Why current-carrying magnetic flux tubes gobble up

arXiv:physics/0301037v1[physics.plasm-ph]16Jan2003toappearinPhysicsofPlasmasWhycurrent-carryingmagneticfluxtubesgobbleupplasmaandbecomethinasaresultP.M.BellanMC128-95Caltech,PasadenaCA91125(AcceptedforpublicationJanuary14,2003)AbstractSupposeanelectriccurrentIflowsalongamagneticfluxtubethathaspoloidalfluxψandradiusa=a(z)wherezistheaxialpositionalongthefluxtube.ThiscurrentcreatesatoroidalmagneticfieldBφ.Itisshownthat,insuchacase,non-linear,non-conservativeJ×Bforcesaccelerateplasmaaxiallyfromregionsofsmallatoregionsoflargeaandthatthisaccelerationisproportionalto∂I2/∂z.Thus,ifacurrent-carryingfluxtubeisbulgedat,say,z=0andconstrictedat,say,z=±h,thenplasmawillbeacceleratedfromz=±htowardsz=0resultinginasituationsimilartotwowaterjetspointedateachother.Theingestedplasmaconvectsembedded,frozen-intoroidalmagneticfluxfromz=±htoz=0.Thecounter-directedflowscollideandstagnateatz=0andinsodoing(i)converttheirtranslationalkineticenergyintoheat,(ii)increasetheplasmadensityatz≈0,and(iii)increasetheembeddedtoroidalfluxdensityatz≈0.Theincreaseintoroidalfluxdensityatz≈0increasesBφandhenceincreasesthemagneticpinchforceatz≈0andsocausesareductionofa(0).Thus,thefluxtubedevelopsanaxiallyuniformcross-section,adecreasedvolume,anincreaseddensity,andanincreasedtemperature.Thismodelisproposedasalikelyhypothesisforthelong-standingmysteryofwhysolarcoronalloopsareobservedtobeaxiallyuniform,hot,andbright.ItisfurthermorearguedthatasmallnumberoftailparticlesbouncingbetweentheapproachingcounterstreamingplasmajetsshouldbeFermiacceleratedtoextremeenergies.Finally,analyticsolutionoftheGrad-Shafranovequationpredictsthatafluxtubebecomesaxiallyuniformwhentheingestedplasmabecomeshotanddenseenoughtohave2μ0nκT/B2pol=(μ0Ia(0)/ψ)2/2;observedcoronalloopparametersareinreasonableagreementwiththisrelationshipwhichisanalogoustohavingβpol=1inatokamak.1I.INTRODUCTIONAlongstandingmysteryinsolarphysicsiswhysolarcoronalloopstypicallyhaveanax-iallyuniformcross-section[1];i.e.,afilamentaryshape.ThisissuehasbeenmadeespeciallypressingbyrecentTRACE(TransitionRegionandCoronalExplorer)spacecraftsoftx-rayimageswhichshowamultitudeofhighly-definedaxiallyuniformloops[2];forexample,seeFig.1.Axialuniformityoffluxtubesisalsocommonlyobservedinlaboratoryexperiments,forexample,inrecentsimulationsofsolarprominences[3].FIG.1:TRACEsoftX-rayphotoshowingaxialuniformityofcoronalloops(imagecourtesyLockheedMartinSpaceandAstrophysicsLab).Asdiscussedinthetext,theslightwrappingoffluxtube#1aroundfluxtube#2indicatesthatnetcurrentsflowalongthefluxtubes.2ThispaperarguesthataxialuniformityistheresultofarathercomplexsequenceofeventswhichoccurwheneveranelectriccurrentIismadetoflowalonganinitiallyaxiallynon-uniform,current-free,axisymmetricmagneticfluxtube(aprocesscorrespondingtoinjectionofmagnetichelicityintothefluxtube).ThesequenceofeventsoccursevenwhenIismodest,i.e.,evenwhenthefluxtubeisonlyslightlytwisted.FIG.2:(a)Potential(i.e.,current-free)coronalloopabovesolarsurfacewithcorrespondingsub-surfacefieldandsourcecurrents.Thecross-sectionoftheloopislargestatthetopoftheloopwherethemagneticfieldisweakest;(b)straightcylindricalrepresentationofcoronalloopusedinmodel.ThetypicalarchedshapeofcoronalloopsisshownschematicallyinFig.2a;tomaketheanalysistractablewewillassumethattheloopisstraightassketchedinFig.2b.3However,inordertoretainanimportantaspectofthearchedshape,wewillallowthelengthofthestraightlooptovarytotakeintoaccountpossiblevariabilityinthelengthofthearchedloop.TheFig.2bgeometrywillbecharacterizedbyastraightcylindricalcoordinatesystem{r,φ,z}wherezreferstothedirectionalongtheloopaxis,φistheazimuthaldirectionabouttheaxis,andristhedistancefromtheaxis.Theφdirectioniscalledthetoroidaldirectionandther,zdirectionsarecalledpoloidal.Fluxcoordinateswillalsobeusedwhenappropriate.Thispoloidal/toroidalnomenclatureisformallythesameasthatusedfortokamaks,buttheconfigurationshouldnotbeconfusedwithatokamakastherearenoclosedpoloidalfieldlines.ThecurrentIwillbeassumedtoberelativelysmallsothatthepoloidalfieldmagnitude∼BzisalwaysmuchlargerthanBφinwhichcasethefluxtubeisonlyslightlytwisted.Wenotethatthisstraightcylindricalapproximationoffluxtubegeometryhasbeenusedinmanypreviousstudiesoffluxtubeequilibria,especiallyforce-freeequilibria(forexample,seeRefs.[4,5,6,7,8,9]).However,theanalysispresentedheredifferssubstantivelyfromthesepreviousstudiesbecauseouranalysisdoesnotbeginbyassumingexistenceofanequilibrium.Instead,ouranalysischaracterizesthedynamicsthatleadtoanequilibriumandshowshowtheresultingequilibriumisintimatelyrelatedtothesedynamics.Furthermore,ouranalysistakesintoaccountthenon-force-freeaspectsoftheequilibrium(i.e.,finitepressuregradients)andshowsthatthesefiniteβaspectsareofvitalimportancetotheaxialuniformityoftheequilibrium.BecauseoftheassumedaxisymmetryinFig.2b,themagneticfieldcanbeexpressedasB=12π(∇ψ×∇φ+μ0I∇φ)(1)where∇φ=ˆφ/randψ(r,z,t)=Zr0Bz(r′,z,t)2πr′dr′(2)isthepoloidalflux.Axialnon-uniformitycorrespondstohaving∂ψ/∂z6=0andaxialbulgingcorrespondstohavingψ−1∂2ψ/∂z20.ThecurrentIissimilarlygivenasI(r,z,t)=Zr0Jz(r′,z,t)2πr′dr′(3)andisrelatedtothetoroi