Tangled Magnetic Fields in Solar Prominences

0.67 MB·English

by  A. A. van Ballegooijen

THEASTROPHYSICALJOURNAL,2010,INPRESS
PreprinttypesetusingLATEXstyleemulateapjv.03/07/07
TANGLEDMAGNETICFIELDSINSOLARPROMINENCES
A.A.VANBALLEGOOIJENANDS.R.CRANMER
Harvard-SmithsonianCenterforAstrophysics
60GardenStreet,MS-15,Cambridge,MA02138,USA
DraftversionJanuary15,2010
ABSTRACT
Solarprominencesareanimportanttoolforstudyingthestructureandevolutionofthecoronalmagneticfield.
0
Here we considerso-called“hedgerow”prominences, whichconsist of thin verticalthreads. We explorethe
1
possibilitythatsuchprominencesaresupportedbytangledmagneticfields. Avarietyofdifferentapproaches
0
areused. First,thedynamicsofplasmawithinatangledfieldisconsidered. Wefindthatthecontortedshape
2
ofthefluxtubessignificantlyreducestheflowvelocitycomparedtothesupersonicfreefallthatwouldoccur
n in a straight vertical tube. Second, linear force-free models of tangled fields are developed, and the elastic
a
responseofsuchfieldstogravitationalforcesisconsidered. Wedemonstratethattheprominenceplasmacan
J
besupportedbythemagneticpressureofatangledfieldthatpervadesnotonlytheobserveddensethreadsbut
5
alsotheirlocalsurroundings. Tangledfieldswithfieldstrengthsofabout10Gareabletosupportprominence
1 threadswithobservedhydrogendensityoftheorderof1011cm- 3. Finally,wesuggestthattheobservedvertical

threadsaretheresultofRayleigh-Taylorinstability.Simulationsofthedensitydistributionwithinaprominence
]
R threadindicatethatthepeakdensityismuchlargerthantheaveragedensity. We concludethattangledfields
provideaviablemechanismformagneticsupportofhedgerowprominences.
S
. Subjectheadings:MHD—Sun:corona—Sun:magneticfields—Sun:prominences
h
p
- 1. INTRODUCTION
o
r Solar prominences (a.k.a. filaments) are relatively cool
t
s structures embedded in the million-degreecorona at heights
a wellabovethechromosphere(seereviewsbyHirayama1985;
[
Zirker 1989; Priest 1990; Tandberg-Hanssen 1995; Heinzel

1 2007). Above the solar limb, prominences appear as bright
v structures against the dark background, but when viewed as
7 filamentsonthesolardisktheycanbebrighterordarkerthan
5 their surroundings, depending on the bandpass used to ob-
7 serve them. Magnetic fields are thought to play an impor-
2 tantroleinsupportingtheprominenceplasmaagainstgravity,
. and in insulating it from the surrounding hot corona. Most
1
0 quiescent prominences exhibit intricate filamentary struc-
0 tures that evolve continually due to plasma flows and heat-
1 ing/cooling processes (see examples in Menzel&Wolbach
: 1960; Engvold 1976; Malherbe 1989; Leroy 1989; Martin
v
1998). In some cases the threads appear to be mostly hori-
i
X zontal,whilein othercasestheyareclearlyradiallyoriented
(“hedgerow”prominences).Figure1showsseveralexamples
r
a ofprominencesobservedinHαattheBigBearSolarObser-
vatory(BBSO)andtheDutchOpenTelescope(DOT).Theex- FIG. 1.—Quiescentprominencesabovethesolarlimb: (a)Hαimageof
prominenceobservedatBigBearSolarObservatory(BBSO),November22,
amplesinFigs.1aand1bshowmainlyverticalthreads,while 1995;(b)Hα,BBSO,1970;(c)Hα,DutchOpenTelescope,September15,
theprominenceinFig.1cshowshorizontalthreads. Off-limb 2006;(d)HeII304Å,STEREO/EUVI,2008April20at00:06UT.
observationsinHeII304Åindicatethattherearehigheralti-
tudepartsthatareopticallythininHαandthereforenotvisi- der 10–70 km s- 1 (Menzel&Wolbach 1960;
bleonthedisk(oratleasthavenotbeenclearlyidentifiedin
Engvold 1976; Zirker,Engvold&Martin 1998;
diskobservations). Figure1dshowsaprominenceabovethe
Kucera,Tovar&DePointieu 2003; Lin,Engvold&Wiik
solarlimbasobservedinHeII304ÅwiththeSECCHI/EUVI 2003; Okamotoetal. 2007; Bergeretal. 2008; Chaeetal.
instrument(Howardetal. 2007) on the STEREO spacecraft. 2008). Recent high-resolution observations of filaments on
Theupperpartsoftheprominenceconsistofverticalthreads thesolardiskindicatethattheyconsistofacollectionofvery
withanintricatefine-scalestructure.Moviesequencesofqui- thinthreadswithwidthsofabout200km,atthelimitofreso-
escentanderuptingprominencescanbefoundattheSTEREO lutionofpresent-daytelescopes(Lin,Engvold&Wiik 2003;
website1. Linetal. 2005a,b; Lin,Martin&Engvold 2008; Linetal.
Prominence plasma is highly dynamic, ex- 2008). Individualthreads have lifetimes of only a few min-
hibiting horizontal and vertical motions of or- utes, but the filament as a whole can live for many days. It
seemslikelythatthesethinthreadsarealignedwiththelocal
1http://stereo.gsfc.nasa.gov/gallery/selects.shtml magneticfield.High-resolutionimagesofprominencesabove
2 VANBALLEGOOIJEN&CRANMER
thelimbhavebeenobtainedwiththeSolarOpticalTelescope ported against gravity? Many authors have suggested
(SOT)onboardHinode. Forexample,Okamotoetal. (2007) that quiescent prominences are embedded in large-scale
observed horizontal threads in a prominence near an active flux ropes that lie horizontally above the polarity in-
region, and studied the oscillatory motions of these threads. version line (Kuperus&Raadu 1974; Pneuman 1983;
Heinzeletal. (2008) observed a hedgerow prominence vanBallegooijen&Martens 1989; Priest,Hood&Anzer
consisting of many thin vertical threads, and they used 1989; Rust&Kumar 1994; Low&Hundhausen 1995;
multi-wavelength observations to estimate the amount of Aulanieretal. 1998; Chaeetal. 2001; Gibson&Fan
absorptionand “emissivity blocking”in the prominenceand 2006; Dudiketal. 2008). The prominence plasma
surrounding cavity. Bergeretal. (2008) observed another is thought to be located near the dips of the helical
hedgerow prominence and found that the prominence sheet field lines. The magnetic field near the dips may be
is structured by both bright quasi-vertical threads and dark deformed by the weight of the prominence plasma
inclusions. The brightstructuresare downflowstreamswith (Kippenhahn&Schlüter 1957; Low&Petrie 2005;
velocity of about 10 km s- 1, and the dark inclusions are Petrie&Low 2005; Heinzel,Anzer&Gunár 2005). Oth-
highly dynamic upflows with velocity of about 20 km s- 1. ers have suggested that the magnetic field in hedgerow
The downflow velocities are much less than the free-fall prominences is vertical along the observed threads, and
speed,indicatingthattheplasmaissomehowbeingsupported that the plasma is supported by MHD waves (Jensen 1983,
against gravity. Bergeretal. (2008) proposed that the dark 1986; Pecseli&Engvold 2000). However, relatively high
plumes contain relatively hot plasma that is driven upward frequencies and wave amplitudes are required, and it is
bybuoyancy.Chaeetal.(2008)observedapersistentflowof unclearwhysuch waveswouldnotlead to strong heatingof
Hαemittingplasmaintoaprominencefromoneside,leading the prominence plasma. Dahlburg,Antiochos&Klimchuk
to the formation of vertical threads. They suggested that (1998) and Antiochosetal. (1999) showed that the promi-
the vertical threads are stacks of plasma supported against nence plasma can be supported by the pressure of a coronal
gravity by the sagging of initially horizontal magnetic field plasma lower down along an inclined field line; however,
lines. this mechanism only works for nearly horizontal field lines
Direct measurements of the prominence magnetic field (also see Mackay&Galsgaard 2001; Karpenetal. 2005;
can be obtained using spectro-polarimetry (see reviews Karpen,Antiochos&Klimchuk 2006; Karpen&Antiochos
by Leroy 1989; Paletou&Aulanier 2003; Paletou 2008; 2008). Furthermore, hedgerow prominences are located
LópezAriste&Aulanier 2007). A comprehensive effort to in coronal cavities where the plasma pressure is very low.
