NASA Technical Reports Server (NTRS) 20110008367: The Structure and Dynamics of the Corona - Heliosphere Connection

1.2 MB·English

by  NASA Technical Reports Server (NTRS)

NonamemanuscriptNo.
(willbeinsertedbytheeditor)
The Structure and Dynamics of the Corona – Heliosphere
Connection
SpiroK.Antiochos · JonA.Linker · Roberto
Lionello · ZoranMikic´ · ViacheslavTitov ·
ThomasH.Zurbuchen
Received:date/Accepted:date
Abstract DeterminingthesourceattheSunoftheslowsolarwindisoneofthemajorun-
solvedproblemsinsolarandheliosphericphysics.First,wereviewtheexistingtheoriesfor
theslowwindandarguethattheyhavedifficultyaccountingforboththeobservedcomposi-
tionofthewindanditslargeangularextent.Anewtheoryinwhichtheslowwindoriginates
fromthecontinuousopeningandclosingofnarrowopenfieldcorridors,theS-Webmodel,
is described. Support for the S-Web model is derived from MHD solutions for the quasi-
steady corona and wind during the time of the August 1, 2008 eclipse. Additionally, we
performfullydynamicnumericalsimulationsofthecoronaandheliosphereinordertotest
the S-Webmodel as well asthe interchange model proposed by Fiskand co-workers. We
discusstheimplicationsofoursimulationsforthecompetingtheoriesandforunderstanding
thecorona–heliosphereconnection,ingeneral.
Keywords Sun:corona·Sun:solarwind
1 Introduction
Since the pioneering theoretical work of Parker (1958, 1963) and the discovery observa-
tionsofNeugebauer&Snyder(1962)ithasbeenknownthattheSun’satmospherestreams
continuouslyoutwardintheformofasupersonicwind.Thiswindcarriesbothplasmaand
magneticfieldtotheboundaryofthesolarsystem,theheliopause.Atabasiclevel,theori-
gins of the wind are straightforward. As argued by Parker, the difference in gas pressure
between the Sun’s hot, 1 MK, corona and the tenuous interstellar gas causes the coronal
S.K.Antiochos
NASAGoddardSpaceFlightCenter,Code674,Greenbelt,MD20771,USA
Tel.:[phone]
Fax:[phone]
E-mail:[email protected]
J.A.Linker,R.Lionello,Z.Mikic´,V.Titov
PredictiveScience,Inc.,9990MesaRimRd.,Ste.170,SanDiego,CA92121,USA
T.H.Zurbuchen
DepartmentofAtmospheric,OceanicandSpaceSciences,CollegeofEngineering,UniversityofMichigan,
2455HaywardSt,AnnArbor,Michigan48109
2
gastoexpandoutward.Iftheheatingtothecoronaisconstant,thenthewindcanadoptthe
steadystatedescribedbyParker’soriginaltheory.
Ofcourse,ontherealSuntherearemanyaddedcomplicationstoParker’ssimplesteady-
statemodel.Thefirstandforemostisthepresenceofthesolarmagneticfield.Thisnaturally
divides the corona into two physically distinct regions. In thoseregions wherethe fieldis
strong,suchasdeepinsideanactiveregion,thefieldpreventstheplasmafromexpanding;
therebyresultinginanapproximatelystaticcoronalplasma.Theseregionsarereferredtoas
“closed”, because all fieldlines areconnected tothe photosphere at twoends. Theclosed
flux is truly coronal in that it does not connect to the heliosphere, but appears instead as
thewell-knownX-raycoronalloops(e.g.,Orrall1981).Ontheotherhand,inthoseregions
wherethefieldisweakandthegasdominates,thegaspressuredragsthefieldlinesoutward
indefinitely.Theseregionsarereferredtoas“open”inthatthefieldlineshaveonlyoneend
connected to the photosphere. The outward mass and energy flow in open regions results
inadecreased densitythere, sothat theyappearas “coronal holes”inX-rayimages (e.g.,
Zirker1977).Notethatforatruesteadystate,thesolarwindandtheheliosphericmagnetic
fluxoriginatesolelyfromphotospheric/coronalopen-fieldregions.
