Table Of ContentTemporal Intermittency of Energy Dissipation in Magnetohydrodynamic Turbulence
Vladimir Zhdankin,1,∗ Dmitri A. Uzdensky,2 and Stanislav Boldyrev1
1Department of Physics, University of Wisconsin,
1150 University Avenue, Madison, Wisconsin 53706, USA
2Center for Integrated Plasma Studies, Physics Department,
UCB-390, University of Colorado, Boulder, CO 80309
(Dated: January 9, 2015)
Energy dissipation in magnetohydrodynamic (MHD) turbulence is known to be highly inter-
mittent in space, being concentrated in sheet-like coherent structures. Much less is known about
intermittency in time, another fundamental aspect of turbulence which has great importance
5
for observations of solar flares and other space/astrophysical phenomena. In this Letter, we
1
investigate the temporal intermittency of energy dissipation in numerical simulations of MHD
0
turbulence. We consider four-dimensional spatiotemporal structures, “flare events”, responsible
2
for a large fraction of the energy dissipation. We find that although the flare events are often
n highlycomplex,theyexhibitrobustpower-law distributionsandscaling relations. Wefindthatthe
a probability distribution of dissipated energy has a power law index close to α ≈ 1.75, similar to
J
observationsofsolarflares,indicatingthatintensedissipativeeventsdominatetheheatingofthesys-
7 tem. Wealsodiscussthetemporalasymmetryofflareeventsasasignatureoftheturbulentcascade.
]
E
PACSnumbers: 52.35.Ra,95.30.Qd,96.60.Iv,52.30.Cv
H
.
h Introduction.—Intermittencyplaysamajorroleintur- termittency in MHD turbulence. A major question is
p
bulence by causing processes such as energy dissipation whether, in the limit of large Reynolds number, energy
-
o and particle acceleration to be highly localized in coher- dissipation is dominated by a few intense, large-scale
r entstructures. Italsoforestallseffortstowardacomplete events or by many weak, small-scale events. A related
t
s theoryofturbulence. Manytools havebeenemployedto question is whether there is an inherent relationship be-
a
study intermittency, including structure functions [1, 2], tween spatial intermittency and temporal intermittency,
[
scale-dependent kurtosis [3], topological methods [4, 5], e.g., whether larger structures retain their coherency in
1
and statistics of discontinuities [6, 7]. However, past time. Thesetemporalaspectsofintermittencyhavebeen
v
studieshavemainlyfocusedonspatialintermittency,giv- practically unexplored in previous MHD studies.
4
6 ing limited information about the dynamics. In order In this Letter, we extend a framework previously de-
6 to understand the temporal aspects of intermittency, in- veloped for the statistical analysis of dissipative struc-
1 cluding characteristic timescales of structures as well as tures [14] into the temporal realm, thereby considering
0
their interactions and stability, a broader framework is 4D spatiotemporal objects representing flare events. We
.
1 needed. apply this novel methodology to study intermittency in
0
A promising new paradigm is the statistical analysis numericalsimulationsofstrongincompressibleMHDtur-
5
of coherent structures, which is robust and informative bulence. Wedescribethedistributions,scalings,andevo-
1
: for studies of intermittency. The occurrence rates, in- lutionofflareeventsbycharacterizingtheirlengthscales,
v
tensities, and morphology of structures yield insight to durations, dissipated energies, and peak energy dissipa-
i
X the inhomogeneity, anisotropy, and characteristic scales tionrates. Thesearethefirstresultsonthefundamental
r ofthedynamics. Coherentstructurescanbesimplyiden- properties of the combined spatial and temporal inter-
a
tified as regions in space bounded by an isosurface of mittency of energy dissipation in 3D MHD turbulence.
some field. This was used to study vorticity filaments TheprimaryquestionsaddressedhereforMHDturbu-
in hydrodynamic turbulence [8–10], magnetic structures lence are also fundamental for the solar corona. In fact,
in the kinematic dynamo [11], and dissipative structures our approach has strong similarities with observational
inmagnetohydrodynamic(MHD)turbulence[12–15]and studies of solar flares [23–32] and stellar flares [33–36],
ambipolardiffusionMHD [16]. Since coherentstructures whichusethetime-seriesofX-rayandextremeUVemis-
andintensedissipativeeventsareexperimentallyobserv- sionstomeasuretheduration,peakintensity,andfluence
able, there are many practical applications including so- of flares, from which dissipated energy is inferred. For
larflares,instabilitiesinfusiondevices[17],andradiative the solar corona, a measurement of central importance
signaturesinopticallythinastrophysicalplasmas,e.g.,in is the probability distribution for dissipated energy, due
black-holeaccretiondiskcoronae[18],hotaccretionflows to its role in assessing the nanoflare model for coronal
[19], and jets [20]; in pulsar wind nebulae [e.g. 21, 22]; heating [37, 38]. This distribution exhibits a power law
and possibly in the hot gas in galaxy clusters. overeightordersofmagnitude,withanindex near 1.8,
−
This Letter addressessomefundamental aspects ofin- somewhatshallowerthanthecriticalindexof 2required
−
2
for nanoflares to dominate the overallheating [39]. giving v ∆t < b /j . The cadences are compara-
rms rms thr
We compare our results with the observed statistical ble to this value and the results show convergence with
propertiesofsolarflares. Westressthatthereareseveral cadence.
