Spontaneous current-layer fragmentation and cascading reconnection in solar flares: II. Relation to observations

0.63 MB·English

by  Miroslav Bárta

Draftversion January18,2011
PreprinttypesetusingLATEXstyleemulateapjv.6/6/04
SPONTANEOUS CURRENT-LAYER FRAGMENTATION AND CASCADING RECONNECTION IN SOLAR
FLARES: II. RELATION TO OBSERVATIONS
Miroslav Ba´rta1,2,3, Jo¨rg Bu¨chner1, Marian Karlicky´2, and Pavel Kotrcˇ2
1MaxPlanckInstitute forSolarSystem Research,D-37191Katlenburg-Lindau,Germany
2AstronomicalInstituteoftheAcademyofSciences oftheCzechRepublic,CZ-25165Ondˇrejov,CzechRepublic
3 ObservatoryVlaˇsim,CZ-25801Vlaˇsim,CzechRepublic
Draft version January 18, 2011
1 ABSTRACT
1
Inthe paperbyB´artaetal. (ApJ, 2010)the authorsaddressedsomeopenquestionsofthe CSHKP
0
scenario of solar flares by means of high-resolution MHD simulations. They focused, in particular,
2
on the problem of energy transfer from large to small scales in decaying flare current sheet (CS).
n Their calculations suggest, that magnetic flux-ropes (plasmoids) are formed in full range of scales by
a
acascadeoftearingandcoalescenceprocesses. Consequently,theinitiallythickcurrentlayerbecomes
J
highly fragmented. Thus, the tearing and coalescence cascade can cause an effective energy transfer
7
acrossthescales. Inthecurrentpaperweinvestigatewhetherthismechanismactuallyappliesinsolar
1
flares. We extend the MHD simulation by deriving model-specific features that can be looked for in

observations. The results of the underlying MHD model showed that the plasmoid cascade creates a
]
R specific hierarchical distribution of non-ideal/acceleration regions embedded in the CS. We therefore
focus on the features associated with the fluxes of energetic particles, in particular on the structure
S
and dynamics of emission regions in flare ribbons. We assume that the structure and dynamics of
.
h diffusion regions embedded in the CS imprint themselves into structure and dynamics of flare-ribbon
p kernels by means of magnetic-field mapping. Using the results of the underlying MHD simulation we
-
derive the expected structure of ribbon emission and we extract selected statistical properties of the
o
r modelled bright kernels. Comparing the predicted emission and its properties with the observedones
t we obtain a good agreement of the two.
s
a Subject headings: Sun: flares — Sun: magnetic reconnection — Sun: electron acceleration
[

2 1. INTRODUCTION tion. Inspired by the “fractal reconnection” conjecture
v
ofShibata & Tanuma (2001) the authorsofPaperI per-
9 For many years the common picture of solar erup-
formeda2.5Dnumericalsimulationcoveringlargerange
6 tions and flares is based on the CSHKP model (see
ofMHDscales. Theyconjectured,that–inanalogywith
0 Magara et al. 1996, and references therein). It in-
the vortex-tube cascade in fluid dynamics – a cascade of
6 volvesreconnectioninaglobalverticalflarecurrentlayer
. formed behind ejected flux-rope/filament (Lin & Forbes magnetic flux-tubes from large to small scale can pro-
1
videthemechanismforenergytransferacrossthe scales.
1 2000, and references therein). The standard model is
Twomechanisms of fragmentationwere identified inPa-
0 ingoodagreementwithobservedlarge-scaledynamicsof
per I: (1) The tearing cascade – in line with concept of
1 eruptiveevents. Severalquestion,however,remainopen.
fractal reconnection by Shibata & Tanuma (2001), de-
: Inparticular,itisnotclear,bywhichmechanismtheen-
v veloped recently into the theory of chain plasmoid in-
ergy accumulated in relatively large-scale ( 1000 km)
i ≈ stability by Loureiro et al. (2007) and Uzdensky et al.