measure prominence magnetic fields was conducted in the Therefore, it seems unlikely that the verticalthreads seen in
1970’s and early 1980’s using the facilities at Pic du Midi hedgerow prominences can be supported by coronal plasma
(France)andSacramentoPeakObservatory(USA).Thiswork pressureonnearlyverticalfieldlines.
showedthat (1) the magneticfield in quiescentprominences In this paper we propose that hedgerow prominences are
has a strength of 3–15 G; (2) the field is mostly horizon- embeddedinmagneticfieldswithacomplex“tangled”struc-
tal and makes an angle of about 40◦ with respect to the ture. Such tangled fields have many dips in the field lines
longaxisoftheprominence(Leroy1989;Bommier&Leroy where the weight of the prominence plasma can be coun-
1998; Paletou&Aulanier 2003); (3) the field strength in- teracted by upward magnetic forces. Our purpose is to
creasesslightlywithheight,indicatingthepresenceofdipped demonstrate that such tangled fields provide a viable mech-
field lines; (4) most prominences have inverse polarity, i.e., anism for prominence support in hedgerow prominences.
thecomponentofmagneticfieldperpendiculartothepromi- Casini,MansoSainz&Low(2009) recentlyinvokedtangled
nence axis has a direction opposite to that of the potential fieldsintheinterpretationofspectropolarimetricobservations
field. Theseearlierdatalikelyincludedavarietyofquiescent of an active region filament. While such filaments are quite
prominences, some with predominantly horizontal threads, different from the hedgerow prominences considered here,
others with more vertical threads. In more recent work, thisworkshowsthattangledfieldshaveimportanteffectson
Paletouetal. (2001) reported full-Stokes observations of a themeasurementofprominencemagneticfields. Sucheffects
limbprominenceinHe I 5876Å (He D ), andderivedmag- willnotbeconsideredinthispaper.
3
netic field strengths of 30–45 G, somewhat larger than the Thepaperisorganizedasfollows.InSection2wepropose
values reported in earlier studies. Casinietal. (2003) pub- that hedgerow prominences are supported by tangled mag-
lishedthefirstvector-fieldmapofaprominencewithaspatial netic fields, and we discuss how such fields may be formed.
resolutionof a few arcseconds. Theyfoundthat the average In Section 3 we presenta simple modelfor the dynamicsof
magneticfieldinthisprominenceismostlyhorizontalwitha plasmaalongthe tangledfield lines, and we show thatweak
strengthofabout20Gandwiththemagneticvectorpointing shock waves naturally occur in such plasmas. In Section 4
20◦ to 30◦ off the prominence axis, consistent with the ear- wedevelopamagnetostaticmodeloftangledfieldsbasedon
lier studies. However,the map also showsclearly organized thelinearforce-freefieldapproximation,andinSection5we
patches where the magnetic field is significantly stronger study the response of such fields to gravitational forces. In
than average, up to 80 G (also see Wiehr&Bianda 2003; Section6wesimulatethedistributionofplasmainacylindri-
LópezAriste&Casini 2002, 2003; LópezAriste&Aulanier calprominencethread. InSection7wediscusstheformation
2007;Casini,Bevilacqua&LópezAriste2005). Itisunclear ofverticalthreadsbyRayleigh-Taylorinstability. Theresults
howthesepatchesarerelatedtothefinethreadsseenathigher oftheinvestigationaresummarizedanddiscussedinSection
spatial resolution. Recently, Merendaetal. (2006) observed 8.
He I 10830 Å in a polar crown prominenceabove the limb,
2. TANGLEDFIELDSINPROMINENCES
andfoundevidenceforfieldsofabout30Gthatareoriented
only25◦fromtheverticaldirection. The spectro-polarimetric observations of prominences de-
How is the plasma in hedgerow prominences sup- scribed in Section 1 are consistent with the idea that quies-
cent prominences are embedded in coronal flux ropes that
TANGLEDMAGNETICFIELDSINPROMINENCES 3
We suggest that the tangled field may be formed as a re-
sult of magnetic reconnection, not the twisting or stress-
ing of field lines. Quiescent prominences are located
above polarity inversion “lines” that are often more like
wide bands of mixed polarity separating regions with dom-
inantly positive and negative polarity. In these mixed-
polarity zones, magnetic flux elements move about ran-
domlyandoppositepolarityelementsmaycanceleachother
(e.g.,Livi,Wang&Martin1985). Newmagneticbipolesfre-
quently emerge from below the photosphere. These pro-
cesses causes the “recycling” of the photosphericflux about
(a) (b) once every 2 to 20 hours (Hagenaar,Schrijver&Title 2003;
Hagenaar,DeRosa&Schrijver2008),andthecoronalfluxis
FIG. 2.— Magnetic support ofsolar prominences: (a) by alarge-scale
coronalfluxrope,(b)byatangledmagneticfieldinacurrentsheet. recycled even faster (Closeetal. 2005). It is likely that the
interactionsbetweenthesefluxelementsproduceacomplex,
lie horizontally above the polarity inversion line (PIL). Fig- non-potentialmagneticfieldinthelowcorona.Withinthisen-
ure 2a shows a vertical cross-section through such a flux vironmentmagneticreconnectionislikelytooccurfrequently
rope. The magnetic field also has a componentinto the im- atmanydifferentsitesinthecoronaabovetheinversionzone.
age plane, so the field lines are helices, and the plasma is Each reconnection event may produce a bundle of twisted
assumed to be located at the dips of these helical windings. field,andthetwistedfieldsfromdifferenteventsmaycollect
A dip is defined as a point where the magnetic field is lo- intolargerconglomeratestoformatangledfield. Thetangled
cally horizontaland curvedupward. As indicatedin the fig- fieldmayrisetolargerheights(asaresultofitsnaturalbuoy-
ure,themagneticfieldmaybedeformedbytheweightofthe ancy), and may collect into a thick sheet that is sandwiched
prominence plasma, creating V-shaped dips. The magnetic betweensmootherfields,asillustratedinFigure2b. Theob-
field near such dips is well described by the Kippenhahn- served prominenceconsists of plasma that is trapped within
Schlüter model (Kippenhahn&Schlüter 1957), and several this sheet. New tangled field is continuallyinjected into the
authors have developed local magnetostatic models of the sheetfrombelow,producingverticalmotionswithinthesheet.
finestructuresobservedinquiescentprominences(e.g.,Low We suggestthatthe “darkplumes”observedbyBergeretal.
1982; Petrie&Low 2005; Heinzel,Anzer&Gunár 2005). (2008)maybeamanifestationofsuchverticalmotionsofthe
However,recentobservationsof“darkplumes”(Bergeretal. tangledfield.
