Table Of Content

Astronomy&Astrophysicsmanuscriptno.arxiv (cid:13)c ESO2013 January22,2013 Nanoflare statistics in an active region 3D MHD coronal model S.BingertandH.Peter MaxPlanckInstituteforSolarSystemResearch(MPS),37191Katlenburg-Lindau,Germany,[bingert,peter]@mps.mpg.de Received.../Accepted... ABSTRACT Aims.Weinvestigatethestatisticsforthespatialandtemporaldistributionoftheenergyinputintothecoronainathree-dimensional 3 magneto-hydrodynamical (3D MHD) model. The model describes the temporal evolution of the corona above an observed active 1 region. The model is driven by photospheric granular motions that braid the magnetic field lines. This induces currents that are 0 dissipated,therebyleadingtotransientheatingofthecoronalplasma.Weevaluatethetransientheatingassubsequentheatingevents 2 andanalyzetheirstatistics.Theresultsaretheninterpretedinthecontextofobservedflarestatisticsandcoronalheatingmechanisms. Observed solar flaresand other smallertransientscover a widerange of energies. Thefrequency distribution of energies follow a n powerlaw,thelowerendofthedistributiongivenbythedetectionlimitofcurrentinstrumentation.Oneparticularheatingmechanism a isbasedontheoccurrenceofso-callednanoflares,i.e.verylow-energydepositionevents. J Methods.Toconductthenumericalexperimentweuseahigh-orderfinite-differencecodethatsolvesthepartialdifferentialequations 1 fortheconservationofmass,themomentumandenergybalance,andtheinductionequation. TheenergybalanceincludesSpitzer 2 heatconductionandopticallythinradiativelossesinthecorona. Results.ThetemporalandspatialdistributionoftheOhmicheatinginthe3DMHDmodelfollowsapowerlawandcanthereforebe ] understoodasasysteminaself-organizedcriticalstate.Theslopesofthepowerlawaresimilartotheresultsbasedonobservationsof R flaresandsmallertransients.Wefindthatthecoronalheatingisdominatedbyeventssimilartotheso-callednanoflareswithenergies S ontheorderof1017Jor1024erg. . h Keywords.Sun:corona—Stars:coronae—Magnetohydrodynamics(MHD)—Methods:numerical p - o r 1. Introduction parameters,such asthe flare peakflux orflare duration,follow t s a powerlaw.A reviewof the statistics of flaresand theirsmall a IntheupperatmosphereoftheSunandstars,energyisreleased counterparts,themicroflares,isgivenbyHannahetal.(2011). [ inatransientfashion.Strongflaresarejustthetipoftheiceberg, wheretheplasmacanbeheatedupto10MK ormoreandpar- Tounderstandthecontributiontotheheatingofthehotcoro- 2 ticles are accelerated to relativistic speeds. More numerousare nal plasma, interpreting the power law is crucial. Following v 7 smallerenergyreleasesthatcausesmallerbrighteningsinX-rays the observation-basedresults, the powerlaw hasa negative ex- 1 andextremeUV,downtothecurrentdetectionlimit.Thereis a ponent: small flares are more numerous than large flares. If 4 continuousconnectionfromthesesmallestobservablebrighten- the frequency of small flares is high enough (the power law 6 ings,oftencalledmicroflares,tothelargestofflaresinthesense slope being steeper than −2), the heating would be dominated . that the distribution in energy of all these events roughly fol- by small events, the nanoflares. These have been proposed by 1 1 lows a power law. Theoretical arguments have been proposed Parker (1988) based on theoretical arguments and are not di- 2 that evensmaller transientreleases of energyhave to exist, the rectlyobservablewithcurrentinstrumentation.Parker(1988)es- 1 nanoflares(Parker1988), which could play a majorrole in the timatedanenergyofroughly1017J(1024erg)andnamedthose : heating of the corona. In this study we investigate the energy nanoflaresbecausetheyaresmallerthantheconventionofami- v distributionoftheselowenergyreleasesastheyarefoundin3D croflare.Thelatterhaveenergiesontheorderof1020Jandthus Xi magnetohydrodynamics(MHD)models. about10−6 timestheenergyofalargeflare.Tounderstandhow muchnanoflarescontributeto the heating orif largerflares are r Rosner&Vaiana (1978) have alreadyinvestigatedthe tran- a sient events of the Sun, flaring stars, and other X-ray sources. composedofsmallerflares,itisimportanttoknowthedistribu- tionoftheenergiesofthesetransienteventsdowntothelowest They found that the energy distribution of these events energylevels. roughly follows a power law. To explain the flare statistics Rosner&Vaiana(1978)providedatheoreticalmodelthatledto The observational approach is hereby limited by a lower apower-lawdistributionoftheflareparameters.Thismodelwas thresholdgivenbythedetectionlimitoftheobservation.Inaddi- basedonthe assumptionsthatflaringisa stochastic relaxation, tion,theflareenergyisonlyaderivedquantityinferredfromthe thewaitingtimebetweentwoeventsisanindependentparame- fluxofextemeUVand/orX-rayphotons,andisthereforeonlyan ter,andtheenergyreleaseduringaflareislargeincomparison indicatorfortheactualenergyinput.Energyisalsolostthrough totheenergycontentaftertheflare.Thepower-lawnatureisnot other processes, such as heat conduction or particle accelera- only observable for X-ray sources, but also solar radio bursts tion,andtherelativemeritofthisenergylossmechanismmight showsuchadistributionasalreadystudiedbyAkabane(1956). varywiththetotalenergyoftherespectiveevent.Thereforeeven These results were followed by numerous observations of thepowerlawindexforthedistributionoftheactualenergyin- solarX-rayflares(Crosbyetal.1993,1998;Christeetal.2008). putandfortheobservedenergylossmightdiffer(foronegiven Theyhavefoundthatnotonlytheflareenergybutalsootherflare channel,e.g.,theX-rayflux). 1 S.BingertandH.Peter:Nanoflarestatisticsinanactiveregion3DMHDcoronalmodel An alternativeapproachis the numericalmodelingof these 50 100 flare statistics. They are based on the theoretical assumptions leading to a power law distribution. A famous model uses the assumption that the system, e.g., the solar corona, is in 40 a self-organized critical state (Lu&Hamilton 1991). The con- 50 G] ceptoftheself-organizedcriticalitywasintroducedbyBaketal. h [ m] gt (tBh1ua9tt8tc7ha)e.nSqbmueeascltloiocmhnpawanrgheeedsthtteoortahtvheaeslacynosctreohmneaswl(mCillhatgarringbegotiencrnfifeuaerultdheeitsrarilen.a2scu0tci0oh1n)as. ntal y [M 30 (a)(d) 0 eld stren stateArneomthaeinrsnoupmeenri(cLaula1p9p9r5o)a.chhasbeenemployedrecentlyby horizo 20 (b) netic fi g Dimitropoulouetal. (2011). They used the cellular automaton −50 ma model to produce an average power law index of -1.80 for the 10 (c) peak energies. For that study they used several different ob- (A)+(B) servedactiveregions.However,intheirmodelthemagneticfield 0 −100 boundary remains constant over the simulation time, and the 0 10 20 30 40 50 drivingisdoneartificially,butbotharenotrealistic. horizontal x [Mm] In this study we present for the first time the frequency distribution of transient heating events in a forward model ap- Fig.1. Map of the vertical magnetic field component in the proach that is based on a 3D MHD model of a solar ac- model photosphere at the lower boundary. Marked are the po- tive region. The hot corona in this model is sustained by sitionsoftheeventsshowninFigs.2and4. Ohmic dissipation of currents, which result from the braiding of magnetic field lines by convectivemotionsprescribedat the lower boundary. Since the first of this type of forward model createdbythebraidingofmagneticfieldlines.Thismechanism (Gudiksen&Nordlund 2002, 2005a,b), it has been shown that of (DC) heating has already been proposed by Parker (1983), these models not only produce a hot loop-dominated corona, where the braiding of field lines is a simplified picture for the but that they also match the (averaged) properties of spectral energy transport due to the Poynting flux and the subsequent EUV observations (Peteretal. 2004, 2006). Furthermore, the dissipation process. The magnetic fields anchored in the pho- synthetic observations show a comparable temporal variability tosphereareshuffledaroundbyphotospheric(granular)motions (Peter 2007), and new suggestions have been made on the na- leading to a build-up of currentsin the upperatmosphere. The ture of the observed net Doppler shifts (?) in the transition re- plasmaisthenheatedbydissipationofthesecurrents.Basicpa- gion and corona (Hansteenetal. 2010; Zachariasetal. 2011). rametersderivedfromthenumericalmodel,suchastheenergy These 3D MHD coronal models show that the heating is con- flux of roughly100 Wm−2 into the corona,match observation- centratedinfield-alignedcurrentstructuresandthattheyfeature allyderivedvalues. astrongtemporalvariability(Bingert&Peter2011).Thespatio- The model describes the solar atmosphere above an ob- temporaldistributionoftheheatinputalsogivesrisetoindivid- served active region depicted in Fig.1. The domain spans over ualcoronalloopsthatappearin(synthesized)extremeUVemis- 50x50Mm2 in the horizontal direction and 30Mm in the ver- sion as havinga constantcrosssection (Peter&Bingert2012), tical direction. The initial condition for the plasma parameters justasinobservations. are given by a hydrostatic plane-parallelatmosphere similar to The success of previous models providesevidence that the solaraverages.Themagneticfieldisinitiallycomputedusinga distributionoftheheatingrateinspaceandtime(ontheresolved potentialfieldextrapolation. scales)isclosetowhatwefindontheSun.Thereforeitistimely The system is then driven by granular motions having the toinvestigatedetailsofthestatisticsoftheenergydepositionin same statistical properties as the photospheric velocities ob- the3DMHDmodels,inparticulartheenergydistributionofthe served on the sun. The evolution of the modelatmosphere fol- heatingevents,whichisthegoalofthisstudy. lows the MHD equations. The lower boundary is prescribed In Sect. 2 we briefly describe the 3D MHD coronal model by the granular motions and a fixed plasma pressure. The top and show in Sect. 3 the transient nature of the self-consistent boundaryishydrostaticwithnoheatflux.Themagneticfieldis coronalheatinput.InSect.4wederivetheenergydistributionof extrapolatedbymatchingapotentialfieldatthetop.Inthehori- theheatingeventsanddiscussinSect.5someobservationalcon- zontaldirectionsthedomainisfullyperiodic. sequences,beforeweinvestigatesomeaspectsofself-organized We ranthemodelforonesolarhourbeforetakinganydata criticalityinoursysteminSect6. tomaketheresultindependentoftheinitialcondition.Afterthat we tooksnapshotswitha cadenceof30secondsforabitmore thanahour. 2. Coronalmodel For the numerical model we employ the Pencil code To understand the heating of the corona, it is crucial to study (Brandenburg&Dobler2002).ItsolvestheMHDequationsfor itsspatialandtemporalevolution.Observationsonlyprovidein- a compressible,magnetized,viscous fluid. These equations are formationaboutconsequencesbutnotdirectlyabouttheheating the conservationof mass, the momentumbalanceequation,the mechanism and its energy deposition. Therefore we employ a energyequation,andtheinductionequation.TheOhmicheating 3DMHDmodelofthesolaratmosphereinordertoanalyzethe termisgivenasηµ j2 with j = ∇× B/µ .Thetime-dependent 0 0 spatialandtemporaldistributionoftheheatinput.Thedataused energy balance in the model includes anisotropic heat conduc- forthepresentpaperisbasedonacoronalmodelintroducedin tion (Spitzer 1962), optically thin radiative losses (Cooketal. Bingert&Peter(2011).Themodelproducesahotcoronaabove 1989), together with the viscous and Ohmic heating. No addi- asmallactiveregionwithself-consistentheating.Theheatingin tionalarbitraryheatingisappliedinthemodelcorona,whichis the modelis due to Ohmic dissipation of free magnetic energy sustainedalmostexclusivelythroughtheOhmicdissipation. 