X structures of magnetic field associated with the flare CS
(2010),supportedfurtherbynumericalMHDsimulations
is transfered towards the small dissipation scales. In-
r by Bhattacharjee et al. (2009), Samtaney et al. (2009),
a deed, the magnetic diffusion in the almost collisionless
and Huang & Bhattacharjee (2010) and, (2) ambient-
solar coronal plasma is an essentially kinetic process at
field driven coalescence of flux-ropes/plasmoids leading
scales of the order d = c/ω (Bu¨chner 2006). Another
i pi
to formation of transversalcurrent sheets subjected fur-
open question is the relation between the well-organised
thertothesamechainofprocessesofcascadingfragmen-
dynamics of solar eruptions observed at large scales and
tation. Extrapolation of the results leads to the picture
the HXR and radio signatures of fragmented energy re-
in which the plasmoid cascade continues down to the
lease (Aschwanden 2002; Karlicky´ et al. 2000). And fi-
kinetic scale where actual magnetic dissipation and par-
nally, the CSHKP model has been put in question by
ticle acceleration occurs, most likely via the kinetic coa-
many authors (e.g. Vlahos 2007), in particular because
lescence of plasmoids (Drake et al. 2005; Karlicky´ et al.
its apparent incapability to explain the large fluxes of
2010). Nevertheless, other processes can appear in the
accelerated particles inferred from HXR observations.
range between MHD and plasma kinetic scales, e.g., the
Recently – in order to address the issue of en-
Hall type of reconnection – see recent simulations by
ergy transfer across the broad range of MHD scales –
Shepherd & Cassak (2010) and Huang et al. (2010).
B´arta et al. (2010) [in further text referred as Paper I]
As it has been shown in the Paper I, cascading re-
investigated magnetic reconnection in an extended cur-
connection forms multiple non-ideal regions (at the res-
rent layer by means of high-resolution MHD simula-
olution limit – see discussion in Paper I) hierarchically
Electronicaddress: [email protected] distributed in the fragmenting current layer. As a re-
2
sult, the open questions of energy transfer, fragmented v as
D
vs. organised flare energy release, and particle accel-
0 : v v
eprraetsieonnceseoefmcatsocabdeincglorseecloynnreelcattieodn.to each other in the η(r,t)=(cid:26) C(vD(r,vt0)−vcr) : ||vDD||≤>vccrr (1)
The questionarises,whetherthis mechanismfoundby
with the threshold v set to the higher, more realistic
simulationsisrelevantforactualsolarflares. Weaddress cr
valuecorrespondingtoincreasedresolution. Theauthors
this topic in our present paper. For this sake we derive
appliedthemodeltothelarge-scaleverticalcurrentsheet
criticalsignaturesspecific to the model developedin Pa-
in solar eruptive flares.
perI,thatallowcomparisonwithobservations. Itiscur-
The simulations inPaperI havebeenperformedin di-
rently impossible to observe small-scale magnetic struc-
mensionless variables: Spatial coordinates x, y, and z
tures, whose formation in the fragmented flare current
has been expressed in units of the current sheet half-
layer is predicted by the cascading-reconnection model,
width L at the photospheric level (z = 0) and the
directly. Almost all information on the impulsive phase A
time has been normalised to the Alfv´en transit time
of flares comes from the radiation emitted by acceler-
ated particles. Therefore we concentrate on such predic- τA = LA/VA,0, where VA,0 = B0/√µ0ρ0 is the asymp-
totic value (x ,z =0) of the Alfv´en speed at t=0.
tionsoftheMHDmodelthatareconnectedwithspecific
→∞
In order to relate the model to real solar conditions ap-
distribution and dynamics of acceleration regions in the
propriate scaling has been adopted in Paper I. Based on
fragmenting flare current layer.
thegravity-introducedscale-heightrelationL =600km
In particular, by magnetic mapping of non-ideal re- A
hasbeenfound. Duetothisrelationboththedimension-
gions embedded in the current sheet to the photo-
lessandSIunitscanbeusedfurtherinourpresentpaper.
sphere we derive expected consequences of the distribu-
In Paper I the authors described how tearing and
tion and dynamics of acceleration regions for flare rib-
drivencoalescenceinstabilitiesleadtothefragmentation
bons. Using the results of underlying MHD model pre-
of an originally unstructured flare current layer. As a
sented in Paper I we qualitatively derive the structure
consequence a cascade of magnetic flux-ropes/plasmoids
and dynamics of expected ribbon emission. We com-
isformedfromlargetoconsecutivelysmallerscales. The
parethemodelledribbonstructurewithobservationsdi-
structuring of the current layer leads to formation of
rectly and by means of its statistical properties used by
multiple thin embedded current sheets. They can host
Nishizuka et al. (2009).
