The alternative paradigm for magnetospheric physics

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 101, NO. A5, PAGES 10,587-10,625,MAY 1, 1996 The alternativeparadigmfor magnetospheric physics E. N. Parker TheEnricoFermiInstitute,University of Chicago, Chicago, Illinois Abstract. This paperemphasizes thatthemacrodynamics of theterrestrial magnetosphere is moreeffectivelytreatedin termsof theprimaryvariablesB andv (the B, v paradigm).The commonpracticeof relatingthe dynamicsto E andj (the E, j paradigm)providesdirectanswersin a varietyof symmetric cases,but breaksdownin evenso simplea staticproblemas a flux bundledisplaced(perhapsby reconnection with the magneticfield in the solarwind) from its normalequilibriumpositionin a staticdipolefield. The essential pointis thata direct derivationfrom the equations of MaxwellandNewtonleadsto field equations written in termsof the continuumfieldsB and v. The equations can be recastin termsof E and j, of course,but theyare thenunwieldy, beingintegrodifferential equations. Hencethe E, j paradigm,whencorrectlyapplied,is seriously limitedin its effectiveness in dynamical problems. Circumventing the limitationswith thecommondeclaration thatE is theprimemover, activelypenetrating from the solarwindintothemagnetosphere, providesdynamics thatis unfortunately at variancewith theresultsthatfollowdirectlyfrom Maxwell andNewton. The paperoutlinesthe standard derivationof the basicfield equations andthengoeson to treat a varietyof circumstances to illustratethe effectiveness of thedeductive B, v paradigmin thecontinuum dynamics of themagnetic fieldandplasma.Thereis no attemptto developa comprehensive modelof magnetospheric activity.Howeverwe suggestthat the ultimatetask is moreeffectivelyattackedwith the B, v paradigm. 1. Introduction So theparadigm,howeverconvenientin simplecases,becomesineffectivefor treatingstrong,nonsymmetric, or time dependentef- Thetheoryof magnetospheric activityiscommonly expressed fects. in terms of theelectric fieldE(r,t) andtheelectric current density The purposeof the presentpaper is to exhibit someof the j(r,t), withtheviewthattheelectricfielddrivesboththeelectric of theE, j paradigm withspecific examples, andthen current andthebulkplasma motion u = cE x B/B 2,whilethecur- limitations thealternative B, v (orB, u) paradigm, working rentcauses themagnetic perturbations AB (r, t). We referto this to elaborate fieldB (r, t) andtheplasma velocity u(r, t) formulation astheE, j paradigm. Theparadigm worksfromthe withthemagnetic equations [Parker,1962,1979,1994; basicprinciples of symmetry, chargeconservation, theunperturbedorv(r, t) in thedynamical 1983].Theessential pointis thatthemagnetospheric sysfieldlinesof a staticmagnetic field[cf.Stern,1992],andAmpere's Siscoe, particles andplaslaw. Theelectricfieldwithinthemagnetosphere isregarded asthe temis drivenby theinternalforcesof trapped activeinwardextensionof the electricfield Es = -rs x Bs/c masandby externalforcesexertedby the solarwind,ratherthan electricfields.Hencethemagnetospheric in the solarwind with velocityv s andmagnetic fieldB s. The by appliedmacroscopic by thedynamical equations relatingmomenparadigmprovidesconvenientrelations,via the Biot- Savartin- systemis described tum and force, that is Newton's equations of motion containing the tegralform of Ampere'slaw, betweenthe electriccurrentsof enappropriate Maxwell, Reynolds, and plasma pressure stresses. The ergeticparticlesandthe associated magneticfieldperturbations in stresses areexpressible directlyin termsof staticconfigurations. Howevertheeffectiveapplication is limited MaxwellandReynolds equations arenaturally formulated by thefactthattheprinciples of symmetry andequipotential field B andv, sothatthedynamical a directdeductive approach to linesdo not applyto nonsymmetric andtime-dependent circum- in termsof B andv andprovide magnetospheric physics. As we shall see, no physics is omitted in stances of magnetospheric activity.Chargeconservation combined failing to mention the electric current in treating the deformation of withtheunperturbed fieldis notalwayssufficient toprovidetheesforces (orthereverse). Theconsiderations oncharge sentialcurrentpathsfor computing thefieldperturbations, evenfor B byparticle playinganessential rolein theE, j paradigm, are arbitrarily smallstaticperturbations of aknownfieldconfiguration.conservation, takencareof by Maxwell'sequations (whichguarFurthermore, theinwardprojection of theinterplanetary electric automatically andby Newton'sequations (whichaufieldalongthemagnetic fieldlinesisnotapplicable unless thecon- anteechargeconservation) tomatically provide ion and electron motions consistent with the vectionof theionosphere is keepingup with thefieldlineconvecelectric current requirements of Ampere's law) on which the B, v tionatthemagnetopause sothatthefieldisstationary (0/0t = 0). paradigmis based. Copyright 1996bytheAmerican Geophysical Union. To statethisin otherterms,themacroscopic behavior(on scaleslargecompared to the thecyclotronradiiof theindividual Papernumber95JA02866. 0148-0227/96/95JA-02866509.00 ionsandelectrons) of themagnetosphere isdescribed bymagnetohydrodynamics (MHD) which we refer to here as the B,v 10,587 10,588 PARKER:ALTERNATIVE PARADIGM fields,notcaughtupin thesolarwind. That paradigm. The equationsof MHD form a completesetof partial roundinglower-latitude differentialequations, providinga deductiveapproach to thetheory is to say,the region 1 currentsfollow from Ampere'slaw in the of magnetospheric activity. The physicalconceptsaccompanying magneticfield deformedby the tailwardtransportof flux from the The region2 Birkelandcurrentsarisein the MHD equationsare mechanicalin nature,involvingthe push sunwardmagnetopause. andpull of the Maxwell stresses (the pressureandtensionof the the region of the returnflow that is forcedby the accumulating magneticfield) againstthefluid motion,with thefield andfluid tied magneticand plasmapressureon the nightside.The returnflow togetherinto a singleelasticcontinuuminsofarasthefluid is unable isblockedfromlow latitudesby theextremeadiabatic compression tosupport anelectric fieldE t initsownmoving frameofreference.of the particlesandplasmathat would arisein bundlesconvected to low latitude[Zhu, 1993]. The intensityof theregion2 current It shouldbe notedthattheMHD equationspermitsurfacesof at any givenlatitudefollowsfrom Ampere'slaw asproportional discontinuityin thecircumstances of theplanetarymagnetosphere, to the gradientin the Maxwell stressnecessary to drivethereturn deformedby a quasi-staticsolarwind. Examplesare the magneflow of the massive v•scous ionosphere. So the Birkeland currents topauseand auroralsheets.The locationof thesediscontinuities is arelocatedwherethemechanics of themagnetospheric convection uniquelydefinedby theB,v (MHD) equations appliedto theconplacesthem. The curvatureandgradientdriftsaredetermined by tinuousfields in the regionsbetween. On the microscopicscales themechanical deformation of themagnetosphere, ratherthanvice (thecyclotronradii) the surfacesof discontinuity havefinitethickversa,andare automatically in compliance with therequirements nessandinternalstructurethatcanbe describedonlyby thekinetic of themechanical deformation of themagneticfield. equationsof plasmaphysics.The macroscopic conditionis simply Theadvantage of thea,v paradigm isthatit proceeds deducthe balanceof the total presureacrossthe surfaceof discontinuity. Thetreatment is analogous to thetreatment of a shocktransi- tively from the field equationsof Newton andMaxwell, automatialongthe centralpathandbypasstion in a macroscopichydrodynamicflow, wherethelocationof the cally steeringthe investigation andcomplicatedindirectrelationsbetweenthe shocktransitionis determinedby the large-scalehydrodynamics, ing thenonproblems with conservation of mass,momentum,andenergy(theRankine- secondaryquantities. Hugoniot relations)acrossthe small thicknessof the shock. The The basicsimplicityof the principalvariationsof the maginternalstructureof the shockcanbe treatedonly with application netospherehasbeennotedfor sometime[cf. Sharmaet a1.,1993], of thecompletekinetictheory. andreferencestherein)from the low dimensionality (,,o 3) of the Now thedynamicalequations of theB,v paradigm can,of observedvariations.This consideration, togetherwith a desirefor course, beexpressed interms ofE andj, replacing thebulkplasma a more deductive,lessad hoc, theoryfor the observedmagneto- velocity byc E x B/B2 andB bytheBiot-Savart integral sphericactivity,againsuggests the B,v paradigmwith its simple mechanicalconceptsandconvenient fieldequations.Fromthe B(r) -1 J(r' )x(r-I3r') c / dar, Ir-r' pointof viewof B•v, geomagnetic activityis themechanical consequence of theturbulentmixingof themagnetopause with thesolar wind andmagneticreconnection acrossthemagnetopause. This takenoverall space.However,theresultis unwieldy, converting grabsgeomagneticflux bundlesand stretchesthemback into the partialdifferential equations intoglobalintegro-differenfial equa- goetailat the sametime that the pressureof the solarwind com,whiletheplasmaandenergetic particles tions.SotheE, j paradigm hasnotractable master fieldequationspressesthemagnetosphere, withininflatethemagnetosphere. In shortthedynamicsof themagtoguidethetheory.Hencein theactiveconvecfing magnetosphere isprimarilya shovingmatchbetweenparticlesandmagtheE, j paradigm is obliged to resortto theadhocintroductionnetosphere of individualeffects.In thehandsof expertstheadhocconstruc- neticfield, thatis betweentheReynoldsstressandparticlepressure tionprovides a surprisingly detailedqualitative modelof theactive on the one hand and the Maxwell stresson the other. The essential stressfieldstranscend the internal magnetosphere [cf. Fejer, 1964;Vasyliunas, 1970],suchastheRice pointis thatthesemacroscopic Convection Model[Wolf,1974,1983;Erickson et al.,1991;Wolfet microscopicdetailsof the plasmaandits electriccurrents.If there al., 1991;Yanget al., 1994].Unfortunately, thephysical explana- is a specialneedto know the microscopicdetails,theyarereadily tionsbecome increasingly complicated andindirect in thelanguage workedout from the macroscopicfields. of theE, j paradigm, andthequantitative errors area stumbling This is not to say that electriccurrentsandparallelelectric blockfor morecomplexsituations. A singleexamplesuffices to fieldsdonotplayakeyrolein specialsituations, forexample creatillustratetheproblem. ingtheauroraandin theexpansion phaseof themagneticsubstorm phenomena aresetupby TheBirkeland currents j_l. areoften"explained" by noting [Zhu,1995]. Howeverthoseimportant conditions developing inB,and thepresence ofEllcan thattheyareassociated withthenonvanishing divergence ofthegra- extreme onlyby firstworkingoutthedeformation of a by dientandcurvature driftsof theionsandelectrons of themagneto- be established forces exerted bytheparticles andplasma. Thenif sphericplasma.The mathematical relationsbetweentheBirkeland themechanical tobesolarge astorequire astrong Ell,theconsequences current density ill andtheplasma presure gradient isthen workedill proves ofEll [Coroniti andKennel, 1973; Schindler eta1.,1991] must be out,withtheconclusion thattheBirkeland currents arecaused by the plasmadistribution.In anotherdirection,it is sometimesstated introduced intothecalculation of a. It isnotpossible topursue the that the region2 Birkelandcurrentsplay the importantrole of shieldingthe low- latitudemagnetosphere fromtheeffectsof the magneticsubstrom,therebyexplainingthe observedabsenceof physicsin theopposite orderexceptwhereobservation provides the substromeffects at low latitudes, but in fact thesestatementsare solution totheproblem ahead oftime.Further commentary onEll is to be found in section 3. The expositionbeginswith a brief reviewof somespecific truisms, missing thefundamental pointthattheregion1 Birkeland limitations oftheE, j paradigm, notalways appreciated along side currents areinduced by themagnetic shearbetween thepolarfields theeffectiveapplications of theparadigm.Thenthewellknowndy(cardedby the solarwind into the antisolardirection)andthesur- namicalequations areworkedoutagainfromMaxwell'sequations , PARKER: ALTERNATIVE PARADIGM 10,589 toillustrate thenatureof theB, v paradigm. Severalstaticmagne- 1961;Kellogg,1962;Walbridge,1967;CoronitiandKennel,1973] tospheric phenomena aretreatedwithbothparadigms in section4, onlyin the specialcasethattheionospheric windsarekeepingup from which the reader can see the individual merits and limitations preciselywith the tailwardtransportof the individualflux bundles undersimplecircumstances. Finally,thepapertreatssometimede- involvedin thefluxtransfer events.However•s hasnopowerto pendentcasesin sections7 and 10, easilyhandledwith B andv maintain thisstationary (0/0l ----0) condition. Forinstance, when butdifficultwithE andj. Thepitfallsof declaring anequivalentthereis no reconnection of field linesbetweenthe magnetosphere electriccircuitfor theE, j paradigm arenotedin sections 10and andthe solarwind, themagnetopause is preciselyanequipotential 11. Section8 develops theB, v paradigm in a partiallyionized surfaceso far as the externalEs is concerned.So therecan be gasappropriatefor the ionosphere. no electrostatic stimulation of theinterior,thatis Es by itselfdoes nothingto createconvection.What little convection theremaybe 2. The E, j Paradigm (~ 102m)isdriven bythesmall-scale fluctuations in themagnetopause[Axford,1962; Parker, 1958, 1967a,b, 1969a;Lerche, As alreadynoted,the]g, j paradigm considers theelectric 1966, 1967;Lercheand Parker, 1967; Eviatar and Wolfe,1968; fieldE withinthemagnetosphere to be a consequence of theac- Vasyliunas,1970;Schieldgeand Siscoe,1971;CoronitiandKentive inwardpenetrationof the electricfield Es = -rs x Bs/c nel, 1973; Tsurutaniet al., 1992]. in thesolarwind,followingtheideasof Alfven[1939,1950,1981], That is to say,it is theMaxwell stressprovidedby a nonvanDungey[1958,1961],Oguti[1971],Heikkila[1974],Banks [1979], magneticfieldBa. thattransmits themomenLysak[1990andref. therein].Theconventional E, j paradigm is ishingperpendicular tum of the solar wind across the magnetopause into the magnetostatedconcisely by McPherron[1991,p. 618] in a recentreviewof sphere.The sameBa. providesa potentialdifference acrossthe substorms. magneticfield,butthestresstransmitted by theelectricfieldis small O(v•/c •) compared tothemagnetic stress. Tsurutani andGonza- ...Sincefield linesare generallyequipotentials, the electricfield of thesolarwindis transmitted intothemagnetosphere andionosphere by theconnected magnetic fieldlines.Thiselectricfieldsets up a convection systemthatmovesmagnetospheric plasmafrom thetail behindtheEarthto thedayside.Theionospheric plasma of the forcearisingfrom outrightreconnection of thegeomagnetic undergoes thesamemotionandin additionconducts electricalcurrentsdrivenby theelectricfield... ward magnetopause. It is this substantial reconnection andthe as- lez,[1995]estimatethattheshearstress exertedbythepassing solar windviathesmallscale fluctuations isapproximately 1- 3 x 10- • fieldlineswitha southward interplanetary magnetic fieldatthesun- sociated largeMaxwelltangential forceonthemagnetosphere that On theotherhand,boththeE, j andB, v paradigms are initiatesthe substorm,of course. sometimes indicated in the contemporary literature on magnetoImagine,then,that a staticmagnetosphere is disturbedby the sphericactivity.We note[Gonzalez et al., 1994p. 5774]where abruptreconnection of a geomagnetic flux bundleintoa southward they statethat ...Theprimarycauses of geomagnetic storms at Eartharestrong dawn-to-dusk electric fieldsassociated withthepassage of southwarddirectedinterplanetary magnetic fields.The solarwindenergytransfermechanism is magneticreconnection betweenthe IMF andtheEarth's magnetic field.Thebasicenergy transfer processin Earth'smagnetosphere is the conversion of directedme- interplanetary magneticfield Bs at the sunwardmagnetopause. Thereconnected fluxbundleisrapidlytransported by thesolarwind into the geomagnetictail, during which time the massiveionosphericfootpointof thereconnected flux bundleresponds butlittle. It follows that the electric field E = - v x B/c within the recon- nectedfluxbundledeclines from]gsatthemagnetopause tozeroat the unmovingionosphere.So the electricfield generallydoesnot map downwardalongthe field linesin the activemagnetosphere, that is the electricfield hasno powerof penetrationto drive the motionof the plasma,contr•aryto the commonassertion.Rather, it is the Maxwell stressexertedon the ionosphereby the tension energyinputrate... andpressureof thedisplacedreconnected magneticfluxbundlethat drivestheionosphere.More simply,it is theaccumulated displaceof the flux bundlethat Another pointtobenotedisthatthebasicroleascribed toj in ment,ratherthanthe rate of displacement, theE, j paradigm, together withthenotion of anapplied electric drivesthe ionosphere.The electricfield mapsalongthe magnetic field]gs,hasledto theidea[Alfven,1981;Heikkila,1974]that field linesonly if andwhentheMaxwell stresses canbringtheionomagnetospheric activitycanbe represented comprehensively and sphericconvectioninto a steadystatethat keepsup with the flux preciselyby simpleelectriccircuits.The notionis a curiousone, transferat the magnetopause. chanical energy fromtheflowof thesolarwindintomagnetic energystored inthemagnetotail, followed byitsconversion intoprimarilythermal mechanical energy intheplasma sheet, auroral particles,tingcurrent, andJouleheating oftheionosphere. Extraction of energyfromthesolarwindrequires a netforcebetween thesolar windandEarth,withforcetimessolarwindspeed givingthe thata dynamical systemwithinfinitelymanydegrees of freedom To considertheproblemfroma differentpointof view,E, canberepresented by a fixedelectriccircuitwithonlya fewloops givenby - v x B/c, cannot bethecompelling physical effect,that (afewdegrees of freedom). In fact,it isoftenpossible toprovide an is theprimemover,because thedynamical equations mustbe coapproximate circuitanalogoncethebasicmechanics is workedout variant. That is to say,the equationsmustbe couchedin termsof fromthedynamical equations. Howeverunfortunately, thecircuit physicalconcepts thatareapplicable in everyinertialframeof ref- analogcanbe established onlyafterthefact,andasnotedin section erence,for exampletheframeof thesolarwind andtheframeof the 3, it ignoresimportantaspects of thefluid motions,soit is of little practicalusein mostcases. magnetosphere. Soif Es wereto beconsidered theprimemover for excitingmagnetospheric activityin the frameof reference of Now the fieldlines[Stern,1994]areequipotentials