2 S.BingertandH.Peter:Nanoflarestatisticsinanactiveregion3DMHDcoronalmodel 6.E−08 3.E−04 3] 3] −m (a) z=28 Mm (b) z=10 Mm −m W W 4.E−08 2.E−04 e [ e [ at at g r 2.E−08 1.E−04g r Fig.2.Temporal evolutionoftheOhmicheat- eatin eatin ginleggraritde-(p∼oinj2ts))aitnfothuermdioffdeerlenctorloocnaat:io(na)sg(sriidn-- h 0.E+00 0.E+00h point with smallest time-averaged heat input, 3] 1.E−06 6.E−063] and(b)withstrongesttime-averagedheatinput −m (c) z=22 Mm (d) z=17 Mm −m inthecoronalvolumeatT>105K;(c)location W W e [ 8.E−07 4.E−06e [ asitztehdectoorponoaflaembriisgshiotnlo,oapndse(edn) iant tthheecsyonrothnea-l at at ng r 4.E−07 2.E−06ng r bcraosseseast mthaerkfotohtepotiinmteosfotfhtehebrsignhaptslhoootps.. TThhee ati ati geometricalheightsofthesefourlocationsare he 0.E+00 0.E+00he indicatedinFig.3.Thelocationsarealsoindi- 0 10 20 30 40 50 60 0 10 20 30 40 50 60 catedinFig.1relativetothephotosphericmag- time [min] time [min] neticfield. 1 3. Sampleevents:nanoflaresandnanoflarestorms (b) (d) (c) (a) The ohmic heating in the modelis transientin time and space. Toillustratethis,weshowinFig.2thetemporalvariationinthe 3] 102 − volumetricheatingrateatfourgridpointsofthenumericalsim- m W ulation. These have been selected using the following criteria: (a)thelowestheatingrateinthebox,(b)thehighestheatingrate e [ 100 at inthecoronalvolume,i.e.,intheregionwherethetemperature g r ethxecseiezdesd)10co6rKon,a(lc)ematisthsieonap,eaxndof(ca)loatopthcelecaorrloynvailsifboloetpinoi(nstyonf- atin 10−2 e thatloop.Obviouslytheselocationsspanawiderangeof(peak) e h heatingrates,coveringfourordersofmagnitude. g 10−4 a In part, this large difference in peak heating is due to the er v a strong drop in the volumetric heating rate with height. In the 10−6 coronal part, the horizontally averaged volumetric heating rate dropsroughlyexponentiallywithascaleheightofabout4Mm, 0 5 10 15 20 25 ascanbeseenfromFig.3(theheatingperparticleshowsapeak height z [Mm] inthetransitionregion;cf.Bingert&Peter2011).Forexample, this drop accounts for a difference in heating rate between lo- Fig.3.HorizontallyandtemporallyaveragedOhmicheatingrate cations(a)and(b)in Fig.2ofaboutafactorof100(cf.dotted (∼ j2).Themarkedpostionsareatabout(a)28Mm,(b)10Mm, lines in Fig. 3). On top of this average drop with height, there (c)22Mm,and(d)17Mmaccordingtothelocationsdepictedin isastrongspatialinhomogeneitycausedbythestructureof the Fig.2. magneticfieldandhowthemagneticfieldlinesarebraided.The magneticfieldgeometryabove4Mm(Bingert&Peter2011)is dominatedbyalargeloopsystemconnectingthemainpolarities heating rate per particle is highest (see Bingert&Peter 2011). (c.f.Fig. 1). Atheightsabove4Mm, almostall field linescon- ThisisshowninFig.4.Lookingataspotwheretheheatingrate necttothesepolarities.Asaresulttheselectedpositionsdonot perparticleisatitspeakgivessomeconfidencethatthisiswhere showanydistinguishedmagneticconfiguration. mostoftheactionis,andthusthisexamplecanbeconsidereda Mostimportantisthatthetemporalvariabilityateachposi- representativeone. tion shows a different nature. Referring to Fig. 2, e.g., at loca- Toestimatetheenergydepositedinonesingleburstofheat- tion (a) the heating is due to short pulses with a lifetime of a ing, the volume covered by this event is needed. The current few minutes, whereasat position (b) only one pulse overmore structuresalignedwiththemagneticfieldaretypicallylimitedby than half an hour seems to occur. At positions (c) and (d), the thespatialresolutionofthenumericalexperiment.Usingahigh- heatingiscomposedofshortpulsesandofpulseswithlifetimes order scheme to solve the MHD equations, these small struc- of ten or more minutes. These fourprofiles representthe com- tureswillhaveanextentofroughly4x4x4gridpoints(seealso pletedomainin thesense thattheenergyisinsertedinmoreor Bingert&Peter 2011, their Fig. 9). This correspondsto a vol- lessrandomlydistributedpulses.Thusitisquitechallengingto umeofveryroughly2Mm3andshouldbeconsideredasalower investigate each of the million gridpoints in the numerical do- limit. main.Thusonehastoemployastatisticalapproach.Beforewe Thetemporalevolutionin Fig. 4 showstwo eventsin close dothisinSect.4,weinvestigateoneparticularlocationinmore succession: event (A) consists of several heating pulses occur- detail. ring shortly one after another so that they merge into one sin- As an illustration we select one spatial location at the very glelongerdurationeventlastingalmost20minutes.Thesecond base of the corona and transition region, where the (average) event(B)consistsofasingleheatingpulsewithalargerampli- 3 S.BingertandH.Peter:Nanoflarestatisticsinanactiveregion3DMHDcoronalmodel V and τ to show that our results do not depend on a particu- 00..00002200 lar choice of V and τ. We choose the subvolumes 0.29Mm3, 0.97Mm3, 2.31Mm3, and 4.51Mm3, which correspondsto 23, 33,43,and53gridpoints.Thegridspacingsaredx=dy=0.39Mm 00..00001155 anddz=0.24Mm.Inthisapproachweuseequal-sizednoncubic −3] volumesratherthansearchingforthespatialextentofeachindi- m W vidualenergydeposition.In Sect.6 weprovidesomemorede- e [ tailsonwhythisapproachisusefulandappropriate.Adecom- at 00..00001100 position in real events is rather challenging and perhaps even g r impossible.Thetransientheatingproduceseventsthatnot only n ati overlapin time but also overlapin space. Additionally,the en- he 00..00000055 ergyreleaseshowsageneraldecreasewithheight,thereforesin- gle events will not be easily detectable, except for a few iso- latedmajorenergyreleases.Toreducetheinvestigationtothose (A) (B) events,whichareclearlyseeninthedata,wouldreducethenum- 00..00000000 berofeventsdramatically,andthestatisticalanalysiswouldfail. 00 1100 2200 3300 4400 5500 6600 7700 time [min] 4.1.EnergydistributionofOhmicdissipationinthewhole Fig.4.Ohmicheatingrate(∼j2)atafixedpointinspaceformore computationaldomain(fromphotospheretocorona) than a solar hour. The data pointsare markedby crosses. Here Theproceduredescribedaboveprovidestheenergiesofalarge a gridpointat the verybase of the coronaand transitionregion numberof events.A commonway to analyzethis is to investi- (≈3Mmheight)waschosenwhichisrepresentativeoftheregion gate the distribution of the energies of these events. If the fre- wheretheheatingrate perparticleis maximal.Thelargecross quency distribution is normalized by the event energies E, we in Fig. 2 indicates the horizontal location. The shaded regions get a frequency-size distribution. In addition we normalize the (A) and (B) indicate a nanoflare storm and a single nanoflare, frequency-sizedistributionbytheareaandtime.Heretheareais seeSect.3. thehorizontalextentofthecomputationaldomain(50x50Mm2), andthetimeisthedurationofthesimulationusedinthisstudy tudelastingonlyafewminutes.(Thesameistruefortheweaker (60min). eventsattimes15minutesand22minutes). Figure5depictsthefrequency-sizedistributions f(E)ofthe When integrating in time and space about 1019J are de- OhmicheatingeventenergiesforseveralcombinationsofV and posited during event (A) and a couple of 1017J are deposited τforthewholecomputationaldomain.Theeventenergiesrange duringevent(B). Theseenergiescoincideroughlywith the en- from1010Jto1025J.Theseareenergiesseveralordersofmag- ergyofatypicalnanoflareof1017J(1024erg)asoriginallysug- nitude below and above the energy of a Parker nanoflare that gested from theoreticalconsiderationsby Parker(1988). Event is about 1017J (1024erg). The frequency distributions have a (B) could then be “classified” as a regular nanoflare. In con- roughlyconstantslopeof s ≈ −1.2overawiderange,irrespec- trast, event (A) could be related to a nanoflare storm, a phe- tive of the choice of the box size V or the temporalbinning τ. nomenon“withdurationsofseveralhundredtoseveralthousand Only for high values of V and τ are the number of events too seconds”(Viall&Klimchuk2011).Originally,thesewereintro- low, and the frequency distribution is not continuous. For in- ducedheuristicallytomodelthelightcurvesofwarmloopswith stanceforthelargestbinninginspaceandtimewehaveroughly thehelpof1Dmodels(e.g.Warrenetal.2002),eventhoughthis 8000 boxes, e.g. events, in comparison to the smallest binning namewascoinedonlylater. withmorethan10millionboxes.Iftheboxsizeincreases,small Our 3D MHD model with self-consistent Ohmic heating eventsare missing andthe size of the biggesteventsincreases. shows that such nanoflare storms are a natural consequence of Thereforethecoveredenergyrangeofthefrequency-sizedistri- the braiding of magnetic field lines. Therefore our results sup- bution clearly dependson the spatial binningbut dependsonly portthe ad-hocuse of such bursts forthe energyrelease in 1D weaklyonthetemporalbinning. loopmodels. As a consistency check,the frequency-sizedistributioncan beusedtoderivethetotalenergybudgetofthesystem.Theto- tal energy flux into the system is given by the integral of the 4. Statisticalanalysisoftheheatingrateintime frequencydistribution Tounderstandthecoronalheatingprocess,itisnotsufficientto studysingleeventsalone.Insteadwehavetoinvestigatestatisti- P= E2 f(E)EdE [Wm−2]. (1) calpropertiesoftheheatinput.Forthiswesubdividethecom- Z E1 putationaldomainintoboxesofequalsizeV,andthetimespan Assuming the frequency distribution to be a power law of the ofthesimulationintotimeintervalsofequallengthτ. form f(E) = AEs,theenergyfluxintherangeofenergiesfrom As one “event” we define the total energy input through E toE isthengivenby Ohmicheatingin oneof these boxeswith volumeV integrated 1 2 overtime τ.Forexample,ifwechooseV = (5×5×5)Mm3 and A τ=5min,wehave10×10×6=600boxesinthe50x50x30Mm3 P= E(2+s)−E(2+s) (2) 2+s 2 1 domainspreadover12binsintimeforacomputationaltimeof (cid:16) (cid:17) 60min. Consequently we will get 600×12 = 7200 individual ifs,−2otherwiseP= Aln(E /E ).Iftheslopesofthepower 2 1 eventsforthisparticularcombinationofV andτ.Wetheninves- lawisflatterthan−2,thentheintegralisdominatedbytheupper tigatethestatisticalpropertiesoftheseevents,inparticulartheir energy limit. Whereas for slopes steeper than −2 the result is energydistribution.Wedothisforavarietyofcombinationsof dominatedbythelowerenergylimit. 4 S.BingertandH.Peter:Nanoflarestatisticsinanactiveregion3DMHDcoronalmodel Table1.SlopesofthedistributionsshowninFig.5usingmaxi- 1015 mumlikelihoodestimations.SeeSect.4.2. 2ms)] 1010 V[Mm3] τ[min] 3 4 5 6 7 −3910/(J 105 s=−1.2 020...932719 111...111996 111...111996 111...111996 111...211096 111...211097 y [ 4.51 1.12 1.12 1.12 1.12 1.12 c n ue 100 q s e e ent fr 10−5 noflar MVLanEdteticmhneiqbuinesfoτrfvoarritohuesecvoemntbsi.nWateionnesgolfecthte5s%paotifalthbeoxloswizeesst Ev 34 mmiinn na energiesfromtheanalysis. 