198lApJ. . .243. .945G The AstrophysicalJournal,243:945-953,1981February1 © 1981.TheAmericanAstronomicalSociety.Allrightsreserved.PrintedinU.S.A. mapped indetail,butmagneticactivityseemstobea Fields of2000gaussormorehavebeenfoundintwo standard featureofstarswithdeepconvectiveenvelopes. Astronomy, Inc.,undercontractwith theNationalScienceFoundation. National ScienceFoundation. which showcyclicvariationsofactivity.Some,perhaps late-type mainsequencestars(Robinson,Worden,and Ca iiemissionsfrommanyG,K,andMstars,someof Harvey 1980).Wilson(1978)hasdetectedchromospheric field iscomparativelyfeeble,allattemptstoexplain all, dMestarsflare,andtheunexpectedlyhighX-ray magnetic activityinstarsarebasedonsolarobservations. luminosities ofKandMstars(e.g.,Vaiana1979)suggest that theyhavestrongmagneticfields.AlthoughtheSun’s netic featuresandcellularconvectioninthephotosphere. have revealedtheinteractionbetweensmall-scalemag- magnetic fieldshasbeenclarifiedbymeasurementsfrom Over thelastdecadestructureoflarge-scalesolar space, whilehighresolutionground-basedobservations Weiss 1977).Nonlinearmagnetoconvectionhasrecently between magneticfieldsandconvection(forareview,see lent convection,itisimportanttostudytheinteraction been investigatedinseveralseriesofnumericalexperi- ments (PeckoverandWeiss1978;GallowayMoore magnetic fieldsatthesurfaceandininteriorofSun and otherlate-typestars. these idealizedmodelcalculationstothestructureof 1979a). Sincemagneticactivityisassociatedwithturbu- 1979; Weiss1981a,b).Theaimofthispaperistorelate 2 1 The Sunisuniqueinhavingamagneticfieldthatcanbe OperatedbytheAssociationof Universities forResearchin TheNationalCenterforAtmospheric Researchissponsoredbythe Theory alsohasmadeconsiderableprogress(Parker © American Astronomical Society • Provided by theNASA Astrophysics Data System 1/2 concentration ofmagneticfluxintoisolatedropesintheturbulentconvectivezonesSunorother Results fromnumericalexperimentsonmagnetoconvectionarepresentedandusedtodiscussthe shredded anddispersedthroughouttheconvectivezone.Theobservedmaximumfieldstrengthsin Magnetic buoyancyleadstotheemergenceofmagneticfluxinactiveregions,butweakerropesare late-type stars.Argumentsaregivenforsitingthesolardynamoatbaseofconvectivezone. late-type starsshouldbecomparablewiththefield(87rp)thatbalancesphotosphericpressure. Subject headings:convection—hydrodynamicsstars:magnetic Recent observationshavedemonstratedtheunityofstudystellarandsolarmagneticfields. I. INTRODUCTION Sun :magneticfields CONVECTION ANDMAGNETICFIELDSINSTARS 1 High AltitudeObservatory,NationalCenterforAtmosphericResearch Received 1980March10;acceptedAugust18 2 Sacramento PeakObservatory D. J.Galloway N. O.Weiss ABSTRACT AND 945 Theory andobservationarebroughttogetherin§IV, models ofhydromagneticconvection.Thestructure where wediscusstheformationofisolatedfluxtubesin turbulent magneticfieldsisthendescribedin§III. where r¡isthemagneticdiffusivity.Ifvelocityu netic fieldBisgovernedbytheinductionequation convection andanexternallyimposedmagneticfield.The idealized geometries.Inaconductingmediumthemag- calculations arerestrictedtoBoussinesqfluidswith field dependsonthemagneticReynoldsnumber, prescribed (thekinematicproblem)thebehaviorof in theSunandotherstarswithvigorousconvectivezones. § V.Finally,weconsidertheoriginofmagneticfields R =UL/r¡,whereU,Lareacharacteristicspeedand programs, coveringstarsofdifferingmass,thatcan the interiorofSun.Photosphericfieldsarecoveredin B bythetwo-dimensionalvelocity better estimatesofthefieldstrengthsinstarspotsand Our conclusionsemphasizetheneedforobservational Figure 1showsthedistortion ofaninitiallyuniformfield length scaleforthemotion(Moffatt1978;Parker1979a). establish thecorrelationbetweenmagneticactivityand areas theyoccupy. rotation orconvectivevelocity;inaddition,weneed m 0 In §IIwesummarizetheresultsobtainedforsimplified In thissectionwetreattheinteractionbetweenlaminar II. CONCENTRATIONOFMAGNETICFLUX u= C/(—sin(nx/L)cos(nz/L), 0, 2 — =curl(«xZ?)