2008)androtationalmotions(Chaeetal.2008)withinpromi-
3. FLOWSALONGTHETANGLEDFIELD
nencesremindusagainthatprominenceshavecomplexinter-
nalmotions,anditisnotclearhowsuchmotionscanbeex- Thespatialdistributionofplasmawithintheprominenceis
plainedintermsofasinglelargefluxrope. Perhapsthemag- determinedinpartbythedynamicsofplasmaalongthetan-
neticstructureofhedgerowprominencesismorecomplicated gledfield lines. Figure3 showsthe contorted(butgenerally
thanthatpredictedbythefluxropemodel(Figure2a). downward)pathofanindividualfieldlineinthetangledfield.
In this paper we propose an alternative model, Notethatthereareseveral“dips”wherethefieldlineishori-
which is illustrated in Figure 2b. Following zontalandcurvedupward,and“peaks”wherethefieldishor-
Kuperus&Tandberg-Hanssen (1967), we suggest that izontaland curveddownward. Tracingthe field line upward
hedgerow prominences are formed in current sheets that from a dip, one always reaches a peak where the field line
overlie certain sections of the PIL on the quiet Sun. Unlike againturnsdownward.Therefore,thequestionariseswhether
those previous authors we suggest that the current sheet theplasmacollectedinthedipswouldremaininthesedipsor
extendsonlytolimitedheight( 100Mm),andmayextend besiphonedoutofthedipsviathepeaksofthefieldlines.
only a limited distance along∼the PIL. Furthermore, we Toanswerthisquestion,weconsiderasimplemodelforthe
proposethattangledmagneticfieldsarepresentwithinthese motion of the prominence plasma along the magnetic field.
current sheets. A tangled field is defined as a magnetic For simplicity we assume that the flow takes place in a thin
structure in which the field lines are woven into an intricate tubesurroundingtheselectedfieldline(i.e.,thedivergenceof
fabric,andindividualfieldlinesfollownearlyrandompaths. neighboringfield lines is neglected), and the cross-sectional
We suggest that the field is tangled on a spatial scale of areaofthistubeistakentobeconstant. Weassumeasteady
0.1–1Mm,comparabletothepressurescaleheightH ofthe flow is established along the tube. Let v(s) and ρ(s) be the
p
prominenceplasma(H 0.2Mm). Theprominenceplasma plasma velocity and density as functionsof position s along
p
is assumed to be locate≈d at the many dips of the tangled the tube, then conservation of mass requires ρv = constant.
field lines. The tangled field is confined horizontallyby the Theequationofmotionoftheplasmais
verticalfieldsoneithersideofthesheet,andverticallybythe dv dp dz
weightoftheprominenceplasma. ρv =- - ρg , (1)
ds ds ds
Akeyfeatureofatangledfieldisthattheplasmaandfield
where p(s)istheplasmapressure,z(s)istheheightabovethe
are in magnetostatic equilibrium, i.e., the Lorentz force is
photosphere,andgistheaccelerationofgravity.Theequation
balanced by the gas pressure gradients and gravity. There-
ofstateiswrittenintheformp=Kργ,whereγandKarecon-
fore, a tangled field is quite differentfrom “turbulent”mag-
stants(weuse γ <5/3todescribenon-adiabaticprocesses).
neticfieldsinwhichlarge-amplitudeAlfvénwavesarepresent
Eliminating p(s) and ρ(s) from equation (1), we obtain the
(e.g.,thesolar wind). Ina tangledfieldthe magneticpertur-
followingequationfortheparallelflowvelocity:
bations do not propagatealong the field lines. In this paper
weexaminethebasicpropertiesoftangledfields,andwein- c2 dv dz
v- =- g , (2)
vestigatetheirabilitytosupporttheprominenceplasma. (cid:18) v (cid:19)ds ds
4 VANBALLEGOOIJEN&CRANMER
(a) (b)
(c) (d)
FIG.4.—Modelforplasmaflowalongasinglefieldlineinatangledfield.
(a)Thesolidcurveshowstheheightz(s)asfunctionofpositionsalongthe
fieldline[inunitsofthedistanceΛbetweenneighboringpeaks,seeequation
(3)].Thedashedcurveshowstheoveralldownwardtrendoftheflowtube.(b)
Thesolidcurveshowstheparallelflowvelocityv(s)(inunitsofc0)forγ=1.5
and Hp,0 =0.15Λ. The dashed curves show the subsonic and supersonic
solutionsofequation(6).Thesonicpointsarelocatedatthepeaksinthefield
line(s=0ands=Λ),andtheshockislocatedats=0.325Λ.(c)Similarplot
forthesoundspeedc(s)(inunitsofc0).(d)Thesolidcurveshowstheplasma
densityρ(s)alongthetube(inarbitraryunits). Thehydrostaticequilibrium
densityisshownbythedashedcurve.
FIG. 3.—Flowsalongahighlydistortedfieldlineinthetangledmagnetic Thesonicpointswillthenbelocatedats=0ands=Λ.Figure
fieldofasolarprominence. Thefieldlineisindicated bythesolidcurve,
andtheblack dotsindicate “dips” inthefieldlinewherecoolprominence 4a shows the height z(s) for A=0.15Λ and φ0 =0.2 rad, so
plasmacancollect. Thearrows show thedirection ofsubsonic(blue) and thatC=0.187. Let p ,ρ andc bethepressure,densityand
0 0 0
supersonic(red)flows. Sonicpointsarelocated atpeaks inthefieldlines soundspeedatthesonicpoints, thenK =γ- 1c2ρ1- γ, andthe
(blackvertical bars), andshockwavesoccurwheresupersonic flowsslow 0 0
downbeforereachingadip(magentabars). soundspeedcanbewrittenas
wherec(s)isthesoundspeed(c2 γp/ρ). Theaboveequa- c(s)=c [ρ(s)/ρ ](γ- 1)/2=c [v(s)/c ]- (γ- 1)/2. (5)
0 0 0 0
≡
tion has a critical point where the flow velocity equals the
Insertingthisexpressionintoequation(2),weobtain
sound speed (v=c). Therefore, a transition from subsonic
to supersonic flow can occur only at points where the RHS du 1 dz
of this equationvanishes, dz/ds=0. These sonic points are (u- u- γ) =- , (6)
ds γH ds
locatedatthepeaksofthefieldlineswheremattercanbesi- p,0
phonedoutofonedipanddepositedintoanotherdipatlower where u(s) v(s)/c , and H p /(ρ g) is the pressure
0 p,0 0 0
≡ ≡
height. The resulting flow pattern is indicated in Figure 3. scaleheightatthesonicpoints.Equation(6)canbeintegrated
Asthesupersonicflowapproachesthenextdip,itmustslow asfollows:
downtosubsonicspeeds,whichcanonlyhappeninashock. 1 z(s)- z(s )
Therefore, the tube has a series of subsonic and supersonic 1[u2(s)- 1]+ u1- γ(s)- 1 =- 0 , (7)
2 γ- 1 γH
flowsseparatedbyshocksandsonicpoints. Theroleofthese (cid:2) (cid:3) p,0
shocksistodissipatethegravitationalenergythatisreleased wheres isthepositionofasonicpoint. Equation(7)canbe
0
bythefallingmatter. solved for u(s) by Newton-Raphson iteration. When γ =1,
Thepositionandstrengthoftheshockscanbecomputedif thereis ananalytic solutionforu(s)in termsof the Lambert
theheightz(s)oftheflowtubeisknown. Neighboringpeaks W function(see Cranmer 2004); in the presentpaper we as-
are generally not at the same height. Therefore, each sec- sumeγ=1.5. Thesupersonicsolutionv (s)iscomputedwith
1
tion between neighboring peaks is approximated as a large- s =0,andthesubsonicsolutionv (s)iscomputedwiths =Λ.