5 min 6 min Having derived a slope is not yet a proof for a power- 10−10 7 min law. Again we follow Virkar&Clauset (2012) and use the 1010 1015 1020 1025 Kolmogorov-Smirnovtest (Pressetal. 1992) to check whether Event energy [J] thedistributionfunctionsmatchthepowerlaws.Theslopes re- trievedbytheMLEgiveaconfidencelevelabove0.9forthetwo Fig.5. Frequency distribution of Ohmic heating event energies distributionfunctionswithsmallestspatialbinnings(c.f.,Fig.5). forfourdifferentspatialboxvolumesVandfivedifferenttempo- Thehighestconfidencelevelswefindareforaslopeof1.20.All ralbinsizesτ.Thetemporalbinningτiscolor-codedandlabeled slopesdiscussedhereare around1.2,andthe confidencelevels inthe figure.Thespatialboxsizes V are0.29Mm3, 0.97Mm3, arehighenoughtoarguethatthedistributionsofOhmicheating 2.31Mm3,and4.51Mm3.Thedistributionwiththesmallestbox eventsareconsistentwithpowerlaws. sizeVisthegroupoflinesstartingatthesmallestenergies(tothe left).Forthetwosmallestbinningsthenumberofeventsislarge enoughthatthefrequencysizedistributioniscontinuous.Forthe 4.3.Heatingstatisticsforthecoronalpartalone two larger binningsthe distribution is interrupted owing to the We now focus on the coronal part. We define the coronal part low numberof events.Overplottedis a powerlaw with a slope eitherthroughathresholdintemperatureorasthevolumeabove of −1.2. The verticaldotted line indicatesthe typicalnanoflare a certain height. Both definitions (when properly chosen) give energyaspredictedbyParker(1988).SeeSect.4.1. similarresults.Thiswillgiveusinformationaboutthedistribu- tionoftheheatinputintothemodelcorona.Therefore,werepeat the analysis from Sect. 4.1 in the subvolume of the computa- The slope of the frequency distribution of Ohmic heating tionaldomainrepresentingthecoronaandthetransitionregion. eventsis−1.2andthusflatterthan−2(c.f.,Fig.5).Whencon- To define the coronal volume, in the first case we choose sidering the whole computational domain including the photo- a temperaturethresholdof logT/[K]=3.9to include everything spheric and chromosphericparts, the majority of heat is there- abovethechromosphere.Inthesecondcasewedefinethecoro- foredepositedineventswithhighenergies. nal volume as being at heights above 4Mm. This corresponds Theenergyfluxintoourmodeledboxderivedfromthefre- to the height where the horizontally averaged temperature ex- quencydistributioninFig.5basedonEq.(1)or(2)isontheor- ceedslogT/[K] = 3.9inourmodel.Itisalsotheheightwhere derofP≈106Wm−2(=109ergs−1cm−2).Thisisthefluxatthe the scale height of the horizontally averaged volumetric heat- photospheric level, which is the lower boundary of our model ing rate increases and becomes constant for the coronal part atmosphere. This represents the average flux over roughly one (see Fig. 3 and Bingert&Peter 2011). The model atmosphere hoursolartime.ItmatchesnicelywithBingert&Peter(2011), isverydynamicsothattheisosurfaceforaconstanttemperature wheretheenergyfluxinputintothecomputationaldomainata ishighlycorrugated.Theheightvariationoftheisothermalsur- fixedtimewasfoundtobeintherangebetween106Wm−2and faceoflogT/[K]=3.9isseveralMm.Thisislargerthantothe 107Wm−2. chromosphericscale height,butmuchsmallerthan the coronal Itshouldbeemphasizedthattheseresultsapplytothewhole scaleheight.Inparticular,thisismuchsmallerthanthevertical computationaldomain,includingthe photosphereand chromo- extentofthecoronalpartofourcomputationaldomain.This is sphere.InSect.4.3wediscusstheenergydistributionandbudget thebasic reason,whythe resultsforthesetwo methodswillbe forthecoronalpartofthedomainalone. essentiallythesame. Whennowconsideringthecoronalvolumealone(usingthe above two definitions), the event energies show a distribution 4.2.Power-lawtestforenergydistribution thatiscomposedoftwopowerlaws,withakneeatabout1017J, EveniftheslopeofthedistributionfunctionsoftheOhmicheat- andaslopeof−1.2belowand−2.5abovetheknee(seeFig.6). ing events looks fairly constant over several orders of magni- The slope at small-event energies is comparable to Fig. 5 as tude, we briefly discuss how to test the validityof the assump- foundforthe wholecomputationaldomain.Forenergiesbelow tion that this is indeed a power law. It can be problematic to 1014Jthedistributionshowsthetypicalcutoffduetolimitedres- use linearregression in a log-logplotto retrieve the slope of a olution and statistics. Here and in the following we only show powerlaw.ThereforewefollowVirkar&Clauset(2012)anduse the results for one particular choice of the spatial box size V amaximum-likelihoodestimation(MLE)toobtaintheslopeof andtemporalbinningτtodeterminethedistributionofeventen- apossiblepowerlaw.AsdiscussedinVirkar&Clauset(2012), ergies (V=0.29Mm3, τ=5min). This is only to not clutter the the result is sensitive to the selection of the lower energy cut- plots, the results are independent of the particular choice of V off. In Table 1 we list the estimated slopes following from the and τ (c.f. Sec. 6). The distribution in Fig. 6 represents a sub- 5 S.BingertandH.Peter:Nanoflarestatisticsinanactiveregion3DMHDcoronalmodel 12 1012 slope=−1.2 2ms)] 1010 −2m] 10 −390/(J 108 n [W 8 1 bi ency [ 106 slope=−2.5 x per 6 u u nt freq 104 ergy fl 4 e logT/[K] > 3.9 n ev 102 z > 4.0 Mm e 2 100 0 1012 1014 1016 1018 1020 1012 1014 1016 1018 1020 event energy [J] energy E [J] 0 Fig.6.DistributionofOhmicheatingeventenergiesinthecoro- Fig.7. Energy fluxes into the corona deposited at different en- nal volume alone. The results are shown for two definitions of ergiesE oftheheatingevents.Derivedfromthedistributionin 0 thecoronalvolume:(a)logT[K]>3.9and(b)heightsz>4[Mm]. Fig.6.Thedistributionisfirstinterpolatedtoallowforan inte- ResultsareshownonlyforV=0.29Mm3,τ=5min).Overplotted grationintervalof0.2dexinlogarithmicvalues.SeeSect.4.3. are lines indicating power laws with slopes of −1.2 and −2.5, respectively.SeeSect.4.3. This result of the distribution of energy fluxes in Fig. 7 showsthattheupperatmosphereinourmodelismainlyheated sampleofthecompletecomputationaldomainsothattherange by events with an energyof a few times 1017J, which roughly ofthecoveredeventenergiesissmallerthanfortheanalysisof matchesthetypicalnanoflareenergy.Inadditiontothequalita- theentiredomainshowninFig.5. tivediscussionofthepositionofthekneeinFig.6,thisquanti- The resultthatthe slope ofthe powerlaw is flatterthan −2 tativeresultalsoprovidesinformationonthe rangeofenergies below1017Jandsteeperthan−2abovehasaverypointedcon- responsiblefortheenergyinput.FromFig.7itisclearthat en- sequence. Below the knee most of the energy is deposited at ergies from 1016 to 1018J (1023 to 1025erg) contribute signifi- higherenergies,i.e.,closeto theknee.Abovethekneemost of cantly. the deposition is at low energies, i.e., again close to the knee. Fromthiswecanconcludethatinour3DMHDmodelthere Thereforemostoftheenergywillbedepositedclosetotheknee isapreferredenergyforthedepositionthroughOhmicheating. at 1017J, which coincides with the originally proposed energy This value is the same orderof magnitudeas the nanoflareen- fornanoflares. ergiesderivedbyParker(1988) ontheoreticalgrounds,despite Amorequantitivewaytolocatethemaincontributiontothe all the limitations in terms of spatial resolution and magnetic heat input in terms of event energies is to evaluate the energy Reynoldsnumberinournumericalexperiments.Thisisreassur- deposited in different energy ranges. For this, we integrate the ing, because our 3D MHD model basically describes the flux- frequencydistributioninFig.6piecewisewithafixedintervalin braidingprocessasproposedbyParker(1988). ∆logE[J]=0.2aroundtheenergyE 0 E0+∆E 5. Consequencesforobservablequantities P(E )= f(E)EdE. (3) 0 Z The frequency-size distributions derived in the previous sec- E0−∆E tion are based on the energy deposition due to Ohmic dissipa- Thisgivesthetotalenergyfluxintothecoronathatisdeposited tion.However,theseeventswithenergiesclosetothevaluesof inarangeofeventenergiescenteredon E .Figure7showsthe nanoflareenergiesarenotdirectlyobservableontheSun.Foran 0 distribution of there energy fluxes P(E ) as a function of the observationalstudyoftheregionwheretheplasmaisheated,the 0 event energy E . The maximum flux is found at an energy of onlysourceofinformationwehaveistheemissioninextemeUV 0 roughly 1017J (1024erg) just as suspected above. The integra- andX-rays.Andthisprovidesonlypartialinformationontheen- tion is performed using an interpolation of initial distribution. ergybudget,becauseheatconduction,particleacceleration,and Butusingtheoriginalresolutioninenergydoesnotalterthere- maybe other processes will also redistribute the energy, which sults,andonlythewidthofthebinswouldbebroader.Theerror willthenbe,e.g.,radiatedinotherplaces. forthepeakfluxisestimatedbythehalfbinwidthofinitialdata, Tostudyhowthedistributionofenergieschangesifonein- e.g,0.16orafactorofroughly1.4 vestigatesthebrighteningsrelatedtotheheatingevents,wesyn- As a sanity check, we can use the result shown in Fig. 7 thesizetheexpectedopticallythincoronalandtransitionregion to compute the total energy input to the model corona by emission fromthe MHD model. Here we follow the procedure integrating over the whole range of energies. We find this outlinedbyPeteretal.(2004,2006).Toderivereliableestimates to be 100Wm−2, which is consistent with the results given ofthecoronalemission,theMHDmodelhastotreattheenergy by Bingert&Peter (2011) and consistent with estimations equationappropriately;inparticularitisessentialtoincludethe of the coronal heating requirements based on observations heatconductiontoself-consistentlysetthepropercoronalpres- (Withbroe&Noyes1977). sure (and thus density). Using the CHIANTI atomic data base 6 S.BingertandH.Peter:Nanoflarestatisticsinanactiveregion3DMHDcoronalmodel aroundthenanoflareenergiescoincidingwiththelocationofthe kneeismainlytransporteddowntowardsthechromosphere,i.e. Si_2 (1533) 4.40000 1015 slope=−0.8 SCi__24 ((11333954)) 44..6905000000 towardsthehigh-energyendofthedistribution,andisthenradi- 2ms)] CCOO____3445 ((((911675437400)81) )4)5 .55.94..001005050000000000 aetnehdanacteloswtheerdteismtrpibeuratitounreastinthethheigtrha-nesniteirognyreengdioonf.tIhnetudrisnt,ritbhuis- −390/(J 1010 ONMe_g_6_8 1( 10(70 (736022))5 55) ..685.2500040000000 tionToofgtheetaraqduiaatnivtietaotuivtpeuets.timateof theflatteningofthepower y [1 law slope from the energy input (−1.2) to the radiative output c (−0.8),we investigatethe scaling laws as already describedby n e Rosneretal.(1978).Thescalinglawsrelatethevolumetricheat equ 105 inputEH (assumedtobeconstant)andthelooplength Ltothe nt fr peaktemperatureT andpressurepofahydrostaticloop(Eq.4.3 e and 4.4 of Rosneretal. 