Simulation of dark lanes in post--flare supra--arcades

0.49 MB·English

by  L. S. Maglione

Astronomy&Astrophysicsmanuscriptno.Maglione (cid:13)c ESO2011
June20,2011
LettertotheEditor
Simulation of dark lanes in post–flare supra–arcades
III. A 2D simulation
L.S.Maglione1,E.M.Schneiter2,A.Costa2,3,4,(cid:63),andS.Elaskar3,4
1 FacultaddeIngenier´ıa,UniversidadNacionaldeR´ıoCuarto,Argentina
2 InstitutodeAstronom´ıaTeo´ricayExperimental,Co´rdoba,Argentina
3 FacultaddeCienciasExactas,F´ısicasyNaturales,UniversidadNacionaldeCo´rdoba,Argentina
4 ConsejoNacionaldeInvestigacionesCient´ıficasyTe´cnicas
Received;accepted
1
1 ABSTRACT
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2 Context.Usingtwosimulationsof1.5D,forthefirsttime,inCostaetal.(2009)andShulzetal.(2010)wenumericallyreproducethe
observationaldarkinflowsdescribedinVerwichteetal.(2005).Weshowthatthedarktrackscanbeexplainedashotplasmavacuums
n
generatedupstreamofaslowmagnetoacousticshockwaveproducedbyalocalizeddepositionofenergy.
a
Aims.Toconfirmthelater‘darklane’interpretationandtoidentifyspecific2Dcontributionstothedescriptionofthephenomenon.
J
Methods.Tosolvetheidealandnon–stationaryMHDequationsweuseda2DRiemannsolverEuleriancodespeciallydesignedto
8 capturesupersonicflowdiscontinuities.
1 Results. Thenumerical2Dresultsareinagreementwiththeobservationalbehaviourhowevertheyshowaslightshiftinthecharac-
teristicparameterwithrespecttothosefoundpreviously.
] Conclusions.Weconfirmqualitativelythebehaviourfoundinthepreviouspapers.Foragivennumericaldomaintheperiodofthe
R
kink–likestructureisafunctionofthemagneticfieldintensity:largerperiodsareassociatedwithlowermagneticfieldintensities.
S Contrarytothe1Dresult-wherethesunwarddynamicisindependentofthemagneticfieldintensityduetoitsexclusivelywaveguide
. role-inthe2Dsimulationthesunwardspeedislargerforlargervaluesofthemagneticfield.Thiscanbeinterpretedasthecapability
h
of the low coronal plasma to collimate the deposition of energy into the magnetic field direction. The moving features consistent
p
oflow–densityandhigh–temperatureplasmacavitieshavelargerinsidevaluesofthestructuringparameterβthantheneighboring
-
o media.Thus,thevoidsseemtobetheemergencestructuresofawholenonlinearinteractingplasmacontextofshocksandwaves
r morethanvoidedplasmaloopsmagneticallystructured.
t
s
Keywords.Corona,magneticfields,shockwaves,instabilities
a
[

1
1. Introduction In Costa et al. (2009) and in Schulz et al. (2010) (hereafter
v
Paper 1 and 2, respectively), by the integration of two 1.5D
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3 Dark sunward sinuous lanes moving along a fan of rays above MHD ideal equations, we presented a new scenario that gives
5 post–flare loops towards a supra–arcade have been extensively account of the observational dark voids described in VNC. We
3 studied(Innesetal.2003a;2003b;McKenzieandSavage2009). simulatedtheeffectsofaninitialimpulsiveandlocalizeddepo-
. The down moving structures observed at [40−60]Mm heights sitionofenergy-supposedtobeassociatedwithabovereconnec-
1
above the top of arcades, with a decelerating speed in the tionprocesses-inaplasmastructuredbysunwardmagneticfield
0 range ∼ [50 − 500]kms−1 were interpreted as sunward voided lines. The impulsive phase was modeled by a pressure pertur-
1
1 flows generated by reconnection processes developed by a cur- bationthatinitiatestwomainprocesses,afundamentallyhydro-
: rentsheetabovetheflarearcade.Anotherconfigurationconsis- dynamic shock pattern directed sunwards and a perpendicular
v tent with the observations is a magnetic flux tube, filled with magnetic shock one, i.e., transverse to the magnetic field. The
Xi flux and very little plasma, shrinking into the post-eruption ar- two patterns were supposed to be independent processes, how-
cade (McKenzie and Hudson 1999; McKenzie 2000). Recently ever linked by their common origin and background magnetic
r
a Lintonetal.(2009),proposedanotherdescriptionwherethedy- and density conditions. The independence of the two dynam-
namic is triggered by a localized reconnection event that pro- icswasjustifiedduetothefarmoreeffectiveconductiveenergy
ducesupanddownflowingreconnectedfluxtubeswhichdecel- transportalongfieldlinesthanacrossthem.