0 2 0
amplitudesinusoidal perturbationsuperposedon a generally ThedashedcurvesinFigure4bshowv (s)andv (s),andFig-
1 2
downwardpath: ure4cshowsthecorrespondingsoundspeedsc (s)andc (s).
1 2
z(s)≈Acos(cid:16)2πΛs - φ0(cid:17)- Cs, (3) Hathmeerpfleliotwwuediesasogsfiuvtmehneedbfiyealds-clainleehdeisigtohrttiHonps,0.=Th0e.1M5Λa,cheqnuuaml btoerthoef
wheresisthepositionalongtheflowtube,Λisthedistance
betweenneighboringpeaks(asmeasuredalongtheflowtube), M(s)=v(s)/c(s)=[u(s)](γ+1)/2. (8)
Aistheamplitudeoftheperturbationinheight,φ isaphase
0 Theshockislocatedatthepoints wheretheMachnumber
angle, and C is the background slope. The phase angle is sh
M beforetheshockandtheMachnumberM aftertheshock
chosensuchthatthepeaksintheflowtube(wheredz/ds=0) 1 2
arelocatedats=0ands=Λ,thentheslopeisgivenby satisfythefollowingrelationship:
C=2πΛAsinφ0. (4) M22= 22γ+M(2γ-- (1γ)M- 112), (9)
1
TANGLEDMAGNETICFIELDSINPROMINENCES 5
whichfollowsfromtheRankine-Hugionotconditionsforpar- Ifgravityandplasmapressuregradientsareneglected,then
allelshocks(Landau&Lifshitz 1959). Therefore,the actual F 0, so the magneticfield B(r) must satisfy the force-free
≈
flowvelocityv(s)betweenthetwosonicpointsisgivenbythe condition:
fullcurvein Figure4b, andthesoundspeedc(s) isgivenby B=αB, (14)
thefullcurveinFigure4c. ∇×
Theplasmadensityρ(s) alongtheflow tubeis determined where α(r) may in generalbe a function of position. In the
bymassconservation(ρv=constant),andisplottedinFigure specialcase thatα is constantthroughoutthe volume, equa-
4d. Note thatthere is a strongpeak in the density at the dip tion (14) becomes a linear equation for B(r), and the solu-
inthefieldline,s 0.56Λ. ThedashedcurveinFigure4d tionsarecalledlinearforcefreefields(LFFF).Woltjer(1958)
dip
showsthedensitypr≈ofilethatwouldexistiftheplasmawere hasshownthatinaclosedmagneticsystemwithaprescribed
inhydrostaticequilibrium(HE): magnetichelicity(H A BdV, whereAisthevectorpo-
≡ ·
tential) the lowest-energRy state is a LFFF. Therefore, in this
ρ (s)=ρ exp - z(s)- z(sdip) , (10) paper only LFFFs are considered, and α is treated as a free
HE dip (cid:20) H (cid:21) parameter. We find that LFFFs can be tangled. The typical
p
lengthscaleofthetangledfieldisgivenbytheinverseofthe
whereρdipandHparethedensityandpressurescaleheightat αparameter,ℓ α- 1. Inthefollowingwefirstconsiderthe
thedip, H =0.254Λ. Thedeviationsfromhydrostaticequi- ≡| |
p casethatℓissmallcomparedtothedomainsizeLinanydi-
librium are significant only in those regions where the flow
rection, and then consider the boundaryeffects. Section 4.3
velocityiscomparabletothesoundspeed.Wedefineanaver-
describestangledfieldsinacylindricaldomain.
ageflowvelocityv¯by
Λρ(s)v(s)ds Λ 4.1. TangledFieldinaLargeVolume
v¯≡ R0 Λρ(s)ds = Λv- 1(s)ds. (11) Intheabsenseofboundaryconditions,thesolutionofequa-
0 0 tion(14)canbewrittenasasuperpositionofplanarmodes:
For the case shownRin Figure4 weRfind v¯=0.6c , so the av-
0 N
erageflowspeedislessthanthesoundspeed. Therefore,the B(r)= B ˆe cos(k r+β )- ˆe sin(k r+β ) , (15)
contortedshapeoftheflowtubesignificantlyreducestheflow n 1,n n· n 2,n n· n
Xn=1 (cid:2) (cid:3)
velocitycomparedtothesupersonicfreefallthatwouldoccur
inastraightverticaltube. whereNisthenumberofmodes,k αˆe isthewavevector
n 3,n
≡
The cooler parts of the prominence are thought to have a (n=1, ,N),B isthemodeamplitude,β isaphaseangle,
n n
temperatureT 104K.Assumingahydrogenionizationfrac- and[ˆe··,·ˆe ,ˆe ]areunitvectorsthataremutuallyorthogo-
1,n 2,n 3,n
∼
tion of 10%, a helium abundance of 10 % and γ =1.5, the nalandformaright-handedbasissystem:
soundspeedc 10kms- 1,andwepredictanaverageflow
velocityv¯ 0.06≈c 6kms- 1. Theverticalcomponentofthis ˆe1,n=cosθn(cosφn ˆy+sinφnˆz)- sinθnˆx, (16)
0
velocity is≈v¯ - C≈v¯ - 1.1 km s- 1, less than the observed ˆe =- sinφ ˆy+cosφ ˆz, (17)
z 2,n n n
verticalveloci≈tiesinp≈rominencethreads(5–10kms- 1). Note ˆe =sinθ (cosφ ˆy+sinφ ˆz)+cosθ ˆx. (18)
3,n n n n n
thatthepredictedvelocityisrelativetothepatternofthetan-
gled field, therefore, if the tangled field expands in the ver- Here θn and φn are the direction angles of the wave vector
tical direction it will push the prominence plasma upward. relativetotheCartesianreferenceframe.
Wespeculatethattheobservedupwardmotionsinhedgerow Figure5showsanexampleofafieldwithN=100modes,
prominences(e.g., Bergeretal. 2008) are due to such large- anisotropicdistributionofdirectionangles(θn,φn),andran-
scalechangesinthetangledfield. domly selected phase angles βn. The starting points of the
field linesare randomlyselected fromthe centralpartofthe
4. LINEARFORCE-FREEFIELDMODELS box, and the field lines are traced until they reach the box
In this section simple models for tangled fields are devel- walls. Note that individual field lines follow random paths,
oped.AvolumeV inthecoronaisconsidered,andtheplasma andthatdifferentfieldlinesaretangledtogether.
insidethisvolumeisassumedtobeinmagnetostaticequilib- Now consider an ensemble β,N of fields with different
E
rium,- p+ρg+F=0,wherepistheplasmapressure,ρisthe (randomlydistributed)phaseanglesβn,butwithafixednum-
density,∇g isthe accelerationofgravity,andF isthe Lorentz berof modesN, and with fixedmode amplitudesBn and di-
force. All quantities are functions of position r within the rection angles (θn,φn). The phase angles are assumed to be
volume.TheLorentzforceisgivenby uniformlydistributedintherange[0,2π],andanglesfromdif-
ferentmodesnandn′ areassumedto beuncorrelated. Then
1 1 theensembleaverageofthemagneticfieldvanishes,
F j B= ( B) B, (12)
≡ c × 4π ∇× ×
<B> =0, (19)
β
where j is the electric currentdensity and B is the magnetic
field. Usingtensornotation,equation(12)canalsobewritten where < > denote the average over phase angles β .
