywords:
charge-transport -- electric-field -- lightning -- sprite
Mathematics -- Dissertations, Academic -- UF
Genre:
Mathematics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract:
Charge rearrangement is analyzed for an intracloud flash on August 24th 2007. The analysis employed data from a balloon borne electric field sensor, or Esonde, data from an exactly Esonde on a mountain, and data from the New Mexico Institute of Mining and Technology Lightning Mapping Array(LMA). A new smooth charge transport model is used to analyze the charge movement, demonstrating high consistency with measured data. Based on the result, 26% of the negative charge was transported to the end of the channel; 36% was deposited along the channel in the positive region; 8% was deposited near the start of the channel in the positive region; and 30% was deposited in another positive region several kilometers beneath the main channel. Charge rearrangement by sprites is also analyzed for a mesoscale convective system(MCS) on July 15th 2010. The electric field data were recorded by Langmuir Electric Field Array(LEFA) and the magnetic field data were recorded by the charge-moment network near Duke University. A high speed video system in Langmuir Laboratory recorded telescopic images of the sprites. For one out of the ten sprites that were recorded, there was a positive hump in electric field a few milliseconds after the +CG return stroke.  The electric field hump is fit by a sprite current that propagates from the ionosphere to about 50 km in altitude. The total charge transport, which was the integral of the current hump, was 23.9 C when the velocity of the current pulse was between 0.25 c and 0.55 c.
General Note:
In the series University of Florida Digital Collections.
General Note:
Includes vita.
Bibliography:
Includes bibliographical references.
Source of Description:
Description based on online resource; title from PDF title page.
Source of Description:
This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility:
by Wei Feng.
Thesis:
Thesis (Ph.D.)--University of Florida, 2013.
Local:
Adviser: HAGER,WILLIAM WARD.
Local:
Co-adviser: PILYUGIN,SERGEI S.

Record Information

Source Institution:
UFRGP
Rights Management:
Applicable rights reserved.
Classification:
lcc - LD1780 2013
System ID:
UFE0046174:00001


This item is only available as the following downloads:


