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arXiv:astro-ph/9703154v225Mar1997THEORYOFPASSIVEMAGNETICFIELDTRANSPORTKRIST´OFPETROVAYE¨otv¨osUniversity,DepartmentofAstronomyBudapest,Ludovikat´er2,H-1083HungaryAbstract.Inrecentyears,ourknowledgeofphotosphericmagneticfieldswentthroughathoroughtransformation—nearlyunnoticedbydynamotheorists.Itisnowpracticallycertainthattheoverwhelmingmajorityoftheunsignedmagneticfluxcrossingthesolarsurfaceisinturbulentform(intranetworkandhiddenfields).Furthermore,therearenowobservationalindications(supportedbytheoreticalargumentsdiscussedinthispaper)thatthenetpolarityimbalanceoftheturbulentfieldmaygiveasignificantorevendominantcontributiontotheweaklarge-scalebackgroundmagneticfieldsoutsideunipolarnetworkareas.Thisturbulentmagneticfieldconsistsoffluxtubeswithmagneticfluxesbelow1010Wb(1018Mx).Themotionofthesethintubesisdominatedbythedragofthesurroundingflows,sothetransportofthiscomponentofthesolarmagneticfieldmustfullybedeterminedbythekinematicsoftheturbulence(i.e.itis“passive”),anditcanbedescribedbyaone-fluidmodellikemean-fieldtheory(MFT).TherecentadvanceinthedirectandindirectobservationofturbulentfieldsisthereforeofgreatimportanceforMFTasthesearethefirst-everobservationsontheSunofafieldMFTmaybeappliedto.However,inordertoutilizetheobservationsofturbulentfieldsandtheirlarge-scalepatternsasapossiblediagnosticofMFTdynamomodels,thetransportmechanismslinkingthesurfacefieldtothedynamolayermustbethoroughlyunderstood.Thispaperreviewsthetheoryofpassivemagneticfieldtransportusingmostlyfirst(andocca-sionallyhigher)ordersmoothingformalism;themostimportanttransporteffectsarehoweveralsoindependentlyderivedusingLagrangiananalysisforasimpletwo-componentflowmodel.Solarapplicationsofthetheoryarealsopresented.Amongsomeothernovelfindings/propositionsitisshownthattheobservedunsignedmagneticfluxdensityinthephotosphererequiresasmall-scaledynamoeffectoperatingintheconvectivezoneanditisproposedthatthenetpolarityimbalanceinturbulent(and,inparticular,hidden)fieldsmaygiveamajorcontributiontotheweaklarge-scalebackgroundmagneticfieldsontheSun.Keywords:solarphysics,magnetism1.Introduction1.1.“Passive”Fieldsvs.“Active”Fields:aHistoricalReviewAbasicruleofthumbinmagnetohydrodynamics(MHD)tellsusthatthecharacteroftheinteractionbetweenmotionsandmagneticfieldsina(highplasmabeta)plasmaisdeterminedbytheratiooftheEMmagneticandEKkineticenergydensities.IfEM≪EKthentheLorentzforcemaybeneglectedintheequationofmotionandourproblemisreducedtothekinematicalcase.If,ontheotherhand,EM≫EKthenthefieldwill“channel”theflowandtheonlypotentialeffectofthemotionsonthefieldisthegenerationofsmall-aplitudeMHDwaves:thisisthestrongfieldcase.Finally,inthehydromagneticcase,whenEM∼EK,thereisacomplicatedinteractionofflowandmagneticfield.OfcoursewemustbeawareofthefactthatIn:SolarSurfaceMagnetism,(R.J.Rutten&C.J.Schrijver,eds.),NATOASISeriesC433,Kluwer1994,p.415-440.2KRIST´OFPETROVAYhB2ihBi2ingeneral,sotheEMtotalmagneticenergydensitymaywellexceedtheenergydensityofthelarge-scalemeanfield.Besides,thevalidityoftheabovesimplerulemaypossiblyalsobelimitedintwodimensionswhereaweakermagneticfield(consistinge.g.ofalowfillingfactorsetofstrongsheets)couldpossiblyalsoin-fluencethemotion,owingtothetopologicalconstraint(theflowcannot“getaround”thesheets).TheselatterpointswererecentlybroughtintofocusbyCattaneoandVainshtein(1991).Nevertheless,apartfromtheseratherobviouscaveats,thesimplerulesummarizedabovecanbeconsideredascorrect.ThissimplenotionformedthebackgroundofMHDthinkinginthe1950’sand60’swhenmean-fieldelectrodynam-icsandmean-fieldMHDweredeveloped(Parker,1955,Steenbecketal.,1966)forthetreatmentofthekinematicandhydromagneticcase,respectively.Thepicturehowevergotmorecomplicatedintheperiodfromthemid-sixtiestothemid-seventieswhensolarobservations(Sheeley,1966,Stenflo,1973,HowardandStenflo,1972)andnumericalexperiments(Weiss,1964)showedthatinthehighlyconductiveturbulentsolarplasmathemagneticfieldisconcentratedintostrongfluxtubeswithverylittlefluxinbetween.TheBtmagneticfluxdensityinsidethetubesisorderof(orgreaterthan)theBeqequipartitionfluxdensitydefinedbyB2eq/2μ=ρv2t/2(1)(SIformula;μisthepermeability,ρisthedensity,vt=hv2i1/2withvtheturbulentvelocity).Asaconsequenceofthisrealization,fluxtubetheorybegantodevelopinthe1970’s(seeParker,1979,forareviewoftheresultsofthisperiod).Accordingtofluxtubetheory,themostimportantforcesactingonamagneticfluxtubearetheFdaerodynamicdrag,theFmmagneticcurvatureforceandtheFbbuoyancy;theirapproximateexpressionsare:Fb∼B2t2μ0HPFm∼B2tμ0RcFd∼ρv2td(2)withHPthepressurescaleheight,dthetubediameterandRcthecurvatureradius;inpractice,onemayputRc∼lwithlthecharacteristicscaleoftheturbulence.Acomparisonoftheseexpressionsshowsthatforasufficientlythinfluxtube,withamagneticfluxΦΦcr=min{l2,H2P}B4eq/B3t,thedragwilldominateandthesurroundingflowwilldeterminethemotion,whilethickertubesmaymovemoreindependentlyofthesurroundingturbulence,undertheactionofdynamicalforces.Thisimpliesthat,foragivenenergydensity(foragivenfmagneticfillingfactororDtypicaltubeseparation),thetransportofthefieldmaybeeitherpassive,i.e.fullydeterminedbytheflowbetweenthetubes,oractive,i.e.toalarg