1978). One can easily combine these v e 100 to relate the density n at the apex to the heat input and loop length, n ∝ E4/7L1/7. Because the dependence on L is weak, H 100 105 1010 1015 1020 we discard this in the following. The energy loss ε due to op- event energy [J] tically thin radiation (per volume and time) is proportional to thedensitysquared,ε ∝ n2,andwiththeabovescalinglawwe find ε ∝ (E )8/7. Therefore the slope of the powerlaw should Fig.8. Event frequency distributions of synthesized emission H change from −1.2 to −1.2×7/8 = −1.05. This change does go lines.Thenameofthelinesaregiveninthelegendalongwiththe in the right direction, but does not quite explain the result of wavelength[Å]andthelineformationtemperatureinlogT/[K]. −0.8 we find in Fig.8. This is not very surprising, because the Rosneretal.(1978)scalinglawsapplyonlytohydrostaticloops withaconstantheatingrate,whileour3DMHDmodelistime- (??), we calculate the emission at each grid point for a num- dependentwithanonconstantheatingrate. ber of emission lines based on the temperature and density at Inconclusion,the energyeventsforthe radiative losses are therespectivegridpointinthemodel.Theresultisthevolumet- still distributed as a powerlaw, but the slope is flatter than the ric energy loss per second through optically thin radiation in a powerlawfortheactualenergyinput.Thisisimportantforob- givenspectralline. servational studies investigating the energy input by analyzing Werepeattheanalysisoftheprecedingsectionandtreatthe theenergylossthroughbrighteningsseeninEUVandX-rays. emissionineachofthelinesinthesamewayastheenergyinput in Sects. 4.1 and 4.3. Figure 8 shows the resulting distribution 6. Self-organizedcriticalityandfractaldimension functionsfortheradiativeenergylossevents.Theenergyrange is shifted to muchlowerenergiesthan in Figs. 5 and 6. Thisis InSect.4wederivedthefrequency-sizedistributionsforOhmic mainly because each line only represents one single emission heatingevents.Thesedistributionsfollowapowerlawoversev- line, and each emission line contributes only a small fraction eralordersof magnitude(c.f. Fig. 5). This powerlaw doesnot of the total energyloss. Nonetheless, the distributionfunctions changeifthebinningofthedatacubeischanged,eitherspatially showsimilarpowerlawsforallemissionlineswithlineforma- ortemporally.Thisbehavioristypicalofapower-lawfrequency- tiontemperaturesrangingfrom30000Kupto106K.Owingto size distribution, so it indicates that our modeled energy input thedifferentformationtemperatures,thelinesoriginatefromdif- also follows a power law. Systems that are characterized by a ferentdensities,sothatthefrequencydistributionshavedifferent powerlaw are called self-organizedcritical systems (Baketal. cut-offsatthehigh-energyend. 1987).Self-organizedcriticalitycanalsobeappliedtoothersys- Most importantly, the power law shows a slope of about temssuchas1/f noise,anditshowsthescaleinvarianceofthe −0.8. This slope is slightly flatter at low energies and a bit system ona macroscopiclevel.Thisscale invariancemanifests steeper for high energies, but is flatter than the slope of −1.2 itself bythefactthatthe slopesofthepowerlawsareindepen- found for the energy input. The reduction of the slope reflects dent of the spatial scale range under consideration, i.e. of the the nonlinearconnectionfrom heat input to the final (optically spatialbinning.Thereforetheapproachofusingaconstantbin- thin) radiative output. There are basically two ways of looking ning for the analysis in Sects. 4.1 and 4.3 does not affect the atthisphenomenon:(1)aheuristicexplanationbasedontheen- results. The constant binning refers to equally sized volumes. ergyredistributionand(2)amoreformalargumentusingscaling Likewisethenoncubicboxesdonotaltertheresultsincewecan laws. usearbitraryvolumesinascale-independentsystem. Atcoronaltemperaturesheatconductiondominatesradiative We find in our analysis that the frequency size distribu- losses (e.g. Priest 1982); the higher the temperature, the more tion of the event energies can be represented by a single slope heatconductiondominates(below107K).Onaverage,theheat alone. This is consistent with other model approaches, e.g, input (per volume and time) is smaller at higher temperatures Dimitropoulouetal. (2011) and Georgoulisetal. (2001), and atlargerheights(Fig.3),wheretheenergyinputislower.Thus with observations (Aschwanden&Freeland 2012). In earlier heatconductionisthemajorlossprocessatthelow-energyend studies Georgoulis&Vlahos (1996) and Georgoulis&Vlahos ofthedistributionofeventenergies.Consequently,there isless (1998) used cellularautomata models and find a double power radiation at the low-energy end, and the distribution of the ra- law behaviorfor the eventsize. That ourdistributionfunctions diative losses will be flatter, in our case the slope is only −0.8 followasinglepowerlawcanbeaccountedforbytheshorttime insteadof−1.2. period of one hour and by the small field of few of our model Thesamemechanismalsoexplainswhythekneeintheen- activeregion. ergy input (Fig. 6; Sect. 4.3) is not clearly visible in the dis- Following Aschwanden (2010) the slope of the power law tribution of the radiative losses (Fig. 8). The energy deposited ofthedistributionfunctionspointtothepropagationproperties 7 S.BingertandH.Peter:Nanoflarestatisticsinanactiveregion3DMHDcoronalmodel of the instability leading to the dissipative events. The fractal MHD models for the flux-braiding mechanism provide a good dimension D is thereby correlated to the slope s by D = 3/s descriptionoftheheatingofthesolarcoronainactiveregions. (Aschwanden 2010; Aschwanden&Freeland 2012). Since we measureaslopeof1.2,thefractaldimensionofthesystemwould References be2.5fortheOhmicdissipationevents.Thismeansthatthefrac- taldimensionisnotboundtothe2Dsurfaceofseparatrixlayers, Akabane,K.1956,PASJ,8,173 andthatthepropagationoftheinstabilitycannotspreadfreelyin Aschwanden,M.J.2010,ArXive-prints allthreedimensions. Aschwanden,M.J.&Freeland,S.L.2012,ApJ,754,112 Bak,P.,Tang,C.,&Wiesenfeld,K.1987,PhysicalReviewLetters,59,381 Formally, using the slope of −1 of the power law for the Bingert,S.&Peter,H.2011,A&A,530,A112+ radiative losses (c.f. Fig. 8), we find a fractal dimension of 3. Brandenburg,A.&Dobler,W.2002,ComputerPhysicsCommunications,147, However,becausetheemissionisjustasecondaryquantityfol- 471 lowingfromtheheatinputandstronglyaffectedbytheheatcon- Charbonneau,P.,McIntosh,S.W.,Liu,H.-L.,&Bogdan,T.J.2001,Sol.Phys., duction,thediagnosticvalueofthisresultislimited,though. 203,321 Christe,S.,Hannah,I.G.,Krucker,S.,McTiernan,J.,&Lin,R.P.2008,ApJ, 677,1385 Cook,J.W.,Cheng,C.,Jacobs,V.L.,&Antiochos,S.K.1989,ApJ,338,1176 7. Conclusions Crosby,N.,Vilmer,N.,Lund,N.,&Sunyaev,R.1998,A&A,334,299 Crosby,N.B.,Aschwanden,M.J.,&Dennis,B.R.1993,Sol.Phys.,143,275 InSec.3wediscussedthetransientnatureoftheheatingbased Dere,K.P.,Landi,E.,Mason,H.E.,MonsignoriFossi,B.C.,&Young,P.R. 1997,VizieROnlineDataCatalog,412,50149 ontimeseriesoftheOhmicheatingatspecificlocations.There Dimitropoulou, M.,Isliker, H.,Vlahos, L.,&Georgoulis, M.K.2011,A&A, the temporal evolution of the heating rate shows a stochastic 529,A101 nature, as well as a qualitative difference at different places. Georgoulis,M.K.,Vilmer,N.,&Crosby,N.B.2001,A&A,367,326 We saw long bursts (10 min and longer), which may consist Georgoulis,M.K.&Vlahos,L.1996,ApJ,469,L135 Georgoulis,M.K.&Vlahos,L.1998,A&A,336,721 of several short pulses. At other locations we saw only short Gudiksen,B.&Nordlund,Å.2002,ApJ,572,L113 pulses (1 min and less). These events could correspond to the Gudiksen,B.&Nordlund,Å.2005a,ApJ,618,1020 nanoflarestormsandnanoflaresoftenassumedin1Dloopmod- Gudiksen,B.&Nordlund,Å.2005b,ApJ,618,1031 els(Viall&Klimchuk2011).Weusedthesetimeseriesonlyfor Hannah,I.G.,Hudson,H.S.,Battaglia, M.,etal.2011,SpaceSci.Rev.,159, aqualitativedescriptionandforaroughestimateofthemagni- 263 Hansteen,V.H.,Hara,H.,DePontieu,B.,&Carlsson,M.2010,ApJ,718,1070 tudeofheatingevents.Wesawthattheseeventsdepositenergies Landi,E.&Phillips,K.J.H.2006,ApJS,166,421 onthetheorderof1017J,whichagreeswiththenanoflareenergy Lu,E.T.1995,ApJ,446,L109 estimatedbyParker(1988). Lu,E.T.&Hamilton,R.J.1991,ApJ,380,L89 We alsopresentedamoresophisticatedapproachusingsta- Parker,E.N.1983,ApJ,264,642 Parker,E.N.1988,ApJ,330,474 tisticalmethodstoanalyzetheseeventsinthecompletedomain. Peter,H.2007,Adv.SpaceRes. 39,1814 The statistical analysis of the coronal heating reveals a power Peter,H.&Bingert,S.2012,A&A,inpress,arXiv:1209.0789 law for the heating event energies. Even though the driver, i.e. Peter,H.,Gudiksen,B.,&Nordlund,Å.2004,ApJ,617,L85 thegranularmotion,doesnotworkonallscales,theslopeofthe Peter,H.,Gudiksen,B.,&Nordlund,Å.2006,ApJ,638,1086 powerlawisconstantoverseveralordersofmagnitudeandflat- Press, W. H., Teukolsky, S. A., Vetterling, W. T., & Flannery, B. P. 1992, NumericalrecipesinC.Theartofscientificcomputing terthan−2.Wefindthatinthetransitionregionandcoronathe Priest,E.R.1982,SolarMagnetohydrodynamics(D.Reidel,Dordrecht) dominantenergyforthe heatingeventsisthe nanoflareenergy, Rosner,R.,Tucker,W.H.,&Vaiana,G.S.1978,ApJ,220,643 i.e. 1017J. Since the higher energy events occur on average at Rosner,R.&Vaiana,G.S.1978,TheAstrophysicalJournal,222,1104 lowerheights,theheatingisfootpoint-dominated.Theeventen- Spitzer, L.1962,Physicsoffullyionized gases(Interscience, NewYork(2nd edition)) ergieswefindinourmodelarebelowthedetectionlimitofcur- Viall,N.M.&Klimchuk,J.A.2011,ApJ,738,24 rentinstrumentationandsupportthescenarioofnanoflareheat- Virkar,Y.&Clauset,A.2012,ArXive-prints ingasproposedbyParker(1988). Warren,H.P.,Winebarger,A.R.,&Hamilton,P.S.2002,ApJ,579,L41 In ourmodelthe energyprovidedby the field-line braiding Withbroe,G.L.&Noyes,R.W.1977,ARA&A,15,363 Zacharias,P.,Bingert,S.,&Peter,H.2011,A&A,531,A97 isconvertedintoheatingthroughOhmicdissipationandnotinto particleacceleration.Thisimpliesthatnonlocaleffectsarenot incorporated.Thisisjustifiedbytheelectronmean-free-pathbe- ing smaller than the grid spacing of the model. The energy is redistributed by advection and conduction or radiated by opti- callythinemission.Fromourmodelatmospherewesynthesized observable extreme UV emission lines. The statistical analysis of these lines shows again a power-law behavior. The slope of this power law is around 1 and smaller than the slope of the heatingprocess.Eventhoughthepowerlawismaintainedover severalordersof magnitude,in contrastto the heating process, theradiativelossesarenotinaself-organizedcriticalstate.This is because the radiative losses are coupled to the heating in a nonlinearly. In this study we showed that a 3D MHD model of the ac- tive region solar corona has a preferred energy for depositing ofheatthroughOhmicdissipationfollowingfield-linebraiding. Thepreferredenergyofabout1017J(1024erg)correspondswell to the value derived by Parker (1988) and thus is yet another piece of evidence that, despite all their shortcomings, the 3D 8

ebook img

Nanoflare statistics in an active region 3D MHD coronal model PDF

0.19 MB·English
by  Sven Bingert
#additional_collections #journals #arxiv
Save to my drive
Quick download
Download
Upgrade Premium
Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.