onlyin Earth,howwouldwe treattheexcitation of thepassing solarwind precisely stationary (O/Or = 0) conditions [Stern, 19771.The by thenonconvecting magnetosphere? Thereisnosignificant elecconceptof equipotential field linescan be appliedto magneto tricfieldwithinthemagnetosphere. Hencetherewouldbenoprime spheric convection [Gold,1959;Dungey,1961;AxfordandHines, moverandhencenoexcitationof thesolarwind. On theotherhand, 10,590 PARKER:ALTERNATIVE PARADIGM which there is then no electric field, there is no basis for activat- The unipolarinductorprovidedby the motionof Io relative to therotatingmagnetosphere of Jupiteris anotherexampleof the ing the magnetosphere. However,therewouldbe an electricfield proper useof theE, j paradigm, withtheprecisely defined motion if the calculation were carried out in the frame of the solar wind, in andLyndon-Bell, 1969]. ]grn= q-vsX B/c in themagnetosphere whichactivates thesolar of Io providingthedrivingemf[Goldreich wind, contraryto the conclusionreachedin the frameof reference On the other hand, it will become clear in the illustrative exof the magnetosphere. amples thatthe]•, j paradigm isoftenhampered by(1)theneedto It shouldnot go unnoticedthat, with the nonrelativisticLo- knowtheperturbed fieldB q-A B in orderto describe theparal- ill toasufficient degree ofprecision and(2)theneed to rentz transformation E• = E q-v x B/c,anelectric fieldEllpar- lelcurrent by analgeallel to themagneticfield, arisingfromintensefieldalignedcurrent knowthefluidmotionv before]• andj canberelated (tensor)Ohm'slaw. Thusit is relativelyeasyto compute E ill'isuniquely defined inallflames ofreference, Ell- Ell,just braic andj onceB andv areworkedoutfromthedynamical equations as B is well definedandinvariant,neglecting termssecondorder inv/c. (whichgenerally donotdepend on]• andj), butit isdifficult togo theotherway.Unfortunately, thesefactsaresometimes overlooked Totreattheotherhalfof the]•, j paradigm consider theoften andthepath ofjlI isbased ontheunperturbed B, while Ohm's law stated concept thatj is thecauseof AB. In theusualmeaning of is appliedto E in thereference frameof thecoordinates ratherthan theword"cause"thisimpliesthattheenergythatgoesintothecre- to the field E • = E + v x B/c in the local frameof the mov- ationof AB, increasing themagnetic energy byB ßAB/47r per ingfluid.In fact,lgt cannot becomputed untilafterv andB have unitvolume,is supplied byj. However, thisisnothowAB comes beendetermined.Thusfor instance,the comprehensive magnetoabout.The energysourceis primarilythe kineticenergyof the solar wind which impactsthe magnetopause, grabsbundlesof flux, and deformsthe elasticterrestrialmagnetosphere. The fluctuating solarwind andthe associated varyingdeformationprovidea vary- derrepresentation of geomagnetic disturbance. However,theycan providethe correctionosphericcurrentpatternonly in the second ing •7 x B, with whichthereis associated incidentally a varying iteration with v deduced from the Lorentz force calculated the first sphericmodelsof Vasyliunas[1970] andWolf[1983]illustratehow farthe]g• j paradigm canbecardedtoprovide anadhocfirstor- j, according toAmpere's law.TheMaxwell equation OE/Ol -- c timearound.A skillfultheoretician canaccomplish theseiterations, V x B - 4•rj tellsusthatanymomentary deviation of 4•rj from butwhy useproxyvariablesanda trickyadhocapproach whenthe cV x B quicklyproduces anE thatcompels j through Newton's B,v paradigm provides thepartialdifferential fieldequatons forthe equations for themotionof theelectronsandionsto becomeequal comprehensive deductiveapproach? Sections6.2 and9 provideiltocV x B/4•r. Theelectric fieldisalsodescribed byFaraday'slustrationsof thedifficulties(1) and(2), respectively. induction equation V x E = -(1/c)0B/0t, of course. The The samegeneralproblemarisesin the popularpracticeof declaringanequivalentelectriccircuitfor a dynamicalsystem,forformationof themagneticfield, asis easilyshownfromPoynting's theorem(section 3). So AB is theenergysource, drivenby the gettingthatthe currentpathsaregenerallynotknownuntilthedyenergyto createthe necessaryelectriccurrentcomesfrom the de- thatthe mechanical forcesof thesolarwind,thatcreates j. Thatis to say, namicalproblemis solvedandB is knownandforgetting forcedependsuponthe motionof the curin the magnetosphere the magneticfield is the causeof thecurrent. effectiveelectromotive rentpathacrossthe magneticfield, whichis notknownuntil v has It shouldbe recognizedthat the situationis differentin the been determined. In fact, the correct electric circuit is distributed laboratoryplasma,wherethe actionis ofteninitiatedby discharg- overv(r, t), anditsproper description isprecisely stated by the ing a condenserbank throughthe plasmaand/orthroughexternal dynamical equations for B andv, fromwhich]• andj areeascurrentcoils. Thereis one uniquereferenceframe,definedby the ily computed onceB andv aredetermined. Section11provides a laboratoryboundaries of theplasma.An enormous potentialdiffer- simpleillustrativeexample. enceis appliedto the boundaryof the plasma,in contrastwith the equipotential magnetopause. The energyis deliveredto thesystem by theappliedandwell definedE, causing electriccurrents toflow anddoingworkat theratej ßE. Themagnetic fieldsarecaused byj, through Amperes law(i.e.,theBiot-Savart integral), andE is 3. The B, v Paradigm Applicationof the B, v paradigmto the activeplanetary magnetosphere [Parker, 1962] is basedon Newton'sequations of clearlytheprime mover. motion,includingtheMaxwell stress,andon Maxwell'sequations, It is interesting to notethattheE, j paradigm is alsoanes- alongthelinesindicatedin thegeneraldevelopment of theparadigm sentialpartof thepicture,alongwith B andthedynamics of the by Lundquist[1952] andElsasser[1954]. The mutualconsistency individualionsand electrons,in the plasmastructureof the ideal of the equationsbecomesevidentuponcloseinspection.For instationarymagnetopause over the cyclotronradiusof theimping- stance,Maxwell's equationsassertthatelectricchargeis conserved, ing solarwindions. The magnetohydrodynamic B,v paradigm is which is tantamountto assertingconservationof particles,and not applicableto suchsmallscales,of course,andtheproblemis hencemass. Then Newton'sequationscanbe appliedto the mononlocalbecausethe impingingsolarwind ionsplay theroleof an tionsof the individualionsandelectronsof a collisionless plasma initial appliedemf, driving currentsalongthe field lines into the to computethe meanelectriccurrentdensity. Substitutingthe reionosphere as an essentialpart of the stationaryequilibriumof the sultingexpressionfor the currentdensityintoMaxwell'sequation magnetopause. Detailedanalysis[Parker, 1967a,b; Lerche,1967; LercheandParker,1967]showsthattheionospheric resistivity dis0E = cV x B- 4•rj (1) sipatesthe currentsso that therecanbe no enduringequilibrium. Ot EviatarandWolf[1968]suggest thatthemicrostructure of themagnetopause is dynamicallyunstablesothatthe absence of stationary yieldsNewton'sequationof motionfor the macroscopic bulk moequilibriumis not confrontedin nature,leavingsomeinteresting tion of the plasma[Parker, 1957]. Thus theparticledynamicsis questionsunanswered. suchas to fulfill Ampere'slaw automatically.This detailedresult, PARKER: ALTERNATIVE PARADIGM 10,591 whichis describedin thenextsection,is impliedby Poynting'sthe- ditionsin inmrplanetary space[Tsurutani et al., 1990;Tsurutaniand orem,of course,showingthattheconceptof energyandmomentum Gonzalez,1993]. For instance,the expansion phaseof a substorm carriesover from mechanicsinto electromagnetism. appears tobethedirect consequence ofswitching onthesame Ell [Zhu, 1995]. That is to say,the In summary, Newton's equations at themicroscopic scaleof thatcausestheauroralenhancement of the tailward the individualions andelectronsautomatically prescribemotions region 1 Birkelandcurrentsheetis a consequence displacement of magnetic flux, and when the current densitybethatproduce theelectriccurrents required byAmpere's law.Comcomes large enough to generate plasma turbulence (anomalous rebiningthisfactwithPoynfing's theorem, it followsthatthemean andelectric double layers) therequired Ell becomes large. macroscopic bulkfluidmotion of theionsandelectrons isalsode- sistivity showthattheEll releases themagnetic fieldin the scribed by Newton's equations in mrmsof themeanmacroscopicCalculations Maxwell stressransot.This shouldcomeasno surprisein view of outermagnetosphere from its line tyingto theionosphere, thereby of thegeotail.Thesudden releaseappears thegeneralcompatibility of classical mechanics andelectromag-facilitatingtheexpansion to be the cause of the Pi 2 pulsations (Zhu, 1995). neticfieldsin whichtheequivalent momentum of a particlewith charge e canbewritten Pi + eAi/c, where Ai isthevector potentialin theLorentzgauge.Nora,then,thatif Ampere's lawwere The directapproach of the u• B paradigmto the driving forces, Pij, Rij, Mij, etc. simplifies manyaspects of thethe- not automaticallysatisfiedin the large scaleby the individualparory of magnetospheric activity. The firstpointis thatthe current ficle motions,it would not be possibleto wrim the Maxwell stress in mrmsof E and B alone. Somefunctionalof theplasmave- j flowsacross B where, andonlywhere, theplasma pushes against thefield. Wheretheplasmadoesnotpushagainst thefield,j is locity v and its derivativeswould appearin the Maxwell's stress. necessarily parallelto B. Soit is theplasmaandparticlepressure Physicswould thenloseits covarianceandwe wouldlive in a difdistribution thatdetermines theelectriccurrentpatterns, whichcanferent world. nototherwisebe construcmd. Hencetheassertion of anequivalent It follows that the time dependentmacroscopic physicsof electriccircuitgenerallycannotbe madeuntiltheplasmapressure magnetospheric activityreducesto thedynamicsof theconrending particle pressure Pij (thermal momentum flux),theMaxwellstress ransot Mij,mo- B•+ Bi Bj (2) andtheReynolds stress tensor Rii, I•ij -- --pUiUj. andMaxwellstress havebeenestablished. Thatmeansa properdynamicalsolutionto theproblemmustbe in hand.For instance,the region1 and 2 Birkelandcurrentsfollow as a directconsequence of Ampere'slaw appliedto themagneticfluxbundlesdisplaced by the solarwind from the sunwardmagnetopause into the geomagnetictail andto theensuingsunwardreturnflow of field aroundthe peripheryof thepolarionosphere [Zhu, 1993]. (3) Quantitatively, thetiltof thepolarmagnetic fieldBp (~ 0.6 Thusfor a plasmawith densityp andmacroscopic bulkmotionui, thedynamicalequationhasthebasicNewtonianform 0 Opij + OR• -•OM• - - G) connectinginto the geotailis observedabovethe ionosphere as thechangeAB in thesunward horizontal component, typically- 2 x 10-3 G(200nT).TheMaxwell stress BpAB/4•rinthean- (4) tisolardirectionon thepolarionosphereis thereforeof theorderof 10-4 dynes/cm 2, initiating ionospheric andmagnetospheric con- vectionin the antisolardirection.Typicalconvectivevelocitiesv to whichmay be addeda gravitational force,viscosity,etc. From in theF layerare 1 km/sin theantisolardirection[Heppner,1977], thefactthatE - -u x B/c in a collisionless plasma it follows representing the magnetospheric convectionabovethepolarionofrom Faraday'slaw of inductionthat sphere.The Maxwell stressdoesworkon theionosphere at theram OBi Ot - vBpAB/4•r~ 10ergs cm-2s -x. Thetotal work over thepolar ionosphere (witharadius of20ølatitude orabout 2 x 108cm) is thenoftheorder of 101Sergs/s -- 10TMW [Zhu,1994a, b]. This powerinputis ultimatelyconsumedby viscousdissipationandrewhichdecrees thatBi is transported bodilywithui. There are occasionalandimportantexceptions, of course,as sistivedissipation,of course(seedetaileddescriptionin section11). alreadynoted.The mostintorestingadditionto theinductionequa- Zhu pointsoutthatthereturn(sunward)magnetospheric convective bythesunward tiltedfieldA B ~ 8 x 10-3 G (800 tionisa significant electric fieldEll parallel toB, arising where flowisdriven the plasmais too tenuousto carry the electriccurrentdemanded nT) observedaroundthe peripheryof the polar ionosphere.The by Ampere'slaw. A casein point arisesin the region 1 Birkeland ionospherearoundthe peripheryis drivensunwardby the magnecurrents in theextremely lowdensity (•<1 electron/cm a) beyondtosphericconvectionat speedsof theorderof 1 km/s,sothepower theplasmapause. Theappearance of themrm--cX7x Ell onthe inputis of theorderof 30 ergscm-2 s-l, overa characteristic widthof 10s cm,sothattheworkdoneisof thegeneral order of 3 x 10xSergs/s3 x 10TMW. right-handside of (5) violamsthe frozen in conditionon the field [Coroniti and Kennel, 1973; Schindleret al., 1991]. If, on the other hand,the plasma,or an un-ionizedgaseousbackground,becomes so densethat thereare manyinmrparticlecollisions,asis the case Applicationof Ampere'slaw to theboundary betweenthean- in theionosphere, a resistive diffusion ransot r]ij isincluded onthe tisolar(polar)magnetictilt andthe peripheralsolarmagnetictilt right-handsideof (5), whichalsogetsawayfrom thefrozenin con- providesthe region 1 inmnseBirkelandcurrentsheet,while the dition. graduallydecliningsolartilt beyondthe peripheryprovidesthe With thesepointsin mind a varietyof magnetospheric phe- morediffuseregion2 Birkelandcurrent,akeadynoted.TheBirkenomenacanbe understood in simplemrmsfromthedisturbedcon- landcurrentsarea simpleanddirectconsequence of Ampere'slaw 10,592 PARKER: ALTERNATIVE PARADIGM andthetailwardtransportof magneticflux bundlesthathaverecon- wheremesubscript _1_ denotes thecomponent perpendicular to B. nectedwith a southwardinterplanetarymagneticfield at the sun- Theplasma pressures P_t.andPll(perpendicular andparallel toB, respectively)are not always equal in a collisionlessplasma,in ward magnetopause. case there isa centrifugal force termK (Pll- Pñ) asa Finally,it should benotedthatthevariables F• andj areeas- which consequence of themotionof particlesalongthecurvedfieldlines. ily computed, whenneeded,f•omu andB, with thewell known The resultis the equationof motion relations E .1.- -u x B/c andj- c•7x B/4•r. du 4. Newton's and Maxwell's Equations Havingasserted thecompatibility of Newton's andMaxwell's equations at boththemicroscopic particlelevelandat themacroscopic fluidlevel,it isimportant tounderstand thetheoretical basis for thissalutary condition. Firstof all, Maxwell'sequation (1) im- Plasma turbulenceexcited by the anisotropicthermal motions pliescharge conservation, asalreadynoted.Thedivergence of (1), reduces P.I_- P[[in allbutthemostrarefied plasmas, so with theadditional relationV ßE -- 47r6,where6 is thecharge rapidly V)B]ñ -v. (p.+ B2 + [(Bß (1+P_t.-Pll ). (7) thatusually density,yields O6 --+ V.j B2 -0 V)B]ñ du•_ ___ -V•_(p•_ + )+ [(B. 47r which is the statementof conservation of charge. Second,if the electric current density isinadequate tosatisfy Ampere's lawj c•7x B/47rbysomesmallamount A j, it follows fxom(1)that 0t (8) for slowbulk motionu. At the sametimetheplasmaparticlesrelax toward auniform distribution along thefield,withEll < < E•_and = 4r/xj at --v"a (0) Theresultis a rapidgrowthof E in thedirection of theinadequacy arethestatement of Newton'slawsof motion,writAj. Thecurrent carrying particles behave in amanner described by Theseequations tenin generaltermsin (4). They areaccompanied by theinduction in suchawayastomakeuptheshortfallin theelectriccurrent.That equation(5) whichfollowsdirectlyfxomFaraday'slaw of induc- Newton's equations, sotheyareaccelerated bytheincrement in E of the is tosay,theexistence of thedisplacement current t2E/Otguar- tionfxomthefactthatE_t.isgivenas-u x B/c interms anteesthat the electriccurrentfollowsAmpere'slaw very closely, electric driftvelocity u = cE_t.x BIB • deduced fxom Newton's with0E/0t generally smalltosecond order in v/c compared to equations. c•7 x B and47rj.It isthissmallt2E/Otthatproduces theparalChew et a1.,[1956] provideanotherimportant derivation of lelelectric fieldEll responsible fortheaurora. Thus, forinstance,themacroscopicmomentum in the absenceof the currentrequiredby Ampere'slaw a parallel plasma,with theprescription electric fieldof 10- :•volts/m, providing anaccelerating potentialequationfor a collisionless of104voltsover10Skm, would arise in10-s sinamagnetic shear where thedirection ofB (0.6G)changes by10-s radian inadis- dpB pñ=0,dt dB 5pll dt p3 =0, tance of2kmacross B(I V x BI= 3 x 10-9 G/cm). Soforallof (10) itsusualdiminutive size,OE/Ot hasprofound effects, providing andparallelpressure components basedonthe theauroraandpreserving Ampere'slawbecause theparticles move for theperpendicular transverse andlongitudinalinvariantsof theparticlemotion. accordingto Newton'sequations. The intimate relation between Newton's and Max- Consider,then,the assertionin section3 that the equationof well'sequations canbe shownin a varietyof ways.For instance, motion(7) can be deducedfrom Maxwell's equation(1) usingthe Newton'sequations canbe deduced fromMaxwell'sequation (1), guidingcenterapproximation for themotionof theindividual partiwhilethemagneticinductionequation(5) followsfxomthemotion clesto represent theelectriccurrent.We startwiththewell-known oftheparticles (inwhose fxame E t - 0),i.e.,fxom Newton's equa- resultWatson [956] thatthemeanmotionv of theguidingcenter tions.Briefly,Watson[1956]andBrueckner and Watson (1956)computed theionandelectron trajectories in amagnetic field B withcharacteristic scaleœlargecompared to thecyclotron radii of theindividual particles.Theyusedtheguidingcenterapproximation andwent on to solvethecollisionless Boltzmannequationfor is 1 v --u+ (•Mwlc/qB4)B xVB•/2 (11) -]-(Mw•c/qB4)B x [(B.V)B] smallperturbations usingthecomputed particletrajectories, which q, andthermalvelocitycompoconstitute thecharacteristics of theBoltzmannequation.Theresult for a particleof massM, charge istheequivalent equation ofmotion fortheelectric drift velocity nents Wl[ and w_t. parallel and perpendicular tothemagnetic field, u =cE x B/B• oftheparticles intheform of(4).Inparticular respectively. Themotion parallel tothefield isdescribed by thereisacontributionffompij arising an anisotropic thermal ß[(B (12) velocity inthe presence offfom the curvature Kofthe field (dv/dt)ll - -( •1wñ:•/B4)B{B ßV)B]} lines. Thedistribution curvature is K -- [(B.V)B]_t. (6) Fora singlyionized gasconsisting of N ionsandN electrons per unit volume,the currentdensityis the differenceof the sumof PARKER: ALTERNATIVE eVion andeVelectronoverall theparticles in a unitvolume,sothat PARADIGM 10,593 Use(16)toeliminate 6 and(17)toeliminate j. Addtheterm BV ßB/4•r to theright-hand sideof (15)topreserve symmetry, with equation(11) for the totalmotionof eachparticle,theresultis with the final result ja. = (c/Be)B dv x[Vpa.