+r}VB,(1) c)B cos (nx/L)sin(nz/L)), (2) 198lApJ. . .243. .945G T/t! =0,1,2,3,4,5,6,7,8,9,10,and20,wheret,5t/8. 0 Fig. 1.—Expulsionofmagneticflux:theevolutionfieldin akinematiccalculationwithR=250.Linesoffereeareshownattimes m © American Astronomical Society • Provided by theNASA Astrophysics Data System 198lApJ. . .243. .945G 1/2 1/3 (0)2 (e)2 (e) when R=250,withboundaryconditionssuchthat at theupperandlowerboundaries.(Forasimilarcalcula- B =0onx,z0,L,sothatlinesofforcecanmovefreely tion withdifferentchoicesofvelocityandboundary magnetic fluxisquicklyconcentratedintosheetsatthe conditions, seeWeiss1966).Astheeddyturnsover, field attheedgesofcellisamplifiedbyafactororder expelled fromtheconvectiveeddy.Infinalstate lateral boundaries,whilethecentralfieldgrowsprogres- sively moredistorteduntilreconnectionoccursandfluxis R (e.g.,ProctorandWeiss1978).Thisamplification field canbeincreasedbyafactorofupto200(Weiss proceeds rapidly:Afteroneturnovertimet=L/U,the R =250,butthistimeincreasesas(Weiss1966). longer: Theprocessiscompletedbyt^5.5twhen m order R(Galloway,Proctor,andWeiss1978;Galloway an isolatedtubeandthefieldisamplifiedbyafactorof In athree-dimensionalflow,thefluxisconcentratedinto pattern ofconvection,soitisnecessarytoconsiderthe and Moore1979). x dynamical problemandsolvetheequationsofmotion. drasekhar 1961;Danielson1961).Intheastrophysically 1966). Ontheotherhand,fluxexpulsiontakesmuch of thediffusivities.Herevisviscousdiffusivity,d Linear theoryhasbeenextensivelydiscussed(Chan- ratios B issufficientlylarge,convectionsetsinasoverstable interesting case,whenthethermaldiffusivitykrjand oscillations. Heattransportbyoscillatoryconvectionis m 0 overstability setsinwhenR=~7i1, layer depth,ßthesuperadiabatictemperaturegradient, described bydimensionlessparameters (the RayleighandChandrasekharnumbers),bythe relatively inefficientsoitisimportanttoascertainwhen steady convectioncanoccur.Theconfigurationis m 0 solution atR=~7r<2. steady solutionsbifurcatefromthestaticconducting m is increased,steadyconvectionfirstappearswithfinite tical steadyconvectionintherangeR<.As periodic boundaryconditions(cf.Weiss1981a).Once amplitude atR=.Figure2showsstreamlinesand lines offorceforsteadytwo-dimensionalconvectionwith within thesesheets,thefieldisstrongenoughtoexclude again, thefluxisconfinedtonarrowsheets,allowing convection totakeplaceinthefield-freecentralregion; 0 the motion.Similarresultshavebeenobtainedforan first appearinnarrowcells,elongated inthedirectionof uniform (GallowayandMoore 1979). a stagnantaxialtubewithin whichthefieldisnearly axisymmetric configuration,where thefluxisconfinedto min min These strongfieldsexertforceswhichmodifythe Nonlinear calculationsconfirmtheexistenceofsubcri- Although lineartheorypredicts thatconvectionshould © American Astronomical Society • Provided by theNASA Astrophysics Data System g =v/kandÇvj/k(4) 4 gocßd KV and Q = CONVECTION ANDMAGNETICFIELDS 2 4nprjv Bod (3) 2(e) convection occursmorereadilyinwidercells,which the fieldR,nonlinearresultsindicatethatsteady that R~7rCQ,asconjecturedbyDanielson concentrated fields.Thenumericalexperimentssuggest allow morespaceforconvectivemotionbetweenthe steady statewithmotionexcludedfromthestagnantfluxsheets. motion ispossiblewhenthefieldexceedsacriticalvalue