β n
···
as F = ∂T /∂x , where T is the magnetic stress tensor, a Also,theaverageofthetensorBBisgivenby
i ij j ij
specialcaseofMaxwell’sstresstensor(Jackson1999):
N
B2 BB <BB> = 1 B2 ˆe ˆe +ˆe ˆe
T - δ + i j. (13) β 2 n 1,n 1,n 2,n 2,n
ij≡ 8π ij 4π Xn=1 (cid:0) (cid:1)
The first term describes magnetic pressure, and the second N
termdescribesmagnetictension. Inatangledfieldbothpres- = 1 B2 I- ˆe ˆe , (20)
2 n 3,n 3,n
sureandtensionforcesareimportant. Xn=1 (cid:0) (cid:1)
6 VANBALLEGOOIJEN&CRANMER
latedwiththedirectionangles,and<B2>=B2/N,whereB
n 0 0
isaconstant.Then<B>=0,sothemeanmagneticfieldvan-
ishes. Furtheraveragingofequation(20)showsthat<BB>
isanisotropictensor:
N
<BB>= 1 <B2> I- <ˆe ˆe > = 1B2I. (22)
2 n 3,n 3,n 3 0
Xn=1 (cid:0) (cid:1)
It follows that <B2 >=B2, so B equals the r.m.s. value of
0 0
thetotalfieldstrength.Theensembleaverageofthemagnetic
stresstensor,equation(13),isgivenby
B2
<T >=- 0 δ , (23)
ij ij
24π
which is also isotropic. Note that the diagonal components
of <T > are negative, so the effects of magnetic pressure
ij
dominateovertheeffectsofmagnetictension. Therefore,the
isotropictangledfield hasa positivemagneticpressure, p =
t
B2/(24π). The averageenergydensity is E =B2/(8π). The
0 t 0
relationship E = 3p is similar to that for a relativistic gas
t t
(e.g.,Weinberg1972).
4.2. BoundaryEffects
The tangled field must be confined within a certain vol-
ume (e.g., a current sheet, see Figure 2b), and the confine-
FIG.5.—Tangledfieldobtainedbysuperpositionof100randomlyselected ment must be effective for a period much longer than the
modesofthelinearforce-freefieldinalargevolume.
Alfven travel time across the volume. What are the condi-
whereIistheunittensor(I=ˆxˆx+ˆyˆy+ˆzˆz). Expression(20)is tionsforsuchconfinement? Toanswerthisquestionwemust
independentofpositionr,sothemagneticfieldisstatistically consider the boundary region between a tangled field and a
homogeneous,butisnotnecessarilyisotropic. smooth field. The tangled field is assumed to be character-
We study the statistical properties of field lines in models izedbyahighvalueof α, andthesmoothfieldpresumably
| |
with different values of the mode number N. For each N hasamuchlowervalueof α. Tolastalongtime,themag-
| |
we construct a series of models (m= 1, ,M) with differ- neticfieldneartheboundarymustbeapproximatelyinequi-
entphaseangles,butwithconstantvalues·o·f·themodeampli- librium(non-linearforce-freefield). Theforce-freecondition
tudesB anddirectionangles(θ ,φ ). Themodeamplitudes (14)impliesthatαisconstantalongfieldlines,sotherecan-
n n n
aretakentobethesameforallmodes(B =1).Foreachreal- not be many field lines that pass from the smooth region to
n
izationmofthephaseanglesβ wetraceoutthefieldlinethat thetangledregion.Therefore,oneimportantconditionforthe
n
startsattheorigin(x=y=z=0),andwemeasurethesquare survivalofthetangledfieldisthatthetworegionsarenearly
oftheradialdistancer2(s)asfunctionofpositionsalongthe disconnectedfromeachothermagnetically. Anotherrequire-
m
fieldline: mentisthatthetworegionsareapproximatelyinpressurebal-
r2(s)=x2(s)+y2(s)+z2(s). (21) ance.
m m m m
Toshowthattheseconditionscanbesatisfied,wenowcon-
WethenaveragethisquantityoverM=100realizationsofthe
sider a simple modelforthe boundaryregion. The interface
phaseanglestoobtainthemeansquaredistancer2(s).ForN= between the tangled and smooth fields is approximatedby a
3 both mutuallyorthogonaldirections(θn,φn) and randomly planesurface,heretakentobetheplanex=0inCartesianco-
chosen directionsare considered. In both cases we find that ordinates.Theabove-mentionedconditiononthelackofcon-
the field lines follow long helical paths, and r2(s) increases nectivity between the smooth and the tangled fields requires
quadraticallywiths. Therefore,forN=3thefieldlinesdonot B (0,y,z)=0at x=0. Thetangledfield inx 0is assumed
x
behaverandomly.ForN 4onlyrandomlychosendirections tobeaLFFFwithaspecifiedvalueofα. The≥solutionofthe
≥
are considered. For N = 4 some of the field lines are long LFFF equation is again written as a superposition of planar
helices, while others have more random paths, and for N = modes. However,in thepresentcase themodesare grouped
5 all field lines seem random, however, in both cases r2(s) into pairs with closely related wave vectors k and k′, and
n n
is not well fit by a power law. True random walk behavior withthesameamplitudeB andphaseβ :
n n
of the field lines, as indicated by a linear dependence of r2 N/2
on distance s, is found only when the number of modes is B(r)= B ˆe cos(k r+β )- ˆe sin(k r+β )
n 1,n n n 2,n n n
increasedtoN 10. InthelimitoflargeN,r2(s) 10s/α. Xn=1 (cid:2) · ·
≥ ≈ | |
phWaseenaonwglecsoβnsnibduertaallsaorgtheeremnosedmeabmlepElNituindewshBicnhanndotdoirnelcytitohne ··· - ˆe′1,ncos(k′n·r+βn)+ˆe2,nsin(k′n·r+βn) . (24)
angles(θn,φn)areallowedtovary.Fromnowon< >will HereN isthetotalnumberofmodes,ˆe1,n andˆe2,nared(cid:3)efined
denoteanaverageoverthislargerensemble.Thedire··c·tionan- in equations (16) and (17), k′ αˆe′ is the modified wave
glesareassumedtohaveanisotropicdistribution,i.e.,thean- vector,andtheunitvectorsˆe′n ≡andˆe3′,n aredefinedby
1,n 3,n
gleφ isuniformlydistributedintherange[0,2π],andcosθ
isuninformintherange[- 1,+1],sothat<cos2θ >= 1. Furn- ˆe′1,n=- cosθn(cosφnˆy+sinφnˆz)- sinθnˆx, (25)
thermore,themodeamplitudesB areassumedtonbeun3corre- ˆe′ = sinθ (cosφ ˆy+sinφ ˆz)- cosθ ˆx. (26)
n 3,n n n n n
TANGLEDMAGNETICFIELDSINPROMINENCES 7
Note that ˆe′ differs from ˆe only in the sign of
3,n 3,n
the x-component, whereas ˆe′ has the sign of the y-
1,n
and z-components reversed. Therefore, the unit vectors
[ˆe′ ,ˆe ,ˆe′ ] again form a right-handed basis system. The
1,n 2,n 3,n
magneticfieldattheboundary(x=0)isgivenby
N/2
B(0,y,z)=2 B cosθ (cosφ ˆy+sinφ ˆz)
n n n n
Xn=1
cos[αsinθ (ycosφ +zsinφ )+β ], (27)
n n n n
×
whichsatisfiesB (0,y,z)=0. Therefore,itispossibletocon-
x
struct a tangled field that is disconnected from its surround-
ings.