Inadditiontointroducing thecomplication oftopology tothecoronaandheliosphere,
themagneticfieldalsoforcesthemtobefullytimedependent.Sincethedistributionofflux
at the photosphere is constantly changing due to flux emergence/cancellation and abroad
rangeofphotosphericflows,thedistributionofopenandclosedfluxmustchange,aswell,
implyingthatthesolarwindisinherentlydynamic.ConsequentlyParker’stheorycanbe,at
best,aquasi-steadyapproximationtotheactualstateofthecoronaandwind.
Thedynamicsintroducedtothesolarwindbythephotosphericfieldevolutionnaturally
breaks upinto different regimesdetermined bythetimerequired for establishingasteady
wind.ThisisoforderafewtraveltimestotheAlfve´nradius,∼20R ,whichimpliesatime
⊙
scaleoftenhoursorso.Sincethephotosphericevolutionhasamore-or-lessconstantspeed
of1km/s orless,thetimescaletranslatesdirectlytoasizescaleforthephotospheric dy-
namics.Large-scalephenomena,suchasthedifferentialrotationortheemergence/dispersal
of large active regions occurs over days, and so could be incorporated within the quasi-
steadyapproximation.Small-scalephenomena,however,suchasthemagneticcarpet(Har-
vey1985;Schrijveretal.1997)orgranularflowshavetimescalestypicallylessthanafew
hoursand,hence,canbeconsideredasaconstantsourceoffluctuationsornoisetothequasi-
steadystate.Intermediate betweenthesespatialscalesisthetypical sizeofsupergranules,
whose lifetimes are of order that required to establish a steady state. It is likely that their
effectonthewindcanbedeterminedproperlyonlybyexplicittime-dependentcalculations.
Intheheliosphere,thephotosphericdynamicsappeartostructurethesolarwindviathe
mediationofthemagneticfieldintotwodistinct forms,theso-calledfastandslowwinds.
Thefastwindhasspeedsgenerallyinexcessof500km/s,butthatisnotitsdistinguishing
feature. This wind has 3 defining features: (a) itstemporal variations, (b) spatial location,
and(c)plasmacomposition.
(a)AsshownclearlybytheUlyssesmeasurements,thehigh-latitudefastwindexhibits
nearconstantspeed(McComasetal.2008)andcomposition(Geissetal.1995;vonSteiger
etal.1995;Zurbuchen2007).ItsobservedvariabilityconsistsprimarilyofAlfve´nicfluctu-
ationsthatmaybedueentirelytotheexpecteddynamicsinducedbythesmall-scalephoto-
sphericevolutiondescribedabove.
(b)Thefastwindoriginatesinsidenon-transient(lifetime>1day)coronalholes,where
thefieldhasbeenopenforasufficientdurationtoestablishasteadystate.
(c)Thefastwindhaselementalabundancesclosetothatofthephotosphere(vonSteiger
etal.1997,2001;Zurbuchenetal.1999,2002).ItdoesnotexhibittheFIPbiasoftheclosed-
3
fieldcorona(Meyer1985;Feldman&Widing2003).Furthermore,itsioniccompositionis
steadyandimpliesafreeze-intemperatureneartheSunaroundorbelow1MK(Zurbuchen
2007).
Fromtheseproperties,weconcludethatthefastwindisthetruequasi-steadywindofthe
originalParkertheory.Theslowwind,ontheotherhand,iscompletelydifferent.Itsspeeds
are generally <500 km/s, but again, this is not the distinguishing feature. The 3 defining
featuresofthiswindaremarkedlydifferentthanthoseabove:
(a) The slow wind is intrinsically variable, both inspeed and, especially, composition
(Zurbuchen &vonSteiger2006; Zurbuchen2007). Thevelocitystructureconsistsofperi-
odsoffastflowsintermingledwithslow.ThisvariationisnotsimplyAlfvenicfluctuations
superimposedonaquasi-steadystate.Certainlythelargecompositionalvariability,bothin
elementalandion-temperatures,cannotbeduetoturbulenceintheflow,butreflectsinstead
anintrinsicdifferenceintheoriginsofthefastandslowwind.
(b) The slow wind is associated with the heliospheric current sheet (HCS), which is
alwaysembeddedinslowwind,notfast(Burlagaetal.2002).Ontheotherhand,theslow
windisobservedtoextendintheheliospheretoanglesfarfromtheHCS,upto30◦orso.