occurs forQphotosphere, thoughtheindividualcellsareephemeral tion formsafairlyordered,cellularpatterninthesolar energy-carrying eddieshaveasimilarstructurewithsome and short-lived.Weinfer,therefore,thatwithinastarthe horizontal scalethatislocallypreferred.Convection driven thermallyand,asthepatternchanges,workdone by pressuregradientsmustultimatelybedissipated molecular viscosity.SincetheReynoldsnumberisenor- mous, theremustexistanenergycascadethrough inertial subrangeassociatedwithsmall-scaleturbulence that isroughlystationaryandhomogeneous.Formally, model, weseparatethevelocity fieldintoanordered this suggeststheuseofatwo-scale analysis.Asasimple motion onthescaleofcellular convectionandinto eddy diffusivity. small-scale turbulencewhose effectisrepresentedbyan So far,wehaveconsideredonlysteady,persistent The velocitystructurewithintheturbulentconvection Vol. 243. 198lApJ. . .243. .945G motion anddiffuseowingtothesmall-scalemotions.The latter bringopposingfieldscloseenoughtogetherfor resistive instabilities,orfastdynamicalreconnection reconnection tooccur,whetherbylaminardiffusion, No. 3,1981 eddy diffusivityiy^0.1w/,whereuand/arethevelocity stood, buttherateofreconnectioniscontrolledbyan for theorderedmotion.Thuseffectivevalueof and thelengthscaleofturbulenteddies,whichare expulsion isapproximatelythatinFigure1,sayabout small comparedwithUandL,thecorrespondingvalues (Parker 1979a).Theseprocessesareimperfectlyunder- flux willbeconfinedtosheetsorropeswithinafew 4t. Ifadiffusefieldentersturbulentregion,therefore, magnetic Reynoldsnumber,R=UL/rj,isprobablyin the range10010Mx)and that therearefundamentaldifferencesbetweenthebeha- (Zwaan 1978).X-rayobservationsfromspace,aswell the distinctionprecise(Golubetal1981).Thesolar smaller features(F<10Mx),thoughitishardtomake convective zone,wherethelarge-scalefieldsoriginate.In dynamo isapparentlylocatednearthebaseof flux atthebaseofconvectivezonesunspotmaxi- and mustvarywiththesolarcycle.Thetotal(unsigned) this region,magneticfieldsarepredominantlytoroidal mum, CD,canbecrudelyestimatedasfollows.Alarge This figureprovidesalowerboundforQ>.However, emerging ina“complexofactivity”(Bumbaand sunspot maycontain3x10Mx,andthetotalflux of 5x10Mxday(Golubetal1974),givingatotal magnetic fluxemergesinactiveregionsatanaveragerate Howard 1965;Svestka1977)maybeasmuch10Mx. an upperboundsincetherecouldbeareservoirofflux of 2x10Mxoverthesolarcycle,andeventhatisnot one mustknowhowmanytimesthesamefluxdoesduty that staysbelowthesurface.Todeduce0)moreprecisely, ~ (2x10/20)=Mx.Theabovearguments as anactiveregion.If,forinstance,wesupposethateach tube emerges10timesasabipolarregion,atdifferent structure ofmotionintheconvectionzoneislargely longitudes, duringthesolarcycle,thenwefindthat theory andlaboratoryexperimentssuggeststhatinthe unknown. Neverthelessallavailableevidencefrom shall adopt10Mxas,onbalance,themostlikelyvalue. suggest thatÓliesbetween10MxandMx;we primarily byeddieswithascaleofatleast100,000km, lower halfoftheconvectionzoneheatistransported motion associatedwiththesegiantcellsextendsrightup comparable tothedepthofwholezone.Whether existence ofgiantcells,with parasitic eddiesdrivenby to thephotosphereremainsunclear.Inorderdescribe the interactionbetweenmagneticfieldsandconvectionin the lowerpartofconvectionzone,weassume for giantcellsiscomparableto thesolarday;thustheyare elongated paralleltotheSun’s axis(cf.Busse1977),so shear instabilities,asdescribed in§HI.Theturnovertime strongly influencedbyrotation andshouldoccurascells Magnetic fluxemergesthroughthesurfaceofSun Apart fromgranulationandsupergranulation,the IV. ORIGINOFDISCRETEFLUXTUBESINTHESUN 