WenowconsiderthestatisticalaverageofthetensorBBat
x=0. Averagingoverphaseanglesβ ,weobtain
n
N/2
<BB> =2 B2cos2θ (cosφ ˆy+sinφ ˆz)
β n n n n
Xn=1
(cosφ ˆy+sinφ ˆz) atx=0, (28)
n n
×
andfurtheraveragingovermodeamplitudesanddirectionan-
glesyields
FIG. 6.—Tangledmagneticfieldsobtainedbysuperpositionofmodesof
<BB>= 16B20(ˆyˆy+ˆzˆz) atx=0. (29) twhiethliinneaarhefdogrceer-ofwreeprfioemldiniennscidee.Tahceylrianddiearl,cwomhipchonreenptreosfetnhtesfiaevlderBtircavlanthisrehaeds
atthecylinderwall. Thethreepanelsshowmodelswithdifferentvaluesof
Hereweassumeanisotropicdistributionofdirectionangles, a≡|α|R,whereαisthetorsionparameterandRisthecylinderradius: (a)
andweuse<B2>=B2/N,whereB isther.m.s.fieldstrength a=3.0;(b)a=4.5;(c)a=6.0.Incase(a)thefieldhasonlyasinglemode,
n 0 0 butiscases(b)and(c)therearemultiplemodes(N=4andN=6),some
intheinteriorofthetangledfield(seeSection4.1). Notethat
ofwhicharenon-axisymmetric,resultingintangledfieldlines. Eachpanel
at the boundary <B2(0,y,z)>=B2/3, while in the interior showsthreefieldlines(red,greenandbluecurves).Inpanels(b)and(c)the
0
<B2 >=B2, so the r.m.s. field strength at the boundary is magentadotsshowdipsinthefieldlines.
0
reduced by a factor 1/√3 compared to that in the interior. 5. DEVIATIONSFROMTHEFORCE-FREECONDITION
Themagneticpressureatx=0isgivenby Theabovemodelsfora tangledfield are purelyforce-free
B2 <B2(0,y,z)> B2 and do not have any magnetic forces to support the promi-
ext = 0 = p, (30) nence plasma against gravity. To include such effects, we
t
8π ≈ 8π 24π
nowconsiderthe“elastic”propertiesofthetangledfield,i.e.,
where pt istheaveragepressureintheinteriorofthetangled itsresponsetoexternalforces. Specifically,theweightofthe
field [see equation(23)]. Equation(30) showsthat itis pos- prominencecauses the tangled field to be compressedin the
sible to maintain pressure balance between the tangled field verticaldirection,resultinginaradiallyoutwardforceonthe
anditssurroundings. plasma. Also, shearing motions may occur within the tan-
gled field as dense plasma moves downward and less dense
4.3. TangledFieldinaCylinder “plumes”moveupward(e.g.,Bergeretal.2008). Thisresults
Here an infinitely long cylinder with radius R is consid- insheardeformationofthetangledfieldandassociatedmag-
ered. We adopt a cylindrical coordinate system (r,φ,z), and neticstressesthatcounteracttheplasmaflows. Inthefollow-
we assume that the radial componentof magnetic field van- ingbothoftheseeffectsareconsideredinsomedetail.
ishes at the cylinder wall, B (R,φ,z) = 0. In the Appendix
r
we analyze the eigenmodesof the LFFF equationin the do- 5.1. CompressionalEffect
mainr Rsubjecttotheaboveboundarycondition. Wefind Wefirstconsidertheeffectsofgravityonalayeroftangled
≤
thatthiseigenvalueproblemhasadiscretesetofmodes,and magneticfield.Themagnetostaticequation(- p+ρg+F=0)
the number of modes depends on the dimensionless param- cannotbesolvedanalyticallyforatangledfie∇ld,sowemake
eter a αR. Figure 6 shows the resulting magnetic fields thefollowingapproximation:
≡| |
fora=3.0,4.5and6.0. Inthefirstcaseonlytheaxisymmet-
ric(Lundquist)modeispresent,sothefieldlinesarehelical. B′(r) 1B(r)e- z/HB = B(r)- 1 ˆz B(r) e- z/HB.
Assumingthecylinderaxisisvertical,therearenodipsinthe ≡∇×(cid:20)α (cid:21) (cid:20) αH × (cid:21)
B
fieldlines.Ifcoolplasmaweretobeinjectedintosuchastruc- (31)
ture, it would spiral down along the field lines and quickly Here B(r) is the LFFF given by equation (15), and H is
B
reachsupersonicspeeds. Incontrast, fora=4.5 anda=6.0 the magnetic scale height of the modified field (we assume
therearemultiplemodesoftheLFFF,andtherandomsuper- αH >2). Note that B′ =0 as required. This modified
B
positionofthesemodescreatesatangledfieldwithmanydips fi|e|ld B′(r) is no longer∇fo·rce-free,but has the followingsta-
where prominence plasma can be supported. The field-line tisticalproperties:
dips(i.e.,siteswhereBz=0andB·∇Bz>0)areindicatedby <B′>=0, (32)
magentadotsinthemiddleandrightpanelsofFigure6. We
willreturntothismodelinSection7. <B′B′>= 1B2 (1+ǫ2)(ˆxˆx+ˆyˆy)+ˆzˆz e- 2z/HB, (33)
3 0
(cid:2) (cid:3)
8 VANBALLEGOOIJEN&CRANMER
where ǫ (αH )- 1 < 1. Therefore, the magnitude of the that create magnetic stresses in the tangled field. The mag-
modified≡fiel|d|drBopsoff2exponentiallywith heightz. Let T′ netic couplingbetweenthe prominenceand its surroundings
ij
bethemagneticstresstensorofthemodifiedfield. Takingits causes the weight of the dense prominenceto be distributed
statisticalaverage,wefindforthenonzerocomponentsofthe overa widerarea. Ineffect, theprominenceplasmaisbeing
stresstensor: supportedby the radial gradientof the magnetic pressure of
the tangled field over this larger area. In the following we
B2
<T′ >=<T′ >=- 0 e- 2z/HB, (34) estimatethemagneticstressesandverticaldisplacementsre-
xx yy 24π sultingfromtheseforces.
B2 The tangled field is modeled either as a vertical slab with
<Tz′z>=- 240π(1+2ǫ2)e- 2z/HB. (35) half-width R, or as a vertical cylinder with radius R. The
prominence is located at the center of this slab or cylinder,
Note thatfor ǫ< 1 the stress tensor is nearlyisotropic. The andhasahalf-widthorradiusr <R. Thentheaverageden-
2 0
net force on the plasma is given by F′ =∂T′/∂x , and the sityinthetangledfieldregionisgivenby
i ij j
averageforcefollowsfromequations(34)and(35):
r n
ρ ρ 0 , (39)
<Fx′>=<Fy′>=0, (36) avg≈ 0(cid:16)R(cid:17)
B2 whereρ0isthedensityoftheprominence,andweneglectthe
<F′>= 0 (1+2ǫ2)e- 2z/HB. (37) mass of the surroundings. The exponent n=1 for the slab
z 12πH
B model or n= 2 for the cylindrical model. As discussed in
Notethattheaverageforceactsinthepositivezdirection,i.e., Section 5.1, observationsof hedgerow prominencesindicate
themagneticforcecounteractstheforceofgravity. Ineffect, ρ0 2 10- 13gcm- 3(e.g.,Engvold1976),andtoexplainthe
≈ ×
theplasmaisbeingsupportedbythemagneticpressureofthe observedheightsofsuchprominenceswithB0=10G,were-
tangledfield. The tangledfield actslike a hotgasthat hasa quireρavg/ρ0<0.05.Accordingtoequation(39),thisimplies
significantpressurebutnomass. Theaveragedensityofthe R/r0 20fortheslabmodel,orR/r0 4.5forthecylindri-
≥ ≥
plasmathatcanbesupportedbythetangledfieldisgivenby calmodel. Theobservedthreadshavewidthsdowntoabout
500km(Engvold1976),whichcorrespondstor 250km.