(c)Theslowwindhaselementalcomposition(FIPbias)closetothatoftheclosedfield
corona. Its ionic composition implies a freeze-in temperature near the Sun (∼1.5 MK),
considerablyhigherthanthatofthefast(Zurbuchenetal.2002).Furthermore,theelemental
andioniccompositionsarehighlyvariable,unlikethesteadycompositionofthefast.Akey
pointisthat,asdefinedbythecomposition,theboundarybetweenthefastandslowwinds
isnarrow,oforderafewdegreesorso(Zurbuchenetal.1999),whichissmallcomparedto
theangularextentofeitherthesloworfastwinds.
The results that the slow wind is associated with the HCS, which maps down to the
Y-line at the top of the helmet streamer belt, and that the slow wind has the composition
oftheclosedcoronasuggestthatitsomehoworiginatesfromnearorinsidetheclosedfield
region. The most obvious scenario is that it is due to the interaction between closed and
openfields,whichreleasesclosedfieldplasmaontoopen fieldlines.Thiswouldnaturally
accountforbothitsobservedvariabilityandcomposition.Theproblem,however,isthatthe
slowwinddoessometimesextendfarfromtheHCSinlatitude,whichimpliesthatitssource
at the Sun is inside the open field region, far from the open-closed field boundary. But in
that case, it is difficult to understand why its composition should resemble that of closed
fieldplasma.Theseapparentlyconflictingobservationshavelongmadetheidentificationof
thesolarsourcesoftheslowwindoneofthemajorunsolvedproblemsinHeliophysics.We
describebelowthebasictheoriesthathavebeenproposedfortheslowwindsources,present
somenumericaltestsofoneofthesetheories(Linkeretal.2011),anddescribeanewtheory
fortheslowwind,theS-Webmodel(Antiochosetal.2011).
2 TheoriesfortheSourcesoftheSlowWind
There areessentiallythreedifferent possibilities for theslow windsource: It originates in
theopen field region, just like thefast; itsomehow originates from insidetheclosed field
region;oritoriginatesfromthestreamertops,attheboundarybetweenopenandclosed.We
discuss eachof them, in turn, below, but focus on the streamertop model and propose an
extensionofthistheory,theS-Webmodel,thatcanreconcilethetheorywithobservations.
4
2.1 TheExpansionFactorModel
Perhaps,thesimplesttheoryfortheslowwindisthatitoriginatesfromopenfieldnearthe
boundary between open and closed (Suess 1979; Kovalenko 1981; Withbroe 1988; Wang
&Sheeley1990;Cranmer&vanBallegooijen2005;Cranmeretal.2007).Thisscenariois
highlyappealinginthatitimpliesaunifiedtheoryfortheoriginsofthefastandslowwinds.
Bothareduetoasinglemechanism:photospherically-drivenMHDwavesdepositheatand
momentum to coronal plasma, resulting in aParker-like solar wind outflow. The key idea
underlyingthemodelisthatthespeedofthewindissensitivetotheexactlocationsofthe
heatandmomentumdepositioninanopenfluxtube;inparticular,heatinglowdownbelow
the critical point leads to a hotter, slower wind (Holzer & Leer 1980). The observed dif-
ferencesbetweenthefastandslowwinds,therefore,mayarisesolelyfromthegeometrical
differencebetweenfluxtubesnearthecoronalholeboundaryversusthosedeepintheinte-
rior.Fluxtubesneartheboundaryexpandsuper-radiallyfromthephotospheretoaheightof
orderafewsolarradii;whereasthoseneartheinteriorexpandradiallyorevensub-radially.
Evenifthephotosphericfluxofwavesintoallfluxtubesinacoronalholeisthesame,the
evolutionofthewavesinthecorona,(theresultingturbulenceanddissipation),willdepend
on thegeometry ofthe fluxtube. Cranmer andco-workers have argued that allthedistin-
guishingfeaturesoftheslowwind,includingthevariabilityandelementalcompositioncan
beexplainedbytheeffectofthefluxtubegeometry,theso-calledexpansionfactor,onthe
waveevolution.