erate due to a underlying magnetic arcade loops. Verwichte et
al. (2005, hereafter VNC) analyzed transverse to the magnetic
We showed that the dark tracks are consistent with plasma
field oscillations associated with sunward dark lanes in a post–
voids generated by the bouncing and interfering of shocks and
flaresupra–arcade.Theyfoundthattheinitialspeedsandthedis-
expansionwavesupstreamtheinitiallocalizeddepositionofen-
placementamplitudes,ofobservationaldarklanesofakink–like
ergy, which is responsible of the two dynamics. The composi-
type,decreaseastheypropagatedownwardswhiletheperiodre-
tion of both, a resulting sunward directed hydrodynamic shock
mainsconstantwithheight.increase.
pattern and a perpendicular magnetic shock one, produces an
overallβ > 1transverselyshakingvoidmovingtowardsthesun
(cid:63) CA:AndreaCosta,[email protected] surface,thatresemblesthekink–likemodedescribedinVNC.
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L.S.Maglioneetal.:Simulationofdarklanesinpost–flaresupra–arcades
Fromthetransversesimulationwefoundthat,inaccordance material locally erasing the vacuum trace. These features -that
withtheshockwavetheoryofuniformmedia(Kirketal.1994), remindobservationalconstrictions,seee.g.,Innesetal.(2003)a;
there is a critical value of the magnetic field beyond which the (2003)b- were found in several cases at different positions and
behaviourofthemagneticshockpatternchanges,themagnetic times, they are the result of nonlinear interactions difficult to
compressionislimitedandthephenomenonisprogressivelysat- predict.Forfurthercomparisonwevariedthebackgroundmag-
urated.Wealsofoundthattheperiodofthekink–likestructure neticfield.Table1showsthenumericalperiods(τ),amplitudes
isafunctionofthemagneticfieldintensitywhiletheamplitude (A), the distances evacuated by the moving perturbations (L ),
v
isafunctionofthetriggeringpressurepulse.Thus,theperiod’s andtheinitialsunwardspeeds(V )obtainedforthe2G,4G,8G
0
constancywithheightfoundinVNCcanbeassociatedwiththe and 20G cases and for P /P = 100. These results are consis-
2 1
almost constancy with height of the background magnetic field tent with characteristic observational values given in literature,
in the region. In accordance with our interpretation that in the e.g.,VNC,McKenzieandHudson(1999).However,inorderto
sunward direction the magnetic field plays the role of being a matchperiodsofdecadesofsecondsandamplitudesofhundreds
wave–guide,wefoundthatthepressurepulsedeterminesanhy- ofkilometerswerequiredpressurepulsesofalmostanorderof
drodynamic shock that moves towards the sun surface at slow magnitudelargerthaninthe1Dcase.Asbefore,wefoundthat
acoustic speed values and the dynamic is independent of the lowervaluesofthemagneticfieldareassociatedwithlargerpe-
magneticfieldintensity. riods (see second column of Table 1). Also, the phenomenon
In this paper we investigate the goodness of the 1D model is progressively saturated resembling the compressional limits
andwediscussthelimitationsandnewcharacteristicsofthephe- (in density and magnetic field intensity) of the HD and trans-
nomenonassociatedtothe2Dstructure. verse MHD shock wave theory in uniform medium (Kirk et al.