non-idealregionswhereparticlescanbeaccelerated. En-
Thepaperisorganisedasfollows: First,inSection2.1,
hanced numerical resolution revealed the structure of
we briefly summarise the results of the underlying high-
dissipation regions: They form many thin channels of
resolution MHD simulation of cascading reconnection.
non-zero magnetic diffusivity. The spatial distribution
Then, in Section 2.2 we present the extension of un-
and dynamics of these diffusivity channels has been fur-
derlying MHD model utilised to produce model-specific
therstudiedvia trackingofthe X-points associatedwith
results/consequencesintheformabletobedirectlycom-
the non-ideal regions. Analysis shows hierarchically-
pared with observations. In particular, we study the
structured grouping of the dissipation channels and in-
distributionanddynamics ofdissipative/accelerationre-
termittency in their life-times and times for which they
gions, represented by the embedded X-points, and their
are magnetically connected to the photosphere.
magnetic mapping to the chromosphere/photosphere of
The structure, distribution and dynamics of the non-
the Sun. For obtained dynamics of acceleration regions
ideal regions should be reflected in specific observable
andtheirmagneticfoot-pointswequalitativelymodelthe
featuresthatwe aregoingto deriveinthe followingsub-
expected emission of flare ribbons (Section 3). We com-
section.
paretheemissionstructureofmodelledandobservedrib-
bons. Furthermore, we extract selected statistical prop-
2.2. Derivation of model-specific observable features
erties of emission kernels embedded in modelled ribbons
and compare them with the results found by analysis of Unfortunately, it is impossible to directly measure the
actual ribbon observations (Nishizuka et al. 2009). Fi- magnetic field or current density in the coronal current
nally, in Section 4 we discuss our results with the in- sheets (and at all not with the resolution reached in the
tention to evaluate the relevance of cascading reconnec- simulation done in Paper I). Hence, we need some in-
tion/fragmentationprocessespresentedinPaperIforac- direct, more subtle comparison between observed and
tual solar flares. modelled quantities. A possible way which may lead to
indirectindicationofcurrent-sheetfragmentationhasre-
2. MODEL
cently been presented by Nishizuka et al. (2009). These
In this section we describe the procedures allowing to authors study the structure of emission in flare rib-
simulate observable signatures specific to cascading re- bons,namelythedistributionanddynamicsofembedded
connection. For clarity we first briefly summarise the bright kernels. They conjecture possible relation of the
main features of current-layerfragmentation model (Pa- resulting power-law distributions found in their statisti-
per I). calanalysis of ribbon-kernelproperties with the concept
of fractal current sheet (Shibata & Tanuma 2001).
2.1. Basic MHD model
We follow this idea from the other end. We start
In Paper I the authors studied cascading reconnection with the structure, distribution, and dynamics of dif-
and energy transfer from accumulation (large) to dissi- fusion/acceleration regions that are specific to the
pation(small) scales. They useda high-resolutionMHD cascading-reconnectionmodel as they has been found in
model with a non-ideal, resistive term depending on the Paper I. Then, we relate these features to the structure
current-carrierdrift velocity anddynamicsofemissioninflareribbons. Weproceedas
3
follows: As shown in Paper I (Fig. 6), the reconnection our model) coordinate in the photosphere. The kernel
X-points are associated with thin channels of magnetic size is set to ∆=0.1L . Since the model is capable nei-
A
diffusivity. Hence,wetaketheX-pointsasgeometricrep- ther to estimate the electron-beam flux nor the energy
resentatives of dissipation regions. In these regions elec- thermalized in the kernel we set the intensity scale to
trons can be accelerated, e.g., by the DC electric fields. I = 1. We will return to this point later in the discus-
0
However,alsodifferentaccelerationmechanismshasbeen sion.