The challenge for the expansion factor theory is that the speed of the solar wind is
sometimesobservedtobeslow,butthewindstillhasthevariability,composition,andother
featuresindicativeofthefastwind.Zhaoetal.(2009)haveshownthatthewindfromsmall
low-latitude coronal holes with large expansion factors is, indeed, slow, ∼500 km/s; but
thiswindstillhasallthetemporalandcompositionalcharacteristicsofthefastwind.These
observations are in direct conflict with any model proposing that the differences between
thefastandslowwindresultsolelyfromdifferencesinfluxtubegeometry.TheZhaoetal.
(2009)resultsdemonstratethatalargeexpansionfactorinanopenfluxtubedoesslowdown
thewind,aspredicted,butitdoesnotleadtothevariabilityandcompositionobservedinthe
slowwind.Weconclude,therefore,thattheexpansionfactormodel,aspresentlydescribed,
isnotconsistentwiththeobservations.
2.2 TheInterchangeModel
AnothertheoryfortheslowwindistheinterchangemodelproposedbyFiskandco-workers
(Fisk et al. 1998; Fisk 2003; Fisk & Zhao 2009), which in many ways is the diametric
oppositeoftheexpansionfactormodel.Intheinterchangemodeltheslowwindispostulated
tooriginatefrom theclosedfieldregionviacontinuous interchange reconnection between
openfieldlinesandtheclosedflux.Consequently,thismodelisintrinsicallydynamic;there
isnosteadystatesolarwind,atleast,fortheslowcomponent.Notealsothatthereisnotruly
closedfieldregion,becausetheopenfluxispostulatedtodiffusethroughouttheapparently
closed field regions outside coronal holes. The key idea underlying the model is that the
evolutionofthecoronalopenfluxisdominatedbythecontinuoussmall-scaledynamicsof
the photosphere, such as emergence/cancellation of magnetic carpet bipoles, which drive
reconnection between open flux and closed. In addition to being fundamentally dynamic
rather than steady, the interchange model also proposes a completely different magnetic
topology than the expansion factor model. In the latter, the topology is smooth with well
5
separatedopenandclosedfieldregions,butintheformerthetopologyisessentiallychaotic
withopenandclosedfluxmixingindiscriminately.
In terms of accounting for slow wind observations, the advantages of the interchange
modelareobvious.Itnaturallyproducesacontinuouslyvariablewindwithclosedfieldcom-
positionandlocatedaroundtheHCS,but withlargeextent.Theprimarychallengeforthe
model is to verify that interchange reconnection induced by photospheric dynamics does,
indeed, produce the required diffusion of open flux into closed field regions. Arguments
have been presented by Antiochos et al. (2007) that basic Lorentz force balance consid-
erations prohibit the mixing of open flux with closed. In fact, Antiochos et al argued that
the open-closed topology remains smooth even during interchange reconnection and have
proposed theorems that severely constrain the possible topologies of the Sun’s open field
regions. Theseauthors, however, didnot perform anactual dynamic calculation oftheef-
fect of interchange reconnection withmagnetic carpet bipoles on open flux evolution. We
presentjustsuchacalculationinthefollowingsection.Aswillbeevidentbelow,theresults
ofthiscalculationdonotsupporttheunderlyingassumptionsoftheinterchangemodel.Our
results, however, do support key aspects of the interchange model in that dynamics and a
statisticalapproacharelikelytobeessentialformodelingtheslowwind.
2.3 TheStreamer-TopModel
Thethirdtheoryforthesourceoftheslowwind,thestreamer-topmodel,isinsomeways,
acompromisebetweentheexpansionfactorandinterchangemodels.Thebasicideaisthat
the boundary between the open and closed flux, the edge of a streamer, is either unstable
(Suess etal. 1996; Endeve et al.2004; Rappazzo et al.2005) orsensitivetoperturbations
and,consequently,undergoescontinuousdynamics(Mikic´ etal.1999).Inresponsetopho-
tospheric changes or other disturbances, closed flux near the streamer boundary expands
outwardandbecomesopenandopenfluxreconnectsattheHCStobecomeclosed.Further-
more,interchangereconnectioncanoccuratthestreamertopsinresponsetophotospheric
changes(e.g.,Wangetal.2000).Theseprocesseswillnaturallyreleaseclosedfieldplasma
intothesolarwindandproduce avariablewindwiththeobservedcomposition. Notethat
inthismodelthesourceoftheslowwindistheboundary regionbetweenopenandclosed
flux, similartotheexpansion factormodel, but theboundary inthiscaseisfullydynamic
andinvolvesthecontinualinterchangeofopenandclosedflux.Inthisrespectthemodelis
similartotheinterchangemodel,buttheopen-closedinterchangeinthestreamer-topmodel
occursonlynearthestreamerboundary. Thereisnodiffusionofopenfluxdeepinsidethe
closed-fieldstreamer.
