Nanoflare statistics in an active region 3D MHD coronal model

0.19 MB·English

by  Sven Bingert

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