+ [(Pll-Pa.)/Be]( B' V)B + pdu/dt] (13) P•- - iv. v, + + (v x e) x v, + (v x 4•r whenall of thegeometricalfactorsaretakenintoaccount.The sum 0ExB ofthefactors •• Mwa. e andM wlle overallparticles ina unit Ot volumeprovidesthe perpendicular pressurePa. andparallelpres- surePll'respectively, irrespective ofthethermal velocity distribution. Substituting thisexpression forja. intoMaxwell's equation (1) yields Ot = -(4a'c/Be)Bx {pdu/dt+ V(pa.+ -[1/4•rq-(pa.- pH)/Be][(B ßV)B]). 4•rc x (18) ' which can be rewritten as dvi Ot 0 Pi P'•-•-+ c• = OMii Ozj (1O) wheretheMaxwellstress tensor Mij is (14) Mij -- --6ij E e +B e 871' + + (20) 4•r Given thatu -- cEa.x B/B eitfollows thatthelefthand side of thisequations is smallcompared tothetermpdu/dt onthefight andPi is thePoyntingvector hand side bythefactor Be/8?rpc eorue/ce,soitmaybesetequal to zero. The resultis themomentumequation(3.4) and(4.2) again [Parker, 1957]. Pi - c½ij•EjB•/4•r, (21) tensor.It is convenient to Theessential pointof thesecalculations is thatthemotionof wherein ½ijk is theusualpermutation theindividualelectronsandionsis automatically suchasto satisfy rewritetheleft-handsideof (18) in termsof theparticlemomentum stress tensor ]•ij givenby(3)sothat Ampere's lawrelating j andV x B (neglecting termsof theorder fluxpoi andtheReynolds ofue/ceorBe/8•rpc ecompared toone) if weaccept thefactthat thebulk motionsatisfiesthe usualmomentumequation. O( Pi) OMij ORij (22) Therearespecialconditions, of course, for exampletheenergeticparticles in theoutermagnetosphere andin thegeotail, where byinspection thatPi/ce represents themomentum theguidingcenterapproximation is notuseful.In suchcircum- It isevident stances a complete formaltreatment of therelationbetween New- density of theelectromagnetic field,whileMij q- Rij represents ton'sandMaxwell'sequations fallsbackon theentirelygeneral thetotalstressor momentumflow field. It mustbe appreciated that considerations of Poynting's theorem, takenupin thenextsection. (22) appliestoeverypointin space,wherep mayormaynotvanish. 5. ElectromagneticForce, Momentum, and Energy The intimaterelationbetweenNewton'sandMaxwell's equationsdescribedin section3 andsection4 is impliedby Poynting's classicaltheorem,establishing theequivalence of mechanical momentumandenergyandelectromagnetic momentum andenergy. Weprovidea briefreviewof thebasicprinciples. Newton's equa- tionforelectrically charged particles inanelectromagnetic fieldEs B can be written dv vxB - The electromagnetic energydensityandenergyflux follow similarlybeginningwith the Newtonianenergystatement 0 1 N(pv - v. =j.E - 4•r xB 4•rc9t .E C[B. V x E- V-(E x B)] 4•' + 0 E Ot 8•r ' as where p(r, t) is themass density and5(r, t) isthecharge den- SincecV x E - -OB/Ot, thiscanberewritten sity. The individualelementary particlesarerepresented by mov- inglocalized maxima in p(r, t) andmoving localized extrema in 6(r, t) coinciding withthemaxima inp. Thus p(r, t) and6(r, t) 5eve+ Ee+ +V.P-0, 0(1 Be) (23) are boundedcontinuousfunctionsof spaceandtime, with characteristicscalesof 10-13cm atthelocationof theindividualelectrons where againP isthePoynting vector cE x B/4•r, givenby(21). It is evident by inspection that the electromagnetic energydensity is andions,andotherwise zero.Thevelocity v(r, t) isdefined only theelectromagnetic energy fluxisrepresented at thelocationof theindividualparticles,withitsvaluein theinter- (E e+ Be)/ 8a'and themechanical sticesof no physicalinterest.We startwith Maxwell'sequations by thePoynfingvector.ThusPoynfingestablished equivalence of electromagnetic andparticlestressandmomentum enumerated in section4. V ßE - 4•r6,V ßB - 0, (16) fields,with theconsequences 0B 0E Ot= -cVxE,4a-j + •- - +cVxB. Note,then,thatj -- v6 ontherighthandsideof (15). NowwhenE = -u x B/c, it isclearthattheelectrical contributionto the momentumandenergyis smallto secondorder (17)in v/c compared tothemagnetic contribution. Neglecting secondorderterms,(20) reducesto (2) whichstatesthatthemagneticfield 10,594 PARKER:ALTERNATIVE PARADIGM possesses anisotropic pressure B2/8•randa tension B2/4•rin direction.It is easyto showthatthe exactdynamicalequations are thesingledirectionalongthemagneticfield. It is easyto showforreallyfromthevirial equations thattheneteffect,averaged overall Ou Bo ObOb P•'4a'Oz' O•= BoOu 0•' threedimensions, is dominated bytheisotropic pressure [Parker, 1953, 1954, 1969b, 1979]. That is to say,a magneticfield, if left to itseft,expandsto fill all of theavailablespace,therebyminimizing its energy.The solidEarthprovidesthe forcesthatanchorthe geomagnetic field,of course,andthe quiet-dayfieldin theregion aroundEarthadjustsitseftintotheminimumenergyconfiguration consistentwith that constraintandthe confiningmagnetopause. (25) Then for the wave u(z,t) -- Usin[w(t- z/C) + •o], where (7-- Bo/(4•rp) -}and T isanarbitrary constant, itfollows that Finally, note that the foregoingresultsapply at everypoint b(z,t) - -u(z,t)(4•rp)«. (27) in the electromagnetic field, andthereforethey applyto any averagingovermanyparticlesto providea macroscopic fluidformula- Themagnetic fieldhasy andz components b andB0, respectively, tion.Theaveraging overthenonlinear terms, forexample Bi Bj, sothatthemagnitudeof thefieldis etc.omitsthesecond orderterms 6Bi • Bj where 6Bi, represents thelocaldeviationfrom themeancausedby theindividualparticle. The standardderivationof Poynting'stheoremin mosttextbooks [cf. Panofsky and Phillips, 1955;Jackson,1975]is formulatedin termsof the integralof themomentumandenergyoveran arbitrary fixedvolumeV. Thematteristreated ascontinuous fluid B(z,t) - [B•+ 4•rpU2(z, t)]«. (28) The energytransported by the wavecanbe computed in threeobviousways.Firstof all, thegroupvelocityis thesameasthephase andp, 6, v, j, E, andB aretreated assmooth continuous func- velocityC, sotheenergyfluxI canbewritten tionsof spaceandtime. Then,sincetheshape,size,andlocationof V is arbitrary, it ispointedoutthattheintegrand, namelyequations <v >+< (19) and (23), mustbe satisfiedat everypoint in space. In thecircumstance thatE = -u x B/c, it follows thatthe representingthe energydensityof the wave propagatingat the Poyntingvectorreducesto Alfven speed,whereangularbracketsindicatethe time average. With (26) and 27) it follows that P- B x (u x B)/4•r = uñBe/4•r. I - p < u2 > C, (24) The Poynfingvectorrepresents the transportof magneticenthalpy 2 Be/4•rwiththeelectric driftvelocity u inthedirection perpendicular to themagneticfield. It illustratesagainthebodilytransportof the field in the movingfluid. On the other hand, the rate at which the Maxwell stressdoeswork acrossanysurfacez -- const is It is instructiveto derive this sameresultdirectly from the mechanical workdoneby theMaxwell stresscarriedin themoving I-- < ub > B/4•r, (30) fluid.Theworkdoneonanelement of areadSj bythemotion ui in oppositonto the Maxwell stressis which comes out the same, of course. Finally, the energytransportcan be computeddirectlyfrom (24) notingthat 4•r ( Beu•B•BJ)dS j -uiMqdSj - uj8•r = wheretheMaxwellstress tensor isgivenby(2),orby(20)uponnot- ingthattheterms inEe aresmall tosecond order inu/ccompared totheterms inBe. Theenthalpy fluxisthesum oftheconvective transport ofmagnetic energy ujBe/8a'and therateatwhich work Thus isdone byuj, sothattheenergy orenthalpy flowis(ujB2 - u•B•Bj)dSj/4•r.Toobtain theenergy flowacross dS•from u• - u-u.B/B - - = %uSo•/S 9'- e•ub/S. (31) uñB•/4•' - eyuBo2/4•r - e,ubB/4•r. (32) (24), note that the velocityparallel to Bi can be written Thetimeaverage of they component is zeroandthez component u}B}Bi/B e. It followsthatuñ canbe written givestheresultakeadyobtained.Thus(24) tellsusthatthemagu}B}Bi/B e, sothat(24)gives thesame result. Thusthetrans- netic energy B0 •/4a'iscarried inthetranverse oscillation inthey portof energyis in thedirectionperpendicular to themagneticfield andincludestheworkdoneby theMaxwellstressin additionto the convectivetransportof magneticenergy. Applicationto a plane transverse Alfven wave in an invis- cidinfinitelyconducting incompressible fluidof uniformdensityp showshowtheenergyis transported in thatsimplecase.Considera direction, butrepresents nonettransport. Thereisanettransport in thez direction because uñ hasa nonnegative forwardpointingz component (eventhoughu doesnot),givenby ub B U •' = --sin2[w(t - z/C) + •]. C (33) plane Alfvenwavewithvelocity u(z, t) andmagnetic fieldb(z,t) Sotheenergyfollowsa serpentine or sinusoidal pathbutalways in the y directionpropagatingalongthe uniformfield B0 in the z progresses in the directionof propagation of thewave. PARKER: ALTERNATIVE PARADIGM 10,595 nevD whereVD is the ion gradientdrift velocitygivenby (11)as 6. Application to Stationary States Theapplications of theE, j andB, v paradigms to simple cp dB (34) casesof stationarycurrentsandmagneticfieldsshowtherelativeef- ficiency of theE, j paradigm in treating certainsymmetric cases, as well as the limitationsanderrorsarisingin nonsymmetric con- = 3cp/ea figurations. In fact,theefficiency of theE, j paradigm arises pre- (35) ciselybecauseit avoidsthe basicdynamics,concentrating on the westward. The azimuthal drift current is simpleconservation laws appliedto the geometryof a symmetric I• - 3cnp/a, (36) unperturbedmagneticfield. As we shallsee,the schemefails in the absenceof symmetrywhenwe cometo computetheperturbed fieldin the field configuration,becausethe field alignedcurrentsarenecessar- aroundthe ring of radiusa, producingtheperturbation of theoriginwith z component ily alignedalongtheunperturbed field.TheB, v paradigm avoids neighborhood thedifficultybecauseit computesthefieldperturbations fromstress ABI> - --27riD/ca balance,finally deducingthecurrentsfrom theperturbedfieldwith theaid of Ampere'slaw. We shallseethatthetopologyof thecur- = -3pAf/a•. rentsisdifferentfromresults obtained withtheE• j paradigm. The applications beginin subsection 6.1 with thewell known axially symmetricproblemof a circularbandof trappedions,that is, a uniformring current,circlingEarth, to illustratethe methods (37) Thereis anadditional contribution AB• fromthediamag- netic moment p ofeach ionintheamount p/a sperparticle, or of thetwoparadigms, showing theefficiency of theE• j approach ABe,- +p.A/'/a 3. relativeto theB• v paradigm in suchsimplecircumstances. Then (38) subsection 6.2 goeson to thenonsymmetric problemof a localized in the neighborhood of the origin. The diamagneticeffectof the equatorialclusterof trappedions,inflatinga singlesmallflux bun- cyclotronmotionof the individualion tendsto excludethe magdle in a dipole field. We employbothparadigmsagain,obtaining neticfieldfrom theregionof theions,therebycompressing thefield resultsthatare quitedifferentwhenit comesto thepathof theperturbed field lines and the associated currents. In view of the im- portanceof this resultthepresentation treatstheproblemin some detail. slightly elsewhere. SoAB• represents a fieldin thepositive z direction.The totalAB in theneighborhood of theoriginis the algebraic sumofABi9 andAB• yielding thereduction Finally, section6.3 considerstheproblemof anelementalflux AB• - -2tt.N'/.• (39) bundleinflatedwithanisotropicparticledistribution withina dipole field. The displacement of the flux bundleis computedto show in the z component of the dipolefield in theneighborhood of the theuseof theopticalanalogyin theB, v paradigm.TheE, j origin. paradigmis not applicablefor the reasonillustratedin sub-section If ABz is a short-lived perturbation, it doesnotpenetrate 6.2. deeplyinto Earth,whichacts,then,asa diamagnetic sphereof ra- 6.1. Equatorial Band of Collisionless Particles dius/•(<< a). Theresult istheperturbation magnetic field Considera uniformthinequatorialbandof energeticionscir- •-• I- -•- co,O cling ataradial distance w -- (z2+ y2)« _ a around athreedimensionalmagneticdipolelocatedat the origin andpointingin thenegative z direction sothatthemagnetic fieldB(w) atthepo- a3 1+• sitionof the ionsis in thepositivez direction.The massof each ionis M andthechargeis e. Theionsall havethesamevelocityto sin0 (40) perpendicular tothemagnetic field (pitch angle «a').There aren insphericalpolar coordinates (r,0,T)inthe vicinity ofEarth (R< ionsperunitlength around theband, foratotalofAf- 2•ran.An r << a). Thereduction ofthehorizontal component ofthefieldof equal number ofcoldelectrons ispresent topreserve charge neu- theequator (z -- 0,0 -- trality. The total kinetic energy ofthe ions is œ - A/' 1• Mw 2 = A/'pB(a) The efficiency oftheE, j paradigm isevident initsdirect some idea of the forces thatdeform the dipole field to achieve the path to the final result. However, itis not without interest to obtain expansion of thefieldin thevicinityof Earth.As a beginning, note that(39)canberewritten as wherep is thediamagnetic moment •1Mw2/B(a)oftheindi- ABz vidual ionandB(a)isthemagnetic field attheradial distance w -- a in theequatorial plane.Elsewhere in theequatorial plane atthesurface ofEarth is3pJV'/a 3. 2œ Bo = 3œE (41) i D 2//3 isthe B(w)- B(a)(a/w) 3.Assume thatœ issufficienfiy smallthat whereœE -- •B magnetic energy intheexternal(r > themagnetic perturbation AB(r) iseverywhere small compared //) unperturbed dipole field withintensity Boattheequator atthe tothe unperturbed field B (r). The easiestway to calculatethemagneticperturbation is to surface ofEarth, r -- R,0 - «7r[Dessler and Parker, 1959, 1968]. In thepresence of thediamagnetic Earth,thefractionalre- usetheE, j paradigm notingthatthetotalazimuthal current is duction of thehorizontal component attheequator atthesurface of 10,596 PARKER: ALTERNATIVE PARADIGM Earthisprecisely œ/œE,thefraction of thetotalmagnetic energy In otherwords,the ring currentI arisesin orderthatthe inward represented by the kineticenergyof the ions. Lorentz forceIB(a)/c balance thediamagnetic repulsion. 1 Consider, then,theB, v paradigm, withitsfocusonMaxwell Nowforanequatorial bandofionswithpitchangle• a',the stressandthegeometricdistortionof themagneticfieldby thepres- pressure tensor has the two nonvanishing components -- Pww, eachof whichis equalto thekineticenergydensureof the energeticions. The direct approachis throughformal Pqoqo solution of themagnetostatic equation, puttingui = 0 in equation sityoftheions. With•vrepresenting radial distance (x:•+ y:•)« (3.4) to obtain fromthez axis,it followsthatr - r t -- -ew •v fortheorigin 0 Oxj (Vii +Mij)-O. (r -- 0). Then B(r) - ezBo(R/w) 3 and B. (r- rt) - 0, (42) so that With axial symmetry this reduces to the quasi-linear Grad-Shafranov equation,whosesolutionis notelementarybecause of the unknown,arbitrary,and generallynonlinearfunctionof the an(o)- exceptin a small vectorpotential. So,aswiththeE, j paradigm, wetakeadvantage Integratingby parts,with Pww nonvanishing of thefact thatthesystemis only slightlyperturbed.Then,applying neighborhood of •v -- a, yields Ampere's lawtotheBiot-Savart integral, theperturbation AB (r) 2œ is aB()- AB(0)--e,,BoR----• i / dar'[V xB(r')] Ix(r-r') ' (45) where /_7dz œ- 2•r The equationfor magnetostatic equilibriumhathe presenceof the forceF perunitvolumeexertedby theparticles andplasma onthe dww pww (46) represents thetotalkineticenergyof theions. Thisresultisjust(41) again. field B is 4•F + (V x B) x B - 0. It is instructiveto look more closelyat the forcesexertedby the equatorialband of ions at w -- a. Denotingthe smallradial ThevectorproductwithB resultsin widthof thebandbyh(< < a), theionsexerta totalradialforce (V x B)a.- 4•rFx B/B 2 inward attheinner edge (w -:r- (,/h)• 1Mw • perunitlength wherethesubscript indicates thecomponent perpendicular to B. Symmetry decrees thatthere isnotorsion inthefield(•7x B)ll 0, sotheBiot-Savart integral forAB(r) becomes a) of thebandandoutwardattheouteredge(•v -- a q-h, assuming thatthe ionsaredistributed uniformlyacrossh). It followsfrom (44) that AB(O) --2a'.•' aB(A)(aq-h)B(a +h) all(r) - f aar B(r')[r(r'). (r- r')](r-r')] I r-r(r')[B(r'). r' I. (43) Specification of theforceF (r) exerted onthefieldsgives thedeformation AB (r) [Parker,1962]. The similarityof themathematics to theBiot-$avart integral 4•rf'h [1+O(h•)] a:•B(a) = 2p.Afla a (47) inwhich p -- «Mw:•/B(a), which isthesame result as(39), of forAB (r) in theE, j paradigm isobvious, of course. In that course.The pointis thatthe net effectof the sameforceper unit directions at a anda + h arises fromthe paradigm wewouldwriteF - B x j/c, fromwhichit follows lengthexertedin opposite andtheweakerfieldata + h. thatj_l. -- cF x B/B 2. Thus, forinstance, aringcurrent I of longercircumference radius a produces amagnetic field2•rI/ca atthecenter oftheting. It is also instructive to examine the forces exerted on the ions In terms of thetotaloutward radialforcef' = IB(a)/c perunit in thesmallangularsectorA•o of theband.At w -- a thereis an lengtharoundtheting, thefieldat theoriginis outward radialforceaA•o.•'. At w -- a + h, theforceisinward andof magnitude (a + h)A•o.T. Thecompressive particle force 2•r•' all(0)- aB(a) •'h is exerted in the azimuthal direction around the radius a of the (44)band,providing anetoutward radial force.]:'(h/a)aA•o.Thusthe Theforce.• is a consequence of thediamagnetic expulsion of the totalradial forceis zero, so that the ionsin the bandarein quasi- equilibrium withthemagnetic fieldB(r), withtheLorentz dipolemomentp of eachion,whichisopposed bytheLorentzforce static nqvD/cbalancing thediamagnetic repulIB(a)/c. Theforcef'i isexerted ona magnetic dipole momentforceofthedriftcurrent Pi bythefieldBj ispj OBi/c9:ej. Inthepresent instance, withp sion. pointing inthez direction, thisispOBw/Oz= 3pB(a)/a per 6.2. An Equatorial Clump of Particles ion, so that .7:'= 3npB(a)/a. As a consequence of thisforce, theion driftsin the azimuthaldirec- Consider a smallcluster ofiV' energetic ionswithmass M, 1 charge e, andvelocity w withpitchangle•r andindividual dia- tionwithavelocity VDsuch thatqvDB(a) -- c.7 z, yielding (35). magneticmomentp at a radial distancea in the equatorialplane PARKER: ALTERNATIVEPARADIGM 10,597 1 of a two dimensional magneticdipolefield. An equalnumberof coldelectrons is assumed aswell asa tenuous background of col- • J flowing awayalong thefieldlines fromthebackend(toward negativex) of the rectangle.It alsofollowsthattwo field aligned Jll converge fromeachsidetoflowintothefrontendof lisionless thermal plasma. Thedipole, withmoment BoRe per currents unitlengthin the z direction,extendsuniformlyalongthex axis therectangle,sketchedin Figure 1. The closureof the currentsysin theneighborhood of thedipoleat the (y = z = 0) andpoints inthepositive z direction, producing the tem acrosstheionosphere origin is assumedto be nondissipative, so thatU is conservedfor a steadystate.