Preview Nanoflare statistics in an active region 3D MHD coronal model

Astronomy&Astrophysicsmanuscriptno.arxiv (cid:13)c ESO2013 January22,2013 Nanoflare statistics in an active region 3D MHD coronal model S.BingertandH.Peter MaxPlanckInstituteforSolarSystemResearch(MPS),37191Katlenburg-Lindau,Germany,[bingert,peter]@mps.mpg.de Received.../Accepted... ABSTRACT Aims.Weinvestigatethestatisticsforthespatialandtemporaldistributionoftheenergyinputintothecoronainathree-dimensional 3 magneto-hydrodynamical (3D MHD) model. The model describes the temporal evolution of the corona above an observed active 1 region. The model is driven by photospheric granular motions that braid the magnetic field lines. This induces currents that are 0 dissipated,therebyleadingtotransientheatingofthecoronalplasma.Weevaluatethetransientheatingassubsequentheatingevents 2 andanalyzetheirstatistics.Theresultsaretheninterpretedinthecontextofobservedflarestatisticsandcoronalheatingmechanisms. Observed solar flaresand other smallertransientscover a widerange of energies. Thefrequency distribution of energies follow a n powerlaw,thelowerendofthedistributiongivenbythedetectionlimitofcurrentinstrumentation.Oneparticularheatingmechanism a isbasedontheoccurrenceofso-callednanoflares,i.e.verylow-energydepositionevents. J Methods.Toconductthenumericalexperimentweuseahigh-orderfinite-differencecodethatsolvesthepartialdifferentialequations 1 fortheconservationofmass,themomentumandenergybalance,andtheinductionequation. TheenergybalanceincludesSpitzer 2 heatconductionandopticallythinradiativelossesinthecorona. Results.ThetemporalandspatialdistributionoftheOhmicheatinginthe3DMHDmodelfollowsapowerlawandcanthereforebe ] understoodasasysteminaself-organizedcriticalstate.Theslopesofthepowerlawaresimilartotheresultsbasedonobservationsof R flaresandsmallertransients.Wefindthatthecoronalheatingisdominatedbyeventssimilartotheso-callednanoflareswithenergies S ontheorderof1017Jor1024erg. . h Keywords.Sun:corona—Stars:coronae—Magnetohydrodynamics(MHD)—Methods:numerical p - o r 1. Introduction parameters,such asthe flare peakflux orflare duration,follow t s a powerlaw.A reviewof the statistics of flaresand theirsmall a IntheupperatmosphereoftheSunandstars,energyisreleased counterparts,themicroflares,isgivenbyHannahetal.(2011). [ inatransientfashion.Strongflaresarejustthetipoftheiceberg, wheretheplasmacanbeheatedupto10MK ormoreandpar- Tounderstandthecontributiontotheheatingofthehotcoro- 2 ticles are accelerated to relativistic speeds. More numerousare nal plasma, interpreting the power law is crucial. Following v 7 smallerenergyreleasesthatcausesmallerbrighteningsinX-rays the observation-basedresults, the powerlaw hasa negative ex- 1 andextremeUV,downtothecurrentdetectionlimit.Thereis a ponent: small flares are more numerous than large flares. If 4 continuousconnectionfromthesesmallestobservablebrighten- the frequency of small flares is high enough (the power law 6 ings,oftencalledmicroflares,tothelargestofflaresinthesense slope being steeper than −2), the heating would be dominated . that the distribution in energy of all these events roughly fol- by small events, the nanoflares. These have been proposed by 1 1 lows a power law. Theoretical arguments have been proposed Parker (1988) based on theoretical arguments and are not di- 2 that evensmaller transientreleases of energyhave to exist, the rectlyobservablewithcurrentinstrumentation.Parker(1988)es- 1 nanoflares(Parker1988), which could play a majorrole in the timatedanenergyofroughly1017J(1024erg)andnamedthose : heating of the corona. In this study we investigate the energy nanoflaresbecausetheyaresmallerthantheconventionofami- v distributionoftheselowenergyreleasesastheyarefoundin3D croflare.Thelatterhaveenergiesontheorderof1020Jandthus Xi magnetohydrodynamics(MHD)models. about10−6 timestheenergyofalargeflare.Tounderstandhow muchnanoflarescontributeto the heating orif largerflares are r Rosner&Vaiana (1978) have alreadyinvestigatedthe tran- a sient events of the Sun, flaring stars, and other X-ray sources. composedofsmallerflares,itisimportanttoknowthedistribu- tionoftheenergiesofthesetransienteventsdowntothelowest They found that the energy distribution of these events energylevels. roughly follows a power law. To explain the flare statistics Rosner&Vaiana(1978)providedatheoreticalmodelthatledto The observational approach is hereby limited by a lower apower-lawdistributionoftheflareparameters.Thismodelwas thresholdgivenbythedetectionlimitoftheobservation.Inaddi- basedonthe assumptionsthatflaringisa stochastic relaxation, tion,theflareenergyisonlyaderivedquantityinferredfromthe thewaitingtimebetweentwoeventsisanindependentparame- fluxofextemeUVand/orX-rayphotons,andisthereforeonlyan ter,andtheenergyreleaseduringaflareislargeincomparison indicatorfortheactualenergyinput.Energyisalsolostthrough totheenergycontentaftertheflare.Thepower-lawnatureisnot other processes, such as heat conduction or particle accelera- only observable for X-ray sources, but also solar radio bursts tion,andtherelativemeritofthisenergylossmechanismmight showsuchadistributionasalreadystudiedbyAkabane(1956). varywiththetotalenergyoftherespectiveevent.Thereforeeven These results were followed by numerous observations of thepowerlawindexforthedistributionoftheactualenergyin- solarX-rayflares(Crosbyetal.1993,1998;Christeetal.2008). putandfortheobservedenergylossmightdiffer(foronegiven Theyhavefoundthatnotonlytheflareenergybutalsootherflare channel,e.g.,theX-rayflux). 1 S.BingertandH.Peter:Nanoflarestatisticsinanactiveregion3DMHDcoronalmodel An alternativeapproachis the numericalmodelingof these 50 100 flare statistics. They are based on the theoretical assumptions leading to a power law distribution. A famous model uses the assumption that the system, e.g., the solar corona, is in 40 a self-organized critical state (Lu&Hamilton 1991). The con- 50 G] ceptoftheself-organizedcriticalitywasintroducedbyBaketal. h [ m] gt (tBh1ua9tt8tc7ha)e.nSqbmueeascltloiocmhnpawanrgheeedsthtteoortahtvheaeslacynosctreohmneaswl(mCillhatgarringbegotiencrnfifeuaerultdheeitsrarilen.a2scu0tci0oh1n)as. ntal y [M 30 (a)(d) 0 eld stren stateArneomthaeinrsnoupmeenri(cLaula1p9p9r5o)a.chhasbeenemployedrecentlyby horizo 20 (b) netic fi g Dimitropoulouetal. (2011). They used the cellular automaton −50 ma model to produce an average power law index of -1.80 for the 10 (c) peak energies. For that study they used several different ob- (A)+(B) servedactiveregions.However,intheirmodelthemagneticfield 0 −100 boundary remains constant over the simulation time, and the 0 10 20 30 40 50 drivingisdoneartificially,butbotharenotrealistic. horizontal x [Mm] In this study we present for the first time the frequency distribution of transient heating events in a forward model ap- Fig.1. Map of the vertical magnetic field component in the proach that is based on a 3D MHD model of a solar ac- model photosphere at the lower boundary. Marked are the po- tive region. The hot corona in this model is sustained by sitionsoftheeventsshowninFigs.2and4. Ohmic dissipation of currents, which result from the braiding of magnetic field lines by convectivemotionsprescribedat the lower boundary. Since the first of this type of forward model createdbythebraidingofmagneticfieldlines.Thismechanism (Gudiksen&Nordlund 2002, 2005a,b), it has been shown that of (DC) heating has already been proposed by Parker (1983), these models not only produce a hot loop-dominated corona, where the braiding of field lines is a simplified picture for the but that they also match the (averaged) properties of spectral energy transport due to the Poynting flux and the subsequent EUV observations (Peteretal. 2004, 2006). Furthermore, the dissipation process. The magnetic fields anchored in the pho- synthetic observations show a comparable temporal variability tosphereareshuffledaroundbyphotospheric(granular)motions (Peter 2007), and new suggestions have been made on the na- leading to a build-up of currentsin the upperatmosphere. The ture of the observed net Doppler shifts (?) in the transition re- plasmaisthenheatedbydissipationofthesecurrents.Basicpa- gion and corona (Hansteenetal. 2010; Zachariasetal. 2011). rametersderivedfromthenumericalmodel,suchastheenergy These 3D MHD coronal models show that the heating is con- flux of roughly100 Wm−2 into the corona,match observation- centratedinfield-alignedcurrentstructuresandthattheyfeature allyderivedvalues. astrongtemporalvariability(Bingert&Peter2011).Thespatio- The model describes the solar atmosphere above an ob- temporaldistributionoftheheatinputalsogivesrisetoindivid- served active region depicted in Fig.1. The domain spans over ualcoronalloopsthatappearin(synthesized)extremeUVemis- 50x50Mm2 in the horizontal direction and 30Mm in the ver- sion as havinga constantcrosssection (Peter&Bingert2012), tical direction. The initial condition for the plasma parameters justasinobservations. are given by a hydrostatic plane-parallelatmosphere similar to The success of previous models providesevidence that the solaraverages.Themagneticfieldisinitiallycomputedusinga distributionoftheheatingrateinspaceandtime(ontheresolved potentialfieldextrapolation. scales)isclosetowhatwefindontheSun.Thereforeitistimely The system is then driven by granular motions having the toinvestigatedetailsofthestatisticsoftheenergydepositionin same statistical properties as the photospheric velocities ob- the3DMHDmodels,inparticulartheenergydistributionofthe served on the sun. The evolution of the modelatmosphere fol- heatingevents,whichisthegoalofthisstudy. lows the MHD equations. The lower boundary is prescribed In Sect. 2 we briefly describe the 3D MHD coronal model by the granular motions and a fixed plasma pressure. The top and show in Sect. 3 the transient nature of the self-consistent boundaryishydrostaticwithnoheatflux.Themagneticfieldis coronalheatinput.InSect.4wederivetheenergydistributionof extrapolatedbymatchingapotentialfieldatthetop.Inthehori- theheatingeventsanddiscussinSect.5someobservationalcon- zontaldirectionsthedomainisfullyperiodic. sequences,beforeweinvestigatesomeaspectsofself-organized We ranthemodelforonesolarhourbeforetakinganydata criticalityinoursysteminSect6. tomaketheresultindependentoftheinitialcondition.Afterthat we tooksnapshotswitha cadenceof30secondsforabitmore thanahour. 2. Coronalmodel For the numerical model we employ the Pencil code To understand the heating of the corona, it is crucial to study (Brandenburg&Dobler2002).ItsolvestheMHDequationsfor itsspatialandtemporalevolution.Observationsonlyprovidein- a compressible,magnetized,viscous fluid. These equations are formationaboutconsequencesbutnotdirectlyabouttheheating the conservationof mass, the momentumbalanceequation,the mechanism and its energy deposition. Therefore we employ a energyequation,andtheinductionequation.TheOhmicheating 3DMHDmodelofthesolaratmosphereinordertoanalyzethe termisgivenasηµ j2 with j = ∇× B/µ .Thetime-dependent 0 0 spatialandtemporaldistributionoftheheatinput.Thedataused energy balance in the model includes anisotropic heat conduc- forthepresentpaperisbasedonacoronalmodelintroducedin tion (Spitzer 1962), optically thin radiative losses (Cooketal. Bingert&Peter(2011).Themodelproducesahotcoronaabove 1989), together with the viscous and Ohmic heating. No addi- asmallactiveregionwithself-consistentheating.Theheatingin tionalarbitraryheatingisappliedinthemodelcorona,whichis the modelis due to Ohmic dissipation of free magnetic energy sustainedalmostexclusivelythroughtheOhmicdissipation. 