ρ (z)= <Fz′> = B20 (1+2ǫ2)e- 2z/HB, (38) Therefore, the magnetic coupling by the tangle0d≈field must
avg g 12πgH extendtoasurroundingdistanceofatleast5Mmfortheslab
B
model,or1.1Mmforthecylindricalmodel. Moregenerally,
wheregistheaccelerationofgravity.
equations(38)and(39)yieldthefollowingexpressionforthe
ThehorizontalcomponentsofLorentzforce,F′andF′,do
x y magneticscaleheightoftangledfield:
not vanish for any particular realization of the tangled field,
andcannotbewrittenasthegradientsofascalarpressure p. B2 R n
H 0 . (40)
Thereasonisthatexpression(31)isnotanexactsolutionof B≈ 12πgρ (cid:18)r (cid:19)
0 0
the magnetostatic equilibrium equation. However, equation
Therefore, the maximum height of the prominence depends
(36) shows that the horizontal forces vanish when averaged
stronglyonthemagneticfieldstrengthB .
overthefluctuationsoftheisotropictangledfield. Therefore, 0
Accordingto the presentmodel, magneticstress buildsup
expression (31) is thoughtto give a good approximationfor
inthetangledfieldasaresultofthedifferenceingravitational
theeffectsofgravityonthetangledfield.
forcesbetweenthethreadanditssurroundings. Canthefield
We now applythe abovemodelto the verticalthreadsob-
supportsuchshearstress? Toanswerthisquestionweexam-
served in hedgerow prominences. To explain the observed
ine the effect of vertical displacements on the tangled field.
heights of such prominences, we require that the magnetic
For simplicity we neglect the mean vertical force given by
scale height H is at least 100 Mm. The size ℓ of the mag-
B
equation (37), and we focus on relative displacements. Let
netic tangles is assumed to be in the range 0.1–1 Mm, so
r′ be the new position of a fluid parcel originally located at
ǫ=ℓ/H 1. For B =10 G and H = 100 Mm, we find
B 0 B
ρhyavdgr≈og1en0-≪d14engsictmy-n3,which5cor1re0s9pocmnd-s3.toTahnisaviseroangley(atobtoaul)t dpeofsoitrimonedr.fieInldthBe′laitmthiteonfeawppeorfseitcitolnyrc′oinsdguicvteinngbpylasma,the
0.05 times the densHit,ayvgρ≈0 ×2 10- 13 g cm- 3 or nH 1011 B′= Bj ∂x′i, (41)
cm- 3 typically observedin≈hed×gerowprominences(E≈ngvold i J ∂x
j
1976,1980;Hirayama1986).Thiscomparisonshowsthatthe
where B(r) is the originalfield, and J is the Jacobian of the
pressureofthetangledfieldinsideaprominencethreadisnot
transformation(e.g.,Priest1982). Weassume
sufficienttosupporttheweightoftheprominenceplasma.To
supportthe plasma with tangled fields, we need to take into x′=x, y′=y, z′=z+h(x,y), (42)
account the magnetic coupling between the vertical thread
whichyields
anditssurroudings. Suchcouplingisneglectedintheabove
plane-parallelmodel. B′ =B , B′ =B , B′ =B +B ∂h+B ∂h, (43)
x x y y z z x∂x y∂y
5.2. ShearStressEffect
where h(x,y) is the verticaldisplacement. The originalfield
We now assume that the tangled field pervades not only B(r) is assumed to be a realization of the isotropic tangled
theobservedverticalthreadsbutalsotheirlocalsurroundings. fieldgivenbyequation(15). Usingequation(22),weobtain
Thedensityinthesurroundingsislessthanthatinthethreads,
′
<B >=0, (44)
so the force of gravity is also much lower. This difference
in gravitational forces leads to vertical motions (downflows <B′B′>= 1B2 ˆxˆx+ˆyˆy+fˆzˆz+∂hˆxˆz+∂hˆyˆz , (45)
in the dense threads, upflows in the tenuous surroundings) 3 0(cid:20) ∂x ∂y (cid:21)
TANGLEDMAGNETICFIELDSINPROMINENCES 9
where
∂h 2 ∂h 2
f(x,y) 1+ + . (46)
≡ (cid:18)∂x(cid:19) (cid:18)∂y(cid:19)
This yields the following expressions for the off-diagonal
componentsofthestresstensor:
B2 ∂h B2 ∂h
<T′ >= 0 , <T′ >= 0 . (47)
xz 12π∂x yz 12π∂y
TheLorentzforceisgivenbyF′=∂T′/∂x ,andsince<T′ >
i ij j zz
isindependentofz,theaverageverticalforceisgivenby
B2 ∂2h ∂2h
<F′>= 0 + =g∆ρ(x,y), (48)
z 12π(cid:18)∂x2 ∂y2(cid:19)
where∆ρ(x,y)isthedensityperturbation(∆ρ ρ- ρ ).Us-
avg
≡
ingthisequation,wecandeterminetheverticaldisplacement
h(x,y) for a given density variation ∆ρ(x,y). In the follow- (a)
ingsubsectionswesolvetheaboveequationfortheslaband
cylindermodels.
5.2.1. SlabModel
Wefirstconsideraslabwithinfiniteextentinthe+y,- yand
+zdirections. Thecoordinatexperpendiculartotheslabisin
therange- R<x<R,whereRisthehalf-widthofthesheet
inwhichthetangledfieldisembedded(seeFigure2b). Then
thedensityperturbationisgivenby
+ρ [1- (r /R)] for x <r ,
∆ρ(x)=(cid:26)- ρ00r0/R 0 othe|rw| ise.0 (49)
Insertingthisexpressionintoequation(48)andsolvingforthe
verticaldisplacement,weobtain
C[1- (r /R)](x2- r2) for x <r ,
h(x)=(cid:26)C(r0/R)0[(R- r0)2- 0(R- x)2] othe|rw| ise,0 (50)
| |
where C 6πgρ B- 2. Here we applied no-stress boundary
condition≡s(dh/dx0=00)atx= R.Notethath(x)anditsderiva- (b)
±
tivearecontinuousattheedgesoftheprominence(x= r ).
± 0 FIG. 7.—Verticaldisplacementsinthetangledmagneticfieldsupporting
Therelativedisplacementacrossthetangledfieldisgivenby
ahedgerowprominence. (a)SlabmodelwithR=10Mm,r0=0.5Mm,and
6πgρ B0=10G.Theverticaldisplacementh(x)isplottedasfunctionofpositionx
∆h h(R)- h(0)= 0r (R- r ). (51) perpendiculartotheverticalslab.Thelocationoftheprominenceisindicated
≡ B2 0 0 bythedottedlines.(b)CylindricalmodelofaprominencethreadwithR=5
0
Mm,r0=1Mm,andB0=3G.Inbothcasestheprominencedensityρ0=
Figure 7a shows the function h(x) for r0 =0.5 Mm, R=10 2×10- 13gcm- 3.
Mm,B =10Gandρ =2 10- 13gcm- 3,sothat∆h=0.491
0 0
×
Mm. Note thata relativelysmalldeformationof the tangled
field (∆h R) is sufficient to redistribute the gravitational
forcesover≪thefullwidthofthetangledfield.However,∆his
largerthanthepressurescaleheightoftheprominenceplasma where B is the magnetic field through the mid-plane of the
x
(H 0.2 Mm). Therefore, the deformation of the tangled prominence (Kippenhahn&Schlüter 1957). Equation (53)
p
≈
fieldintheneighborhoodoftheprominencemayhaveasig- describes the angle of the field lines in the flux rope model
nificant effect on the distribution of the prominenceplasma. showninFigure2a. Therefore,theflux-ropeandtangledfield
ThisissuewillbefurtherdiscussedinSection7. modelsaresimilarintheirabilitytoexplainthemagneticsup-
Forcomparisonofthetangledfieldslabmodelwiththeflux portoftheprominenceplasma,providedthehalf-widthR of
ropemodel(Figure2), we define the averagesag angleθ of the tangled field region is similar to the radius R of the flux
theprominencerelativetoitssurroundings: rope.