proposed – e.g. Drake et al. (2005) suggest Fermi-type Letus nowapply thisprocedurestothe resultsofsim-
acceleration in contracting plasmoids. We can include ulations presented in Paper I.
this type of acceleration into account by following con-
sideration: Extrapolating the results of numerical sim- 3. RELATIONTOOBSERVATIONS
ulations of reconnection cascade to smaller scales one Fig. 1 depicts the above-described procedure of map-
can imagine, that each dissipative region surrounding pingofX-pointstothephotospherealongmagneticsepa-
studied X-point in fact contains many unresolved small- ratrices and flare-ribbon emission calculation. Obtained
scale magnetic islands. Electron acceleration in these results are shown for t=391.5τ . Panels (a) – (c) show
A
non-ideal regions is then performed (in line with ideas how three dissipative regions/X-points embedded in the
by Drake et al. 2005) via coalescence and shrinkage of fragmented current layer are magnetically connected to
thesemicro-plasmoids. Anyway,electronsacceleratedby thebottomboundaryandmappedbyseparatricestothe
Fermi-type mechanism inside the plasmoids are trapped positionsx 2800km(one kernel)andx 3750km
andcanbereleasedtotheopenfield(whichconnectsCS (two mutua≈lly±close kernels). ≈±
to chromosphere) again in the vicinity of the X-points, A view at the flare ribbons obtained by applying
where they become demagnetized. For further drelated the simple model described by Eq. (2) is presented in
discussion on electron acceleration in magnetic islands Fig. 1(d). As the figure indicates, in some cases one
see, e.g. Oka et al. (2010). can observe even multiple pairs of flare ribbons. This is
Inourstudy wefocusonelectronsasweareinterested closetotheideaofchromosphericre-brighteningstudied
inopticalandUV/EUVchromosphericresponseascribed by Miklenic et al. (2010). The inner separatrix associ-
toelectronbeams. Thesearethentransportedalongthe atedwiththe internalpairofribbonsisconnectedtothe
magnetic field lines either up-ward to the solar corona X-point which appeared temporarily in the transversal
or downward, until they reach dense layers of the solar currentsheet formedbetween merging plasmoid and the
atmosphere, where their energy is thermalized. Here we looparcade(seealsoFig.4inPaperI).Figs.1(e)and(f)
concentrate on the down-ward transported electrons. In show the emission intensity profile along the x-axis cal-
orderto study the positions and dynamics of the points, culated according to the model in Eq. (2). The enlarged
where these electrons reach the chromosphere, i.e. the detail shownin Fig. 1(f) revealsan internal double-peak
expected bright kernels, we map found X-points to the structureoftheouterribbons. Note,thatasimilarstruc-
bottom boundary of the simulation box using magnetic turing is seen in observed H ribbons (Fig. 2).
α
field. The magnetic field lines that go through given X- In order to study the dynamics of kernels and associ-
point represent magnetic separatrices in our 2D model. ated X-points during the entire recordedinterval of evo-
Therefore, we are seeking for intersections of magnetic lution(300τ —400τ )wetrackedthemovingpositions
A A
separatriceswith the bottom boundary. In the following ofallX-pointsandthemagnetically-associated(mapped)
we will refer to this footpoints at the bottom boundary kernels. Kinematicsofthekernels(foot-pointsofsepara-
as to kernels. Thus, from knownposition of all X-points trices) is depicted in Fig 3. Panel 3(a) presents a global
(Fig. 7 in Paper I) and knowing the magnetic field we pictureofkerneldynamicsovertheentirerecordedinter-
can calculate the positions xk(t) of all kernels k for each val. Because of the CS symmetry only the right ribbon
recorded time step t of the simulation. Note, that due positions are displayed. The spatial coordinate x, the
to the 2D geometry of the model the foot-point/kernel distancefromthepolarity-inversionline(PIL),islimited
positionisuniquelygivenbyitsx-coordinate(seePaperI totheintervalwherethekernelsactuallyoccur. Fig3(b)
for details). showsadetailedviewoftheareaselectedinpanel(a). In
The calculations of the chromospheric emission in a order to reach higher zoom the global trend (motion) of
certain spectral line (e.g. Hα or C iv) in response to thegroupofkernels(i.e. theincreasingribbonseparation
the bombardment of electrons accelerated in a distant as the flare proceeds) has been subtracted by applying
reconnection region is a difficult task (Kaˇsparov´aet al. the transformation x/L = x/L 0.0125(t/τ 300).