(In fact,the terrestrialionosphere is dissipativeand magneticfield theenergyœis slowlydegraded [Parker, 1966]).In thepresent B==- +Bo sin9o, B• - -Bo idealized(dissipationless) state,it followsthat cos9o (48) JII- cU/aB(a). (57) incylindrical polar coordinates w -- (y2q_z2)« and 9o(measured Stern [1992]suggests thattheregion1 Birkeland currents may aroundthe x axisfrom the y axis). In Cartesiancoordinates, beprecisely thisJIIproduced byenergetic particles trapped inthe geomagnetictail. 2BoR2yz BoR2(z • - y2) The next questionis the natureof the geomagnetic perturba- (49) tionproduced byJ.l_andJl['Thecurrent Jllflows along thefield lineson thesurfacez = 0, I betweenthetwo circles It is convenient to thinkof Bo asthepolarfieldat thesurface of a two-dimensional Earth,w --/i•. Themagnitude of thefieldis .- w -- acos9o, w -- (a + h)cos9o. (58) WithJII flowing along thefield,it follows (seeFigure 1)thatthe ribbon ofcurrent Jllhasawidth h(w) such thath(w)B(9o)- (50) const,whileconservation of current requires h(w)Jll(W ) = and the field lines are the circles const. w = 2• cost Thus (51) Jll(W)- Jil(a)(a/w) • withradiusA andthecenters on they axisat y = A sothatthey (59) aretangentto thez axisat y = z = 0. Forh, l • • a thefieldaround JII isthefieldprøduced•y two Suppose, then,thatthe.iV'ionsarespread uniformly overa smallrectangle in theequatorial plane(z = 0) atx = 0, y = a. antiparallel ribbons ofelectric current ofwidthh(•v) separated by a fixeddistance l, withdh/dw = 2h/w • 1. The magneticfield of two antiparallelribbonsof surfacecur- Therectangle hassideh in theradialy direction andsideI in the rentdensity JIIwithwidth 2bandseparation 2disreadily expressed x direction (h, l < < a) witha surface number density •, sothat in termsof thelocalCartesian coordinates (•, r/,•), sketched in •r = •lh. Themagnetic fieldattheposition of theparticles is Figure1, where( represents distance parallel to theribbons, r/is B(a) = Bot•2/a• andpoints inthenegative z direction. distance measured fromthemidplanebetweenthetwoand• isdisTheE,j paradigm beginsby noting[Stern, 1992]thatthe ionshavethe gradientdrift tancemeasured acrossthewidth2b of theribbonfromtheline up themiddleof eachribbon.Theresultis thescalarmagnetic potential - l B'(a) l ,B(.) 2cp vv - q--- (52) (53) _Ts.{(o - )en + +(, -- + (0 + g),+ (. + in thenegative z direction. Thesurface current density withinthe rectangleof ionsis rl• +tanJ- yevv (54) rl+• +tan rl+ 2cU J- aB(a) (55)describing thefieldatposition (•, r/)for•:•,r? < < a:•. Atgreater in thenegativex directionwhereU is theparticlekineticenergy per unit area, widthh with( (alongthefieldlines)mustbetakenintoaccount. Thetworibbons ofJlllielocally intheplanes r/= 4-d(/?= 2d) andtheedges oftheribbons areat• = +b(h(w) = 2b).Thefield component Bgthrough thespace between theribbons is U-•vMw • = •,pB(a). distancefrom the ribbon the curvatureand the slow variationof the (56) Thetotalparticleenergyœis thenhlU. Assuming that there are enough backgroundthermal electrons,the electrostatic forcesmaintainlocalchargeneutrality. If followsfromconservation of chargethatthedriftcurrentJ bifur- cates intotwofield-aligned currents, each ofsurface density Jll - B•- JIIx ½ b+ • 'tan1•q-tan-1 b- •rI dd-q •q-tan _lb-• d+q d•q-tan-1 bq_ •rI ]' (60) 10,598 PARKER:ALTERNATIVEPARADIGM Figure1. A schematic drawing oftheunperturbed fieldlines rv = a coscp andzv = (a + h) cos•patx = 0, l for thetwo-dimensional dipole attheorigin.Therectangle h x I ofionsata < y < a -Jr-h, z = 0 provides thesurface current J, which divides intothefield-aligned currents Jllflowing away inboth directions fromtheback andintothefront ends ofthecluster ofions. Theindicated local coordinate system (•, r/,•) isused in(60)and(61)fortreating themagnetic fields produced byJl[. cause J(rv) -,- B(rv), sothatO(w) - O(a) = 0. Note,then, Onthemidplane (r/-- O)between thecurrent sheets thatthefieldis deflected abruptly by thesurface currentdensityJ in theamount20 whereit passes throughtherectangle of energetic ionsin the equatorialplane. 2Jll (lan-l b- •-llan -lb+•) c d d ' (61) The nextquestionis thepathof theperturbedfield lines,particularlythoseonwhichtheclusterof equatorialparticlesis trapped. thathasserved sowelluptothispoint, withBn - 0 fromsymmetry considerations. In thesimple case It isherethatE, j paradigm, runs into difficult. The field lines in the midplane r/-- 0 in theinthatœ< < h (i.e.d < < b),wehaveBn - 0 and teriorof theregionarerepresented by thefamilyof solutions of the differentialequation B•(w) - 4•rJil(w)/c throughout theinteriorof thethinregionbetween thetwoparallel d( ribbons of-4-Jll. Close totheedges oftheribbon thefielddeclines, of course.The field is negligiblein the exteriorspace,exceptnear B 4•rJll cB theedges, and declines asB•(•) ,,, (4Jll/c)bd/•2 for•2 >> b2. It followsthatthemagnetic fieldextending lengthwise along =0 theribbons ofJll(w) isdeflected bythesmall angle O(w),where forb > > d(h > > œ).It follows thatin a distance oftheorder of b/0, thefieldlineleaves theregion between theribbons, thatis, I • ]> b. Withsufficiently smallh(= 2b)thisdistance issmall compared tothelength O(a) ofthefieldline.For• > > b, O(w)-B•(w) B(w) 4•rJll(W) cB(w) 4•rJIl(a) inorderofmagnitude. Thedeflection fallsrapidly below0 in• > > b. Thetotaltransverse •/b displacement increases asymptotically cB(a) = 4•rU (63) at the radial distance cv along the unperturbedfield line w -- acos•o.The deflection is constant alongthefieldlinebe- as(3v9• lb)«.Theperturbed field lines have theform sketched in Figure2 wheretheypassthroughbetweenthetwo ribbonsof current. This resultis incorrect,(1) becausethe perturbedfield lines fail to intersecttheequatorialclusterof particlesthataresupposed PARKER: ALTERNATIVE PARADIGM 10,599 Theoutward displacement of theionsattheequator (•o -- O)is, therefore, H(O)- aO•pn, (6,5) = 4•rVTs/B(a)•, 4•rUtpR a4 = Bn2R4 , (66) Note thatthewidth ofeach ribbon ofJIIdeclines aszv2 orsin2 •r -- 9) asT increases toward •r/2, sothattheidealization thatthewidthh(T) oftheribbons islargecompared totheirfixed 1 separation œrequires thatTR notbesonearto•r astoviolate cos2TR • h(O)/œ.It isevident thatthiscalculation isnot self-consistent, because theangulardeflection • indicates thatthe fieldlinesaredisplaced a totaldistance of theorderof aO from their unperturbed positions,carryingtheminto regionswherethe fielddiffersby AB ~ OB. However,thiserroris aslargeasthe perturbationdescribedby (62), sowe cannotexpecttheresultto be quantitativelycorrect. Soconsider theB, v paradigm, which avoids these problemsby computingtheperturbedfield directlyfrom theequations of stressbalance,usingthe opticalanalogy.The electriccurrents thenfollow from Ampere'slaw, andwe shahfind themto be substantially different fromthosecomputed usingtheE, j paradigm andsketchedin Figure 1. To pursuetheB• v paradigm,we beginby notingthatthe effect of the trappedparticlesis a slightinflationof the elemental fluxbundleat theequatorialplane.Hencetheelementalfluxbundle is expelledmorestronglyat theequatorby thepressuregradientin the surroundingmagneticfield, that is, by the diamagneticrepulsion. The normalequilibriumbalancebetweenthe outwardforce -K7B2/8•roneach elemental fluxbundle andtheinward tension force(B ßV')B/4•r - TK isupset, where T isthemagnetic tension B•/4•r andK isthecurvature ofthefieldlines.TheinFigure 1. A schematic drawing of theunperturbed fieldlines• = flaredflux bundleis displacedoutwardalongitsentirelengthsothat a co$• and•x7= (a -{-h) co$•o atz = 0, I forthetwo-dimensional thesharpcurvatureacrosstheequatorialapexof thebundle(where dipoleattheorigin. Therectangle h X l ofionsata < y < a+h, z = 0 theparticlesaretrapped)affordsan equilibriumbalanceof tension provides thesurface current J, which divides intothefield-aligned currentsagainstexpulsion,sketchedin Figure3. Theeasiestwayto compute Jllflowing away inboth directions from theback and intothefront ends of the forcewith which the ambientpressuregradienttendsto expel thecluster ofions. Theindicated local coordinate system (•, r/,½)isused the ionsis to notethatthediamagneticmomentper unitareavp is in (60)and(61)fortreating themagnetic fields produced byJll' repelledby theambientdipolefieldby a forcef' perunitareagiven by •to be thecauseof their deflection,and(2) becausefield linesin static OB• •'POz = 2Via. (67) equilibrium aredeflected in themannershownonlyuponcrossing This forceof expulsionis opposedby the magnetictensionin the aregion ofenhanced pressure, thatis,enhanced B. SoJllmust be displacedflux bundleon eachsideof theclusterof ionsin theequa- tiedto theperturbed field,ratherthantheunperturbed field. That torialplane.Thefieldis deflected by anangle0 fromthez direc- istosay,inthecorrect solution thecurrent Jllpartially follows the tiononeach side, sketched inFigure 4. Sothetension B(a)2/4•r perturbed fieldlines,providing theuniformdeflection givenby (63) provides a totalinward force OB(a)2/2•rfrombothsides ofthe everywhere alongtheperturbed field. The erroris qualitative be- equatorial plane.Equatingthisto f' yields(63) at •v = a. On this causeof thearbitrarilysmallwidthof theperturbed flux bundle. point the two paradigmsagree. Tocompute theconsequences tofirstorderin theperturbation Now considerthe path of the flux bundlefrom its displaced notethatarclengthds alongthecircle•v = acostpis givenby apexat the equatorialplaneinwardto the dipoleat the origin. In ds = adtp.Theseparation H (9) of theperturbed fieldlinefrom theB, v paradigmthepathis calculated fromtheopticalanalogy theunperturbed satisfies thecondition dH = Ods.If thefieldline [Parker, 1981a,b, 1989,1991,1994]. Themethodis notrestricted is fixedin therigidconducting Earthat • = R = acos•oR,•o= in any way to small deflectionsand displacements,althoughwe •OR,it follows thatH(•on) - 0 andfor0 < •o< TR, shallcarrythroughthe computationin the limit of smalldeflection tocompare directlywiththeresults of theE, j paradigm. Sothe H(7•) -- a0(7•- 70. employing theexactmethod of theopticalanalogy, (64) B,v paradigm, 10,600 PARKER: ALTERNATIVE PARADIGM Figure3. A sketch oftheambient field (solid lines) asabackdrop forthefluxbundle (arrows) displaced bythepresence of energetic ionstrapped at theequatorial plane. avoids questions ofJII along perturbed orunperturbed fieldlines,describedby Fermat'sprinciple withJ computed onlyafterthefieldisfullydetermined. Theessential pointis thatthepathof anyelemental fluxbundle,however it maybedisplaced in theambient field,isthesameas anoptical raypath(geodesic) in anindexofrefraction proportionalEuler's equationreducesto 6/dsB-O. tothemagnitude B ~ (a/w) 2oftheambient field. Inthepresent case, pressure balance guarantees thatB withintheelemental flux K - OlnB/Osñ, (68) bundleis the sameas the ambientfield outside,exceptat the locationof theionsin theequatorial plane.Sothepathof thebundleis where K is curvatureof the field and s.l_ denotesdistanceper- o+H.L(O) F a F 1 Aa F' Figure4. Adrawing ofthedisplace fluxbundle ADEB (given by(71))against theambient fieldline(dashed curve) through thepoint E. The center ofthe field line ADA ofradius a lies atF while the center ofthe displaced flux bundle lies at/pt. PARKER: ALTERNATIVE PARADIGM 10,601 pendicularto the ray path. This formalstatement is nothingmore requiringthat than the requirementalready mentioned,that the transversegra- ! dientOP/Os.[ of theambient pressure P isbalanced bytheop- Aa--•0-• 1-•- posingtransverse magnetic tension forceTK. In thepresent case (72) T = B2/47randP = B2/87r,which yields (68).Forthepresentto thefirstorderin 0, givenby (63). In thelimita > > R, these problem, B ~ rv- • fromwhich it follows thattheraypaths are results reduce to eithercirclesrv = consl concentricabouttheorigin,orradiallines T = coast emanatingfrom theorigin,or circlespassingthrough Aa--Ac--•aO(-•) theoriginrv = Acos(T-- X) (seeAppendix A). Neither of the first two is appropriate,so the perturbedfield linesextendingbe- tween thesurface of Earth(rv = R) andtheionsattheequator aA (rv = a) mustbe arcsof circlesthatpassthroughthe origin. The unperturbed field linesarethecircles b. ThusAa andAc arelarger thanAbbythelarge factor 2aiR. y-•a2+z•_•a1•, (1) Thepicture isclearinthelimitoflarge a/R, then.Thecircle, sketched in Figure4, is rotatedawayfromtangency to thez axis bytheangle0 andthediameter ofthecircleisincreased byAa, so of course,tangentto the z axisat theorigin. The perturbedfield thattolowest order inR/a theequatorial (T = 0) apex oftheflux linesare not necessarilytangentto the z axis. bundle isdisplaced outward thedistance H(0) = Aa. Tolowest Now the flux bundlecontainingthe ionsis displacedoutward orderthecenterof thecircleremainsonthey axis,butis displaced andit is convenientto thinkof eachpoint(y, z) on theunperturbed outward adistance «Aa. InFigure 4theunperturbed field lineis circularfield line movingradiallyoutward(from thecenterof the a circleof diametera withitscenterat thepointF onthey axis. circle)to thenewposition(y + Ay, z + Az). Forsucha radial The perturbedfield line hasa diametera + Aa with centerat the displacement, Az Ay pointFt. ThelineOF t isinclined atanangle 0 tothey axis.The z - , ß y- •a perturbedfield line intersects the y axisat pointB, at a distance H(T = 0) beyond pointA, where it isinclined bytheangle 0 to thez direction. ThetwopointsD andD t represent thefixed Thetransverse radialdisplacement H of thefluxbundleis terrestrialfootpoints of theperturbed fieldline at w -- R. H- [(Ay)• + (Az)2] « Analytically,it followsfrom(71) that œ = Ay 5a y- «a' (70) Ay-- «a • (yAa +2zAb) The perturbationintroduced by theionsdisplaces thecenterof the and from (70) that circle fromy -- •a, z -- 0 tosome newposition y1 -- - i Aa), z -- Ab. Theradius ofthecircle becomes •(a )Xc. Neglecting termssecond orderin Aa, Ay, etc.,theequation for thedisplacedcirclereducesto 1 1 H(y, z)--al-(yAa +2zAb) (73) ontheupper half(z > 0) of theunperturbed fieldline(y = 2(y-•a)Ay+ 2zAz --yAa+ 2zAb + •a(Ac-Aa) wala,z -- w(1-- wa/aa)«)sothatH(T - 0) - Aaat theapex (w = y = a, z = T = 0). Thatistosay, thepressure fortheupper(z > 0) halfof thefieldline.Using(69)toeliminate of theionscauses anoutward displacement Aa of theapexof the A z, theresultis 1 fieldlinefromy = a toy = a + Aa = a + H(• = 0) where 2 ! •Ay 1 y--5. a - yAa + 2zAb+ «a(Ac- Aa). (71) Threeconditions areimposed on theperturbed circle.Thefirstis thatthecirclepassthroughtheorigin,sothatAy = Az = 0 wherey = z = 0, requiringAc = Aa. The second condition H(T-0)-0• 1-•! B(a) • R 1- •5- ß (74) isthatd(y + Ay)/dz = -0 asz declines tozeroattheposition w = a of thetrapped ions.Sincedy/dz ~ z (vanishing as Thisresult istobecompared with(66)giving H(T ) attheapex Thefactor Ta in(65)isreplaced bythe z --• 0),differentiation of(71)forAyyields Ab-- ---}aO. The fromtheE,j paradigm. thirdcondition is thatthefieldlineis fixed(Ay = Az = 0) at the factor (a/R)(1- R2/a2)«. Inthelimit oflarge a/R,Tagoes 1 solidEarth,represented by to •a', whichisreplaced bythelargenumber air in theB, v paradigm. Thedifference isqualitative inthelimitoflarge acosta, T = Ta, Y= R•/a, z = R(1- R•/a•)«, It is essential to understand preciselywherethe difference arises.Bothmethods providethesameinclination 0, givenby(63) at thefixedfootpoint of thefieldlineandat theapex.TheE, j 10,602 PARKER:ALTERNATIVEPARADIGM paradigm givesuniformt9everywhere alongtheunperturbed field line. It is easyto showthatthe samecalculationin thepresentcase givesquitea differentresult,with - ,in(v) ds ,in(v) a dT Noting thaty -- a cos2 T andz -- a sinT cosT along theunpertarbed field line, it follows from (73) that 1[Aasin2T - 2Ab cos 29] or, with zv -- a cosT, [( 0(zv)-0(a)2 1-•- • 1-•- ] -I--2•--1 withO(a) representing theinclination attheapex,givenby(63). It isevident that0(]/) -- O(a),so0 hasthesame valueatboth ends, already noted. Thedifference liesinthefactor (w/•)(1 -- w:•/a:•)« inthebrackets ontheright-hand side, increasing tothe Figure6. A drawing oftheambient anddisplaced fields intheneighboroftheequatorial plane, again showing thecurrent Jstopoint halfway large value (2«/3)a/1•atitsmaximum atw/a -- 1/3« -- hood between the two fields. 