Full Text

PAGE 1

CHARGETRANSPORTANALYSISFORLIGHTNINGByWEIFENGADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2013

PAGE 2

c2013WeiFeng 2

PAGE 3

Tomyparents,Iloveyouallthetime 3

PAGE 4

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

PAGE 5

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

PAGE 6

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

PAGE 7

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

PAGE 8

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

PAGE 9

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

PAGE 10

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

PAGE 11

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

PAGE 12

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

PAGE 13

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

PAGE 14

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

PAGE 15

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

PAGE 16

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

PAGE 17

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

PAGE 18

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

PAGE 19

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

PAGE 20

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

PAGE 21

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

PAGE 22

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

PAGE 23

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

PAGE 24

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

PAGE 25

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

PAGE 26

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

PAGE 27

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

PAGE 28

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

PAGE 29

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

PAGE 30

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

PAGE 31

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

PAGE 32

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

PAGE 33

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

PAGE 34

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

PAGE 35

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

PAGE 36

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

PAGE 37

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

PAGE 38

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

PAGE 39

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

PAGE 40

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

PAGE 41

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

PAGE 42

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

PAGE 43

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

PAGE 44

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

PAGE 45

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

PAGE 46

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

PAGE 47

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

PAGE 48

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

PAGE 49

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

PAGE 50

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

PAGE 51

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

PAGE 52

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

PAGE 53

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

PAGE 54

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

PAGE 55

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

PAGE 56

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

PAGE 57

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

PAGE 58

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

PAGE 59

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

PAGE 60

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

PAGE 61

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

PAGE 62

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

PAGE 63

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

PAGE 64

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

PAGE 65

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

PAGE 66

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

PAGE 67

REFERENCES Adlerman,E.J.,andE.R.Williams(1996),Seanonalvariationoftheglobal electricalcircuit,J.Geophys.Res.,101,29,679-29,288. Bell,T.F.,etal.(1996),Runawayelectronsasasourceofredspritesinthe mesosphere,Geophys.Res.Lett.,22,21272130. Boccippio,D.J.,etal.(1995)Sprites,ELFtransients,andpositivegroundstrokes, Science,269,10881091. Chiu,C.S.(1978),Numericalstudyofcloudelectricationinanaxisymmetric,time dependentcloudmodel,J.Geophys.Res.,83,5025-5049. Coleman,L.M.,etal.(2003),Effectsofchargeandelectromagnetic potentialonlightningpropagation,J.Geophys.Res.,108108(D9),4298, doi:10.1029/2002JD002718. Cummer,S.A.(2003),Currentmomentinsprite-producinglightning,J.Atmos. SolarTerr.Phys.,65,499508. Cummer,S.A.,andU.S.Inan(1997),Measurementofchargetransferin sprite-producinglightningusingELFradioatmospherics,Geophys.Res. Lett.,24,17311734. Cummer,S.A.,etal.(1998),ELFradiationproducedbyelectricalcurrentsin sprites,Geophys.Res.Lett.,25,12811284. Cummer,S.A.,andU.S.Inan(2000),ModelingELFradioatmospheric propagationandextractinglightningcurrentsfromELFobservations, RadioSci.,35,385394. Cummer,S.A.,andW.A.Lyons(2005),Implicationsoflightningcharge momentchangesforspriteinitiation,J.Geophys.Res.,33,L04104. doi:10.1029/2005GL024969. Cummer,S.A.,etal.(2006),Submillisecondimagingofspritedevelopmentand structure,J.Geophys.Res.,110,A04304,doi:10.1029/2004JA010812. Cummins,K.L.,etal.(1998),AcombinedTOA/MDFtechnologyupgradeof theU.S.NationalLightningDetectionNetwork,J.Geophys.Res.,103, 90359044. Davis,T.A.,andW.W.Hager(2009),DynamicsupernodesinsparseCholesky update/downdateandtriangularsolves,ACMTrans.Math.Softw.,35. Franz,R.C.,etal.(1990),Televisionofalargeupwardelectricaldischargeabove athunderstormsystem,Science,249,4851. 67

PAGE 68

Few,A.A.(1970),Lightningchannelreconstructionfromthundermeasurements, J.Geophys.Res.,75,75177523. Hager,W.W.(1989),Updatingtheinverseofamatrix,SIAMRev,31,221. Hager,W.W.,etal.(1989),Theevolutionanddischargeofelectriceldswithina thunderstorm,J.Comput.Phys.,82,193-217 Hager,W.W.,etal.(2007),Analysisofchargetransportduringlightningusing balloonborneelectriceldsensorsandLMA,J.Geophys.Res.,112, D18204,doi:10.1029/2006JD008187. Hager,W.W.,etal.(2010),Threedimensionalchargestructureofamountain thunderstorm,J.Geophys.Res.,115,D12119,doi:10.1029/2009jd013241. Hager,W.W.,etal.(2012),Chargerearrangementbyspritesoveranorth Texasmesoscaleconvectivesystem,J.Geophys.Res.,117,D22101, doi.10.1029/2012JD018309. Hager,W.W.,andW.Feng(2013),ChargeRearrangementDeducedfromNearby ElectricFieldMeasurementsofanIntracloudFlashwithK-Changes,J.Geophys.Res.,118,doi.10.1002/jgrd.50782 Helsdon,J.H.,Jr.,(1980),ChargeChaffseedingeffectsinadynamical-electrical cloudmodel,J.Appl.Meteor.,,19,1101-1125 Hu,W.,andS.A.Cummer(2006),AnFDTDmodelforlowandhighaltitude lightning-generatedEMelds,IEEETrans.AntennasPropag.,54(5), 1513)]TJ /F1 11.955 Tf 9.3 0 Td[(1522. Jacobson,E.A.,andE.P.Krider(1976),Electrostaticeldchangesproducedby Floridalightning,J.Atmos.Sci.,33,103117. Karypis,G.,andV.Kumar(1998),AAfastandhighqualitymultilevelschemefor partitioningirregulargraphs,SIAMJ.Sci.Comput.,20,359392. Klemp,J.B.,andR.B.Wilhelmson(1978),Thesimulationofthree-dimensional convectivestormdynamics,J.Atmos.Sci.,35,1070-1096 Koshak,W.J.,andE.P.Krider(1989),Analysisoflightningeldchangesduring activeFloridathunderstorms,J.Geophys.Res.,94,11651186. Koshak,W.J.,andE.P.Krider(1994),Alinearmethodforanalyzinglightningeld changes,J.Atmos.Sci.,51,473488. Krehbiel,P.R.(1981),Ananalysisoftheelectriceldchangeproduced bylightning,Ph.D.thesis,Univ.ofManchesterInst.ofScie.andTechnol., Manchester,England. 68