∆h 6πgρ r
tanθ 0 0, (52) 5.2.2. CylindricalModel
≡ R ≈ B2
0
We now consider a cylindrical model for a prominence
where B is the r.m.s. field strength of the tangled field, and
0 threadwithrthedistancefromthe(vertical)threadaxis. The
weassumedr R. Thisexpressionissimilartothatderived
0≪ densityperturbationisgivenby
fortheKippenhahn-Schlütermodel:
tanθ= 4πgρ0r0, (53) ∆ρ(r)= +ρ0[1- (r0/R)2] forr<r0, (54)
B2x (cid:26)- ρ0(r0/R)2 forr0<r<R.
10 VANBALLEGOOIJEN&CRANMER
The vertical displacement is obtained by solving equation withvelocitiesoftheorderof10-30kms- 1(e.g.,Bergeretal.
(48),whichyields 2008;Chaeetal.2008). Ifthethreadsandtheirsurroundings
are indeed coupled via tangled fields, these relative motions
h(r)= h0[(r/r0)2- 1] forr<r0, implythatthefieldiscontinuallybeingstretchedintheverti-
(cid:26)h0[2R2ln(r/r0)- r2+r02]/(R2- r02) forr0<r<R, caldirection.Therefore,theshearstresscontinuallyincreases
(55) withtime,unlessthereisinternalreconnectionthatcausesthe
where shearstresstobereduced.
h 3πgρ r2 1- r02 , (56) Wespeculatethattangledfieldshaveatendencytorelaxto
0≡ B2 0 0(cid:18) R2(cid:19) theLFFFviainternalreconnection. Asimilarrelaxationpro-
0
cessesoccursinthereversedfieldpinchandotherlaboratory
andweappliedno-stressboundaryconditionsatr=R. Then
plasma physics devices (Taylor 1974). Therefore, the long-
thetotaldisplacementacrossthetangledfieldis
term evolution of prominence threads likely involves small-
6πgρ R scalereconnectionwithinthetangledfield. Thetangledfield
∆h h(R)- h(0)= 0r2ln . (57) maybehavemorelikea“plastic”mediumthatisirreversibly
≡ B2 0 (cid:18)r (cid:19)
0 0 deformed when subjected to shear stress. Such plasticity
Figure7bshowstheverticaldisplacementh(r)forR=5Mm, makes it possible to understand how the dense threads can
r =1Mm,B =3Gandρ =2 10- 13gcm- 3,whichyields movedownwardrelativetheirthesurroundingsatasmallbut
0 0 0
h =0.551Mmand∆h=1.847M×m.Inthiscase∆hissignif- constantspeed. Theseflowssignificantlydeformthetangled
0
icantlylargerthanthe pressurescale height(H 0.2Mm), field, butthe field is neverthelessable to supportthe plasma
p
mainly because of the lower field strength comp≈ared to the againstgravity.Adetailedanalysisofreconnectionintangled
caseshowninFigure7a. InSection7weconsidertheeffect fieldsand its effectonthe prominenceplasma is beyondthe
ofsuchdeformationonthefield-linedips, andonthespatial scopeofthepresentpaper.
distributionoftheprominenceplasma. Theobservedverticalstructureslikelyreflectthenon-linear
Theaboveanalysesonlyprovideanroughestimateforthe developmentoftheRT instabilityinhedgerowprominences.
density of prominence plasma that can be supported by the To establish a vertical column of mass resembling a promi-
tangledfield.Theactualdensitydistributionρ(r,t)islikelyto nence thread will likely require vertical motions over a sig-
bemuchmorecomplexforseveralreasons.First,plasmawill nificantheightrange(tensofMm). Startingfromahomoge-
tendtocollectatthedipsofthefieldlines,sothedensitywill neous density distribution, it may take several hours for the
varyonthespatialscaleℓofthetangledfieldandonthescale threadstoformbyRTinstability.
ofH ;thiseffectwillbeconsideredinmoredetailinSection
p 7. MODELFORAPROMINENCETHREAD
7. Second, the densitywillvarywith time becausethereare
flowsalongthefieldlines(seeSection3)andtheseflowsare Wenowconstructamodelforthedensitydistributionina
likely to be non-steady. Also, the magnetic structure is not fullyformed(vertical)prominencethreadsupportedbyatan-
fixedandwillcontinuallyevolveasdippedfieldlinesaredis- gledfield. ItisassumedthattheRT instabilityhasproduced
torted by the weight of the prominence plasma. To predict averticalthreadthatisclearlyseparatedfromtherestofthe
the actual density will require numerical simulations of the prominence plasma. Therefore, only a single thread and its
interaction of tangled fields with prominenceplasma, which localsurroundingsareconsidered,andthetangledfieldisas-
isbeyondthescopeofthepresentpaper. sumedtobecontainedinaverticalcylinderwithradiusR=5
Mm. AsdiscussedinSection3,therewillingeneralbemass
6. FORMATIONOFVERTICALTHREADSBYRAYLEIGH-TAYLOR flowsalongthetangledfieldlines, butforthepurposeofthe
INSTABILITY presentmodelweneglectsuchflowsandweassumethatthe
Accordingtothepresenttheory,hedgerowprominencesare plasmaisinhydrostaticequilibriumalongthefieldlines.
supportedbythepressureofa tangledmagneticfield, which To constructthe density model, we first computea partic-
acts like a tenuous gas and is naturally buoyant. It is well ular realization of the LFFF with αR=9 (see Appendix for
known that a tenuous medium supporting a dense medium details).Toaccountfortheweightoftheprominenceplasma,
issubjecttoRayleigh-Taylor(RT)instability(Chandrasekhar this field is further deformed as described by equation (55).
1961). Therefore, we suggest that the observed vertical The deformation parameters are r =1.25 Mm and h =1.5
0 0
threadsmaybeaconsequenceofRTinstabilityactingonthe Mm,whichyields∆h=4.44Mm, somewhatlargerthanthe
tangled field and the plasma contained within it. As cool valuesusedinFigure7b. AsshowninSection5.1,theweight
plasma collects in certain regions of the tangled field, the of the prominenceplasma causes the strength of the tangled
weight of the plasma deforms the surrounding field, which fieldtodecreasewithheight[seeequation(31)],butforsim-
causesevenmoreplasmatoflowintotheseregions. plicity this gradient is neglected here. The pressure scale
Adetailedanalysisoftheformationofprominencethreads height is assumed to be constant, H =0.2 Mm, which cor-
p
byRTinstabilityiscomplicatedbythefactthatwepresently responds to a temperature of about 8000 K, typical for Hα
donotunderstandhowatangledfieldrespondstosheardefor- emittingplasmainprominences.
mation. InSection5.2weestimatedtherelativeverticaldis- We introduce cartesian coordinates (x,y,z) with the z axis
placement∆hoftheprominenceplasmaassumingnorecon- along the cylinder axis; the x and y coordinates are in the
nection occurs during the deformation of the magnetic field range[- R,+R],andtheheightzisintherange[0,10R]. The
by gravity forces [see equations(51) and (57)]. In this case density ρ(x,y,z) in this volume is computed on a grid with
the tangled field behaves as an “elastic” medium with mag- 200 200 1000gridpoints,usingthefollowingmethod.We
× ×
netic forces proportional to the displacement. However, it randomlyselectalargenumberofpointswithinthecylinder
is notclear thatthis approximationis valid. High-resolution andtraceoutthefieldlinesthatpassthroughthesepoints.For
observations of prominences indicate that the dense threads eachfieldline weplottheheightz(s)asfunctionofposition
movedownwardrelative to their moretenuoussurroundings s along the field line, and we find the peaks and dips in the