0.577, beforefallingto zeroat :v ---- a. The question arisesas tohowt9canbecome solargebetween theendpoints.Theanswer is, of course, that it is the inclination of the flux bundle relative to theambientunperturbed fieldalongtheperturbed paththatmatters. position E. It isobvious byinspection thattheangleof intersection Thedashed curvein Figure4 represents a fieldline w ----a cosT isof theorder of O(a). Theformal calculation iscarded through of theunperturbed field,passingby theperturbed fluxbundleat the in AppendixB showingthattheinclination of theperturbed field linesrelativeto the local unperturbed ambientfield hastheuniform valueO(a),givenby(63),along theentire length oftheperturbed fieldline. The errorin computingthedisplacement of thefieldline through thelg, j paradigm arises inplacing JIIalong theunperturbed field lines. In fact, the currentpattemis qualitativelydifferentfrom that employed in thelg, j paradigm. Weneedonlycompute X7x B to seehow thecurrentflows. The essentialpointis thatthesurface currentJ, associated with thedisplaced flux bundleflowsin the directionhalfway betweenthe externalambientfield andthe field of thedisplacedflux. Hencethecurrentscloselocally,asindicated in Figure5. To runquicklythroughtheformof thecurrents,consider Figure 6 whichis an idealizedsketchof the displacedflux bundlein- clinedrelativeto theambient fieldby theangleO whereit crosses the equatorialplane shownagainstthe backgroundambientfield B. Thesurface current J, isinclined totheambient fieldby•10. The forcef' per unitareain theequatorial planeresponsible for relocating thefluxbundle involves theLorentz forceJBz/c (opposingthe diamagneticrepulsion)in the equatorialplane. That is to say,thediamagneticrepulsionis opposedby theMaxwellstress ByBz/4a'inthefluxbundle oneach sideoftheequatorial plane. Hence JB• = 2ByB• c 47r (75) Figure5. A skematic drawing oftheflattened anddisplaced fluxbundle it follows thatthesurface current J(= cB/2a')flows passing between thefieldlinesoftheambient fieldoneitherside.Theclosed fromwhich pathofthesurface current Js flowing around thefluxbundle isindicatedin the equatorialplaneacrossthe field and out of the planeof the currentdensityJs arisingin the bytheheavy lines.Thedirection of Js oneach sideofthefluxbundle is pagein Figure6. Now thesurface halfwaybetweentheperturbed andunperturbed magnetic field. shearplanebetweentwo magneticfieldswhosedirectionsdifferby PARKER: ALTERNATIVE PARADIGM theangle0 followsfromAmpere's lawas 10,603 to zer• aszv nearsR. Thepurpose of theexampleis to illustrate theutilityof theopticalanalogyin theB, v paradigm. Js- •cSsin2 For thetwo-dimensional dipoletheambientfieldis againpro- videdby (50). Themagnetic fieldb(w) withintheinflated flux with thecomponent bundlefollows from the conditionof pressureequilibriumacross thebundle,yielding i cB J,cos•O - •--•sinO b(w)- Bo(R4/w 4-/3) « cBy where ,3-- 8•rp/Bo • represents theplasma fl atthesurface w -- 4•r _ 1j 2 (77) R. Thefieldstrength b(w) withintheinflated fluxbundle provides the effective index of refraction for that field. The field lines are the (76)geodesicsdefinedby perpendicular to theequatorial planeflowingawayin bothdirections from the sheet of ions at z -- 0. Hence current is conserved. However,it mustbe emphasized againthatconservation of current computedfrom Ampere'slaw is nothingmorethana checkon the arithmeticof thecomputation. The fieldsaredetermined by stress balance,andAmpere'slaw guarantees conservation of currentfor anyfielddeformation. Soin placeof Stem'ssuggestion thatthere- •/ dwœ(w, 9,9')- 0 where now E(w,9') - (1+ =29'2)«(R4/w4 - gion1Birkland currents areprimarily aconsequence ofJ IIfromen- with9' -- dg/dw.Then since 0œ/09-- 0,Euler's equation ergeticparticles trapped in thegeomagnetic tail,theB, v paradigm statesthatregion1 currentsarisesimplyfromAmpere'slaw in the magneticshem'betweenthefieldof thegeotailandtheclosedfield linesat low magneticlatitudes.Themagnetic fieldof thegeotailis maintained in itsextended postureby itsconnections intothesolar =0 dw canbe integratedto give wind,by intrusion of thesolarwindintothemagnetopause, andby the inflationof an internalpopulationof energetic particles,each 0œ in variableandgenerallyunknownproportions. At anymomentin - 1/'• timethedistribution of theshearandof theregion1 currents over latitudeisdetermined by detailedcombination of theseseveral dy- where• is a constant withdimensions of inverse length.Thiscan namicaleffectsdistributed alongthegeotail. be written as An essential resultof theB, v paradigm istheflattening of thedisplacedflux bundleby the pressure of the ambientfieldon eachside[Parker, 1981a,b], shownschematically in Figure5. The fieldoutsidethedisplaced fluxbundleis hardlyperturbed at all,whereas withinthefluxbundlethefieldall deviates by theangle 0 fromthe ambientfielddirection.Thisextends downto the •g'(R4- •74)« -- (1-I-•9'•) «. (78) - 4-1 (79) or dissipationless ionosphere employedin thecalculation.The result is a decrease of thehorizontal component Bocos9oin theamount of OBosin9oat thefootpoints of thedisplace fluxbundle, and where• -- •w and zeroelsewhere.The flattening of thedisplaced bundleprovides a footpointin theformof a narrownorth-south strip. • (l+4n)«::F1 , The flatteningof the flux bundleimpliesthatthe clusterof ionsat theapexin theequatorial planeis alsoflattened to somedegree.Thedynamics of theflattening of theclusterpresents anintern- flt•4/R4. estingproblem,particularly whencombined withthefield-aligned Equation(79) reduces to thequadrature currentsandtheassociated ionospheric dissipation. 6.3. Isotropic Particle Distribution Considerbriefly the problemof computingthe deformation (80) (81) ,3«(9-91) -- •[(•12 _•2)(•2 2q-•2)]•(82) of a dipolefieldby variousparticledistributions. $ckopke[1966] where theintegration constant 91 ischosen such thattheapex • = providesa generalresultfor a axiallysymmetricequilibriumshell •l ofthepath occurs at9 -- 9•. Theelliptic integral isastandard of energeticionswith arbitrarypitchangledistribution.Here we form, from which it follows that takeup the exampleof a singleelementalflux bundlein a two dimensional magneticdipoleinflatedby plasmawith isotropicthermalvelocity,for whichtheplasmapressure p is uniformalongthe (1+ 4n)¬(9 - 91)-- F(cos-•/•, k), (83) fluxbundle.We neglectthesmalllossconethatarisesin thepres- enceof a solidEarthatw = R, causing theplasma pressure tofall = cn-•(,•/,•l,k), 10,604 PARKER: ALTERNATIVE where the modules k is PARADIGM In ordertocompare thisresult with(74)(L • a), notethat thethermal energy density of a monammic gasis •p withp -- flBoe/87r - fl[B2(L)/87r] L4/R4. Thearea between thefield k- •-r 1- 2• . (1+ 4n)« by unitdistance at theequator canbecomputed to (84) linesseparated firstorderin X fromtheunperturbed field,forwhichtheseparation H(T ) ofthelinesisproportional toy/L, where y = L cos2T n«[3 I- •nq--•- --n 3q31n2 1•7 O(n4)] ' alongthefieldline.Thenwitharclengthds = Ld•o,it isreadily shown that thearea isL(TR+ sinTRcosTR ) ~ L[cos-• R/L+ Then (R/L)(1- R2/L2)«].It follows that thethermal energy U in thefluxbundlewithunitcrosssectional areain theequatorial plane • -- •cn[(1+4n)¬(T-T1),k]. (T = 0)is U- •pL[cos• q-•(1- •-•)«]. Placing theapex ofthefieldline atw -- L, wheredw/dT0, it followsfrom (78) that Usingthisrelationto express p or/3 in termsof U, it followsthat t•- LIRa(1- X)« (85) 16•-U where X = flL4/R4 ( 1. Notethatif X > 1, theplasma 3LBo•[cos-•R/L + (R/L)(1- Ra/L2)«]' pressureat the apex exceedsthe ambientpressureand thereis no equilibrium solution, except forradiallines(dT/dw = 0). 4•rL4U[cos-11•/L q-«(R/L)(1- R2/L2)«] Bo2RS[cos-iR/L + (R/L)(1- R2/L•)«] H It is sufficientfor presentpurposesto treat the caseof weak inflation,X (( 1, for whichit is convenientto notethe exact relafionsfit•L • - X/(1-X), t•• - x/(lq-x), • - 15X/(1- where Boisthefieldintensity B(R) atw - /i•. In terms ofthe X), •22-/•/(1- X), n - X/(1- X)•,(1 + 4n){ - (1+ field B(L), X)/(1 - X). Forafield•newitha•x at•e equaWr T = •1 = 0, ß e fieldfine•rsects w = R at •e latin& TR givenby (83)as F(cos-•R/L k) ' (1+ 4n)¬ H __ 4•rUL [cos-•R/L+ «(R/L)(1R:•/L:•)« ] B(L)2• co8-112•/L q-(R/L)(1- R2/L')« ' (86) In thelimitof smallR/L, 32U L 4X3cos•Zq- 1- •-•)« ---- - (87) 3LBo •' H e,• reaches zv--/i• atthelatitude cos-• I•/L. fromwhich it isclear This is the sameas (74). (88) L n ThefieldlinewiththesameapexradiusL in theabsence of inflation that the foot pointsof the inflatedfield lineshave a latitudethatis lower by the amount 4•-U That is to say,inflationof a flux bundle withanisotropic plasmaof energyU causes theapexof theinflated flux bundleto moveoutwardby aboutthe samedistanceasthesame 1 energy inparticles withpitchangle5a'attheequatorial plane. It shouldbe notedagain,that the displacedflux bundletends to flattenfrom thepressureof the ambientfield acrosswhichit extends.The foot point is againa narrownorth-southstrip. thantheambientfieldlines.Thatis to say,theinflatedfieldlinesare distended,aswe wouldexpect.Thisisbestillustratedby computing 7. Flux Transfer the changein the radial distanceto the apex as a consequence of a smallinflation/? orX whileT•t remains fixed.Thentolowestorder 7.1. Individual Flux Bundles n • k2 -• X. Use(87)tocompute c9L/c9 X, withtheresult thatL changes by thesmallamountH, givenby H 4n x, The transferof geomagnetic fluxbundles intothegeomagnetictail,followingreconnection withtheinterplanetary magnetic fieldatthesunward magnetopause, is anintegral partof magnetosphericconvection at timesof geomagnetic substorms andstorms. Relativelylittleattention hasbeenpaidtothedynamics of theindividualfluxbundlesin theprocess of transfer.Thissectiontreatsthe = to firstorderin X- subject brieflyto showhowtheB, v paradigm handles theproblem. As we shallsee,thebasicconcepts areclearenough whereas thedetailsof thephenomenon aresufficiently chaotic astobequan- •-ff)[36os-•+•(1-•-•) ] titativelyintractable. Considera flux transferevent,involvingthepickupof a magnetosphericflux bundle from the sunwardmagnetopause from PARKER: ALTERNATIVE PARADIGM 10,605 (dz/dy)distheslope ofthedisplaced bundle given by(90) whereit is carriedin thesolarwindin themagnetosheath backinto where the geotail.A typicalflux transferbundleis observedto havea di- and(dz/dy)a istheslope oftheambient fieldgiven by(91).The ameterof theorderof 1/i•E andaninternalmagnetic fieldof the electriccurrentpatternaroundandalongthedisplaced fluxbundleis orderof 10-4 13,fromwhichoneinfers a totalmagnetic fluxof describedin subsection 6.2 andisillustratedin Figure5. Theinabiltheorderof 4 x 10la Maxwells.The flux bundleconnects intoan ityof theE, j paradigm to address theproblem of finding thepath ionospheric footpointwithanareaof7x 10la cm2 (forB - 0.6 of a widelydisplaced fluxbundleis obvious.Notein particular that G), equivalentto a square80 km on a side. The outerendof the themotionof thelowerendof theflux bundleis determinedlargely fluxbundleat themagnetopause is transported a distance of theor- bythemechanical motion oftheionosphere andhasnodirect conderof2x 101øcrn(30RE)fromthesunward magnetopause into nectionto theelectricfield in therapidlymovingouterendof the the geotailin the 200 km/s solarwind in the magnetosheath in a flux bundle.As notedearlier,theforceexertedon theionosphere timeoftheorder of 10a s.Thegreat mass oftheionospheric foot bythedisplaced fluxbundle isproportional tothedisplacement of pointsof thefluxbundlesuggests thatthefootpointsdonotmove significantly in thattime. On theotherhand,theAlfvenspeedin the the outer end of the flux bundle rather than to the instantaneous rate of displacement to whichtheelectricfieldis rela•ed,illustrated in outer magnetosphere isoftheorder of10akm/s, asaconsequence subsection ofthelowambient plasma density (N _<1atom/cm a)beyond the 7.2. It should benoted/hat thebasicphysical principles arethe themagnetopause suggests thatthedisplaced fluxbundle isnotfa• samein a three-dimensionaldipole, but the computationis much fromquasi-static equilibrium withintheoutermagnetosphere dur- more.complicated. First,theEulerequationfor theequilibriumpath ingthetranspori intothegeotail.Hence,except neartheionosphere, is of the form of Abel's equationandhasno elementarySolutions. plasmapause. The relativelyslowtransport velocityof 200 km/sat theinstantaneous pathof a movingfluxbundlecanbeapproximated Second,theouterendof thedisplaced flux bundleat themagneby the staticequilibriumpathbetweentheouterend,movingwith topause is transported by thesolarwind. In twodimensions thisis thesolarwind,andthefixedionospheric footpoint.In thes•ple parallelto themeridionalplanesin whichthefieldlineslie. Howcase ofatwo-dimensional model ofthegeomagnetic dipole, treated ever,in threedimensions thedisplacement generallyhasa nonva- in subsections 6.2 and6.3, it followsfromtheopticalanalogythat nishingcomponent perpendicular to thelocalgeomagnetic fieldat theflux bundlelies alonga circlethatintersects theorigin.Denotthemagnetopause. The flux bundleslidesfreelyonlyin thedikecingtheionospheric footpointby St-- a, z = b andtheouterend tothegeomagnetic field,butnotacross thegeomagatthemagnetopause bySt-- Y (t), z = Z(/), it isreadily shown tionparallel neticfield, so the motionperpendicular to the geomagnetic field thatthepathof the bundleis givenby thecircle strongly deforms thegeomagnetic fieldatthemagnetopause, prob- (y:•+ z:•)(bV- aZ) + y[a:•Z- bY:•+ bZ(B- Z)] -z[b2Y- aZ:•+ aY(a- Y)] - 0 ablyresultingin localreconnection andgenerallycreatinga messy theoretical problem. Theessential pointisthattheproblem canbe (89) statedeasilyin termsof theB, v paradigmevenif thedynamics at any instantin time. The motionu of thedisplacedflux bundleat anypointdeter- is toocomplicated to allowa formalsolution. 7.2. IonosphericMotion mines the local electric field E = - u x BIc within the flux bundle. The surfacecurrentdensityaroundthedisplaced fluxbundleisread- Nowby thetimea largebodyof displaced magnetic fluxa½. ilycomputed fromtheinclination ofthedisplaced bundle relative to cumulates in thegeotail(asduringtheonsetof a substorm), the theambientmagneticfield. in particular,it followsfrom(89) that ionosphere begins to moveandit isof interest tonotetheformulation of this dynamicalproblem.The succeeding sections treatthe thepartially ionized stateoftheionodz_ a2Z - bY :•+bZ(b - Z)+2(bY - aZ)y(90)detailedproblem,recognizing dy - b2Y- aZ2+ aY(a - Y) - 2(bY- aZ)z sphere.However,aspreparation for thecomplications OftheHall alongthepathof the displacedflux bundle.For the ambientfield linewithanapexat St-- L, z - O,we have Thereis nothingparticularlynew hereexcepttheemphasis on the directapproach to thedynamics, describing themomentum in terms of theplasmavelocityandthe stressin termsof themagneticfield. y2-yL + z• -0 dz dy L- 2y 2z and Pedersenconductivities,it is useful to treat an idealized caseto enumerate theelementary stress balance andmomentum transfer. Consider,then,the simpleidealizedplanepolarionosphere andmagnetosphere modelneglecting all resistivityandviscosity. of uniformdensityp confinedbetweenthe Hence, fortheambient fieldlinethrough thepoint(y, z) it follows Imaginean ionosphere that planesz -- 0 andz -- L. Consider theidealizedcasein which thenonconducting atmosphere below(z < 0) hasnoviscosity so -- o L- Y•+Z• Y dz theionosphere slidesfieelyaboutexceptfor itsowninertia.The magnetosphere inthespace L < Z < A(A > > L) hasneg- z• - •!• d'•= 2Stz (91) ligiblematerialdensityandessentiallyinfiniteAlfvenspeed.The at thatpoint.Theanglebetween thedisplaced fluxbundleandthe ambient field is wholesysiem is threaded by a uniform vertical magnetic fieldB andthemagnetopause atz -- A issubject toarbitrary horizontal displacement, aswhenthefieldispickedupby thesolarwindand transported a largedistance H alongthemagnetopause in, say,the O(y, z)=tan -1(d•yy) -tan-1 (d•yy)(92) d a Stdirection.Thelayoutis sketched in Figure7. 10,606 PARKER:ALTERNATIVEPARADIGM B z-A z-L P• y Figure7. A schematic drawing ofthesheared field(0,Bb(z,0,B) associated withthedisplacement •(z,t)relative tothe footprint atthemagnetopause z -- A. DenotetheLagrangiandisplacement of anelementof fluidin t) theionosphere by•(z, t), assuming that0/0y -- 0. Thefluid in thei9nosphere, wheretheintegration constant is zerowiththe velocity v(•, t) intheionosphere isO•/Ot,introducing themag- choice• = 0 for b = 0. Substituting into(94) gives neticfield'Bb•z,t) in theF direction attheratedescribed bythe inductionequation Ob Ov Ot Or 0• Ot• c•O• • = 0. (97) • - Dcos • coscot, (98) Consider the solution 0• = OzOt' (93) In the presentidealizedexampleof a verticalfield withoutlocal whereD is anarbitraryconstant. It followsfrom(96) that compression thedisplacements arehorizontalanduniformoverthe horizontalplanesothattheyproduceno magneticperturbation be- b- D• sin•coscot. low.the ionosphere. Theappropriate boundary condition forb(z,t) is b(0,t) = 0 withb = 0 throughout thenonconducting atmo- (99) spherein z < 0. The conducting Earthplaysa roleonlywhere Substituting(98) and(99) into (95), theresultis O/Oy# o. col L =•. L Ctan• TheMaxwell stress Myz -- .Bb/4•rreacts back onthefluid for whichthe momentumequation•s 0• B• Ob POt2 = 4n'Oz (100) It is clearthatin thelimitof smallL/A, thequantity coLICis (94) eithersmallcomparedto one,sothat TheAlfven speed B/(4•rp) « intheionosphere isdenoted byC. co• The Allyen speedis .essentiallyinfinitein the tenuousmagneto- c (101) sphere (L .• z < A), sothatif thefieldisheldfixedatthemag- netopause (z = A), it is convenient to measure thedisplacementor • fromthei•osition'at which thefieldisvertical (b = 0). Hence, ' t)- A-L co~-l-n•r1-tn•r2A•'" wheren = 1, 2, 3,-. ßWe areconcerned hereonlywiththefundamentalfrequency,givenby (101). ~ 1 = A t) (95) (z Suppose, then,thatthetimet = 0 findsthemagnetopause = A) displaced a distance H in thenegative y - direction andb(z,t) = b(L,t) throughout L < z < A. Equation (93)can andthereafterheldfixed.The systemis withoutmotionat thetime be integratedonceto give t = 0, butthenonvanishing Maxwellstress accelerates theiono- PARKER: ALTERNATIVE PARADIGM 10,607 spherefromrest. Theinitialvalueof • is thenH throughout the sphericmotionandviceversa.The ionosphere movesin response ionosphereandthereis a kink in thefield at z = L, with a deflec- to the Maxwell stresses. tiontan-1H/A. Thekinkprovides aninitial sharp transient, of It is easyto extendthecalculations to includeviscosity in the ionosphere and in the non-ionized atmosphere (z < 0) below the tweenz = 0 andz = A withpartial reflection ateachcrossing of aswell asa conducting Earthat z - œetc. However, z = L sothattheinitialpulserapidly degrades intoanincreasingionosphere, for purposes of ilnumberof pulsesof diminishing amplitude.Theprincipalmotion the foregoingidealizedcalculationis adequate no particularphysicalinterest,thatpropagates backandforthbe- is the fundamental oscillation lustratingthebasicprinciples.We turnnextto a formulationof the z •(z, t) - Zeos B, v paradigm in theionosphere, takingaccount of themassive ct neutral atomic constituent. (1;a) • 8. Ct Hcos• Considera planestratifiedionosphere composed of N neutral (œA)« H z Ct - (/;a)« Hz --'• -- --C08 Ct • in theionosphere, neglecting terms O(L/A) compared toone.It follows that CH Partially Ionized Gases Ct v(t) "• (LA) «sin (LA)--'--• The electric current and electric field are uniform across the iono- atoms percm3, andn electrons andionspercm3, theionsbeing singlycharged.The meaneffectivecollisiontime for the individual ion with the ambientneutralatomsis denotedby ri and the collision time for an electronwith the neutral atomsis re. The ion-electroncollisionsprovidean equivalentcollisiontime r. The electron-electron andion-ioncollisiontimesarecomparable toeach otherbut of no particularinterest.The local meanvelocityof the ion gasis denotedby w andof the electrongasby u, with the v representing thelocalmeanvelocityof theneutralatoms.Thecollisionsprovidea meandragforceon eachconstituent, sothatfor cold electronsandsinglychargedions,themomentumequations are spherewith the instantaneous values dw cB Ob j(t) -- -ex---4•r Oz cBH Ct • +e•4•rLA cos (LA) « BCH , (102) wxn) :U(w-v) re(w-u) c Ti M-•- - -e E+ vB E(t) - -e•-- +e E+ - c + re . 7' (103) c Ct • +ex c(LA)« sin(LA)----• The equationof motionfor theneutralatomsis NMdv - v) + nm(u - v) (104) d'-[ = -V'p+ nM(w ri re intheionosphere (0 < z < L). Inthemagnetosphere bisuniform withthevalueb(L,t) givenabove, whilethevelocity is where/9is thepressure. Viscous andgravitational termscanbeaddedif desired. Note that theelectric current density j is t) - v(t)(a- j - ne(w- u) Theelectricfieldis accordingly (105) c E(z, t)- E(t) (A- z)/(A- L), = 4-•Vx B, (106) vanishingat the magnetopause z -- A, fromwhichit is clearthat thesecondequalitybeingAmpere'slaw.TherelationcanbesolvedforU, yielding theelectricfieldat themagnetopause hasnothing to do withthe motionwithinthemagnetopause andin theionosphere. Theinitial, relatively rapiddisplacement of themagnetopause byadistance H in thenegative y direction at timet = 0 canbeadequately described bythevelocity -HS(t) atz = A with t) - - ;)/(A - c u-w 4rrneV x B. (107) Consider, then,thecaseof weakionization n < < N, appropriate for theterrestrial ionosphere. The inertiaof theionsandelectrons is smallcompared to theinertiaof theneutralgas.TheMaxwell stressesare exerted on the ions and electrons and are transferred throughoutthe magnetosphere. The associated electricfield is the to theneutralgasthrough collisions. Therelatively massive neushortintensepulse tral gasresponds onlysluggishly to thecollisions, sothaton that timescale theleft-hand sides of (102)and(103)arenegligible, that is,theinertiaoftheelectrons andionsisnegligible compared tothe inertiaof theneutral gas.Theionsandelectrons areinquasi-static whichvanishes beforetheionosphere hasbegunto move.Soagain, equilibrium between theMaxwellstresses andthecollisional drag t) - ;)/(A- the electricfield at the magnetopause is not relatedto the iono- of theneutralgas. 