A A A
2009; Varady et al. 2010). In order to obtain an at least X-points that are separated only−by tiny magn−etic is-
qualitativeoutputthatcouldbelatercomparedwithob- lands can map practically to the same kernel. They
servationsweuseforribbon-intensitydistributiontheex- then cannot be distinguished even in the zoomed dis-
pression play. Therefore, for each time we take a set of kernel
positions (its size varies with time) and display every
(x x (t))2
I (x,y,t)=I exp − k kernelin different colour. Fig. 3 shows various life-times
k 0 − ∆2 ! of the kernels and other aspects of their dynamics, like
bifurcation/merging. Note, that the life-time of the ker-
I(x,y,t)= I (x,y,t) , (2)
k nel is not determined solely by the life-time of the asso-
k ciated distant X-point but also by the processes in the
X
instead of description of all the complicated processes of current layer that can change the magnetic connectivity
electron transport, energy deposition, chromospheric re- of diffusion regions embedded in the currentlayer to the
sponse,andradiativetransfer. TheintensityIissummed photosphere (Fig. 7 in Paper I).
over all kernels k; y is the second (however invariant in We have further extracted selected characteristics of
4
a) b) x/LA c) x/LA d) x/LA
t=391.5 τ
A
z/L
A
m
k
m y/
k
z/
m x/km
k
z/ e) x/LA f) x/LA
nits]
u
y
bitrar
y [ar
y/LA x/LA Intensit
x/km x/km x/km x/km
Fig. 1.— Model-specific consequences of cascading magnetic reconnection. (a) Global 3D magnetic and current-density structures at
t=391.5τA. Themagneticfield(blacklines),theseparatricesthatmapthediffusiveregionstothebottomboundary(greenlines),andthe
current density (redcolour scale) areshown. (b) Projectionof (a) intothe xz-plane. The x-axisshows positions both inthe units of LA
(top)andinkilometresaccordingtothescalingadoptedinSection2.1. (c)Enlargedviewoftheselectedrectanglerevealsdouble-structure
oftheouter pairof separatriceshitting thebottom boundarynear x±3750km. (d)Modelledviewattwopairsofflareemissionribbons
(inverted colour scale, darker means higher intensity here). (e) Modelled emission profiles across the ribbons. (f) Detailed view of the
outer-ribbonprofilerevealsitsinternaldouble-peaked structure.
the modelled kernel dynamics and compare them with The simulations presented in Paper I showed, that
the observationsofNishizuka et al.(2009). Forthissake a cascading reconnection resulting from the formation
we processed our model results in the same manner as andinteractionofplasmoids/flux-ropescanaddressthese
those authors by binning photosphere along the x-axis. three fundamental questions at once and that it might
For the bin size we chose ∆x = 0.05L . For each time provide viable scenario for magnetic energy dissipation
A
we integrated the intensity given by Eq. (2) over each in large-scale systems, like solar (eruptive) flares.
bin. Thus we obtained a proxy for “light-curves” of all In the present paper we evaluated the relevance of
bins. Fig. 4 shows three examples of such light-curves this model for actual solar flares. For this sake we de-
forbinscentredaroundpositionsx=4.58L (2750km), rived observable model predictions that are specific for
A
x = 4.93L (2950 km), and x = 5.73L (3435 km). the cascading-reconnection scenario, and searched for
A A
Narrowpeaks coming from the short-living X-points are the predicted features in observed data. Since multi-
superimposed over the longer living bright kernels (c.f. ple acceleration regions embedded in the global flare
Fig. 2(b) in Nishizuka et al. 2009). current layer are – via magnetic field-line mapping –
As in Nishizuka et al. (2009), we then determined the inherently connected with the structured emission in
peaks at each light-curve and recorded their time of oc- the flare ribbons, detailed study of the ribbon kernels
currence and their maximum intensity. We calculated might bring some evidence for the cascading processes
the statistical distributions of the peak intensities and in the current layer above the flaring region. This
the time-intervals between two subsequent peaks. The idea is illustrated by Fig. 1 which shows the struc-
results of this analysis are presented in Fig. 5. Despite turing of modelled emission in ribbons. Fig. 1 fur-
ofthe poorstatistics(forour1Dmeshweidentifiedonly ther indicates, that the larger-scale plasmoid interact-
68peaks)we obtained spectralindices s= 1.48for the ing with the flare-loop arcade can temporarily form a
−
peak-intensity (panel 5(a)) and s = 1.73 for the time second pair of ribbons. The plasmoid/loop-arcade in-
−
interval between subsequent peaks (Fig. 5(b)). teraction may play significant role in flare dynamics.