2 S.BingertandH.Peter:Nanoflarestatisticsinanactiveregion3DMHDcoronalmodel 6.E−08 3.E−04 3] 3] −m (a) z=28 Mm (b) z=10 Mm −m W W 4.E−08 2.E−04 e [ e [ at at g r 2.E−08 1.E−04g r Fig.2.Temporal evolutionoftheOhmicheat- eatin eatin ginleggraritde-(p∼oinj2ts))aitnfothuermdioffdeerlenctorloocnaat:io(na)sg(sriidn-- h 0.E+00 0.E+00h point with smallest time-averaged heat input, 3] 1.E−06 6.E−063] and(b)withstrongesttime-averagedheatinput −m (c) z=22 Mm (d) z=17 Mm −m inthecoronalvolumeatT>105K;(c)location W W e [ 8.E−07 4.E−06e [ asitztehdectoorponoaflaembriisgshiotnlo,oapndse(edn) iant tthheecsyonrothnea-l at at ng r 4.E−07 2.E−06ng r bcraosseseast mthaerkfotohtepotiinmteosfotfhtehebrsignhaptslhoootps.. TThhee ati ati geometricalheightsofthesefourlocationsare he 0.E+00 0.E+00he indicatedinFig.3.Thelocationsarealsoindi- 0 10 20 30 40 50 60 0 10 20 30 40 50 60 catedinFig.1relativetothephotosphericmag- time [min] time [min] neticfield. 1 3. Sampleevents:nanoflaresandnanoflarestorms (b) (d) (c) (a) The ohmic heating in the modelis transientin time and space. Toillustratethis,weshowinFig.2thetemporalvariationinthe 3] 102 − volumetricheatingrateatfourgridpointsofthenumericalsim- m W ulation. These have been selected using the following criteria: (a)thelowestheatingrateinthebox,(b)thehighestheatingrate e [ 100 at inthecoronalvolume,i.e.,intheregionwherethetemperature g r ethxecseiezdesd)10co6rKon,a(lc)ematisthsieonap,eaxndof(ca)loatopthcelecaorrloynvailsifboloetpinoi(nstyonf- atin 10−2 e thatloop.Obviouslytheselocationsspanawiderangeof(peak) e h heatingrates,coveringfourordersofmagnitude. g 10−4 a In part, this large difference in peak heating is due to the er v a strong drop in the volumetric heating rate with height. In the 10−6 coronal part, the horizontally averaged volumetric heating rate dropsroughlyexponentiallywithascaleheightofabout4Mm, 0 5 10 15 20 25 ascanbeseenfromFig.3(theheatingperparticleshowsapeak height z [Mm] inthetransitionregion;cf.Bingert&Peter2011).Forexample, this drop accounts for a difference in heating rate between lo- Fig.3.HorizontallyandtemporallyaveragedOhmicheatingrate cations(a)and(b)in Fig.2ofaboutafactorof100(cf.dotted (∼ j2).Themarkedpostionsareatabout(a)28Mm,(b)10Mm, lines in Fig. 3). On top of this average drop with height, there (c)22Mm,and(d)17Mmaccordingtothelocationsdepictedin isastrongspatialinhomogeneitycausedbythestructureof the Fig.2. magneticfieldandhowthemagneticfieldlinesarebraided.The magneticfieldgeometryabove4Mm(Bingert&Peter2011)is dominatedbyalargeloopsystemconnectingthemainpolarities heating rate per particle is highest (see Bingert&Peter 2011). (c.f.Fig. 1). Atheightsabove4Mm, almostall field linescon- ThisisshowninFig.4.Lookingataspotwheretheheatingrate necttothesepolarities.Asaresulttheselectedpositionsdonot perparticleisatitspeakgivessomeconfidencethatthisiswhere showanydistinguishedmagneticconfiguration. mostoftheactionis,andthusthisexamplecanbeconsidereda Mostimportantisthatthetemporalvariabilityateachposi- representativeone. tion shows a different nature. Referring to Fig. 2, e.g., at loca- Toestimatetheenergydepositedinonesingleburstofheat- tion (a) the heating is due to short pulses with a lifetime of a ing, the volume covered by this event is needed. The current few minutes, whereasat position (b) only one pulse overmore structuresalignedwiththemagneticfieldaretypicallylimitedby than half an hour seems to occur. At positions (c) and (d), the thespatialresolutionofthenumericalexperiment.Usingahigh- heatingiscomposedofshortpulsesandofpulseswithlifetimes order scheme to solve the MHD equations, these small struc- of ten or more minutes. These fourprofiles representthe com- tureswillhaveanextentofroughly4x4x4gridpoints(seealso pletedomainin thesense thattheenergyisinsertedinmoreor Bingert&Peter 2011, their Fig. 9). This correspondsto a vol- lessrandomlydistributedpulses.Thusitisquitechallengingto umeofveryroughly2Mm3andshouldbeconsideredasalower investigate each of the million gridpoints in the numerical do- limit. main.Thusonehastoemployastatisticalapproach.Beforewe Thetemporalevolutionin Fig. 4 showstwo eventsin close dothisinSect.4,weinvestigateoneparticularlocationinmore succession: event (A) consists of several heating pulses occur- detail. ring shortly one after another so that they merge into one sin- As an illustration we select one spatial location at the very glelongerdurationeventlastingalmost20minutes.Thesecond base of the corona and transition region, where the (average) event(B)consistsofasingleheatingpulsewithalargerampli- 3 S.BingertandH.Peter:Nanoflarestatisticsinanactiveregion3DMHDcoronalmodel V and τ to show that our results do not depend on a particu- 00..00002200 lar choice of V and τ. We choose the subvolumes 0.29Mm3, 0.97Mm3, 2.31Mm3, and 4.51Mm3, which correspondsto 23, 33,43,and53gridpoints.Thegridspacingsaredx=dy=0.39Mm 00..00001155 anddz=0.24Mm.Inthisapproachweuseequal-sizednoncubic −3] volumesratherthansearchingforthespatialextentofeachindi- m W vidualenergydeposition.In Sect.6 weprovidesomemorede- e [ tailsonwhythisapproachisusefulandappropriate.Adecom- at 00..00001100 position in real events is rather challenging and perhaps even g r impossible.Thetransientheatingproduceseventsthatnot only n ati overlapin time but also overlapin space. Additionally,the en- he 00..00000055 ergyreleaseshowsageneraldecreasewithheight,thereforesin- gle events will not be easily detectable, except for a few iso- latedmajorenergyreleases.Toreducetheinvestigationtothose (A) (B) events,whichareclearlyseeninthedata,wouldreducethenum- 00..00000000 berofeventsdramatically,andthestatisticalanalysiswouldfail. 00 1100 2200 3300 4400 5500 6600 7700 time [min] 4.1.EnergydistributionofOhmicdissipationinthewhole Fig.4.Ohmicheatingrate(∼j2)atafixedpointinspaceformore computationaldomain(fromphotospheretocorona) than a solar hour. The data pointsare markedby crosses. Here Theproceduredescribedaboveprovidestheenergiesofalarge a gridpointat the verybase of the coronaand transitionregion numberof events.A commonway to analyzethis is to investi- (≈3Mmheight)waschosenwhichisrepresentativeoftheregion gate the distribution of the energies of these events. If the fre- wheretheheatingrate perparticleis maximal.Thelargecross quency distribution is normalized by the event energies E, we in Fig. 2 indicates the horizontal location. The shaded regions get a frequency-size distribution. In addition we normalize the (A) and (B) indicate a nanoflare storm and a single nanoflare, frequency-sizedistributionbytheareaandtime.Heretheareais seeSect.3. thehorizontalextentofthecomputationaldomain(50x50Mm2), andthetimeisthedurationofthesimulationusedinthisstudy tudelastingonlyafewminutes.(Thesameistruefortheweaker (60min). eventsattimes15minutesand22minutes). Figure5depictsthefrequency-sizedistributions f(E)ofthe When integrating in time and space about 1019J are de- OhmicheatingeventenergiesforseveralcombinationsofV and posited during event (A) and a couple of 1017J are deposited τforthewholecomputationaldomain.Theeventenergiesrange duringevent(B). Theseenergiescoincideroughlywith the en- from1010Jto1025J.Theseareenergiesseveralordersofmag- ergyofatypicalnanoflareof1017J(1024erg)asoriginallysug- nitude below and above the energy of a Parker nanoflare that gested from theoreticalconsiderationsby Parker(1988). Event is about 1017J (1024erg). The frequency distributions have a (B) could then be “classified” as a regular nanoflare. In con- roughlyconstantslopeof s ≈ −1.2overawiderange,irrespec- trast, event (A) could be related to a nanoflare storm, a phe- tive of the choice of the box size V or the temporalbinning τ. nomenon“withdurationsofseveralhundredtoseveralthousand Only for high values of V and τ are the number of events too seconds”(Viall&Klimchuk2011).Originally,thesewereintro- low, and the frequency distribution is not continuous. For in- ducedheuristicallytomodelthelightcurvesofwarmloopswith stanceforthelargestbinninginspaceandtimewehaveroughly thehelpof1Dmodels(e.g.Warrenetal.2002),eventhoughthis 8000 boxes, e.g. events, in comparison to the smallest binning namewascoinedonlylater. withmorethan10millionboxes.Iftheboxsizeincreases,small Our 3D MHD model with self-consistent Ohmic heating eventsare missing andthe size of the biggesteventsincreases. shows that such nanoflare storms are a natural consequence of Thereforethecoveredenergyrangeofthefrequency-sizedistri- the braiding of magnetic field lines. Therefore our results sup- bution clearly dependson the spatial binningbut dependsonly portthe ad-hocuse of such bursts forthe energyrelease in 1D weaklyonthetemporalbinning. loopmodels. As a consistency check,the frequency-sizedistributioncan beusedtoderivethetotalenergybudgetofthesystem.Theto- tal energy flux into the system is given by the integral of the 4. Statisticalanalysisoftheheatingrateintime frequencydistribution Tounderstandthecoronalheatingprocess,itisnotsufficientto studysingleeventsalone.Insteadwehavetoinvestigatestatisti- P= E2 f(E)EdE [Wm−2]. (1) calpropertiesoftheheatinput.Forthiswesubdividethecom- Z E1 putationaldomainintoboxesofequalsizeV,andthetimespan Assuming the frequency distribution to be a power law of the ofthesimulationintotimeintervalsofequallengthτ. form f(E) = AEs,theenergyfluxintherangeofenergiesfrom As one “event” we define the total energy input through E toE isthengivenby Ohmicheatingin oneof these boxeswith volumeV integrated 1 2 overtime τ.Forexample,ifwechooseV = (5×5×5)Mm3 and A τ=5min,wehave10×10×6=600boxesinthe50x50x30Mm3 P= E(2+s)−E(2+s) (2) 2+s 2 1 domainspreadover12binsintimeforacomputationaltimeof (cid:16) (cid:17) 60min. Consequently we will get 600×12 = 7200 individual ifs,−2otherwiseP= Aln(E /E ).Iftheslopesofthepower 2 1 eventsforthisparticularcombinationofV andτ.Wetheninves- lawisflatterthan−2,thentheintegralisdominatedbytheupper tigatethestatisticalpropertiesoftheseevents,inparticulartheir energy limit. Whereas for slopes steeper than −2 the result is energydistribution.Wedothisforavarietyofcombinationsof dominatedbythelowerenergylimit. 4 S.BingertandH.Peter:Nanoflarestatisticsinanactiveregion3DMHDcoronalmodel Table1.SlopesofthedistributionsshowninFig.5usingmaxi- 1015 mumlikelihoodestimations.SeeSect.4.2. 2ms)] 1010 V[Mm3] τ[min] 3 4 5 6 7 −3910/(J 105 s=−1.2 020...932719 111...111996 111...111996 111...111996 111...211096 111...211097 y [ 4.51 1.12 1.12 1.12 1.12 1.12 c n ue 100 q s e e ent fr 10−5 noflar MVLanEdteticmhneiqbuinesfoτrfvoarritohuesecvoemntbsi.nWateionnesgolfecthte5s%paotifalthbeoxloswizeesst Ev 34 mmiinn na energiesfromtheanalysis. 