10,608 PARKER: ALTERNATIVE PARADIGM Neglectthe left-handsidesadd(102) and(103) andthenmultiply by n, with the resultthat j xB= nM(w v)+nm(uc ri v) re - B{-vxb ½ (118) (108)+a(Vxb)a.x b+(,+b2/5)(Vxb)a.+,(Vxb)11} where thesubscripts .1.andII Note that (117) can be written B that NMdV B)xB •- - -Vp+ (Vx4a' (109) ½ x {-/•[(V x b) x b]x b+a(V x b) x b+ r/V x b} (119) plus whatever viscosityand gravitationeffects are appropriate. This, of course,is the usualMHD momentumequation,with the where E'- E + (S/c)v x • fluidpushedaround by •7p andby theLorentzforce,thatis,by the divergenceof the Maxwell stresstensor. is the electric field in the frame of referenceof the movingneutral atoms. Equation(119) is the form of Ohm's law for a parfially ionizedgas,of course,but it cannotbe usedeffectivelyuntil thevelocityv hasbeendeterminedfromthedynamicalequations. Nextsolve(107)and(108)foru andw in termsofv,X7 x B, etc., with the result w- v+ (V x B) x B emir, 47rnQ+ 47rneQ •7xB u-v + (VxB)xB cM/ri 47rnQ- 47rneQ V'xB Thisdifficultyis sometimes overlookedin applications of theEs (110)j paradigm.Illustrativeexamples areprovided in sections 9-11. Substituting(117) and(118) into (111) 0B where M ot m Q = -- + -ri re m•e components perpendicular andparalleltob. The right-handsideof thisrelationrepresents thefrictiontermson the right-handsideof (104) with the result,usingAmpere'slaw, (112) =-cV xE yieldstheinductionequation It is thena straightforward calculationto solveeither(102) or (103) 0b for E, obtaining o-F = v x (vx •) •, - .x• c [(vxn) xn]xn +v x {a[(v x b) x b]x b- •(v x b) x b- . 4•ncQ M/ri - m/re (VxB)xB 4•rneQ =Vx(vxb)-V (113) x{(.+b•/?)(Vxb)a.+a(Vxb)a_ xb+.(Vxb)11} ' (121) It is notwithoutinterest to express E in termsof theiondriftve- Q c [(•Uln)(ml•,) +•]Vxn' +4•ne 2 locityw ortheelectron driftvelocity u. Solving (110)and(111) forv andsubstituting into(113)yields thetwoexpressions It isconvenient todefinethediffusion coefficients c•,/• andr/, with cB(M/ri -+m/re) a - 4•ne(M/ri m/r•) representing theHall resistivediffusioncoefficient, xb+•(Vxb)x b+;/Vxb} (122) c 4•rneQ (114)E- B{-w cBm/r, E- B---{-u xb-•(Vxb) xb+;lVxb} (123) c 4•rneQ so•at •e •ducfionequation c• • expressed a•ord•gly. B2 1•= 4•rn (M/ri+m/re) x b} (120) Them• (115) sen• •e g•s• asthe Pedersenresistivediffusioncoefficient,and • V x (v x b) on•e fight-h•dsideof(120)repre- of fieldwi• •e movingfluid,•at is,•e smnd•d •ducfioneffect.Therem•g m•s represent d•sion •d dissipationand•e HaH effect. R is •s•cfive rl= 47me2 M/ri+ as the Ohmic resistivediffusioncoefficient,where B is a charac- teristicmagnetic fieldstrength.Fejer [1953]andothersprovide conductivities as a functionof altitudein theionosphere. Write b - B/B, sothat E- •t-v xb c -/•[(V x b) x b] x b+a(V x b) x b + r/Vx b} (117) • •m out •e en- ergyequation • s• •e nat•e of •e dissipation. For•g •e scalg produce ofb wi• (118)•d using•e v•r idenfi• A.VxD-D.VxA+V.(DxA), it can be shown that 0 b• b• Ot8•-+ v. (v•.•) = -f. v - 4•r/?f • - q(V x b)•/4•r -V. [/•b2f + af x b + ;If], (124) PARKER:ALTERNATIVE PARADIGM where f istheLorentz force divided byB2, f= (Vxb) 4•rxb The convective transport of 10,609 toeliminate V (125) Usingthisexpression x f from(128)yields magnetic enthalpy is v.t.b2/4;r.Thefirsttermontheright-hand side istherateatwhich the fluid does work on the field. (130) 5--•= V x(vxw)+4a'C2V xf. ß The third term is the usual Joule (131) dissipation term, which canbewritten ,/2/•rffdesired. Thesecond It follows thatthevector b+aw/C • iscarried bodily withthefluid motionv, asdistinctfromthetransport of b withv in theabsence vergence of/•b2f+ af x b + •/f represents onlyaredistribution term is thePedersendissipation.The final term, involvingthe di- of the Hall effect. So the conceptof a "frozenin" field remains. of magneticenergywithintheregion.It providesnonetdissipation, The field is neitherb, that is "frozenin" in the absenceof theHall as is obviousfrom the fact thatit vanisheswhenintegratedover a resistivity,nor thevorticityin the absenceof a magneticfield,but a volumesufficienfiy largethatf vanishes on thesurface.TheHall linear combination b + (a/C2)coofthetwo. effectprovidesno netdissipation, of course,althoughit providesan energyredistributionflux af x b- -ab2(V x b)a. Note,then,thatsincem/re < < M/ri in theionosphere, (114)yieldsa • eB/47rne,where n represents theionandelec- (126) tronnumberdensities.It followsthatthecoefficient a is givenby a/6 '2'"'N/nai within the regionof nonvanishing Lorentzforce. The momentum (132) (109) can be rewritten NMdv d-•' = -Vp+B2f where•i -- eB/Mc represents theioncyclotron frequency in thecharacteristic fieldB.Thus,with[w[ ~ 1/tw,where tw isthe (127)characteristicrotationor shearingtime of the fluid motion,it follows that theratio ofaco / C2tob isoftheorder of(N/n)/ flit•. TheratioN/n islarge(~ 10a atanaltitude of 300krn)and Qi ~ 3 x 102/sec fora single ionized oxygen atomin 0.6G. Sotheratioaco/C2b issmall formotions withcharacteristic time to whichviscousterms,etc. may be addedif needed. Simultaneous solutionof (120) and (127), alongwith some statementon thepressurep, providesa completedescription of the macroscopic behaviorof the ionosphere.In fact, the dynamicsof largecompared to say,10s,andit is b thatis tiedto v to a first AsN/n declines upward intothemagnetosphere, theionosphere involvesonlysmallperturbations AB of thequiet approximation. the ratio becomessmaller, and the frozen in conditionobtains(in the absenceof significantdiffusion).However,idealor not, the in- day field so that the equationscan be linearizedto goodapproximation. The linearform is studiedbrieflyin thenextsection.The essentialpointis that,with thebasicequationsformulatedin terms of the variablesv andB, thetime dependent casecanbe treated directly. Electricfieldscan be calculatedafterthe fact ff they are needed. Chargeconservationis automatic,and electriccurrents take duetion equation shows thatb + aco/C '2isaphysical concept that is usefulin treatingdynamics. As is well known,thepertinent quantity is theratioof the Pedersen andHallcoefficients t3/a. Toobtainsomeideaof the care of themselves. magnitude of/3 anda in theionosphere, recallthatthemeanfree paths for ions and electrons are comparable inorder ofmagnitude. The momentumequation(127) is the familiar momentum Hence ri/re~ (M/m)« and m/reissmaller than M/ri bya equationof magnetohydrodynamics. The induction equation (120) is similarto the inductionequationof MHD. It is evidentfrominspectionof the right-handsideof (121) thatthe totalresistivedif- factor oftheorder of(m/M)«.Itfollows from (114) and (120) that cB fusion coefficient of(X7x b)[[(representing diffusion offield perpendicular ofb) isr/,while thecoefficient of(V'x b)a.isr/+b•/•. 4•'ne C/nfl. So theresistivediffusionis anisotropic, but stillof thenatureof resistive diffusion. The Hall term, whose coefficientis c•, is more B2r• interesting.Neglectingresistivediffusion,write (120) as 4;rnM •b 0--•= V x (vxb)- 4•rVx af, (128) • C2r•N/ra, where 6' -- B/(4•'mm)« isagain thecharacteristic Alfven wheref isthedimensionless Lorentz forcedefined by(125).Then speed.Hence we havethe familiar resultthat write themomentumequation(127) as 0v 1 1 O--•+co xv - N-MVP - V•v2+4•-C2f,(129)in orderof magnitude,showingthatresistivediffusionis moreimwhere co-- X7x v isthevorticity and 6' -- B/ (4•rNM) «isthe Alfven speedin thecharacteristic fieldB. Considertheidealizedsituationin whichtheneutralgasdensityNM andtheHall coefficient a are locallyuniform. Thenthecurlof themomentum equation yieldsthevorfieityequation: portant thanthecorrection c•co/C 2tobintheinduction equation. Note thatin the sameapproximation rac 2(• + -1), rl-• 4•ne2 7' 10,610 PARKER: ALTERNATIVE PARADIGM ofz. Theessential pointisthat(133)-(136) provide acomplete set so that offourequations fortime dependent v•,vy,b•,and If theelectric current density j isneeded forsome purpose, it followssimplyas cB Oby J•'- in orderof magnitude. cB Ob• 4•rOz jy- q-4-•0--•-' (138) The electricfield followsfrom the linearizedform of (117) with 9. The Plane Polar Ionosphere Beforetakingupsome diffusive dynamical problems ofionospheric convection in thesucceeding sections, it isuseful tonote briefly some elementary idealized circumstances inwhich anionosphere without diffusion maybesetin closed circulation bythe - -- - % + a • 77-,- (•/+/•) 77J' (139) - -- q-v•q-(r/q-/•)•zz q-a•zzJ' (140) Maxwellstressassociated with magnetospheric convection. The c response ofthemassive ionosphere isslow, ofcourse, soitisalso of interest to seehowthestressed magnetic fieldarranges itselfacross from which it follows that the electricfield E t in the frameof refis a simple uniform ionospheric model whenthemotion oftheiono- erenceof the movingionosphere sphere canbeneglected. Wecitea variety of simple solutions to (120) and (129). 9.1. -_ . -(.+•) (141) Plane Model Consider thepolarionosphere in theapproximation thatthe 4•r •)j•+ajy] C2 [(r/+ • meanfield is verticalwith a uniformstrengthB. Abovea plane Earththeionosphere is setin motion bythetransport of magnetic fluxbundles fromthedaysidemagnetosphere intothemagnetotail asa consequence of reconnection atthemagnetopause, etc. The neteffectis to tilt thepolarfieldawayfromthevertical, sothat (142) 47r thereis a Lorentzforceexertedontheionosphere by thetension in c•-[ - "j•+(•1 +/?)Jy ]. thetiltedfield,assketched in Figure7. In thesimplest case,letthe tilt of thefieldbeindependent of thehorizontal coordinates x andy. Solving these twoexpressions forj• andjy yields thefamiliar reTheresultof thetilt is a timedependent magnetic fieldin anaccel- suit erated ionosphere (v•, Vy• 0). Thehorizontal magnetic perturbations B• andBy aresmall compared tothevertical component, ofcourse, sothatthequantities b•c• B•c/Bandby-- By/B arebothsmallcompared to one.Notethatthehorizontal unifor- (X7x b)ñ = -ezOby/Oz + eyOb•/Oz. Equation (120)or c• (.+ •)•'• + (.+ (143) (144) JY- 4•r (r/+/•)2 +c• 2 mityyields(•7 x b)z = 0 whilethetransverse component is (121) reducesto the linearizedform c• (r/q-/•)E'• - aE• J•- 4-• (r/q-/•)•q-c• • ' It is well known,andobviousby inspection of 133)-(136), thatthereareAlfvenwaves,modified by theHall resistivity a and dissipated bytheOhmic andPedersen resistivity r/q- fl. Timedependent solutions ezpi(cot + kz) in a uniform medium provide (133) Otz q-aOby] '•z-z J' Obz ~0-• 0[vrq-(r/q-/3)• Obz ot+ ('•+/•)•;J' ot, y_•o-• o[vy- "3-; oo• ot, y] (134) neglecting terms second orderinbxandby.Themomentum equa- theusualdispersion relation •-• - •-• U[i(r/+/•) 4-a]- I - 0, (145) so that tion (127) becomes 1 +•-• •c- + •+Y•[•"+i•+•]"• • [+"+i•+•)] ot =C• 35-z + •O Oz]' Ov• Ob• (•'Ov• • (135) wherethesign4-a is independent of the4- in frontof thebraces, that is, thereare four modes. The PedersenandOhmicresistive ot =c2 •zz +•O Oz? 0% Oby (•'0% • (136) diffusion coefficients (fl q- r/) provide diffusion whereas theHall resistive diffusion coefficient a couples b• andby.WithHallre- inthesame approximation, noting that(v. •7)v -- 0. Thequantitysistivity alone (fl q-r/-- 0) theresult is Cisagain thecharacteristic Alfven speed B/(4a'NM) « and we haveincluded a viscosity/•.TheLorentzforce,givenby (125),is 1 db•, 1 dby f•- 4•rdz fy- 4-•d•-' co=4-1+ 4-2C ' kC (137) Note,then,thatthecoefficients, a, t, •/,/•, andC maybefunctions ak •+1+76+ • ß PARKER: ALTERNATIVE PARADIGM Forak/2C > > 1, therearetherespective fastandslowmodes 10,611 vz(z t)_az[t+(rl+•) r/+•] , kC=+ 1+ +... kC=+ • 1- • +.... C• 0 C• ' Vy (Z,t) --ax•-•- •-ffo Note then thatVyisindependent oftime. Itisnonvanishing only in sofarasc•/C2varies withz. Theelectric fieldfollows fromthe Ontheotherhand,in anionosphere sodense thatc• < < r/q- fl (139)and(140)asthegrowing uniform field theequations reduceto theusualresistivemagnetohydrodynamic equations for a scalarfield.Thereisonlya singlewavemodepropagatingin eachdirectionalongB. B , oEy-ax-(•-•- o+t . (150) E•- az--(•c c The electric current is 9.2. SteadyAccelerationof the Ionosphere cBax jx- O,jy- 4rrC2(z). Suppose thattheMaxwell stress Bgbz/4rrismaintained at a constant valueatthetopof theionosphere whileby = 0. The (151) asymptoticform of the motion,after initial transients havedied Thetotalintegrated current isjust(c/4•r)Bbx(L)fromAmperes law, of course. Note that applying theelectric fieldEy tothelump(146) away,is vz = a•t + U•(z), edionospheric conductivity gives thewrong result because itisE• Vy- ayt+ Uy(z), (147) thatdrives thecurrent rather thanEy. Notealsothattheproblem of ionospheric acceleration by a specified Maxwellstress cannotbeadwhere az andayareconstants. Presumably c•,r/,t, andtheAlfven dressedby prescribinganappliedelectricfieldat themagnetopause speedC areall functionsof heightz. Thenif viscosityis neglected, because the electric field is not the driver and F, is not known at the ionosphereuntil the dynamicalequationshavebeensolved.Using (135) and(136) canbe integratedimmediatelyto give theB, v paradigmto deducethesolutionmakesit possible to go backandrestatethedynamics of theionosphere in termsof theE, zC(z,):•, dzt bz(z)azj•o j paradigm of course, butthisadhocexercise serves noobvious (148) zC(zt):z, dzt by(z)ayj•o (149) purpose. The ion andelectrondrift velocitiescanbecomputedfromthe linearizedformsof (110) and(111) if theyareneeded.Theelectric magnetosphere (z > L) is satisfyingthe conditionthat the Maxwell stressacrossthe lower drift(plasma)velocityin thetenuous surface (z = 0) of theionosphere is zero.However, if thereis noapplied by(L),it follows thatay = 0. Hence by(z) = 0 ltx __--C . B' throughout 0 < z < L. Thestatement is then,thattheMaxwell stress B2b•(z)/4a' atthelevelz provides theacceleration a• of (152) --az[(rlC-½2'8)o the mass azS3- W lty --' --C" up to thatlevel, sincethereis no Maxwell stressacrossthelower B' boundary (z -- 0) andwehaveomitted theviscous stress in the = -az presentillustrativeexample. ß (153) 0 The inductionequations(133) and (134) can be integrated over z, with the result Therateatwhich theMaxwell stress exB:•bx(L)/4rr does work Ux(z)+ (rI + fl)az/C:z-' D1, perunitareaontheionosphere (0 < z < L) is p - u•B•b•,(L)/4• - Uy(z)- aaz/C:•= D:•, where D1 andD2areconstants. Working inthefixedframe of reference inwhich Ux(0) - Uy(0) - 0,it follows that The work goesinto accelerating the ionosphere whosekineticenergydensityis r/+ flh ]0 Uy(z)- az Babx(L)ax [(rl +fl'• t] (154) 4• C2 ]o qr- -- , + so that o wherethesubscript zerodenotes thevalueat z -- 0. Thefinal result is dT It+\(r/+ •)o r/+ •] d-7=pa: (155) 10,612 PARKER: ALTERNATIVE PARADIGM The work also goes into resistive dissipation ofdensity Dwith D/B 2 fromAmpere'slaw (138)that given bythetwo terms 4•r•f2and t/(Vx b2)/4•r onthe right- i•(z) = + handsideof (!24). It followsthat c•b.(•) (•+•)•'•(•) - •'•(C) [(r/+/•)a+ aa][El(L)2 + (156) D - a•• (,•+/•)p/c • ß 4-• The total is (166) &(•) - •s • (•+•)r•(•)+•(•) Ca +t (157) W• •( )[(.+e)•+•][r•(•)•+•(•)•' dz•+D = a• 4• 0 (167) Which isprecisely eqtial •otheenergy input rate Pofcourse. The Theelectric fi61d follows from(139)and(140)as Lorentz forceœisgivenby(137). E•= Bb•(L) •a(L) c [z](t,) + z•,(t,)]' 93. Asymptotic FieldNeglecting Ionospheric Wind (168) In theidealizedcaseof anionosphere thatis somassive that v• andv• arenegligible compared totherateofdiffusioi• ofthe s,,= •(•) •,•(•) c •(L)a +Ea(L)a.(169) fieldput v• -- vy-- 0in(133) and (134). After atime ofthe ørder ofLa/(r/+ t) thefieldrelaxes asymptotically toa finalsteadyOnenotesthatthetotalintegrated Loreritzforceisequaltothetotal state (O/Or-- 0) inwhich (133jand(134)canbeintegrated onceMaxwellstress exerted ontheionosphere whichfollowsfrom(137) to yield db• as db• (•/+j•)-•zz + a-•zz = C•, db• db• -a-•zz + (r/+/•)•z-z - Ca, (158) • d•f•- •b•(•). (159)Therateat whichtheMaxwellstressdoesworkon theionosphere whereCx andCa areconstants. Thesetwoequations canbesolved fordb•/dzanddb•/dzandintegrated again yielding isjusttheMaxwellstress multiplied bythespeed of themagnetosphere cE•/B above thetopoftheionosphere. Thepower input is then b•(•) = C•Z•(•) - C,Z•(•), (160) •(•) = c•r•(z) + c• •(•), (161) wheretheintegrals •1 and•a aredefinedas •, = •(;)a•/4,•, = 4•[r•(•)• Bab•'(L)aZ•(L) (170) + r•(•)•]' Turningto theenergyequation (124)thetwodissipative termson thefight sideare r•(•)= •(•&(. ++•).•' +•)• Ea(z) -- (r/+/•)a +aa, (162) 4•r/•fa+r/ Vxb (163) andrepresent theheight integrated conductivities. Withb•(L) given andb•(L) = 0 it follows that c• = •(•)r•(•)/[r•(•)• + r•(•)•], /4•' •T + \(db• •T/ a] +•-•-(J• ), _4_•[(db•)a 4•rrl. a+s•.a b•(;)•(, + •) (171) 4•r[•(L) + •](L)][(r/+/•)a + aa]' Multiply byBa todimensionalize theresult toergs/cm 3 secand integrate overz from0 toL. TheresultisjustP of course. so that 9.4. FieldRelaxationNeglecting Ionospheric Wind b•(z) =b•(L) Z•(L)P"•!z)+ Za(,L)•a(z) (164) •(•;)• + •(•)• ' Thenextproblem isthetime-dependent relaxation ofthemag- neticfield in an idealizedmassiveionosphere whosebulk motion canbeneglected. It is sufficient forthispreliminary studytotreat •(•)_ •.(;)•,(;)•(•) r•(;)r•(z) the idealized case in which the ionosphere is a uniform slabof thickZl(L)a +Za,(L)a. (165) nessL withc•, /J, andr/uniformacross L. In theidealizedcircumstance of a verticallyuniformionosphere it follows that •(•) = •(•)•/•, •(•) = 0. If for somereasonwe needthe electriccurrentdensityit follows Thecalculation begins with(133)and(134)wherevx andvy areputequalto zero.Theequations canbewritten •0- (rI +/•)•-z2 Oz a, (172) 0•]b•- +aO•by. PARKER: ALTERNATIVE PARADIGM 0 subsequently returns asymptotically tozero ast- 1/2with 02 ] • - (rl+/•)•-•z • bu- -o•oZ •, (173) inauniform ionosphere. Hence bothbzandbysatisfy •e equation 0 0•]• 10,613 b•(L,t)~bo • !