Another indirect evidence of such interaction based on
4. DISCUSSIONANDCONCLUSIONS
analysis of series of X-ray images of the limb flare has
The ’standard’ CSHKP picture of solar eruptive flares been presented by Kol oman´ski & Karlicky´ (2007) and
involvestheformationofaglobalflarecurrentsheet(CS) Milligan et al. (2010).
behindaejectedfilamentfollowedbymagneticreconnec- Similar structuring of H emission as presented in
α
tion in this CS. Nevertheless, deeper study of this sce- Figs.1(d)–(f)is,indeed,visibleinFig.2. Yetanotherin-
nario invokes further questions: 1) How is the energy terestingfeature in Fig.2 is the darkfilament-like struc-
accumulated in flares in relatively large-scale structures ture (indicated by an arrow). Since the original large-
(global flare current layer) transferred to the dissipative scale filament whose eruption initiated the flare was al-
(kinetic) scales, 2) How can we observe regular dynam- ready far away at the time of observation, the absorp-
ics in eruption/flare on large scales, just in line with the tion feature can be interpretedas a manifestationof one
CSHKP scenario, and the signatures of fragmented en- of the secondary plasmoids formed in the global current
ergy release at the same moment, and 3) How can the layer in line with the presented cascading-fragmentation
large fluxes of accelerated electrons inferred from HXR scenario. The secondary plasmoids represent enhanced-
observations be reconciled with a single, relatively small density structures. Density increase in connection with
diffusion region assumed in classical CSHKP model.
5
x/km
a)
a)
b)
s]
nit
u
y
ar
bitr
ar
y [ Fig. 3.—Kinematicsofflareribbonkernelsobtainedbymagnetic
ntensit 4mth0ae0ppτpAhino.gtAoosbfpshdceiissrsseai.ppa(otais)oitnViorineewsgiiononfksaililonlmkceeartsnrceeasld(ipnuogpspirteeirocnosnscanilenec)tti=aonn3d0d0inowτuAnni—ttos
αH i rohfoLmAb.oid()b.) TDheetagilleodbavliemwotoiofnthoef athreeagsreoluepcteodf kinern(ael)s(htahsebbeleune
subtracted in order to reach larger zoom. For each time instant
eachkernel isdepicted byuniquecolor inordertodistinguishbe-
tweenclose(almostoverlapping)kernels/foot-poits.
b) In our present paper we have established the connec-
Pixel position along the path tionbetweenthe hierarchicaldistributionofdiffusionre-
gions formed by cascading reconnection and the statis-
Fig. 2.— Observation of flare Hα emission ribbons (a) and tical properties of emission kernels from first principles.
profiles of Hα intensity (b) along selected paths (red lines in the
We havestudiedthe dynamics (positions,andthe mech-
panel (a)). The ribbon emission is structured (double-peaked?)
– c.f. Fig. 1. The green arrow points to the filament-like dark anisms and times of creation/annihilation)of O- and X-
structurewhichprojectstothespacebetweenflareribbons. type points in the current layer as well as of the kernel-
points associated with the diffusion regions (X-points)
by means of magnetic field-line mapping. We then asso-
ciated each found kernel with the emission according to
consequentfaster radiativecooling mightlead to the de- Eq. (2). We chose this simple model as we do not have
tectable H absorption. The secondary plasmoid can be anyinformationabouttheacceleratedelectronfluxesab-
α
consequentlymanifestedas adarkerfeature atthe back- sorbedinthekernel,norcanweeasilycalculatethechro-
ground of relatively brighter chromosphere. mospheric response in terms of (e.g., H ) emission. Our
α
A sophisticated study of the structure of flare rib- choice of the unified peak intensity I =1 for all kernels
0
bons was presented by Nishizuka et al. (2009). These can be justified following way: After all, the particle ac-
authorsstudiedthestatisticalpropertiesofemissionker- celerationisperformedbykineticprocessesatthe small-
nels found inside the ribbons and they found a power- scale end of the cascade. Suggested mechanism (see e.g.