PAGE 69

Krehbiel,P.R.(1986),Theelectricalstructureofthunderstorms,inTheEarths ElectricalEnvironment,,pp.90113,NationalAcademyPress, Washington,D.C. Krehbiel,P.R.,etal.(1979)Ananalysisofthechargestructureoflightning dischargestoground,J.Geophys.Res.,84,24322456. Li,J.,etal.(2008),Coordinatedanalysisofdelayedspriteswithhigh-speed imagesandremoteelectromagneticelds,J.Geophys.Res.,113,D20206, doi:10.1029/2008JD010008. Liu,X.,andP.R.Krehbiel(1985)Theinitialstreamerofintracloudlightning ashes,J.Geophys.Res.,90,62116218. Lu,G.,etal.(2011),Chargetransferduringintracloudlightningfrom atime-dependentmultidipolemodel,,J.Geophys.Res.,116,D03209, doi:10.1029/2010JD014495. Marshall,T.C.,andW.D.Rust(1991),Electriceldsoundingsthrough thunderstorms,J.Geophys.Res.,96,22,29722,306. Marshall,T.C.,andW.D.Rust(1993),Twotypesofverticalelectricalstructures instratiformprecipitationregionsofmesoscaleconvectivesystems,Bull.Am. Meteorol.Soc.,74,21592170. Mazur,V.,andL.H.Ruhnke(1998),Modelofelectricchargesinthunderstorms andassociatedlightning,J.Geophys.Res.,103,23,299-23,308 McHarg,M.G.,etal.(2010),Streamertipsplittinginsprites,J.Geophys.Res., 115,A00E53,doi:10.1029/2009JA014850. Murphy,M.J.,etal.(1996),LightningchargeanalysesinsmallConvectionand PrecipitationElectrication(CaPE)experimentstorms,J.Geophys.Res.,101, 29,61529,626. Nisbet,J.S.(1983),Adynamicmodelofthundercloudelectricelds, J.Atmos.Sci.40:2855-73. Pasko,V.P.,etal(1997)Spritesproducedbyquasi-electrostaticheatingand ionizationinthelowerionosphere,J.Geophys.Res.,102,45294561. Pawar,S.D.,etal(2009),Effectofrelativehumidityandsealevelpressureon electricalconductivityofairoverIndianOcean,J.Geophys.Res.,114, D02205,doi:10.1029/2007JD009716. Proctor,D.E.(1981),VHFradiopicturesofcloudashes,J.Geophys.Res.,86, 40414071. 69

