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1、TopologicalConstraintsonMagneticRelaxationA.R.Yeates,∗G.Hornig,andA.L.Wilmot-SmithDivisionofMathematics,UniversityofDundee,Dundee,DD14HN,UK(Dated:August6,2018)AbstractThefinalstateofturbulentmagneticrelaxationinareversedfieldpinchiswellexplainedbyTaylor’shypothesis.However
2、,recentresistive-magnetohydrodynamicsimulationsoftherelaxationofbraidedsolarcoronalloopshaveledtorelaxedfieldsfarfromtheTaylorstate,despitetheconservationofhelicity.Wepointouttheexistenceofanadditionaltopologicalinvariantinanyfluxtubewithnon-zerofield:thetopologicaldegreeof
3、thefieldlinemapping.Weconjecturethatthisconstrainstherelaxation,explainingwhyonlyoneofthreeexamplesimulationsreachestheTaylorstate.PACSnumbers:52.30.Cv,52.65.Kj,52.35.Vd,96.60.HvarXiv:1007.4925v1[astro-ph.SR]28Jul20101ThelandmarkpaperofJ.B.Taylor[1]showedhowthefinal,relaxe
4、d,magneticfieldinaturbulentplasmaexperimentcouldbepredictedtheoreticallybyassumingconservationRofthetotalmagnetichelicityH=A·Bd3x,whereB=∇×A.TaylorhypothesizedVthat,inaturbulentplasmawithsmallbutnon-vanishingresistivity,Histheonlyrelevantconstraintontherelaxation.Theresul
5、tingminimumenergystateisthenalinearforce-freefield[2],∇×B=αB,whereαisaconstantdependingonthevalueofH.ThesuccessofthistheoryinpredictingthefinalstateofrelaxationinareversedfieldpinchhasledtospeculationthatTaylor’shypothesismightapplymoregenerally,forexample,tomagneticstructu
6、resintheSun’satmosphere[3–6].Here,thenatureofthefinalstateisofgreatinterestbecauseitlimitsthemagneticenergyavailableforconversiontoheatduringtherelaxation.ThisiskeytounderstandinghowmagneticfieldsproducetheextremecoronaltemperaturesintheSunandotherstars.ToapplyTaylor’shypo
7、thesisinsuchmagneticallyopendomains,HisreplacedbytherelativehelicityHRwithrespecttoareferencefield[4,7],typicallyapotentialfield.Alocaldomain,suchasanisolatedloop,istaken(thecoronacannotbegloballyalinearforce-freefield)andthehypothesishasmetwithsomenotablesuccess[6].However
8、,counter-examplesareknown[8,9]anditnowseemslikelythatconstraintsbeyondtheconservationofglobalmagnetiche