1994), i.e., the augmentation of the magnetic field from 4G to
6Gimpliesadecreaseof55secintheperiodwhilethechangeof
2. NumericalCodeandInitialConditions magneticfieldfrom8Gto20Gimpliesadecreaseofonly10sec
intheperiod.Thereisnovisiblerelationbetweenthemagnetic
WeintegratetheMHDequationsusingthetwo-dimensionalver-
fieldandtheamplitude.
sion of the ‘Mezcal’ code, an Eulerian Godunov MHD code
Figure1crepresentsthevoiddensityforthelongitudinaldy-
(De Colle and Raga 2005, 2005, De Colle and al. 2008). The
namic of the 20G magnetic field. Contrary to the sunward be-
MHD Riemann solver uses a standard second-order Runge-
haviour found in the 1D simulations, when comparing Fig. 1c
Kutta method for the time integration and a spatially second-
withthe4GcaseofFig.1b,thevoidassociatedwithlargermag-
order reconstruction of the primitive variables at the interfaces
neticfieldintensitiestravelsalargerdistanceinthesunwarddi-
(exceptinshocks).Theconstrainedtransportmethod(e.g.,To´th
rection(seeTable1,fourthcolumn).InPaper2,thehypothesis
2000) is used to conserve ∇· B = 0 to machine accuracy. The
of independence between the two 1D simulations was justified
codehasbeenextensivelytestedagainststandardproblems(De
by the anisotropy imposed by the magnetic field, thus its role
Colle2005).
was solely to wave guide the longitudinal dynamic. In the 2D
All calculations were performed using a numerical grid of
simulationthedepositionofenergyisdistributedradially,from
(x,y)=(1600,620)grid–pointscorrespondingtoaphysicalsize
the location of the initial perturbation, to all directions of the
of (40,9)Mm, chosen in accordance with VNC observations.
(x,y)plane.However,thefreezinginoftheplasmatothemag-
Thecoordinate xrepresentsthesunwarddirectionandtheyco-
neticfieldinducesthecollimationofpartoftheenergytowards
ordinatethetransversetothemagneticfieldone,asseenbythe
thelongitudinaldirection.Themoreintensemagneticfieldsare
lineofsight.Weassumedaconstantradialinitialmagneticfield
themoreefficientinline–tyingtheenergy,thus,givingimpulse
structure,atypicalbackgroundtemperatureofT = 3.0×106K
tothefinalresultingsunwardmotion.Hence,thelargerthemag-
andanumericaldensityvalueofρ=0.46×109 cm−3.Aspher-
neticfieldthemoreaccuratethe1Ddescriptionwillbe.
ical(circularin2D)relativetriggeringpressurepulseP /P (P
2 1 2 Figure2a-dshows,respectivelythe‘darklane’densitystruc-
is the triggering pressure pulse and P is the background gas
1 ture,thetemperature,theparameterβandthetotalpressuredis-
pressure of the corona) of radius 0.6Mm was localized in the
playedinthewholenumericaldomainfor4GandP /P = 100
position(1520,434)equivalentto(38,6.3)Mm. 2 1
att = 200sec.AsinPapers2,inaccordancewithobservational
Severalsimulationwerecarriedout,varyingtheradial(nor-
data, the vacuum density is almost an order of magnitude less
maltothesun’ssurface)magneticfield,B,andthelocalizedde-
thantheexternalmedium(seeFig.3a),whilethevacuumtem-
position of energy, modelled as the relative triggering pressure
peratureisalmostanorderofmagnitudehigherthanthetemper-
pulse.AsinPaper1and2,theboundaryconditionswerefixed,
ature in the surroundings. When plotting β we see an extended
allowingreboundsinthelateraltransversedirectionsandalsoin
channel (not confined to the vacuum zone) of inner β values
theupperradialdirection,resemblingtheactionofthereconnec-
lowerthantheoutsideones.Thisbehaviouroccursfortheother
tionsite.Weassumedthattheperturbationsareabsorbedinthe
cases,withdifferentmagneticfieldintensities.Figure3ashows
sunwarddirection.Thecharacteristicparameterswerechosenin
theratioofanaverageinnertoanaverageouterβvalueforthe
accordancewithtypicalobserveddarklanestructures,e.g.,VNC
differentcasesstudiedatafixheightof35Mm.Thisindicates,as
dataandMcKenzieandHudson(1999).