law distribution of the studied kernel characteristics. Drake et al. 2005) involves, e.g. kinetic coalescence and
A power-law distribution can be a signature of self- shrinkage of small-scale magnetic islands/plasmoids. In
organisedcriticalityevolution(Aschwanden2002;Vlahos the dissipative range of scales the slope of the energy
2007). It can also indicate a fractal current sheet decay spectrum(seeFig.1inPaperI)isverysteep. Hence,the
(Shibata & Tanuma 2001). The latter scenario is in fact size of plasmoids at kinetic scale can be expected more
favoured by the subsequent analysis of Nishizuka et al. or less uniform as indicated also by PIC simulations. In
(2010) which relates the energy release measured by such case the elementary injections of accelerated parti-
means of HXR flux and ejection of multiple plasmoids cles are, perhaps, roughly of the same magnitude. The
formed in the fragmenting current sheet. intermittent, spiky structure of observed emission pro-
6
a)
4.58 L
A −1.48
4.93 LA ~I
5.73 L
A
Fig. 4.—Modelledlight-curvesforthreeselectedbinsatpositions
x = 4.58LA (2750 km), x = 4.93LA (2950 km), and x = 5.73LA
(3435km).
b)
files is then the result of overlapping of many of these
elementary injection peaks (Aschwanden 2002). Indeed,
weobservethisfeatureinexamplesofthemodelledlight- −1.74
~(∆t)
curves obtained at fixed points at the bottom boundary
(Fig. 4). Despite of the intensity produced by the ele-
mentary particle injection being normalised to I = 1,
0
the overlapping elementary peaks produce spiky inten-
sity profiles ranging from I =0 to I >3.
In order to directly compare our results with the find-
ings of Nishizuka et al. (2009) we have performed the
same statistical analysis of the peaks in our artificial
light-curves. We have found the statistical distributions
of the emission peak intensities and of the time-lag be-
tween subsequent emission peaks (Fig. 5). Because of
the 2-D geometry of underlying MHD model we have
obtained 1-D distribution of emission kernels. Conse- Fig. 5.— Statistical distribution of the modelled intensity (a)
and the time-interval between subsequent emission peaks (b) for
quentlywehavefoundsmallernumberofthe peaks. De-
accumulated light-curves from all mesh-boxes. N indicates the
spite of the resulting poor statistics we could conclude, numberofpeakswiththepropertywithinthebin.
that the slopes of the power-law distributions obtained
by our model agree surprisingly well with those found
by Nishizuka et al. (2009) for observed light-curves: For
nores structuring of the current sheet in the y-direction
thestatisticaldistributionofthepeakintensitieswehave
(along the PIL). This result might indicate that modes
found spectral index s = 1.48 (Nishizuka et al. (2009)
obtained s = 1.5) and fo−r the time-lag between peaks with ky > 0 (in the off-plane direction) may not play
− too significant role in current-layer fragmentation in so-
we obtained s = 1.74 (s = 1.8 in Nishizuka et al.
− − larflares,oratleast,thatthey donotreachthe smallest
(2009)). We are aware that our statistics is, however,
scales to substantially influence the dynamics (life-time)
ratherpoorasitoperateswithonly68peaksfoundinto-
of kinetic-scale diffusion regions. This issue can be ad-
tal. Thus, it would be premature to take this agreement
dressed,however,onlywithinthe frameworkofafull3D
as conclusive. Nevertheless the result indicates that the
MHD model.