There area number of observations that provide compelling support for the streamer-
top model. Movies of coronal evolution often show the upward expansion and eventual
opening of closed loops (Hundhausen et al. 1984; Howard et al. 1985; Sheeley & Wang
2002).Conversely,coronagraphobservationsfrequentlyshowblobsstreamingoutwardnear
the current sheet and reconnection events at the HCS (Sheeley 1997), as required for the
streamer-top model. In the heliosphere the HCS is observed to be clearly dynamic with
no evidence for a simple field reversal as would be the case for a quasi-steady wind. We
conclude, therefore, that the boundary between the open and closed flux in the corona is,
indeed,highlydynamicand,therefore,willbethesourceofavariableslowwind.
Theproblem,however,isthatthesedynamicsareexpectedtobeconfinedtoarelatively
narrowregionaboutthestreamerboundary,orequivalently,abouttheHCS.Assumingthat
theopen-closedboundaryis“blurred”bythephotosphericdynamicsoverascaleofordera
6
supergranuleradius,∼30,000km,wederiveanangularextentforthisdynamicwindofno
morethan5◦.Infact,thiswidthiswhatisobservedforthestreamerstalksemanatingfrom
thetopsofstreamersandfortheso-calledplasmasheetintheheliosphere(Winterhalteret
al.1994;Wangetal.2007).Butthisangularwidthisfartoosmalltoexplaintheslowwind
whichhasbeenobservedtoextendoutto30◦ fromtheHCS.Explainingthelargeangular
extent of the slow wind is the fundamental challenge for the streamer-top model. Since
the boundary between open and closed flux maps directly to the HCS, it seems unlikely
that blurring this boundary by a small distance on the Sun would ever produce effects in
the heliosphere far from the HCS. We describe below our theory, the S-Web model, that
accomplishesexactlythisunlikelyresult.
3 TheS-WebModel
In previous work (Antiochos et al. 2007) we proposed the uniqueness conjecture, which
states that any unipolar region on the photosphere can contain at most one coronal hole.
However,low-latitudecoronalholesthatappeartobedisconnectedfromtheircorresponding
polarholearefrequentlyobservedontheSun(e.g.,Kahler&Hudson2002).Wearguedthat
inthesecases,thelow-latitudesatelliteholeisactuallyconnectedtothemainpolarholeby
anarrowopenfieldcorridorwhosewidthisbelowthespatialresolutionoftheobservations.
Notealsothatifthecorridorissufficientlynarrow,itwillbeobscuredbyneighboringclosed
fieldregionswithmuchhigherbrightness.
Let us consider how such a corridor would map into the heliosphere. For illustrative
purposes, Figure 1 shows such a mapping in the extreme case where the polar hole and
satelliteholeshavenearequalflux.Theinnerhemispherecorresponds tothephotosphere,
with the dark yellow region representing the closed field and the light blue representing
the two coronal holes connected by an open-field corridor. The light orange hemisphere
correspondstosomesurfaceintheheliospherewherethefieldisallopen,sayat10R .On
⊙
thissurfacetheradialfluxisapproximatelyuniformlydistributed,asintherealheliosphere,
andthereisasingleHCS,indicatedbytheblacklinerunningaroundtheequatorofthe10
R surface. Note that this HCS maps down to the boundary of the single, (topologically
⊙
connected), openfieldregiononthephotosphere. Fourfieldlineswithfootpoints nearthe
“endpoints”oftheopenfieldcorridoraredrawn,illustratingthismappingfromtheHCSto
theopenfieldboundary.
Iftheholeshaveroughly equalflux,theirfluxmustdividethe10R surfaceintotwo
⊙
nearequalhalves.InFig. 1,thesatelliteholemapstothenearhalfandthepolartothefar.