PAGE 70

Proctor,D.E.(1997),Lightningasheswithhighorigins,J.Geophys.Res.,102, 16931706. Rakov,V.A.,andM.A.Uman(2003),Lightning,PhysicsandEffects,Cambridge Univ.Press,Cambridge,U.K. Rison,W.,etal.(1999),AGPS-basedthree-dimensionallightningmapping system:InitialobservationsincentralNewMexico,Geophys.Res.Lett.,26, 3573-3576. Rison,W.,etal.(2003),NewMexicoTechLightningMappingArray:Real-Time SystemMonitoringandDataDisplay,AmericanGeophysicalUnion,Fall Meeting2003,abstractAE22A-1108 Simpson,G.C.,andG.D.Robinson(1941)Thedistributionofelectricityin thunderclouds,II,,,Proc.R.Soc.,Ser.A,,177,281329. Simpson,G.C.,andF.J.Scrase(1937)Thedistributionofelectricityin thunderclouds,Proc.R.Soc.,Ser.A,161,309352. Sonnenfeld,R.G.,etal.(2006),ComparingEeldchangesalofttolightning mappingdata,J.Geophys.Res.,111,D20209,doi:10.1029/2006JD007242. StanleyM,etal.(1999),Highspeedvideoofinitialspritedevelopment,Geophys. Res.Lett.,26,32013204.doi:10.1029/1999GL010673 Stanley,M.,etal.(2000),DetectionofdaytimespritesviaauniquespriteELF signature,Geophys.Res.Lett.,27(6),871874,doi:10.1029/1999GL010769. Stolzenburg,M.,andT.C.Marshall(1994),Testingmodelsofthunderstorm chargedistributionswithCoulombslaw,J.Geophys.Res.,99,25,92125,932. Stolzenburg,etal.(1994),Horizontaldistributionofelectricalandmeteorological conditionsacrossthestratiformregionofamesoscaleconvectivesystem, Mon.Wea.Rev.,122,17771797,MWRSpecialIssue. Stolzenburg,M.,etal.(1998a),Electricalstructureinthunderstormconvective regions2.Isolatedstorms,J.Geophys.Res.,103,14,07914,096. Stolzenburg,M.,etal.(1998b),Electricalstructureinthunderstormconvective regions3.Synthesis,J.Geophys.Res.,103,14,09714,108. Stolzenburg,M.,etal.(1998c),Electricalstructureinthunderstormconvective regions1.Mesoscaleconvectivesystems,J.Geophys.Res.,103, 14,05914,078. Stolzenburg,M.,etal.(2001),Serialsoundingsofelectriceldthrougha mesoscaleconvectivesystem,J.Geophys,Res.,106(D12),12,37112,380. 70

PAGE 71

Stolzenburg,M.,etal.(2002),Twosimultaneouschargestructures inthunderstormconvection,J.Geophys.Res.,107(D18),4352, doi:10.1029/2001JD000904. Thomas,R.J.,etal.(2001),ObservationsofVHFsourcepowersradiatedby lightning,,Geophys.Res.Lett.,28,143146. Thomas,R.J.,etal.(2004),Accuracyofthelightningmappingarray, J.Geophys.Res.,109,D14207,doi:10.1029/2004JD004549. Uman,M.A.,andD.K.McLain.(1969),Magneticeldofthelightningreturn stroke,Am.J.Phys.,74,6899-6910. Uman,M.A.,andD.K.McLain.(1970),Radiationeldandcurrentofthelightning steppedleader,J.Geophys.Res.,75,1058-1066. Uman,M.A.,etal.(1975),Theelectromagneticradiationfromaniteantenna, Am.J.Phys.,43,33-38 U.S.AirForce(1960),HandbookofGeophysics,MacMillam,NewYork. Weber,M.E.,etal.(1982),Athundercloudelectriceldsounding:Charge distributionandlightning,J.Geophys.Res.,87,71587169. Wilson,C.T.R.(1916),Onsomedeterminationsofthesignandmagnitudeof electricdischargesinlightningashes,Proc.R.Soc.Ser.A,92,555574. Wilson,C.T.R.(1920),III.Investigationsonlightningdischargesandonthe electriceldofthunderstorms,Phil.Trans.R.Soc.,Ser.A,221,73115. Wilson,C.T.R.(1925)Theelectriceldofathundercloudandsomeofitseffects, Proc.R.Soc.,Ser.D,37,3237. Winn,W.P.,etal.(2011),Lightningleaderstepping,Kchanges,andother observationsnearanintracloudash,J.Geophys.Res.,116,D23115, doi:10.1029/2011JD015998. Workman,E.J.,andR.E.Holzer(1939),Quantitiesofchargetransfersin lightningdischarges,JPhys.Rev.,55,598. Ziv,A.,andZ.Levin(1974),Thundercloudelectrication:Cloudgrowthand electricaldevelopment,J.Atmos.Sci.,31,1652-1661. 71

PAGE 72

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