-•+'" (179) forlarget. Theresulting fieldor tilt b, throughout 0 < z </• is • -0 (174) Thegeneralsolution of theseequations isexpressible in termsof an infinitesumovertheseparable solutions. Weconsider somespecial : bo [err[(L+ z)/(4rlt)«]-erf[(Lz)/(4rlt)«]] (180) cases to illustrate theseparate contributions fromc•and/3+ r/beforetreatingthe generalcase.Considerfirsttheidealizedsituation ~ (•rrlt)« 1- 12tit+... , in whichtheionosphere issodensethat•i ri, •e re < < 1, where (181) fli andf•e are.theionandelectron cyclotron frequencies eB/Mc asymptotically asz/t • inresponse tothedeclining tilt andeB/mc, respectively. (Recallthattheionandelectron mean declining theionofreepathsfor collisionwithneutralatomsarecomparable in or- appliedat z = L butvaryinglinearlywithz upacross Theelectric current isgiven by(cB/47r)Ob•/Oz ff it is derofmagnitude). Hence ri/re ~ O(M«/m«)and f•ir/is sphere. TheLoreritz forceexerted bytheMaxwell stress Bbm/47r small compared to•erebyO(m«/M«).Neglecting m/re= needed. toc3bm/Oz sothatit isuniform across theheight of (eB/c)/nere compared toM/ri - (eB/c)/niri in (114)- isproportional Otherexamples aregivenin Appendix C. (116)yieldsfl/a • •ivi, a/rl --• •ere.) Thena,• << r/ theionosphere. In the opposite extreme (a > > r/+/3) for a tenuous ionoand (174) reducesto spherethefieldequation(174) approximate s to 0 0• [•. - r/•-•z•]•b - 0. (175) Ot•-F •-z•-O. (182) Noteagainthatb• andbyvanish atthetop(z = 0) of themas- hastheseparable solutions expi(cot + kx) withco-sivenonconducting atmosphere. Themagnetosphere of collision- Thisequation 2 representing waves withaphase velocity vq•----co/klessplasma (r/+/3 = 0) liesabove z = L andhasnegligibleq-c•k 4-c•k and a group velocity dco/dk q-2ak. There aresimilarity mass. Thusthemagnetic fieldin z > L canchange quickly with timeasa consequence of thetransport of magnetic fieldat thedis- solutions oftheform ta9(•) where • ----z/t« and 9 satisfies tantmagnetopause. (z = A > > L). Soit isinteresting toseethe a2m•v+7, -- +(i-a)( ,•2,H consequences ofvarious time-dependent forms ofthetiltbz(L) of thefieldapplied tothetop(z = L) oftheionosphere. 9z+ a(a- 1)9 - 0. (183) wheretheromannumeralsuperscripts indicatederivatives. Suppose thenthatbit(L) - 0 forallt whileb•(L) - 0 Setting a -- --« yields thetwosolutions fort < O,jumpingsuddenly tothefixedvalueb0at timet - O. Mainta. iningba•= b0at z - L fort > 0 requires horizontal 22 displacement in thex direction atthemagnetopause z = A. The fieldfort > 0 isreadi'ly shown tobe Z2 9 - sin•-•,cos4at, thefirstofwhich satisfies thebounda•. condition thatb• - by-- 0 atz -- 0. Therefore wehavefrom(172)and(173)(with. r/- • - 0) thesolution -- bo+-71'n= Z(-1)nsin n•rz(- n2•r20t -•-exp L2 )] (176) 1 andforut > > L, z2 z2 b•- +t-•/2sin•t-•, by - +t-1/.•cos 4at' andthecomplement.ary solution b•(z, t),,,bo - -•sin-•exp -• b•=+t-1/2cos•t-.•, by --t,l!2sin z2 4at' asthefieldrelaxes totheasymptotic static formboz/L. Butonlyoneof b•, bycanbemadetovanish atz = 0 in this Alternatively, suppose thatthetiltbz(L) ofthefieldisswitch, case. Polynomial solutions in/•areeasily constmc .ted for(182)and edonsuddenly at timet -- 0 in z > L butthereafter relaxes back thefirstfeworders areexhibited in Appendix D. Thepolynomials to zero.A simplecaseis givenby theapplied. transverse component b•(L,t) intheform thatcontain onlythefirstpow eroft vanish atz -- 0 satisfying theboundary conditions onb• andby.Polynomials ofsecond or higherorderin t do notvanishat z = 0 andsoareunsuitablein the - into(172) and(173) (178) presentcontext.It is obviousby substitution that one solution is for t > 0 whereerf denotesthe errorfunction.Thensince erf(oo) -- 1 thetiltbxjumpsfrom0 tob0attimet - 0 and = (184) 10,614 PARKER: ALTERNATIVE PARADIGM dvy forwhichbz = by= 0 atz -- O.Thecomplementary solution is bz(z, t)-• •z • , by(z,t)- - azt (185) C2by4-P'•zz-- V4 (192) wherev isthekinematic viscosity lt/p andwhere thefour14are constants, to be evaluatedby applyingtheboundaryconditions. Therotation of thevector(bx, by)overheight z isaconsequence of the Hall effect, of course. Nowby(L) = 0 and,since there isnoviscous stress inz > thatdv•/dz = dvy/dz= 0, asz increases toL. Thegeneral case,retaining botha andr/4-t, employs thefull L, it follows equation (174).Theseparable solution ezpi(wt4- kz) providesHence(191) and(192) yield thedispersion relation w = k214-a 4- i(r/4- fi)]. Thesimilar- • = C2b•(œ), • = o. itysolutions taG(•) aredescribed bythefourth order differential (193) equation Use(189)and(190)to eliminate vz andvy from(191)and(192), +(•/+ fi)•GH' + [¬•' - 2(a- 1)(•/+fi)]GH (186) obtaining b•- v(•1 av Va C4-15) • d2b•, dz • C • d2by dx • -_ C•, +(• - a)•G'+ a(a- 1)G- o avd2bz whichwe noteis invariantunderthe transformation • --• --•. 3 a -- •, thesolution G -- • leads to (195) Then let • - +.zt, % - z[•z'+ (. + b•- • + Q•expqz, (187) V4 andthecomplementary solution by-- • + Q•expqz, • - z[•1z 2 + (. +/•)t], %of (172) and (173). v(•l+ 15)d2by _ V4 C2dz2 [-by- C2 dz2 - C•. There is againa variety of polynomialsolutions.For instanceif (194) (196) (197) (188) whereQ1 andQ2 areconstants. Theresultis These solutions satisfy the boun- [C=/v- (rl+ lS)q2]Q1 - aq=Q= = O, daryconditionthatbz = by = 0atz = • = 0. Theyex- (198) hibit a Maxwell stress(tilt of the field) growinglinearlywith time .q•Ql 4-[C2/v- (rl4-13)q•]Q• - O. (199) 1 3 , by- 0 attime starting frombx- 0, by-- •123 orbx- •'z t = 0. Additionalpolynomialsolutions areindicatedin Appendix Settingthedeterminant equalto zeroyieldsthetwopossibilities D. The purposeof the foregoingarrayof mathematical solutions C•(.+/• 4-i.) hasbeento illustratetheapplication of theB,v paradigm to mo- _= v •[(•+•)•-5•]' q•,'= •(• +• •:i•)' (•00) tion of a partiallyionizedmedium.The nextsectiontakesup the combinedmotionanddiffusion,goingto a steadystateto minimize thecomputation. As thereadercanreadilydemonstrate, approach- sothattherearefourroots4-q• and4-q2,withq2= q• and ingtheproblem withtheE, j paradigm isproblematical unless the solutionfromtheB, v paradigm is available asa guide. C q•- v• It/+fl- ia]«' 10. SteadyMotion of the Polar Ionosphere C(cosO+ isinO) If thesteady Maxwell stress B2b(L)/4•' ofthetilted magneticfieldis applied attheuppersurface z = L of theionosphere for a sufficientlylongtime,themassiveidealizedionosphere even- (201) -«[(. +/•)• + with tuallyachieves a steady motion vz(z), vv(z) limited onlybythe viscous dragon thelowerboundary z = 0. Viscosity maydrag thenonconducting atmosphere in z < 0 alongwiththeionosphere to somesmalldegreesowe work againin theframeof reference of themotion atz = 0, putting vz = vy = bz = by = 0 at z = 0. Understationary conditions (O/Or= 0) andtheidealizationthata, t, •/, p, and(7 areindependent of z, (133)- (136)can be integratedover z to give dbz+ adbv •: + (• + /•)-•z • _ v• dbz dby_ v•- "27,+ ("+ •s) • dvz C=•' + V'•z-z - Va 1 2,cos0 - 1+ [(r/+ f/)2 +a=]«' (202) •+/• }«' (203) 2«sinO - 1- [(r/+/•)2 +a2]] Forq = +q, wehaveQ2 = -iQ•, whileforq = -l-q2,Q2 = (189) +iQ•. The completesolutionhasthe form (190) Va bx(z) -- • + Klexpql z (191) 4-K2exp(-qlz)+ K3expq2z + K4exp(-q2z), (204) PARKER: ALTERNATIVE PARADIGM 10,615 Equations(204) and(205) cannow be written or(z)-C2 iKlezpqlz- bz(z) iK2ezp(-qlz) + iKaezpq•z+ iK4ezp(-q•z), (2o5) wheretheKi arearbitrary constants. Thevelocity (vz, vy) followsfrom(189)and(190) b•(L)1- sinhql 2sinhqlL 2sinhq•Lz)] ' (217) (L- z) sinhq•(L- Theboundary conditions thatvz -- vy -- b: - by- 0 at z - 0 requirethefourrelations sinhq2L ' (218) =•ib•(L)[ sinhql(Lz) sinhq•(L - z)] sinhqlL The velocitycomponents follow from (189) and(190) with uV•/C• + i(K• - K2)/q•- i(Ks- K4)/q•- O, (207) V3/C2+ K1+ K2+ Ks+ K4- O, (208) V4/C• - i(K14-K2)4-i(Ka4-K4)- 0, (209) vz(z) -- C2bx(L) [coshql L--coshql (L--z) 2u qlsinhqlL q•sinhq2L coshq2L - coshq•(L - z)]' (219) v2(z) -- iC2bx(L) 2y [--coshql LqxsinhqxL - coshql (L- z) coshq2 Lq2sinhq•L - coshq2(L - z)]' (22o) respectively, uponmakinguseof thefact that Nowtheboundary conditions atz -- L arebv(L) - 0 with dv•,/dz - dvy/dz - 0, yielding (193).There isnoviscous stress inz > L andnoacceleration inz < L, sod2vz/dz• = d2vy/dz • - 0 asz approaches L, requiring TheLorentz force(V x b) x b/4a'exerted onthefluidis Kl ezpqlL + K•ezp(-ql L) L- +Ksezpq•L + K4ezp(-q•L) - 0, 1 db• 4•' dz ' (210) ' (221) bx(L) [qlcoshql(Lz)+ q•coshq•(Lz)] K1ezpqlL + K2ezp(-ql L) 8•r -Ksexpq•L- K4exp(-q•L) - O. sinhql L (211) i dby fY- 4•rdz' - It follows from (211) and (212) that 8•' sinhq•L ' ' (222) _qlcoshql (L- z)+ q•coshq2(L - z)] (212) K• - -K1 exp2qlL, K4 -- -Kaexp2q•L ibz(L) sinhqlL For therecordthecurrentdensityis Thesetworelationscombinedwith(208) and(209) yield Vs + iV4 cB dby K1----2C2(1 - ezp2qlL)' 4•r dz ' = -cW, (213) 2(1 - exp2qlL) ' (223) cB db•, 4•r dz ' Vs-iV4 = +cBf•. (224) Ks= - 2C•(1 - ezp2q•L)' _2(1 - b•(L) (214)Theelectricfieldfollowsfrom(113)andthe(189)and(190)as ezp2q•L)' dby from (206) and (207) as V1-- C2bz(L) 2// (cothqlL ql + cothq•L) q• , (215) 2u - ql + q• , (216) = -V•B/c, (225) -- 4-ViBIc, (226) C - B[v' V2iC2br(L) ( cothqlL cothq2L) - dbz, E.- B[- V__y _(q+/?)-•-z +o• , -27J uponusing(193)for Vs andV4. Theconstants V1 andV2 follow whereV1 andV2 aregivenby(215)and(216). 10,616 PARKER:ALTERNATIVE PARADIGM The motionof the magnetosphere (z > L) is givenby cE x BIB 2,or = V•, (227) vy- -eEl/B, = +;4. (233) Note from (219) and(220) thatthemotionof theneutralgasat the upperboundary of theionosphere (z -- L) is 2•, (1 qlsinhql Lq- 1) C2b•(L) q2sinhq2L , (229) (234) vy(œ)- v2 2y q•sinhq• iC2b•(L)( i L _q•sinhq• i L )' (230) Comparing (229)and(230)with(227)and(228)it isevidentthat themagnetospheric plasmadoesnotmovewiththesamevelocity astheneutralgasat thetopof theionosphere. The motionof the magnetospheric plasmaisdifferent fromthedriftof theionsandthe electrons, too.Themagnetospheric plasmamoveswiththeelectric driftvelocity,whereas theionospheric constituents all driftslowly (235) in variouswaysrelativeto thatframeof reference asa consequence of the collisionalforcesandthe viscousdrag. Thedriving forceB2b•(L)/4 •r isin thex direction. The y component of theionospheric velocityvy is nonvanishing asa consequence of theviscosity andcollisional dragthrough theHall = V2+•bx(l) sinhq•L i [(•l +ia2)q• coshq• (L- z) effect,of course.This followsformallyuponnotingfrom(201) - sinhq2L - z)] ' _(,- ia2q•coshq•(L (236) (203)thatputtingc• -- 0 leadsto q• -- q2. Equation (216)gives V2- 0 sothat(225)yields Ex - 0 and(220)yields vy(z)- O. Newton'sthirdlaw requiresthattheviscousdragacross thelower With theseresultsit is easyto showfrom(105) or (106) that surface(z -- 0) of the ionosphere is alsoin the x directionand icB equaltothedrivingforceattheuppersurface of themagnetosphere. j•-- 8•b•(L) Differentiation of(220)establishes thatdvy/dzvanishes atz -- 0, whiledifferentiation of(219)yields theviscous stress pdvx/dzat z - 0 asbeingpreciselyequalto theappliedMaxwellstress dv• B2b•(L) P dz - 4• x JY- +cBb 8•r•,(l) To examine the solutionsin more detail, note that the ion and electron drift velocitiesfollow from (110), (111), (189), and(190). It is convenient sinhq•L coshq•(Lz)- q•coshq•(L - z)]' (237) x to write q•coshq•(Lsinhq• L z)q-q•coshq•(L sinhq2L - z)]. (238) Sinceq2 - q•*, it isobvious thatwx,wy,ux,uy,jx, jy areall a•= cBM/ri 4•rneQ 'a•-=cBmlr, 4•rneQ' (231)real quantities. so that Whenql,aL < < 1,it isreadily shown from(217)and(218) that z b(z)= Then,usingprimesto denotedifferentiation with respectto z, the ion drift velocityis - - •z) x[1- 3y[(r/+/•]2 + c•• +04(q•,2L)], (239) PARKER: ALTERNATIVE PARADIGM 10,617 The energy delivered into the ionosphereby the Maxwell stress,that is, by the Poyntingvector,is dissipated into heatvia viscosityandvia thecombinedOhmicandPedersen resis- b•(L) b'z(z)- L tivities. ThetermB2f. v ontheright-hand side of(247)represents C2(? +•)(L - 3Lz +•z•)]_04(q•,2L)] x[13y[(7 +• •)2 + a2] , (240) therate at whichtheLorentzforceB2f doesworkon the neutral atmosphere in opposition to theviscosity.It is readilyshownthat by(z)• -b•(L) B2 «z) + [1+ 02(q1,2L)], (241) + B2f'v- •-•[Vlb• + V2by - (7+•)(b•+ by)],(249) by(z)- b•(L)C2a(L• - 3Lz+ •z•) L 3y[(7+ B)2+ a2] wherein b•and b•follow from (217) and (218) while theremaining termson the right-handsideof (247) reduceto the resistivedissipation [1-902(q1,2L)].(242) 2 B2147r•f 2-97(Axb)2/47r] - •-• B (7.9/•)(b• ,2-9by ,2). (250) It follows from (215) and (217) that 04(ql,2L)] , (243) L (•+f;)[1+3•(• +2 f;)-9 V1 -•b.(L) C2L dissipation rate(ergs / cm-3s)is The sumreduces,then,to the totalgivenby (248). The viscous V2- bx(L) c•[1 +O4(ql 2L)] L ' ' .vl (244) B2C2 -9b•x, respectively, through Arepete's law,ofcourse. The energybudgetof theionosphere is easilyelaborated.The energyflux is given by the z componentof the Poyntingvector )- sinhql(L z)sinhq2(L-z) ----br(1) 2 4•r y , Theelectric current densities j• and jy areproportional to-by and + - sinhqlL sinhq2L ' (251) sothatthetotalviscousdissipation across(0, L) is •o Le(vi+v.,2)__ evaluated at z = L, thatis,by theMaxwellstress multplied by the electricdrift velocity,sothat 8 [(7+/•)2 +a2] b•(L)2[qlcøthql L_q2cothq2L]. (252) CB Pz(z) - -•--7•-w (E•,by - Eyb•,), Theresistivedissipation is S2 47r (Vlb• + V2by). (245) B2 B2 4•r(7+•)(b•+by) - 4•r At z = L this reduces to S2 P•(L)- -•-• Vlb•(L). (7 -9]3)qlq2b•(L) 2 coshql (L- z)coshq2(L-z), (253) sinhql L sinhq2L (246) It is evidentfrom (227) that this is just the x-componentof the velocityof the magnetosphere multipliedby the appliedMaxwell andtheintegratedtotalis stress[Zhu,1994a].It is, of course,alsoequalto E-J, whereJ is thetotalintegratedcurrentacrosstheheightof theionosphere. Theenergyiswithdrawn fromthePoynting vectorin theiono- B2 ]o Ldz(b• -9b•2) __ lB2 (7-9 ]•)C2bx(L)2 [½o•thqlL ½ø•thq2L] (254) .•--•w (7-9• ) sphere attherategivenby theright-hand sideof (124).Notingthat thedivergence termisidentically zerobecause O/Ox-- O/Oy-0,therateD ergs / cm-3sis 8•r ay ql q2 It is straightforward toshowthatthesumoftheviscousandresistive D- B2[f.v + 4?r/•f 2+ 7(A x b)2/4?r], (247) dissipation is equalto thetotalenergyinput,givenby (246),with V1 givenby (215). S2 47r [b•v•-9by vy-9(f•-97)(b•-9b•2)], It isevident fromthisexample thatthereisnooverallalgebraic relation (i.e.,noOhmslaw)between F, andj because thevelocity of theionosphere. Thereis (ll7), but since fi - b'i/47r. Eliminating vxandvythrough (189)and(190), v variesacrossthethickness it cannotbe useduntilv(z) is known.So,asalreadynoted,it is notpossible to compute thecurrent flowin theionosphere fromthe electricfieldappliedattheuppersurface of theionosphere without firstsolving thedynamical equations forv(z). TheB•v paradigm Comparing thisresult with(24:5) forPz(z), itisobvious thatdPz/ beginswithexactlythatdynamical problem,andthesolution of the dz - -D. ThePoynting vector fallstozeroatthelower boundarydynamical equations provides bothF,(z)andj(z)ff wewishtoknow it follows that D- -•-•-• (Vlbi -9V2 B•). (248) (z -- 0)oftheionosphere where bz- by- 0,ofcourse. them. 10,618 PARKER: ALTERNATIVE 11. Equivalent Electric Circuit PARADIGM fact, a smallfield componentin boththex directionandthe z dition. they to œ), The E, j paradigm hasled to theideathatvariousaspects sotheycanbe neglected.Linearizingthedynamical equations for of the magnetospherecan be representedby a fixed electric circuit, to which Kirchhoff's laws can be appliedto deducethe dy- b andv, andneglecting all termsin vz, bz, andall termsof second order in O/Ox and v andb,theresult forstationary flowisthe namicalproperties of themagnetosphere. It is declared thatj is velocity channeledthroughthepathof leastionospheric resistivity,andit is declaredthat a changein the parametersanywherearoundthe cir' cuit necessarilyeffectsthecurrentflow everywhereelsearoundthe circuit,just as it would in a rigid electricalloop circuitof wires, inthey direction throughout -- A ( resistors,inductances,appliedemfs,etc. in the laboratory.Unfor- In0(z<L, tunately,the magnetosphere is not a rigid body so that the circuit Ov analogyhas only limited use. The electriccurrentis determined z)= 4•'pu force F - B x (X7x B) exerted onthefieldbythegaswith (55) z ( 0 sothatv(z, -- A) = 0. + by thedeformation X7 x B of themagnetic field,thatis, by the +z) -0 (25) C20b 0 v- 0 •z + U•z• (V x B) •_ - F x B/B•. Thishasnogeneral tendency tocoincideprecisely withtheregion of minimum (r/ q- •). A more (257) fundamentaldifficulty is createdby the fact that the currentpath where C isagain theAlfven speed B/(4a'p)«. Asingle integrais distributedbroadlyover a rangeof motionv, with varyinglocal tion yields electric fieldE t(-- E q-v x B/c). Thereisnouniquely defined "voltage"to apply to the circuit. Consequently, differentportions of thecircuitbehavemoreor lessindependently of eachother,their connectionbeingonly the dynamicalonedescribed by (4) and(5) ratherthanby the simultaneityimpliedby Kirchhoff'slaws. The analogous electriccircuitcanbe constructed, if at all, only afterthe Ob v+ - (58) Ov C2b +U•z- V4 (259) whereV2 andV4 areconstants in directanalogy with(190)and dynamical problemhasbeensolvedto determine B andv. (192). The boundaryconditionsat z -- 0 are b The generalaprioriinapplicabilityof an electriccircuitcon- viscousstress equalto theappliedMaxwellstress, ceptcanbeseendirectlyfromtheinductionequation(5) andthemomentumequation(4), pluswhateverheatflow conditionsareneeded Ov B2b(x,L) PV•z- to specifyPij. Thetwoequations together determine thelocaldynamicalevolutionof v andB. Giventheinitialv andB through- 0, with the 4•r ' (260) outtheinterior ofaclosed surface $, thesubsequent dynamical and the velocity fitting to(255) atz -- 0.Atz -- L,b(z,z)beevolution of v and B throughout the interior of $ is uniquely decomes equal to the applied b(z,L) whileOr/ termined by thevaluesof v andB on $, regardless of whathapOz = 0 sincethereis no viscousstress exertedon z = L from penselsewherealong the currentpathsextendingoutwardacross z > L. It followsfrom (259) that the conditionat z = 0 and $. Andthatis nota property of a localportionof a simpleelectric z = L bothrequire circuit,where the interruptionof the currentanywherearoundthe circuitinstantaneously affectsthecurrenteverywhere.Onemayintroducemorecomplicatedcircuitsincludingmovingloadedlinesto simulateAlfven wave propagation,of course,but suchcircuitsbecomepreciseonly in thelimit of a continuous distributionof circuit branches. However the correct behavior V4= C•b(x,L). Thenuse(256)toeliminate Ov/Ozfrom(259)obtaining of the continuous distributionof circuitbranchesis describedexactlyby (4) and (5) and cannotbe determinedby othermeansin the generalcase. •z 2 k•b+ k•b(z,L) = 0 It is sufficient for present purposes of illustration of electric where currentflow to considerthesimplecaseof a viscousnonconducting (r/= cx>)atmosphere extending fromthefixedsurface z: k = --A to (261) z -- 0. In0 < z < L there isanidealized ionosphere inwhichIt follows that thereis a resistivediffusioncoefficientr/and a uniformviscosity kt ----p•,. Abovez = L thereis onlya tenuous highlyconducting plasma(r/: 0), extending upto z = A. Thesystem ispenetrated by a uniformmagneticfield B in the z direction.The systemis b(z,z) =b(z,L) [1-sin hk(Lz)] sin hkL (262) drivenfrom above,as in section10, by movingthe foot pointsof themagnetic fieldB at themagnetopause (z = A) in they direc- (:263) tion,producing a y component Bb(x, L) throughout thetenuous magnetosphere L < z < A, asin section10. Thereis,however, an importantdifferenceand thatis thatin thepresentcaseit is as- with sumed thatb(x,L) varies slowly withz, ona scale I >> L,A. Theneteffectis to drivethesystemin they directionwiththespeed V2 - b(x,L) (-• +•lkcot hkL). v(x,z) under theapplied Maxwell stress B2b(z,L)/4•r. Asa consequence of theslowvariation of b(x,L) withx, there is,in inorder tosatisfy (255)atz -' 0. Theelectric current is PARKER: ALTERNATIVE •cos 10,619 in theframeof reference of thefluid.ThenE• follows asj•/•r, c BOb J•--4-• 0'•' given by (264). The termsdependingupon z cancelout of this relation,of course, because y7x E = 0, leavingtheconstant terms, hk(L- z), 4•r sin hkL PARADIGM (264) - ---' cB Ob + •kcot hkL . (269) If the fixed circuit analoguwere correctin somedirectway, jz 4•rOz' onewouldexpecttofindj• -- •rE•. However, ofcourse, thisisnot 4-•Ob(z,L) •':• [1- sin sin 'hkL z) cB hk(L]' (266) Thus jz issmall O(1/kl) compared toja•.Note, then, that theprincipal current ja•declines upward across 0 < z < L in response tothedeclining Loreritz force while theresistivity r/is correct,becausethe fluid is movingandtherelevantelectricfield fordriving thecurrent isE•, asnoted insections 9 and10.Soeven in so simplea systemas the one-dimensional circuitconstructed here,the construction canbe carriedthroughonly aftertheproblem is solved. However, now considerthe consequences of the slow varia- uniform. Thisillustrates thepointmade earlier thattheionospherictionof thedriving fieldb(x,L) withx. Theresult isa smalljx, flowingupwardfromz = L alongtheperturbed currents arenotsimply concentrated atthelevelwhere r/isamini- hithertoneglected, mum.Thereareotherinfluences aswell,forexample thevariation fieldandclosingat z -- A wherethedrivingforcethatcauses the of E • withthefluidvelocity v. Appendix E provides anillustra- Maxwell stress B2b(z,L)/47risapplied. Suppose, asanexample, that tiveexample oftheeffect ofthevertical variation ofr/(z),showing again that there isnodirect local relation between ja•(z)and r/(z) b(x,L) = bo+ b•cosKz (270) alone. Nowtheprincipal electric current isinmenegative x direction,providing aone-dimensional electric circuit extending (iffor where b• ( boandK(= 1/1) issmall compared tok. Then jx- -cBb•Ksin Kz[1 - sin hk(Lz)] (271) themoment weneglect theslowvariation withx) fromx 4•r sin hkL tox = +o0. If maybeimagined thatthespace isperiodic, sothat thecircuitcloses atx -- -4-o0.Thetotalcurrent Jx perunitlength Theresultisa seriesof sideloopssketched in Figure8. Theconvenin they directionis tionalelectriccircuitanalogcanbe awareof theseauxiliary loops J• - only by considering the stressbalance,andit cantreatthemquantitativelyonly aftercarryingthroughthe aboveformalsolutionto thedynamicalproblem,or its equivalent. dzj• (267) 471' Asa finalpoint, notethatforb(x,L) • C 1, treated here, themagneticconditions at anypositionx dependonlyon thelocal whichis simplyanexpression of Ampere's law. Theelectricfield b(z, L). Thesmall fieldperturbation elsewhere hasnofirst-order E• isreadilycomputed fromOhm'slaw,whichiswritten ja• -- effectonconditions at x. Thustheideathata changein conditions •rE'•(a= c2/4•rr/)and E• istheelectric field = E, + at onelocationin anelectriccircuitimpactsconditions everywhere (6s) )B/c aroundthecircuitis inapplicable, showing thatthecircuitanalog conveysthewrongphysics.Localconditions areexactlythat,and z=A z=L • _2•- • • -•' +•' • z=O +2•' Kx Figure8. A schematic drawing oftheelectric circuit analogue withthemaincurrent described by(264)andtheauxiliary currentloopsindicated by equation(271). 10,620 PARKER: ALTERNATIVE PARADIGM theysprouttheirownauxiliarycurrentloopsasneededto remain localized. dw(l+w2(p'2)« - 0. In the morecomplexsituation whenthe appliedforceis strongly inhomogeneous andafunction oftime,thevarious parts of thesystem communicate witheachotheronlyontheAlfvenwave Integrationyields transit time,whichmaybeslowcompared tothecharacteristic rate atwhichtheapplied forcechanges. Theinstantaneous circuit isentirelyinappropriate andsomesuitable loaded lineWOuld benecessarytorepresent theeffect.However, againacorrect circuitanalog where k isanarbitrary constant. If k = 0, theresult isthefamily canbe setup onlyaftertheappropriate fieldequations for v and ofradial lines, forwhich (p•= 0. Intheopposite extreme that(p•is B areproperly solved. Thatistosay, theactivity ofthemagnetoitfollows thatrv = ,• = const,providing circles concentric sphere ismechanical innatwe,andelectric circuit analogs canbe large, yields constructed onlyafter.thefact.Thequestion, [hen,iswhatistobe abouttheorigin.For0 < krv < 1, integration gainedby theirconstruction? In conclusion, thesolution to a problem in magnetospheric kw - sin((p- (p•. ) plasma liesin thedynamicalstress andmomentum balanceandthe associated deformation of B. Electric circuit analogies canbeconThisprovides a circleof diameter structed onceiheproblem issolved, butthereseems tobelittletobe where(Pl isanarbitraryconstant. gained byit since theelectric circuit deals onlywiththeperipheral1/Itthrough theorigin. Thecenter ofthecircle liesatrv-- 1/2k, quantities E andj, whicharedirectly available fromB andv. 12. ConcludingRemarks AppendixB: Angular Deflectionof Flux Bundle Thereis little to sayin conclusion, unlesswe areto repeatthe generalremarksin the introductorysection.The idealizedexamplesmakingup the bulk of the text illustratethedeductivemethod givenby thecircles Thefieldlinesof theambient two-dimensional dipoleare of theB, v paradigm andthegeneralapplicability to themacro physicsof themagnetosPhere. Theparadigmfollowsdirectlyfrom theequations of NewtonandMaxwell,yieldinga setof partialdifferentialequationsthat are sel.f-consistent andcompletein them- (y-«a) 2+z •-- ¬a•, selves.ThustheB, v paradigm worksdirectlywiththemechani- for cialpushing andpullingthatmakesupmacroscopic magnetospheric which dz _ = dy activity,thereby providing a'directdeductive approach to thedynamicsof the magnetosphere, and avoidingthe difficultiesof the •, j paradigm, whichfocuses onsecondary quantities. It- «a z . The diameterof thecircleis a andthecenteris locatedon the As thesubjectof magnetospheric physicsadvances intomore complexdynamicalproblems,thedeductiveapproach becomes im- y axis aty -- «a.Theperturbed field line, described by(71), can perative.Theprinciples anddeclarations thatcarrythe•, j para- be written digmforwardarealreadyin somedegreeof errorin thesimpleex(V- •a')• + (z- a•) • -- • -'• (•3) amplescitedin thetext: The definitionof parallelcurrentsin terms of theunperturbed field,theideaof thedynamicalinwardpenetrathediameter, withthecenter at tionof theinterplanetary electricfieldalongthemagneticfieldlines, wherea' = a 4- Aa represents andthenotionthatsimpleelectriccircuitanalogsdefinemagneto- y -- «a',z -- Ab,with Aaand Abgiven by(72)and ---}aO, spheric dynamics beforethefact,areall cases in point.It canbe respectively. It followsthat argued, of course, thattheE, j paradigm canbemadetowork with sufficientcare and foresight.However,thisis equivalentto the statementthat, if the correctsolutionto a dynamicalproblem dz its tractableandcompletesetof partialdifferentialequations, providesthe meansfor deducingthe correctdynamicalsolutionfrom theboundaryconditions andthebasicphysicalprinciplesembodied in Newton'sandMaxwe•'s equations.' Only with a purelydeduc- We areinterested in theangulardifferencein thedirectionof specify thelocation ofthepoint ofintersection, forwhich z(> 0) for a specifiedcircumstance. =0 z -- +(ay- y:•)« from(B1). Substituting thisresult into(B3)yields ! In polarcoordinates (w, (p)Fermat's principle canbewritten 6/ dw (1+w•(p'g) « (B4) thetwocircles wheretheyintersect at some point(y,z). Lety tive procedure canoneproceedwith assurance of a correctresult follows as AppendixA: Path of Flux Bundles (y - •yy =- z-Ab ' isknownin someway,thenthedescription of thedynamics iseasily translated intotheE, j paradigm, whichwehaverepeatedly emphasized: Theessential pointis thattheB, v paradigm, with in termsof thelocatio n y of thepointof intersection. Theslope (dz/dy)a oftheambient field line and (dz/dy)p where (pt• d(p/dwand theindex ofreffaction B is!/w •. Sinceof theperturbed fieldlinefollowfrom(B2)and(B4).Theirdiffer(pdoesnotappear 'm.theintegrand, Euler'sequation reduces to ence is PARKER: ALTERNATIVE PARADIGM 10,621 [4(rl+/3)t]« [y(a-- y)]«([y(a- y)]«- +[4(.+ •3)t] « (86) 71' 1 --•-z•b 5a y(a- y) (•7) 4(rl+ •3)t 4(rl+ •3)t ' Then for4(rl+ •3)t> > L2, upon neglecting Abcompared toy«(a - y)«. Denotethesmallanglebetween thetwoslopes by •P.Let b•(z, t)~bo • 1- [•r(rl +•3)t]« '" ' - tan(O+ •b). (dz)_tano,(d•yy) Thefieldrelaxesto a linearincrease upwardthroughtheionosphere. Since tan(O+ •b)- tanO"• •Plcos20, As a finalexample,suppose thatthefieldappliedat z -- L (s8) growslinearlywith time in the beginningbut aftera timecompa- rable tothecharacteristic diffusion timeL2/(rl+/3), slacks offto anincrease proportional onlytot «. Let it follows from (B7) and (B8) that • -• -2Ab/a -• 0. (B9) b•(L, t)- bo •-• 1+ [•r(rl +•3)t]« exp- (rI+ Sothedeflection 0 relativetotheambient fieldisindependent of positionalongtheperturbed fieldlineandhasthesamevalueas 2L2 « -[1+ (rl•)t]erfc[(rl+t3) }. givenby (63). AppendixC' Ionosphere WithoutHall Effecta - O: L For(rl+ •3)t<< L2, As anotherexampleof thediffusionof fieldin a motionless o•(L,t)~ Ooz•, ionosphere with c• -- 0, suppose thatthefieldor tilt appliedat z-Lis while for(rl+ fl)t > > L2 b•(L, t)- boerfc [(rl +•3)t]« b•(L,t) ,,,bo [(r/+ fl)t] « •rL 2 •rL 2 «(1-exp[-(rl+fl)t])]' +[(.+/•)t] ß Thefieldthroughout 0 < z < L followsas Then for(rl+ fl)t < < L2, bz(z {[2(rl+ (L-z)2 fc[4(rl+ L-•3)t] z« ' t)- bø(rl L+ 2•3)t •3)t+l]er b•(L t)~bo [(rl +l•)t ]{ (L+z) 2+l]erfc L+z L+z with •e app• field •cm•g ast • •omzero attime t -- 0.At - 2(. +/•)t [4(. +/•)tl« + [•r(. +/•)tl« (z+ exp- 4(. +z)"] •)t , =L • !•ge t, b•(L,t) • bo1- =(? +•)t •"' [•r(r l + •3)t]« expThefieldapplied totheupper surface (z -- L) oftheionosphere grows initially ast -}and subsequently, after atime comparable to Thenfor(rl + •3)t> > L, thecharacteristic diffusion timeL2/ (rl+j•),levels offtoauniform 4(rl+ •3)t ' valuebo.Thefieldthroughout theionosphere 0 < z < L follows from (153) as b:(z, t)- •-•(L+z)er fc [4(rl +•3)t]« o•(z, t)~•o• • (.+•)t andagain bz(z, t) increases linearly upward across theionosphere. The linearvariationis a consequence of thevanishingor declining 10,622 PARKER: ALTERNATIVE PARADIGM timerateof increase of bz(L,t) in thelimitoflargetimet. The localLorentz forceintheionosphere isproportional tooqb/oqz and The series terminates after the m + n term with the choice a -- «n 4- 2m 4- «. is asymptotically uniformover z. (D14) AppendixD: PolynomialSolutionsto Equations(9.51) The linearityof (183) permitssuperposition of thesemanyindivid- and (9.54) ual forms, of course. In a similarvein polynomialsolutions of (185) canbe generIt is a simplematterto generate polynomialsolutions to (183) I 3 ated. For instance, the solution and(186). Generally,the solutionscanbe writtenasinfiniteseries, givingrise to the solutionsz and(186) and(187). The quadratic but with properchoiceof a, the seriescanbe terminatedat anydeform q• siredpoint.S•g withthesimplest solutions, consider ß = •o, fora = 1,andq•= •2 + 2(r/+/3) fora = 2. Hence fora = 1, satisfying(183) for a -- 0, 1. Theresultingsolutions are bz = Do, by= D• + D2z, bz= z• + 4(r/+/3)t. (D!) (D15) Substituting thisinto (172) and(173) yields Dot, by= Doz2/2a, (D2) whereDo, Dx, andD2 arearbitraryconstants. Thecomplemen- tarysolutions areobtained byinterchanging bzandbyandchanging the signof oneof them. by =rl+l•z2 - 2a [1-(rl+l•)•]t (D16) together withthecomplementary solutions. Fora = 2, If • -- •, then a -- «, •. Thesolutions are bz= z2t4-2(, 4-fl)t2 b• = z, by= 0, (D3) bx - zt, by- za/6a. (D4) with by= 12a - a[1- (rl z4+ (rI +•)z2t +•)t2] (D18) If • = •2, thena = 1, 2. Thesolutions are bz= z2, by= -2at, (D5) bz-- z2t,by-- z4/12a- at2. (D6) (D17) etc. Note, however,that thesesolutionsdo not individuallyor in combination satisfy theboundary condition thatbz = by = 0 at Z:0. Thehigher-order polynomial solutions canbegenerated along The solution(D5) isjustthecomplementary solutionto (D2). If 9 _ •a,then a -- •, •. Thesolutions are AppendixE: bx-- Z3, by-- -6c•zt, (D7) b• = fit, by= zS/20a- 3azt2. (D8) The solution(D7) is the solutioncomplementary to (D4). For higherorderpolynomials, write • = •n 4-al• n+44-a2•n+84--'.. (D9) The cross-fieldcurrentsare determined by the forceexerted on thefield by the fluid, sothatthecrossfieldcurrentscanbe determinedonly afterconsideration of thedynamicalequations.Thereforethereis no simplealgebraicrelationbetweenthecrossfieldcurrentdensityandthelocalresistivediffusioncoefficient,althoughin the final asymptoticsteadystateof ionospheric motiondrivenby Maxwell stresses transmitteddownwardby a deformedmagnetic field, theLorentzforcedependsin a functionalway upontheform ofthevertical variation oftheresistive diffusion coefficient r/(z). inthepresent case. Writeb(z)inplace ofb(z,z) andimag(D10) interest ine thatagainthemotionin the y directionat theunderside z = 0 ofthemagnetosphere hassome such valuev(0) asgiven by(255) withu = u(0) throughout --A < z < 0. Withp, u andr/as sothatn = 0, 1, 2, 3. Then a2- a(n+ 1)+ ¬n(n+ 2) a•= -a2(n+ 4)(n+ 3)(n+ 2)(n+ 1)' a2 - a(n+ 5) + ¬n(n+ 4)(n+ 6) (Dll) functionsof heightz throughtout the ionosphere 0 < z < L, rewrite (256) and (257) as dv (D12) a2---a• a2(n+8)(n+7)(n+6)(n+5) ' am+l CrossField Currents To treata singlesimpleillustrativeexample,considerthecasetaken up in section11 ignoringthe slowvariationwith z, whichis of no Substitution into (182) yieldsthe indicialequation n(n- 1)(n - 2)(n - 3) = 0, the lines indicated for (182). dq+ d db B 2 db d dv - O,T;4 + - o. (El) Integrationover z yields ----am a2- a(n+ 4m+ 1)+ ¬(n+ 4m)(n+ 4m+ 2) (n+ 4m)(n+ 4m- 1)(n+ 4m- 2)(n+ 4m- 3)' (D13) db B2 dv (E2) whereV2 andV4 areindependent of z, withl/• againequaltothe PARKER: ALTERNATIVE PARADIGM 10,623 Beri Maxwell stress Beb(L)/4•r,asrequired bytheboundary condi- l•"' 4a'nm tions at z -- L where the viscous stress falls to zero. Introduce the (El4) variable (given by(115)withm/re << M/r) along withr/,write ½(z)with the dimensionsof inversevelocityand definethe function f(•) as[p(z)v(z)/p(O)v(z)]«, with Ce-- Be/4a'p(0) again. = 1/NAiwi Equation(E2) becomes (ELS) wherewi is theion thermalvelocity,andAi is thecrosssection (E4) for collisionof an ionwitha neutralatom.TheAlfvenspeedC v + db/d•- Ve,Ceb+ fedv/d•- C2b(L) isessentially B/(4a'NM)« where M istaken tobethemass of a neutralair atom as well as an ion. Approximatethe kinematic from which it follows that viscosity v by«,•wiwhere Aisthemean freepath1/NA forcollisionof neutralatomswith neutralatoms,with a crosssectionA. d2 fe(•) Ce][b(z)-b(L)]-O. d•2 Then withp(z)/p(O),,, 1,itfollows thatf2 __v/l•(l• >> r/). (E5)Putting •(L) = L/I•, it follows that,inorder ofmagnitude, In thesimplest case suppose thatf(() isaconstant. Then b(z) - b(L) 1 - sinh•[C(L) -C(z)] } sinh•C(L) . (E6) The velocityfollowsfrom (E4) andtheboundaryconditionat z -0 from (255), with theresult cc() ~ I - = (3LenNAiA) «. (El6) Itisevident atonce that theeffect declines upward as(nN)«,so thattheionospheric D layerappears tobetheprincipal contributor. Thenwithn -- 105electrons/cm a andN - 10le atoms/cm a, Ai - A - 10-16cm e, andthecharacteristic scale L - 107 v(z) - Cb(L){ CA +1 f (1- cosh. f +sinhC•Z)]} . (E7) [cothC'C(L) C'C(z)) velocities (wi•lOScm/s) intheD region (T ,,, 500K)Aimay cmfor theD regiontheresultis approximately 0.5. However,the crosssectionfor ionson neutralatomsis substantially largerthan thegeometrical atomicdimension because of thedeformation of the neutralatomby theelectrostatic fieldof thepassingion. At thelow The cross field current is be10-14 cme orlarger. Thevalue ofC{(L)/f istenormore c timeslarger,thatis 5 or more.Theresultis a strongmodulation of db jr(Z)- - •-•Bd• //(z)-• bythe factor cosa[C[C(L)-C(z)l/f]/sinh[CC(L)/f] on theright handsideof (E8). cSb(;)c 4•rf•7(z) sinh •((L) It iseasytoconstruct morerealistic examples. Instead of f -- (E8) const,thechoicef ,-, •a or exp,c• yieldssolutions to (ES)in termsof Besselfunctionswith imagineryarguments, etc.However, The electric field follows from Ohm's law as - 4a' = velocityvariesstrongly upward through theD layer,withtheresult Bv _ thatjr does notvarysimply as[r/(z)-- /•(z)]-1 upward across c ycothf A weakionosphere, C((L)/f theessential pointisthesame, thatC•(L)/f•O(1) sothatthe the ionosphere. .(o)' Acknowledgements. The authorexpresseshis gratitudeto manycolleaguesfor discussion andcomment.Discussion with Xi- aomingZhu aroused hisinterestin magnetospheric physics again, << 1,yields withZhu'sremarkably clearreduction of theexpansion phase of (El0) - the substorm andthepolarconfinement of magnetospheric convectionto basicphysicalprinicples.Many thanksto David Stem for informative discussions of magnetospheric modelsandthecurrentsystems produced by trappedenergetic particles.GeorgeSiscoegavea carefulreadingandsuppliedmanyvaluablecomments andcriticisms thatgreatlyhelpedin puttingtheproblems in proper perspective. T. Gombosisetup a discussion withhisgraduate studentsAaronRidley,NathanSchwadron, andAnnamarie Sinkovics c•+ C((z) ((z)] 'f [1_ •(L) -!-ßßß (Ell) cS,(c) 4•r((L)r/(z) {1+•-ff[((L)c2 ((z)]e+...} (El2) thestudent aswell astheexpertcouldseetheissues clearly.Finally,it should beunderstood thatanyerrors andmisconceptions in Er-- Bb(L)C c [cdc) 1 y(o)' (El3) thatprovided a varietyof modifications of theexposition sothat thepaperareentirelythoseof theauthorandin nowayreflectthe scientific views of the aforementioned individuals. This work was Thevelocityvariesbutlittleacross theionosphere andjn isdirectly supported in partby theNationalAeronautics andSpaceAdminis- proportional to1/r/(z) tolowest order asaconsequence. trationthroughNASA grantNAGW 2122. On the otherhand,the terrestrialionosphere is notweakin thisrespect.IncludingthePedersen resistivediffusion paper. 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