Separatingthesehalvesisathinarc,bluecurveonthe10R surface,thatmapstotheopen
⊙
fieldcorridor.Notethatthisarcdividestheheliosphericsurfaceand,hence,mustextendto
near90◦fromtheHCS.Iftheopen-fieldcorridorisverynarrow,thenthefield-linemapping
from thebluearcdown tothesolarsurface isquasi-singular, sotopologically, thearcisa
so-calledquasi-separatrixlayer(Priest&De´moulin1995;De´moulinetal.1996;Titovetal.
2002). The HCS,ontheother hand, isatrueseparatrix, becausethefield-line mapping is
singularthere.
Figure 1 illustrates a steady model, but to obtain the slow wind we need to add the
temporalvariability.Assumethatduetotherandomphotosphericevolution,theopen-closed
boundaryatthephotospherebecomesdynamicallyblurredbycontinuousfieldlineopening
and closing over some finite width, such as a supergranule scale. This narrow boundary
regionisnowasourceofslowwind.Duetothefieldlinemapping,theHCSalsoacquires
afiniteangularwidthoforderseveraldegrees,andbecomesalocationforslowwindinthe
7
Fig.1 Illustrationofanopenfieldcorridorconnectingtwocoronalholes(blueshading)atthephotosphere
(yellowinnersurface),anditsmagneticfieldline(reddashed)mappingtoaheliosphericsurface(paleor-
ange).
heliosphere.Butiftheopenfieldcorridorisnarrowerthanasupergranulescale,thenallthe
fluxinthecorridor willbedynamically blurred byfieldlineopening andclosingandwill
beasourceofslowwind.Sincethecorridor fluxmapstothebluearc,thismustalsobea
location ofslow windintheheliosphere. Thekeypoint isthat thearcextends well above
theHCS,sothat inthecaseofFig. 1,slowwindoccurs upto90◦ abovetheHCS,atthe
heliosphericpole!
Notethataslongasitisnarrowerthanasupergranulescale,thewidthoftheopen-field
corridorisirrelevanttoourarguments.Dependingonthedistributionofphotosphericflux,
Titovetal.(2011)haveshownthatthecorridorcanshrinkdowntothepointitbecomesthe
zero-widthfootprint ofaseparatrixdome.Regardlessofwhetherafinitecorridororasin-
gularityispresent,thisisaregionwhereelectriccurrentconcentrationswouldbeexpected
toformwhenthefootpoints ofthefieldsarestressedbyphotospheric motions,andwhere
themagneticfieldismostsusceptibletoreconnection.Reconnectioninturnallowsplasma
formedonclosedfieldlinestoaccessopenfieldsandescapeaspartoftheslowsolarwind.
Notethattheinstantaneousopen-fieldboundary andresultingHCSarealsoofzerowidth,
andyet,theyproduceaslowwindregionoffiniteextent.
We conclude from the discussion above that open-field corridors and the topology of
Fig.1maybeabletoreconciletheobservedextentoftheslowwindwiththestreamertop
model.Ifthisisthecase,themainchallengetothemodelwouldbeovercome.Ofcourse,for
asinglecorridorasinFig.1,theslowwindoccursonlyinanarrowarcofseveraldegrees
width,whichisnotsufficienttoaccountfortheobservedslowwind,irrespectiveofwhere
thearclies.Theimportantquestion,thereforeisthenumberofsuchcorridorsandarcsfora
truesolarphotosphericfluxdistribution.
Toanswerthisquestionwecalculatedwithveryhigh numerical resolution, thesteady
state MHD solution for the corona and wind during a Carrington rotation centered about
the August 1, 2008 eclipse(Rusˇinet al. 2010). Figure 2 shows the open and closed field
distributionatthephotospherecalculatedfromthemodel,alongwiththepolarityinversion
8
Fig. 2 Distribution of open magnetic flux (black) and closed (grey) at the photosphere for a Carrington
rotationcenteredaboutAug.1,2008.Alsoshownisthepolarityinversionlineonasurfaceslightlyabovethe
photosphere,(adaptedfromAntiochosetal2011).
line slightly above the photosphere (for clarity). It is evident that during this time period
therewerenumeroussatelliteholesextendingfromthemainpolarholes.