statistical properties of emission kernels resulting from
cascading reconnection and of those observed are – at
least – not in direct contradiction. This researchwas performed under the support of the
Resultsofpresentedcomparisonbetweenthemodelled European Commission through the SOLAIRE Network
featuresattributedspecificallytocascadingreconnection (MTRN-CT-2006-035484)andthegrantsP209/10/1680,
andtheirobservedcounterpartssupports–inourview– 205/09/1469, 205/09/1705, P209/10/1706 of the Grant
therelevanceofcascadingreconnectionprocessesstudied Agency of the Czech Republic, by the grant 300030701
in Paper I for reconnection in solar atmosphere. Agree- of the Grant Agency of the Czech Academy of Science,
ments found between the statistical properties of mod- andthe researchprojectAV0Z10030501ofAstronomical
elledandobservedemissionkernelsinvokes,however,fur- Institute of the Czech Academy of Science.
ther questions. The underlying 2.5D MHD model ig- Theauthorsthankstoanonymousreferee,whosecom-
7
ments whereveryhelpful inimprovingquality ofthe pa- per.
REFERENCES
Aschwanden, M.J.2002,SpaceSci.Rev.,101,1 Milligan,R.O.,McAteer,R.T.J.,Dennis,B.R.,&Young,C.A.
B´arta, M., Bu¨chner, J., Karlicky´, M., & Ska´la, J. 2010, ApJ, 2010,ApJ,713,1292,1003.0665
submitted, 1011.4035 Nishizuka,N.,Asai,A.,Takasaki,H.,Kurokawa,H.,&Shibata,K.
Bhattacharjee,A.,Huang,Y.,Yang,H.,&Rogers,B.2009,Physics 2009,ApJ,694,L74
ofPlasmas,16,112102, 0906.5599 Nishizuka, N., Takasaki, H., Asai, A., & Shibata, K. 2010, ApJ,
Bu¨chner,J.2006,SpaceScienceReviews,124,345 711,1062
Drake, J. F., Shay, M. A., Thongthai, W., & Swisdak, M. 2005, Oka,M.,Phan,T.,Krucker,S.,Fujimoto,M.,&Shinohara,I.2010,
PhysicalReviewLetters,94,095001.1 ApJ,714,915,1004.1154
Huang, Y., & Bhattacharjee, A. 2010, Physics of Plasmas, 17, Samtaney, R., Loureiro, N. F., Uzdensky, D. A., Schekochihin,
062104, 1003.5951 A. A., & Cowley, S. C. 2009, Physical Review Letters, 103,
Huang, Y.-M., Bhattacharjee, A., & Sullivan, B. P. 2010, ArXiv 105004, 0903.0542
e-prints,1010.5284 Shepherd, L. S., & Cassak, P. A. 2010, Physical Review Letters,
Karlicky´,M.,B´arta,M.,&Ryb´ak,J.2010,A&A,514,A28+ 105,015004, 1006.1883
Karlicky´,M.,Jiˇriˇcka,K.,&Sobotka,M.2000,Sol.Phys.,195,165 Shibata, K., & Tanuma, S. 2001, Earth, Planets, and Space, 53,
Kaˇsparova´, J., Varady, M.,Heinzel, P.,Karlicky´, M.,& Moravec, 473,arXiv:astro-ph/0101008
Z.2009,A&A,499,923,0904.2084 Uzdensky, D. A., Loureiro, N. F., & Schekochihin, A. A. 2010,
Ko loman´ski,S.,&Karlicky´,M.2007,A&A,475,685 PhysicalReviewLetters,105,235002, 1008.3330
Lin,J.,&Forbes,T.G.2000,J.Geophys. Res.,105,2375 Varady, M., Kaˇsparova´, J., Moravec, Z., Heinzel, P., & Karlicky´,
Loureiro,N.F.,Schekochihin,A.A.,&Cowley,S.C.2007,Physics M.2010,IEEETransactions onPlasmaScience,38,2249
ofPlasmas,14,100703, arXiv:astro-ph/0703631 Vlahos, L. 2007, in Lecture Notes in Physics, Berlin Springer
Magara,T.,Mineshige,S.,Yokoyama,T.,&Shibata,K.1996,ApJ, Verlag, Vol. 725, Lecture Notes in Physics, Berlin Springer
466,1054 Verlag,ed.K.-L.Klein&A.L.MacKinnon,15–31
Miklenic, C. H., Veronig, A. M., Vrˇsnak, B., & B´arta, M. 2010,
ApJ,719,1750