inPaper2,thattheinnermagneticpressureisnotresponsibleof
preventingthecollapseofthevacuumzone.Furthermore,from
Fig. 2d, it is possible to see that the features that characterize
3. ResultsandDiscussion
thevacuumzonehavedisappearedmeaningthatthecontourof
Figures1a,bshowthedensitypatterninthecentraltransversedi- thecavityis,onaverage,intotalpressureequilibrium.Thiscon-
rectionandrespectively,inthecentrallongitudinaldirection,as firmsthatthevoiddynamicsmustbesustainedbytheinteraction
afunctionoftimeforP /P =100andB=4G.Thefiguresare ofnonlinearwavesandshocksactingintimescomparabletothe
2 1
qualitatively similar to the 1D simulations (see Paper 1 and 2). observations.ForamoredetaileddescriptionFig.3bshowsre-
Thetransversekink–likestructureisdiscontinuedatatimestep spectively, the gas, the magnetic and the total pressure at each
∼ 80sec apparently due to a strong shock front that has swept point corresponding to the line traced in Fig. 2a. Note that the
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L.S.Maglioneetal.:Simulationofdarklanesinpost–flaresupra–arcades
B[G] τ[sec] A[km] L [Mm] V [km/sec]
v o
4 80 725 22 200
6 25 825 24 250
8 20 825 26 259
20 10 750 40 280
Table1. Numerical2Dparameters. Bthemagneticfieldinthe
radial direction, τ the period, A the amplitude, L the distance
v
evacuatedbytheperturbationsandV theinitialsunwardspeed.
o
P /P =100G.
2 1
P /P τ[sec] A[km] L [Mm] V [km/sec]
2 1 v o
50 20 375 12 140
100 25 1000 24 259
150 25 750 23 280
200 −− 1875 32 318
Table2. Numerical2Dparameters. P /P thetriggeringpres-
2 1
surepulse,τtheperiod,Atheamplitude,V theinitialsunward
o
speedandL thedistanceevacuatedbytheperturbations.Forall
v
casesB=8G.
almostconstancyofthetotalpressureisduetothegaspressure
increase (which can only be accomplished through large tem-
perature values) accompanied of a magnetic pressure decrease
throughthevacuumregion.
InPaper2wefoundthatthecasesstudiedhadβ>1,whereas
inthe2Dsimulationsonlythecaseswithmagneticfieldslower
than3Ghaveβ > 1.However,asinthe1Dcase,thebehaviour
is explained by the fact that β is larger inside than outside the
voided cavity, i.e., the cavity is a result of dynamic nonlinear
interactingwavesandshockssustainingthewholepattern.Note
thedifferentiatedshapeoftheFig.3acurve.Thealmostconstant
valueofthecurveforlargevaluesofthemagneticfieldintensity
indicateauniformbehaviourwithinthisrange.Thesteepnega-
tiveslopeoflargeinnervaluesofβandsmallvaluesofthemag-
neticfield,uptoB∼3G,exhibitadynamicwheretheanisotropy
imposed by the magnetic field is softened and 2D features be-
comemoresignificant.However,magneticfieldintensitieslower
than B ∼ 2.5Ghavenon–perturbedβ > 1valueswhicharenot
realisticparametersforthelowcoronadescription.Weconclude
thatthetwo1Dsimulationsgiveagoodapproximatedescription Fig.1.Densitypatterns(gcm−3)a)transversekink–likeoscilla-
ofthephenomenon. tionofthedarklaneasafunctionoftimefor4G.b)andc)sun-
Fourruns,withdifferentP /P ,werecarriedouttoanalyze wardmotionasafunctionoftimefor4Gand20G,respectively.