Figure 3showsthedistributionofthesquashingfactor,Q,ontheheliosphericsurfaceat
10R .Thesquashingfactorisarobusttopologicalmeasurethatallowsforstraightforward
⊙
identificationofquasi-separatrixlayersandseparatrices(Titovetal.2002, 2008).Wenote
thatsurrounding andconnected totheHCS(thickdarkredline)isadensewebofhighQ
layers,theS-Web.
TheS-WebofFig. 3isexactlywhatisneededtoaccountfortheslowwindobservations.
At some locations it extends up to 30◦ from theHCS, which explains theobservations of
slowwindathighlatitudes.TheHCSisnotalwayssymmetricallylocatedinsidetheS-Web
butcanlienearoneedgeand,infact,theHCSisoftenobservedintheheliospheretooccur
closetooneoftheslowwindboundariesratherthansymmetricallybetweenthem(Burlaga
et al. 2002). Although the Q layers are densely spaced, they are not space filling, so we
expectthatthewindintheS-Webregionwillactuallyconsistofamixtureofslowandfast,
asobserved.Notealso,thattheS-Webhasaclearlydefinedboundaryseparatingitfromthe
polar flux regions, which can account for the observed sharp transition between slow and
fastwind(Zurbuchenetal.1999).
Weconcludethat,atleast,forthetimeperiodofAugust2008,theopen-fieldcorridors
atthephotosphereandtheS-Webintheheliospherehaveallthenecessarytopologicalstruc-
turetoproducetheobservedslowwind.Ofcourse,theunderlyingassumptionisthatwhen
thephotosphericdynamicsareaddedtothemodel,magneticfieldwillopenandcloseasre-
quired,andasufficientamountofclosedfieldplasmawillbereleasedintotheheliosphere.
Wedescribebelowasimulationwhichrepresentsourfirststeptowardsaddingdynamicsto
theS-webmodel.
4 CoronalandHeliosphericDynamics
Itisnotyetpossibletoperformafullytime-dependentcalculationthatadequatelyresolves
dynamics such as reconnection occurring on the smallest scales shown in Fig. 2. Even
relaxing this configuration to asteady statestrained our computational resources! Adding
the magnetic carpet and its complex evolution to the calculation is well beyond present
capabilities.Therefore,weconsiderinsteadasimplifiedscenarioinwhichwebeginwithan
9
Fig.3 DistributionofQ-factoronaheliosphericsurfaceat10R⊙forthecoronalholesystemofFig. 2.The
HCSappearsasthedarkredlineandthecontoursofhighQaswhite,(adaptedfromAntiochosetal.2011)
observedphotosphericfluxdistribution,butlessstructuredthaninFig. 2,andtwo“magnetic
carpet” bipoles. Rather than attempting to emerge and submerge the flux, which is also
difficultcomputationally,weonlyconvectthebipoleswithasimplephotosphericmotion.
AlthoughthissimulationdoesnotallowforafulltestoftheS-webmodel,itdoestest
whetheropen-fieldcorridorsformandcoronalholesstayconnectedinafullydynamicevo-
lution. It also determines whether the field opens and closes in response to photospheric
dynamics,asrequired.Furthermoreitprovides aseveretestoftheinterchangemodel.Ac-
cording to this model, interchange reconnection leads to the diffusion of open flux into
theclosed.Weexpectthattherewillbesubstantialinterchangereconnectionasthebipoles
move; hence, if the interchange model is correct, the open and closed field regions will
becomehighlymixed.
4.1 TheNumericalModel
Weusesphericalcoordinatesandadvanceintimethefollowingsetofviscousandresistive
MHDequations(incgsunits):
4p
(cid:209) ×B = J, (1)
c
1¶ B
(cid:209) ×E =− , (2)
c ¶ t
v×B
E+ =h J, (3)
c
¶r
+(cid:209) ·(r v) =0, (4)
¶ t
1 ¶ T
+v·(cid:209) T =−T(cid:209) ·v, (5)
g −1(cid:18)¶ t (cid:19)
¶ v 1
r +v·(cid:209) v = J×B−(cid:209) p+r g+(cid:209) ·(nr (cid:209) v), (6)
(cid:18)¶ t (cid:19) c
where B is the magnetic field, J is the electric current density, and E is the electric field.