5 min 6 min Having derived a slope is not yet a proof for a power- 10−10 7 min law. Again we follow Virkar&Clauset (2012) and use the 1010 1015 1020 1025 Kolmogorov-Smirnovtest (Pressetal. 1992) to check whether Event energy [J] thedistributionfunctionsmatchthepowerlaws.Theslopes re- trievedbytheMLEgiveaconfidencelevelabove0.9forthetwo Fig.5. Frequency distribution of Ohmic heating event energies distributionfunctionswithsmallestspatialbinnings(c.f.,Fig.5). forfourdifferentspatialboxvolumesVandfivedifferenttempo- Thehighestconfidencelevelswefindareforaslopeof1.20.All ralbinsizesτ.Thetemporalbinningτiscolor-codedandlabeled slopesdiscussedhereare around1.2,andthe confidencelevels inthe figure.Thespatialboxsizes V are0.29Mm3, 0.97Mm3, arehighenoughtoarguethatthedistributionsofOhmicheating 2.31Mm3,and4.51Mm3.Thedistributionwiththesmallestbox eventsareconsistentwithpowerlaws. sizeVisthegroupoflinesstartingatthesmallestenergies(tothe left).Forthetwosmallestbinningsthenumberofeventsislarge enoughthatthefrequencysizedistributioniscontinuous.Forthe 4.3.Heatingstatisticsforthecoronalpartalone two larger binningsthe distribution is interrupted owing to the We now focus on the coronal part. We define the coronal part low numberof events.Overplottedis a powerlaw with a slope eitherthroughathresholdintemperatureorasthevolumeabove of −1.2. The verticaldotted line indicatesthe typicalnanoflare a certain height. Both definitions (when properly chosen) give energyaspredictedbyParker(1988).SeeSect.4.1. similarresults.Thiswillgiveusinformationaboutthedistribu- tionoftheheatinputintothemodelcorona.Therefore,werepeat the analysis from Sect. 4.1 in the subvolume of the computa- The slope of the frequency distribution of Ohmic heating tionaldomainrepresentingthecoronaandthetransitionregion. eventsis−1.2andthusflatterthan−2(c.f.,Fig.5).Whencon- To define the coronal volume, in the first case we choose sidering the whole computational domain including the photo- a temperaturethresholdof logT/[K]=3.9to include everything spheric and chromosphericparts, the majority of heat is there- abovethechromosphere.Inthesecondcasewedefinethecoro- foredepositedineventswithhighenergies. nal volume as being at heights above 4Mm. This corresponds Theenergyfluxintoourmodeledboxderivedfromthefre- to the height where the horizontally averaged temperature ex- quencydistributioninFig.5basedonEq.(1)or(2)isontheor- ceedslogT/[K] = 3.9inourmodel.Itisalsotheheightwhere derofP≈106Wm−2(=109ergs−1cm−2).Thisisthefluxatthe the scale height of the horizontally averaged volumetric heat- photospheric level, which is the lower boundary of our model ing rate increases and becomes constant for the coronal part atmosphere. This represents the average flux over roughly one (see Fig. 3 and Bingert&Peter 2011). The model atmosphere hoursolartime.ItmatchesnicelywithBingert&Peter(2011), isverydynamicsothattheisosurfaceforaconstanttemperature wheretheenergyfluxinputintothecomputationaldomainata ishighlycorrugated.Theheightvariationoftheisothermalsur- fixedtimewasfoundtobeintherangebetween106Wm−2and faceoflogT/[K]=3.9isseveralMm.Thisislargerthantothe 107Wm−2. chromosphericscale height,butmuchsmallerthan the coronal Itshouldbeemphasizedthattheseresultsapplytothewhole scaleheight.Inparticular,thisismuchsmallerthanthevertical computationaldomain,includingthe photosphereand chromo- extentofthecoronalpartofourcomputationaldomain.This is sphere.InSect.4.3wediscusstheenergydistributionandbudget thebasic reason,whythe resultsforthesetwo methodswillbe forthecoronalpartofthedomainalone. essentiallythesame. Whennowconsideringthecoronalvolumealone(usingthe above two definitions), the event energies show a distribution 4.2.Power-lawtestforenergydistribution thatiscomposedoftwopowerlaws,withakneeatabout1017J, EveniftheslopeofthedistributionfunctionsoftheOhmicheat- andaslopeof−1.2belowand−2.5abovetheknee(seeFig.6). ing events looks fairly constant over several orders of magni- The slope at small-event energies is comparable to Fig. 5 as tude, we briefly discuss how to test the validityof the assump- foundforthe wholecomputationaldomain.Forenergiesbelow tion that this is indeed a power law. It can be problematic to 1014Jthedistributionshowsthetypicalcutoffduetolimitedres- use linearregression in a log-logplotto retrieve the slope of a olution and statistics. Here and in the following we only show powerlaw.ThereforewefollowVirkar&Clauset(2012)anduse the results for one particular choice of the spatial box size V amaximum-likelihoodestimation(MLE)toobtaintheslopeof andtemporalbinningτtodeterminethedistributionofeventen- apossiblepowerlaw.AsdiscussedinVirkar&Clauset(2012), ergies (V=0.29Mm3, τ=5min). This is only to not clutter the the result is sensitive to the selection of the lower energy cut- plots, the results are independent of the particular choice of V off. In Table 1 we list the estimated slopes following from the and τ (c.f. Sec. 6). The distribution in Fig. 6 represents a sub- 5 S.BingertandH.Peter:Nanoflarestatisticsinanactiveregion3DMHDcoronalmodel 12 1012 slope=−1.2 2ms)] 1010 −2m] 10 −390/(J 108 n [W 8 1 bi ency [ 106 slope=−2.5 x per 6 u u nt freq 104 ergy fl 4 e logT/[K] > 3.9 n ev 102 z > 4.0 Mm e 2 100 0 1012 1014 1016 1018 1020 1012 1014 1016 1018 1020 event energy [J] energy E [J] 0 Fig.6.DistributionofOhmicheatingeventenergiesinthecoro- Fig.7. Energy fluxes into the corona deposited at different en- nal volume alone. The results are shown for two definitions of ergiesE oftheheatingevents.Derivedfromthedistributionin 0 thecoronalvolume:(a)logT[K]>3.9and(b)heightsz>4[Mm]. Fig.6.Thedistributionisfirstinterpolatedtoallowforan inte- ResultsareshownonlyforV=0.29Mm3,τ=5min).Overplotted grationintervalof0.2dexinlogarithmicvalues.SeeSect.4.3. are lines indicating power laws with slopes of −1.2 and −2.5, respectively.SeeSect.4.3. This result of the distribution of energy fluxes in Fig. 7 showsthattheupperatmosphereinourmodelismainlyheated sampleofthecompletecomputationaldomainsothattherange by events with an energyof a few times 1017J, which roughly ofthecoveredeventenergiesissmallerthanfortheanalysisof matchesthetypicalnanoflareenergy.Inadditiontothequalita- theentiredomainshowninFig.5. tivediscussionofthepositionofthekneeinFig.6,thisquanti- The resultthatthe slope ofthe powerlaw is flatterthan −2 tativeresultalsoprovidesinformationonthe rangeofenergies below1017Jandsteeperthan−2abovehasaverypointedcon- responsiblefortheenergyinput.FromFig.7itisclearthat en- sequence. Below the knee most of the energy is deposited at ergies from 1016 to 1018J (1023 to 1025erg) contribute signifi- higherenergies,i.e.,closeto theknee.Abovethekneemost of cantly. the deposition is at low energies, i.e., again close to the knee. Fromthiswecanconcludethatinour3DMHDmodelthere Thereforemostoftheenergywillbedepositedclosetotheknee isapreferredenergyforthedepositionthroughOhmicheating. at 1017J, which coincides with the originally proposed energy This value is the same orderof magnitudeas the nanoflareen- fornanoflares. ergiesderivedbyParker(1988) ontheoreticalgrounds,despite Amorequantitivewaytolocatethemaincontributiontothe all the limitations in terms of spatial resolution and magnetic heat input in terms of event energies is to evaluate the energy Reynoldsnumberinournumericalexperiments.Thisisreassur- deposited in different energy ranges. For this, we integrate the ing, because our 3D MHD model basically describes the flux- frequencydistributioninFig.6piecewisewithafixedintervalin braidingprocessasproposedbyParker(1988). ∆logE[J]=0.2aroundtheenergyE 0 E0+∆E 5. Consequencesforobservablequantities P(E )= f(E)EdE. (3) 0 Z The frequency-size distributions derived in the previous sec- E0−∆E tion are based on the energy deposition due to Ohmic dissipa- Thisgivesthetotalenergyfluxintothecoronathatisdeposited tion.However,theseeventswithenergiesclosetothevaluesof inarangeofeventenergiescenteredon E .Figure7showsthe nanoflareenergiesarenotdirectlyobservableontheSun.Foran 0 distribution of there energy fluxes P(E ) as a function of the observationalstudyoftheregionwheretheplasmaisheated,the 0 event energy E . The maximum flux is found at an energy of onlysourceofinformationwehaveistheemissioninextemeUV 0 roughly 1017J (1024erg) just as suspected above. The integra- andX-rays.Andthisprovidesonlypartialinformationontheen- tion is performed using an interpolation of initial distribution. ergybudget,becauseheatconduction,particleacceleration,and Butusingtheoriginalresolutioninenergydoesnotalterthere- maybe other processes will also redistribute the energy, which sults,andonlythewidthofthebinswouldbebroader.Theerror willthenbe,e.g.,radiatedinotherplaces. forthepeakfluxisestimatedbythehalfbinwidthofinitialdata, Tostudyhowthedistributionofenergieschangesifonein- e.g,0.16orafactorofroughly1.4 vestigatesthebrighteningsrelatedtotheheatingevents,wesyn- As a sanity check, we can use the result shown in Fig. 7 thesizetheexpectedopticallythincoronalandtransitionregion to compute the total energy input to the model corona by emission fromthe MHD model. Here we follow the procedure integrating over the whole range of energies. We find this outlinedbyPeteretal.(2004,2006).Toderivereliableestimates to be 100Wm−2, which is consistent with the results given ofthecoronalemission,theMHDmodelhastotreattheenergy by Bingert&Peter (2011) and consistent with estimations equationappropriately;inparticularitisessentialtoincludethe of the coronal heating requirements based on observations heatconductiontoself-consistentlysetthepropercoronalpres- (Withbroe&Noyes1977). sure (and thus density). Using the CHIANTI atomic data base 6 S.BingertandH.Peter:Nanoflarestatisticsinanactiveregion3DMHDcoronalmodel aroundthenanoflareenergiescoincidingwiththelocationofthe kneeismainlytransporteddowntowardsthechromosphere,i.e. Si_2 (1533) 4.40000 1015 slope=−0.8 SCi__24 ((11333954)) 44..6905000000 towardsthehigh-energyendofthedistribution,andisthenradi- 2ms)] CCOO____3445 ((((911675437400)81) )4)5 .55.94..001005050000000000 aetnehdanacteloswtheerdteismtrpibeuratitounreastinthethheigtrha-nesniteirognyreengdioonf.tIhnetudrisnt,ritbhuis- −390/(J 1010 ONMe_g_6_8 1( 10(70 (736022))5 55) ..685.2500040000000 tionToofgtheetaraqduiaatnivtietaotuivtpeuets.timateof theflatteningofthepower y [1 law slope from the energy input (−1.2) to the radiative output c (−0.8),we investigatethe scaling laws as already describedby n e Rosneretal.(1978).Thescalinglawsrelatethevolumetricheat equ 105 inputEH (assumedtobeconstant)andthelooplength Ltothe nt fr peaktemperatureT andpressurepofahydrostaticloop(Eq.4.3 e and 4.4 of Rosneretal. 