949 198lApJ. . .243. .945G 13-1 2 24 24 23 4 netic fieldresemblesthatinthetwo-dimensional that theorientationofvelocityrelativetomag- calculations. which themotionisexcluded?Suchropesmustfloat cal regime.Isthisfluxconfinedtoisolatedropesfrom 2 x10Mxcm;thisissufficienttobeinthedynami- upward owingtomagneticbuoyancy(Parker1979a), unless thelocalmotiondragsthemdown.Considera longitude wheretheradialvelocityisdownwardand extending between±30°inlatitude.Forafluxof suppose thatthetotalfluxfillsahomogeneouslayer from asinglecomplexofactivitywouldfillthewhole km thick;ifthefieldwereaslow100gauss,flux convective zone.Alternatively,supposethatthefieldis + 30°,thefluxperunitlengthalongameridianisabout 950 confined toisolatedtubes,eachcontaining10Mx flux O=10Mxcannotbefittedintothelowerpartof toroidal fluxatsunspotmaximumiscontainedina 10 Mxandafieldofgauss,thislayermustbe20,000 flux inthislayervarieswiththesolarcycleand vective zoneandtheradiativebelow.Theamountof ference withconvectiondoes,ofcourse,becomeless the convectivezoneunlessfillingfactorisaround shallow layersituatedattheinterfacebetweencon- 20%, whichwouldseverelyhamperheattransport.(Inter- (corresponding toatypicalactiveregion);thenthetotal serious ifapproachesitsminimumvalueof10Mx.) From theseconsiderationsitseemsmostlikelythatthe is alsolocatedinthismagneticlayer.Thenflux dal fluxispartofthedynamoprocessthatmaintains changes canproducefluctuationsoforder0.1%insolar toroidal fieldreversesaroundsunspotminimum:These luminosity (SpiegelandWeiss1980).Generationoftoroi- is generatedwithstrongfields(around10gauss)and solar cycle;itisnaturaltosupposethatthedynamoitself bulk oftheconvectivezoneatanystagesolarcycle. there isnoneedtointroduceweak(100gauss)fieldsinthe instabilities drivenbymagneticbuoyancy(Parker1979a) layer ismaintained.Aswehaveseen,itmaybeunstable and developcorrugations;inaddition,itisliableto emerge asactiveregions,whichshowthesystematic guided bythepatternofconvection.Theseloopscanthen so thatnarrowloopsoffluxriseupwardtothesurface, zone, buoyancy-driveninstabilitiesmaybepartiallyin- pattern ofbehaviorthatdefinesthesolarcycle.Provided that thefluxisconfinedtoalayerbeneathconvective field tosurviveforanappreciablefractionofthesolar convection zone,itwillriseatleastasfastthelocalgas cycle. If,ontheotherhand,fluxiswithin hibited byrotation(Acheson1979a,b),soenablingthe and sobelostwithinamonth(Parker1979a). These features,whichareprevalent atalllatitudes,seem formed higherintheconvective zone(Golubeial1981). different patternofbehavior, which suggeststhattheyare regions orX-raybrightpoints(Golubetal1977)showa to correspondtheintermittent magneticfieldsthat should beexpectedintheconvective zone.Largerflux In thesunspotzones,extendingbetweenlatitudesof More workisneededinordertoestablishhowsucha The smallerfluxtubesthatemergeasephemeralactive © American Astronomical Society • Provided by theNASA Astrophysics Data System GALLOWAY ANDWEISS 1 2_1 1/2 21 Thus thesmall-scale,turbulentmagneticfieldsareonly 20 weakly relatedtothedynamothatdrivessolarcycle. ropes whichcanbemaintainedbytheconvectivemotion. tubes areshreddedbyturbulenteddies,formingsmaller ropes aretooweaktowithstandconvection;conversely, be estimatedfromequation(7)oncetheappropriate an isolatedfluxtubecansurvivelongerifitcontains factor isoforderunity.Thevaluefjcanbeestimated where fjistheturbulentdiffusivity,sincelogarithmic diffusivities areknown.Withinthefluxropethereisstill magnetic waves,andturbulenceismoreeffectivethan enough fluxtoexcludethemotion.Thecriticalcan below): Alltheseestimatesyieldavaluearound2x10 from microturbulentvelocitiesinsunspots(Beckers