InpracticethevectorpotentialAisadvanced, withB=(cid:209) ×A.Thevariablesr ,v, p,and
T are the plasma mass density, velocity, pressure, and temperature, g=−g rˆ/r2 is the
0
10
gravitational acceleration, h the resistivity, n is the kinematic viscosity, and g =1.05 is
thepolytropicindex.Thepolytropicapproximationisadequateforthisstudy,sinceweare
interestedprimarilyinthemagneticfieldevolutionratherthanthedetailedplasmaenergetics
asinLionelloetal.(2009).TheboundaryconditionsarediscussedbyLinker&Mikic´(1997)
and Linker et al. (1999). Atthe inner radial boundary, afixed temperature of 1.8×106 K
andanelectrondensityof108cm−3areprescribed.Thecomponentofthevelocityalongthe
magneticfieldisnotspecifiedbutcalculatedfromthecharacteristicequations.Attheouter
radialboundarytheflowissupersonicandsuper-Alfve´nic,andvariablesarecomputedwith
theaidofthecharacteristicequations.
Thegridisnonuniforminr×q ×f of151×191×291points,withD r≈2.6×10−3R =
⊙
1.8MmatthelowerradialboundaryandD r≈0.75R at20R .Thelatitudinalmeshvaries
⊙ ⊙
between Dq ≈3.7◦ at the poles and Dq ≈0.5◦ near the equator. The azimuthal (longitu-
dinal)meshvariesbetweenDf ≈0.5◦ intheprimaryregionofstudytoDf ≈3.0◦ further
away.Thesimulationdomainextendsoutto20R .TheAlfve´ntraveltimeatthebaseofthe
⊙
corona(t =R /V )for|B|=2.205Gandn =108cm−3,whicharetypicalreferenceval-
A ⊙ A 0
ues,is24minutes(Alfve´nspeedV =480km/s).Auniformresistivityischosensuchthat
A
theLundquistnumbert /t is1×105,wheret istheresistivediffusiontime.Auniform
R A R
viscosityn isalsoused,correspondingtoaviscousdiffusiontimetn suchthattn /tA=500.
Again,thisvalueischosentodissipateunresolvedscaleswithoutsubstantiallyaffectingthe
globalsolution.Duringthephasesofthesimulationwherereconnectionoccurs,thelength
scalesareconsiderablysmallerthan1R andthenumericaldissipationexceedsthespecified
S
h , sothe Lundquist number is consequently smaller in the regions where these dynamics
occur.
On the solar surface, the bottom boundary, we prescribe for the magnetic flux distri-
bution a smoothed NSO Kitt Peak map for Carrington Rotation (CR) 1913 (August 22 –
September18,1996).Theresultingcoronalholepatternshowsalongsouthwardextension
ofthenorthern coronal hole, theso-calledElephant’s Trunk (Gibson etal. 1999) that was
visibleduringthistimeperiod,Figure 4.Tostudytheeffectsofphotosphericdynamicson
this system, we add two small bipoles in the area around the Elephant’s Trunk, top panel
inFig. 4.Thenorthernbipoleproducesasmallclosedfieldregioninsidethecoronalhole,
while the southern bipole results in a small satellitecoronal hole connected to the rest of
the Elephant Trunk by an open field corridor, Fig. 4. Note that although it has consid-
erable fine-scale structure, the open field topology in Fig. 4 is topologically smooth and
well-connected.
Wethendrivethissystemwithasurfaceflowthatisuniforminlongitudeanddirected
fromwesttoeast,vf 0≈−1km/sbetween30.5◦≤Lat.≤36.0◦,andbetween7.8◦≤Lat.≤
15.3◦.Thisflowcausesthebipolestomovethroughtheopenandclosedfieldregionsand,
inparticular,acrosstheopen-closedboundaries.Westoptheflowafter70hoursandallow
thesystemtorelaxfurtherfor14hours.
4.2 ResultsandConclusions
Fig. 4showstheopenandclosedfluxatthephotosphereatfourtimesduringtheevolution.
The dynamics of this evolution are dominated by two basic processes: flux opening and
closing at the HCS and interchange reconnection at the moving bipoles. Each of them is
discussed,inturn,below.