1978). One can easily combine these v e 100 to relate the density n at the apex to the heat input and loop length, n ∝ E4/7L1/7. Because the dependence on L is weak, H 100 105 1010 1015 1020 we discard this in the following. The energy loss ε due to op- event energy [J] tically thin radiation (per volume and time) is proportional to thedensitysquared,ε ∝ n2,andwiththeabovescalinglawwe find ε ∝ (E )8/7. Therefore the slope of the powerlaw should Fig.8. Event frequency distributions of synthesized emission H change from −1.2 to −1.2×7/8 = −1.05. This change does go lines.Thenameofthelinesaregiveninthelegendalongwiththe in the right direction, but does not quite explain the result of wavelength[Å]andthelineformationtemperatureinlogT/[K]. −0.8 we find in Fig.8. This is not very surprising, because the Rosneretal.(1978)scalinglawsapplyonlytohydrostaticloops withaconstantheatingrate,whileour3DMHDmodelistime- (??), we calculate the emission at each grid point for a num- dependentwithanonconstantheatingrate. ber of emission lines based on the temperature and density at Inconclusion,the energyeventsforthe radiative losses are therespectivegridpointinthemodel.Theresultisthevolumet- still distributed as a powerlaw, but the slope is flatter than the ric energy loss per second through optically thin radiation in a powerlawfortheactualenergyinput.Thisisimportantforob- givenspectralline. servational studies investigating the energy input by analyzing Werepeattheanalysisoftheprecedingsectionandtreatthe theenergylossthroughbrighteningsseeninEUVandX-rays. emissionineachofthelinesinthesamewayastheenergyinput in Sects. 4.1 and 4.3. Figure 8 shows the resulting distribution 6. Self-organizedcriticalityandfractaldimension functionsfortheradiativeenergylossevents.Theenergyrange is shifted to muchlowerenergiesthan in Figs. 5 and 6. Thisis InSect.4wederivedthefrequency-sizedistributionsforOhmic mainly because each line only represents one single emission heatingevents.Thesedistributionsfollowapowerlawoversev- line, and each emission line contributes only a small fraction eralordersof magnitude(c.f. Fig. 5). This powerlaw doesnot of the total energyloss. Nonetheless, the distributionfunctions changeifthebinningofthedatacubeischanged,eitherspatially showsimilarpowerlawsforallemissionlineswithlineforma- ortemporally.Thisbehavioristypicalofapower-lawfrequency- tiontemperaturesrangingfrom30000Kupto106K.Owingto size distribution, so it indicates that our modeled energy input thedifferentformationtemperatures,thelinesoriginatefromdif- also follows a power law. Systems that are characterized by a ferentdensities,sothatthefrequencydistributionshavedifferent powerlaw are called self-organizedcritical systems (Baketal. cut-offsatthehigh-energyend. 1987).Self-organizedcriticalitycanalsobeappliedtoothersys- Most importantly, the power law shows a slope of about temssuchas1/f noise,anditshowsthescaleinvarianceofthe −0.8. This slope is slightly flatter at low energies and a bit system ona macroscopiclevel.Thisscale invariancemanifests steeper for high energies, but is flatter than the slope of −1.2 itself bythefactthatthe slopesofthepowerlawsareindepen- found for the energy input. The reduction of the slope reflects dent of the spatial scale range under consideration, i.e. of the the nonlinearconnectionfrom heat input to the final (optically spatialbinning.Thereforetheapproachofusingaconstantbin- thin) radiative output. There are basically two ways of looking ning for the analysis in Sects. 4.1 and 4.3 does not affect the atthisphenomenon:(1)aheuristicexplanationbasedontheen- results. The constant binning refers to equally sized volumes. ergyredistributionand(2)amoreformalargumentusingscaling Likewisethenoncubicboxesdonotaltertheresultsincewecan laws. usearbitraryvolumesinascale-independentsystem. Atcoronaltemperaturesheatconductiondominatesradiative We find in our analysis that the frequency size distribu- losses (e.g. Priest 1982); the higher the temperature, the more tion of the event energies can be represented by a single slope heatconductiondominates(below107K).Onaverage,theheat alone. This is consistent with other model approaches, e.g, input (per volume and time) is smaller at higher temperatures Dimitropoulouetal. (2011) and Georgoulisetal. (2001), and atlargerheights(Fig.3),wheretheenergyinputislower.Thus with observations (Aschwanden&Freeland 2012). In earlier heatconductionisthemajorlossprocessatthelow-energyend studies Georgoulis&Vlahos (1996) and Georgoulis&Vlahos ofthedistributionofeventenergies.Consequently,there isless (1998) used cellularautomata models and find a double power radiation at the low-energy end, and the distribution of the ra- law behaviorfor the eventsize. That ourdistributionfunctions diative losses will be flatter, in our case the slope is only −0.8 followasinglepowerlawcanbeaccountedforbytheshorttime insteadof−1.2. period of one hour and by the small field of few of our model Thesamemechanismalsoexplainswhythekneeintheen- activeregion. ergy input (Fig. 6; Sect. 4.3) is not clearly visible in the dis- Following Aschwanden (2010) the slope of the power law tribution of the radiative losses (Fig. 8). The energy deposited ofthedistributionfunctionspointtothepropagationproperties 7 S.BingertandH.Peter:Nanoflarestatisticsinanactiveregion3DMHDcoronalmodel of the instability leading to the dissipative events. The fractal MHD models for the flux-braiding mechanism provide a good dimension D is thereby correlated to the slope s by D = 3/s descriptionoftheheatingofthesolarcoronainactiveregions. (Aschwanden 2010; Aschwanden&Freeland 2012). Since we measureaslopeof1.2,thefractaldimensionofthesystemwould References be2.5fortheOhmicdissipationevents.Thismeansthatthefrac- taldimensionisnotboundtothe2Dsurfaceofseparatrixlayers, Akabane,K.1956,PASJ,8,173 andthatthepropagationoftheinstabilitycannotspreadfreelyin Aschwanden,M.J.2010,ArXive-prints allthreedimensions. Aschwanden,M.J.&Freeland,S.L.2012,ApJ,754,112 Bak,P.,Tang,C.,&Wiesenfeld,K.1987,PhysicalReviewLetters,59,381 Formally, using the slope of −1 of the power law for the Bingert,S.&Peter,H.2011,A&A,530,A112+ radiative losses (c.f. Fig. 8), we find a fractal dimension of 3. Brandenburg,A.&Dobler,W.2002,ComputerPhysicsCommunications,147, However,becausetheemissionisjustasecondaryquantityfol- 471 lowingfromtheheatinputandstronglyaffectedbytheheatcon- Charbonneau,P.,McIntosh,S.W.,Liu,H.-L.,&Bogdan,T.J.2001,Sol.Phys., duction,thediagnosticvalueofthisresultislimited,though. 203,321 Christe,S.,Hannah,I.G.,Krucker,S.,McTiernan,J.,&Lin,R.P.2008,ApJ, 677,1385 Cook,J.W.,Cheng,C.,Jacobs,V.L.,&Antiochos,S.K.1989,ApJ,338,1176 7. Conclusions Crosby,N.,Vilmer,N.,Lund,N.,&Sunyaev,R.1998,A&A,334,299 Crosby,N.B.,Aschwanden,M.J.,&Dennis,B.R.1993,Sol.Phys.,143,275 InSec.3wediscussedthetransientnatureoftheheatingbased Dere,K.P.,Landi,E.,Mason,H.E.,MonsignoriFossi,B.C.,&Young,P.R. 1997,VizieROnlineDataCatalog,412,50149 ontimeseriesoftheOhmicheatingatspecificlocations.There Dimitropoulou, M.,Isliker, H.,Vlahos, L.,&Georgoulis, M.K.2011,A&A, the temporal evolution of the heating rate shows a stochastic 529,A101 nature, as well as a qualitative difference at different places. Georgoulis,M.K.,Vilmer,N.,&Crosby,N.B.2001,A&A,367,326 We saw long bursts (10 min and longer), which may consist Georgoulis,M.K.&Vlahos,L.1996,ApJ,469,L135 Georgoulis,M.K.&Vlahos,L.1998,A&A,336,721 of several short pulses. At other locations we saw only short Gudiksen,B.&Nordlund,Å.2002,ApJ,572,L113 pulses (1 min and less). These events could correspond to the Gudiksen,B.&Nordlund,Å.2005a,ApJ,618,1020 nanoflarestormsandnanoflaresoftenassumedin1Dloopmod- Gudiksen,B.&Nordlund,Å.2005b,ApJ,618,1031 els(Viall&Klimchuk2011).Weusedthesetimeseriesonlyfor Hannah,I.G.,Hudson,H.S.,Battaglia, M.,etal.2011,SpaceSci.Rev.,159, aqualitativedescriptionandforaroughestimateofthemagni- 263 Hansteen,V.H.,Hara,H.,DePontieu,B.,&Carlsson,M.2010,ApJ,718,1070 tudeofheatingevents.Wesawthattheseeventsdepositenergies Landi,E.&Phillips,K.J.H.2006,ApJS,166,421 onthetheorderof1017J,whichagreeswiththenanoflareenergy Lu,E.T.1995,ApJ,446,L109 estimatedbyParker(1988). Lu,E.T.&Hamilton,R.J.1991,ApJ,380,L89 We alsopresentedamoresophisticatedapproachusingsta- Parker,E.N.1983,ApJ,264,642 Parker,E.N.1988,ApJ,330,474 tisticalmethodstoanalyzetheseeventsinthecompletedomain. Peter,H.2007,Adv.SpaceRes. 39,1814 The statistical analysis of the coronal heating reveals a power Peter,H.&Bingert,S.2012,A&A,inpress,arXiv:1209.0789 law for the heating event energies. Even though the driver, i.e. Peter,H.,Gudiksen,B.,&Nordlund,Å.2004,ApJ,617,L85 thegranularmotion,doesnotworkonallscales,theslopeofthe Peter,H.,Gudiksen,B.,&Nordlund,Å.2006,ApJ,638,1086 powerlawisconstantoverseveralordersofmagnitudeandflat- Press, W. H., Teukolsky, S. A., Vetterling, W. T., & Flannery, B. P. 1992, NumericalrecipesinC.Theartofscientificcomputing terthan−2.Wefindthatinthetransitionregionandcoronathe Priest,E.R.1982,SolarMagnetohydrodynamics(D.Reidel,Dordrecht) dominantenergyforthe heatingeventsisthe nanoflareenergy, Rosner,R.,Tucker,W.H.,&Vaiana,G.S.1978,ApJ,220,643 i.e. 1017J. Since the higher energy events occur on average at Rosner,R.&Vaiana,G.S.1978,TheAstrophysicalJournal,222,1104 lowerheights,theheatingisfootpoint-dominated.Theeventen- Spitzer, L.1962,Physicsoffullyionized gases(Interscience, NewYork(2nd edition)) ergieswefindinourmodelarebelowthedetectionlimitofcur- Viall,N.M.&Klimchuk,J.A.2011,ApJ,738,24 rentinstrumentationandsupportthescenarioofnanoflareheat- Virkar,Y.&Clauset,A.2012,ArXive-prints ingasproposedbyParker(1988). Warren,H.P.,Winebarger,A.R.,&Hamilton,P.S.2002,ApJ,579,L41 In ourmodelthe energyprovidedby the field-line braiding Withbroe,G.L.&Noyes,R.W.1977,ARA&A,15,363 Zacharias,P.,Bingert,S.,&Peter,H.2011,A&A,531,A97 isconvertedintoheatingthroughOhmicdissipationandnotinto particleacceleration.Thisimpliesthatnonlocaleffectsarenot incorporated.Thisisjustifiedbytheelectronmean-free-pathbe- ing smaller than the grid spacing of the model. The energy is redistributed by advection and conduction or radiated by opti- callythinemission.Fromourmodelatmospherewesynthesized observable extreme UV emission lines. The statistical analysis of these lines shows again a power-law behavior. The slope of this power law is around 1 and smaller than the slope of the heatingprocess.Eventhoughthepowerlawismaintainedover severalordersof magnitude,in contrastto the heating process, theradiativelossesarenotinaself-organizedcriticalstate.This is because the radiative losses are coupled to the heating in a nonlinearly. In this study we showed that a 3D MHD model of the ac- tive region solar corona has a preferred energy for depositing ofheatthroughOhmicdissipationfollowingfield-linebraiding. Thepreferredenergyofabout1017J(1024erg)correspondswell to the value derived by Parker (1988) and thus is yet another piece of evidence that, despite all their shortcomings, the 3D 8

See more

The list of books you might like

Upgrade Premium
Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.
DMCA & Copyright: Dear all, most of the website is community built, users are uploading hundred of books everyday, which makes really hard for us to identify copyrighted material, please contact us if you want any material removed.
Copyright @ PDFdrive.to, 2022 | We love our users