2 1
thebehaviourofthevoids.Table2showstheperiodsτ,theam- P /P =100G.
2 1
plitudes A,thedistancesevacuatedbytheperturbations L ,and
v
the initial speed for a magnetic field intensity of 8G as a func-
tion of P /P . The obtained parameters are in correspondence
2 1 verseandlongitudinalpatternsobtainedfromthe2Dsimulation
with typical observed parameter ranges (see Table 1 in VNC).
areingoodagreementwiththeobservations,confirmingthatthe
NotethatfromTable2weseethat,asinPaper2,theincreaseof
two1Ddescriptionissuitableforthisphenomenon.
P /P ,impliestheincreaseoftheinitialspeedandthedistance
2 1 The features, travelling decades of Mm, are voided cavities
evacuated by the perturbation, whereas the periods and ampli-
thatelongatetowardsthesundeceleratingatsoundsspeeds,i.e.,
tudesshownoregularvariation.
at hundreds of kms−1. The 2D results shown in Table 1 repro-
ducetypicalperiods,amplitudes,distancestravelledbytheper-
turbations and their initial speeds. However, the pressure pulse
4. Conclusions
requiredisalmostanorderofmagnitudehigherthaninthe1D
Weperformeda2Dnumericalsimulationtoreproduceobserved simulations.Thisseemsreasonableduetothefactthatthepulse
darksinuouslanesmovingsunwardlytowardssupra–arcades.In musttriggerthedynamicinalltheradialdirectionsandnotonly
previouspaperswereproduceobservationalfeaturesofthephe- intwomainones.AsshowninTable2,theincreaseofthepres-
nomenonusingtwo1Dsimulations-supposedtobeindependent sure pulse that triggers the phenomenon augments the initial
duetothefarmoreeffectivetransportalongthefieldlinesthan speedandthedistancetravelledbythefrontofthevoidedcavity.
acrossthem-thatweretriggeredbyapressurepulseactingonthe Linton et al. (2009) also reproduce the sunward motion though
transverseandlongitudinaldirections.Thecorrespondenttrans- theyassumeopenlateralboundariesandabottomreconnection
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L.S.Maglioneetal.:Simulationofdarklanesinpost–flaresupra–arcades
Fig.2.a)density,b)temperature,c)βparameterandd)totalpressureofthedarklanepattern.Forallthecasesweusedthe(x,y)
griddefinedasinVNC, x theradialdirectionandythetransversedirection,themagneticfieldis4G,thetimeis200sec,andthe
triggeringpulseisP /P =100G.
2 1
site. A main difference with our work is that the reconnection theinteractionofnonlinearwavesandshocksthatreboundinthe
process is intrinsic to their model as the voids are formed and lateraldensermedium,and2)theinteractionofnonlinearwaves
confined by a current sheet. From their work is not clear if an andshocksthatupwardlyreboundandareabsorbedsunwardly.
oscillatingbehaviour,asdescribedinVNC,isobtained.
Asinearlierworks,wefoundthatlowervaluesofthemag-
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neticfieldvalue(between4Gand6G),thephenomenonispro- DeColle,F.,Raga,A.C.,MNRAS359,164,2005
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Duetothefactthatβislargerinsidethanoutsidethevoided
cavity, and that its contour is in total pressure equilibrium we
concludethattheinternalmagneticpressurecannotberesponsi-
bleofpreventingthecollapseofthevacuumzone.
The2Dsimulationshows,asinrecentpapers,thatthesun-
wardlymovingshakingvoidsareproducedandsustainedbytwo
main processes that can be viewed as almost independent phe-
nomenon if the magnetic field intensity is sufficiently high: 1)
4
L.S.Maglioneetal.:Simulationofdarklanesinpost–flaresupra–arcades
Fig.3.a)βaverageinnertoouterratiovaluesfordifferentmag-
netic field intensity cases. b) gas, magnetic and total pressure
values for the grid positions of the line marked in Fig. 2a and
B= 4G.ForallcasesthetriggeringpulseisP /P = 100Gand
2 1
τ=200.
5