from thedecayoflong-livedsunspots(Meyeretal.1974), some motion,correspondingtooverstablehydro- flux ropes(cf.Spruit1977),andweadoptthisvalueforfj cm s,lessthan1%oftheeddydiffusivitiesoutside 17 laminar diffusion.Theviscousandmagneticeddydiffu- throughout theconvectivezone.Fromequation(7) to sivities shouldberoughlyequal,soequation(8)reduces field: corresponding fieldisapproximatelytheequipartition 1976), orfromdowndraftsinslenderfluxtubes(see Galloway, Proctor,andWeiss(1977)obtainedanupper bound forBbyequatingtherateofworkdonein containing themagneticfieldtorateofturbulent ratio oftheturbulentdiffusivities,wasaround100; dissipation inthewholeconvectivecell,sothat(v/fj), the fluxrope,sothat(v/fj)isoforderunity,and,therefore, that vandfjshouldbothbeevaluatedattheedgeof arguments ofGalloway,Proctor,andWeiss(1978)imply maintained byexternalconvection. give amorestringentupperlimittothefieldsthatcanbe maximum fieldB,derivedfromequations(9)and(10),at netic pressurebalancestheexternalgasPwhen convective zone,computedbySpruit(1977).Themag- levels correspondingtogranules,supergranules,and m giant cells;thevaluesofpandUaretakenfromamodel greater thanexceptnearthephotosphere.Thus the fieldreachesastrengthB=(8tlP)andismuch estimates ofBgiveninTable1shouldbevalidforfields with fluxesgreaterthanabout10Mx(corresponding granular magneticfields.Wesee,moreover,thatropes between supergranulesandgiantcellsbutnotforinter- zone withoutbeingpenetratedbyexternalmotions,while to asmallactiveregion)canrisethroughtheconvective milarly, ropeswithfluxesgreater than10Mxcanresist smaller fluxtubesaremorelikelytobeshredded.Si- super granularmotionswhile flux tubeswithmorethan m the fieldsthatemergeasephemeral activeregions,while starting deepintheconvective zone,therefore,cansupply p m 10 Mxmaystanduptogranules. Smallfluxropes, Turbulent shreddingproceedsmostrapidlywhenflux Table 1givesestimatesofthecriticalfluxFand c ll2 1/2 F ~(4np)fjL,(9) c B ~(4np)U.(10) e Vol. 243 198lApJ. . .243. .945G 21 s 21 No. 3,1981 density becomescomparablewiththethermalenergyof or abovethephotosphere,wheremagneticenergy to bevalid:Acorrecttreatmentoftheinteractionbe- the gas.TheBoussinesqapproximationthereforeceases tween magneticfieldsandconvectionrequiresafully compressible model.Untilsuchacalculationiscarried port thetraditionalpictureoforiginactiveregions sunspots. Theargumentsdevelopedherethereforesup- larger ropescanremaincoherentandemergetoform Boussinesq resultsoronsomeparametricrepresentation out, wehavetorelyeitheronboldextrapolationfromthe Parker (1979a,b)hasrecentlydeveloped. of turbulentconvection.Weshallseethattheseover- simplified modelsareinadequate. (e.g., Babcock1961)ratherthantherivalviewsthat convection byalarge-scalemagneticfield,extendingover a regionwhosehorizontaldimensionsaremuchgreater than thediameterofatypicalconvectioncell.Insolar convection inaporeorsunspot.Thenonlinearstudiesof context thiscorrespondstothesuppressionofgranular when B=~(gpotßd),fromequation(5).Accord- from normalconvectiontooscillatorymotionoccurs Boussinesq magnetoconveetionimplythatthetransition ing tomixinglengththeory,thevelocityisjust netic energydensityexceedsthekineticof normal convectionisindeedsuppressedwhenthemag- Bmax icomparabletotheequipartitionfieldB,and reduced free-fallspeed,sothatU~(gocßd);hence Bmax becomecomparabletothepressure-balancingfield the motion(Cowling1976).Atphotosphere,Band B, andtheBoussinesqapproximationisnolongervalid. drop inphotospherictemperature,whichincreasesthe with aradiativeboundaryconditionatthephotosphere. Moreover, areductioninconvectivetransportleadsto superadiabatic gradient.Thusapropertreatmentofthis problem requiresacalculationforcompressiblegas max gauss, andapproximatelyequaltoB. However, thecriticalfieldBcannotexceed:In than thefieldstrengths(around 1500gauss)thathave calculations suggestavalueof orderB,whichislower strength ofintergranularmagneticfields.TheBoussinesq e rough estimateandBmight exceed Bbysomefactorof been observed(Harvey1977). Tobesure,thisisonlya order unity.Nevertheless,it isclearthatBoussinesq Sun, thecriticalfield,atedgeofapore,isaround1500 e p p maxp e m e All observationsnecessarilyrecordmagneticfieldsin Consider firsttheinhibitionofnormalphotospheric Similar difficultiesariseinattemptingtoestimatethe © American Astronomical Society • Provided by theNASA Astrophysics Data System 61 1207 3195 Granules 500100010“27x107005000 Giant cells150,000100,00010"0.072x1090002.5 Supergranules 10,00015,00010"0.43x1045005 V. PHOTOSPHERICMAGNETICFIELDS 3 Scale Depth(km)I(km)p(gcm)U(kmsF(Mx)ß(G)B cmP Critical FluxesandMagneticFieldsintheSun’sConvectiveZone CONVECTION ANDMAGNETICFIELDS TABLE 1 from therope,thenafurtherthermalinstabilityoccurs models areinadequate:Ifthefieldisconfinedtoarope, partially evacuated,allowingthefieldtoapproachB collapses untilitreachesastableequilibriuminwhichis and ifthefluxexceedsF,sothatconvectionisexcluded Ando 1979),andnonlinearaspectshavebeeninvestigated by Spruit(1979).Heassumes,however,thatconvection and Roberts1978;SpruitZweibel1979;Unno several authorsinthethintubeapproximation(Webb (Parker 1979a). (Galloway, Proctor,andWeiss1977).Thefluxtube can beparametrizedaccordingtomixinglengththeory magnetic fieldsandindividualconvectioncells. and doesnotattempttorepresenttheinteractionbetween photosphere. Therearestrongreasonsforsupposingthat concentrated intoropeswhenitemergesthroughthe observations indicatethatthefluxhasalreadybeen ing, downwardflow(Weiss1978).Thesolarfiligree these ropesrunbetweengranulesinregionsofconverg- by granules.Moreover,inmomentsofexceptionalseeing the crinklescanberesolvedintoseparatefeaturesthat these individualelementsareshiftedaboutandbuffeted sheetlike crinklesinthedarkintergranularlanes,and cation) asthoughafluxsheetattheedgeofconvection outline smallmicropores(R.B.Dunn,privatecommuni- p c lent magneticfieldsdescribedin§IIIabove,andthey observations areallconsistentwiththepatternofturbu- cell hadseparatedintoseveralisolatedtubes.These (Dunn andZirker1974)ismadeupofbrightpoints suggest thatthereisanintimaterelationshipbetween necessary thatthereshouldbenosignificantflowacross cells. small-scale magneticfieldsandindividualconvection flux elementsatthesolarsurface(GiovanelliandSlaugh- fected bythesurroundingpatternofconvection,itis the field.Thesystematicdowndraftsobservedinsmall unless thereissubstantialmotionacrossthelinesofforce. ter 1978)are,however,difficulttoexplaintheoretically overstable oscillationswithavelocity-brightnesscor- ward velocityincreaserapidly withdepth,scale- demonstrate thiseffect.Both thedensityanddown- culations bySpruitandGallowayhavesofarfailedto relation intheobservablelayers,butpreliminarycal- heights around250kmatthe photosphere. Theradiusof Spruit (1979)hassuggestedthattheycouldresultfrom field showsnocomparablevariation withheight.Henceit the fluxtubecannotchange much,sincethemagnetic The lineartheoryofthiscollapsehasbeenstudiedby Small-scale solarmagneticfieldsareintermittent,and For thestructureofanisolatedfluxtubetobeunaf- 951 198lApJ. . .243. .945G 1 12- 1/2 is impossibletoprovideasteadyfluxofmatterfrom described locallybyanaxisymmetric,solenoidalvelocity field lines(Giovanelli1977).Thismotioncanbe above: Continuitydemandsaradialinflowacrossthe with U=1.6kms~andif250km,referredto 952 cylindrical polarcoordinates(r,star, Bismuchlessthan,thefieldstrengthforwhich leads toacollapseandthefieldapproachesB.The magnetic fieldscannotbedescribedadequatelyineither (Giovanelli 1977),foreveninaturbulentmedium,flow strength increasestowardtheaxisofalargefluxconcen- the reductionofconvectivetransportwithinafluxtube is neededafullycompressiblecalculationthatcan 1 tration, owingtogeometriceffectsandthedepressionof the Boussinesqorthinfluxtubeapproximation.What describe boththethermodynamicstructureofflux tube anditsrelationtoexternalconvection,coupledwith priate boundaryconditions.Thisisstillatallorder. greater insightintoturbulentdiffusionandtheappro- ep e ep p We havearguedthatmagneticfluxemergingthrough Turbulent convectionconcentratesmagneticfluxuntil In thissectionwehaveshownthatphotospheric © American Astronomical Society • Provided by theNASA Astrophysics Data System VI. STELLARMAGNETICFIELDS 2 B =B*exp(—Ur/2fjH),(12) u =U(—r/H,0,2z/H),(11) GALLOWAY ANDWEISS REFERENCES surfaces ofconstantopticaldepth,butsurfacefields main sequencestars,fieldstrengthsshouldbesimilarto B. Sincethesurfacepressurechangesslowlyinlate-type should notgreatlyexceedB(0),thephotosphericvalueof quite different.Theobservationsoffields2500gaussin those observedintheSun,thoughtotalfluxesmaybe a G8Vstarareconsistentwiththefieldsinspotsand active regionsontheSun(Robinson,Worden,and lie betweenB(0),insmallfluxelements,and2B(0) Harvey 1980).Thusitseemsthataveragefieldstrengths large magneticfeatures.Thiswouldallowfieldsof5000 gauss indMstars.Ontheotherhand,redgiantshave —.1977,inProblemsofStellarConvection,ed.E.A.Spiegeland such distendedatmospheresthatB(0)~100gaussand surface magneticfieldsarelikelytobesmall. Chandrasekhar, S.1961,Hydrodynamic andHydromagneticStability magnetic fieldifthereisnovigorousconvectioninits p p ropes thatrunthroughtheconvectivezoneortoalayerat outer layers.Otherwisemagneticfluxwillbeconfinedto The dynamothatisresponsibleforthesolarcycle Cowling, T.G.1976,Magnetohydrodynamics (Bristol:Hilger). probably locatedatthebaseofconvectivezoneandis the interfacebetweenconvectiveandradiativezones. Dunn, R.B.,andZirker,J.B.1974, Solar Phys.,33,281. Danielson, R.E.1961,Ap.J.,134,289. maintained bythecombinedeffectsofthermaland Similar dynamosshouldoperateinotherGandKstars, whose activityisrelatedtotheirrotationrate.InMstars magnetic buoyancy,interactingwiththeCoriolisforces. p convective. Iftheinnerradiusofconvectivezoneis convection extendsdeeper,untilthewholestarisfully to asphericalshell;starswithJf<0.5^#probably have adifferenttypeofdynamo,dependingonfluxropes less thanascaleheight,thefieldcanscarcelybeconfined p emissions allsuggestthatMdwarfshavestronglarge- that crossthewholeconvectivezone.Theappearanceof activity inflarestars,andtheenhancedcoronalX-ray spots inRSCVnandBYDrastars,theincidenceof suppress convectionlocallyand,therefore,mayallow more efficientthanthemechanismoperatinginSun. scale magneticfields.Fluxropedynamosareapparently however, tobeexpectedthatfieldsaregeneratedintheir convection layersarerelativelythinandineffectual, abundance anomaliestopersist(e.g.,Michaud1976).The convective cores,thoughthemagneticfluxistrappedat the fieldsareprobablyprimordial(Mestel1976).Itis, o interstellar magneticfieldsareobservedtogetherwith the outerboundaryofcore.Finally,wenotethat field structuremustbeveryintermittent. similar (Spitzer1978),anditseemsinevitablethatthe turbulent motion.Themagneticandkineticenergiesare Knobloch, E.N.Parker,M.R.Proctor,Rosner,H. 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