basic differences between incompressible MHD turbu-
lenceandthesolarcorona. Incontrasttovolumetrically-
TABLE I.Simulations
driven turbulence, the solar corona is modeled by force-
Sim. Res. Re ∆t−1 Time interval
free MHD with slowly-driven, line-tied boundaries. Fur-
1 2563 800 64 10.0
thermore, kinetic effects may become important during
2 5123 800 32 12.2
reconnection. Despite these differences betweenour sim-
ulations and the solar corona, we find that the statisti- 3 5123 1250 64 15.6
calproperties of flareevents have multiple similarities in 4 5123 1800 32 12.2
bothcases. ThissuggeststhatMHDturbulencemayplay
a role in the energetics of the corona [30, 40], a possibil- We refer to spatial dissipative structures in a given
ity that should be investigated more carefully in future time snapshot as states; they are identified as spatially-
studies. connected sets of points with current densities j(x) ex-
| |
ceeding a fixed threshold, j [43]. Typical states are
thr
Method.— We perform simulations of reduced MHD, thin, ribbon-like current sheets aligned with the z direc-
applicable since the uniform background magnetic field tion. They occupy a small fraction of volume but ac-
B0 = B0zˆ is strong relative to turbulent fluctuations, count for a large fraction of the overall resistive energy
B0 5brms. The equations are [41] dissipation. Their lengths and widths span the inertial
≈
∂ range, while thicknesses are localized inside the dissipa-
(cid:18)∂t ∓VA·∇k+z∓·∇⊥(cid:19)z± =−∇⊥P +ν∇2⊥z±+f±⊥ tion range [13, 14].
Ourpresentworkextendsthisprocedureintothetem-
z± =0 (1)
∇⊥· poral realm by applying a similar threshold criterion to
where z± = v b are the Els¨asser variables (perpen- the 4D space-time field j(x,t). We refer to the result-
dicular to B ),±v is the velocity field, b is the fluc- ing 4D spatiotemporal structures as processes or flare
0
tuating magnetic field (in units of the Alfv´en velocity, events. Ournumericalalgorithmfirstidentifiesthestates
V = B /√4πρ , where ρ is the uniform plasma den- ineachsnapshotasdescribedabove. Itthenconnectsthe
A 0 0 0
sity), P is the total pressure, and f± is the external states in time by finding, for each given state, any other
⊥
forcing. We use uniform fluid viscosity ν and magnetic states in the two adjacent (past and future) snapshots
diffusivity η with ν = η. We consider structures in the having points connected to the same spatial region. In
current density j =j =zˆ b; the resistive energy general, the states in one snapshot are not in bijective
z ⊥
·∇ ×
dissipation rate per unit volume is ηj2. (i.e., one-to-one) correspondence with the states in the
Equations (1) are solved using a fully dealiased 3D adjacent snapshot. This is due to interactions between
pseudo-spectralalgorithm(for details, see [42]). The pe- structures, including mergersand divisions, and also the
riodic box is elongated in zˆ by a factor of L /L = 6, spontaneous formation of new states and destruction of
k ⊥
where L = 2π is the box size in simulation units. old states. We refer to a sequence of states representing
⊥
Timescales are in units of eddy turnover times, τ = bijective evolution of a structure, beginning and ending
eddy
L /(2πv ) 1. The turbulence is driven at large with interactions, formation or destruction, as a path.
⊥ rms
≈
scales by colliding Alfv´en waves, generated from statis- Processes are then obtained as sets of paths connected
tically independent random forces f± at wave-numbers via interactions. The natural and conservative approach
⊥
2π/L k 2(2π/L ), k = 2π/L . The forc- for a temporal analysis is to study processes rather than
⊥ x,y ⊥ z k
≤ ≤
ing is solenoidal, has random Fourier coefficients taking individual paths, which become ambiguous upon inter-
Gaussianvaluesthatarerefreshedindependentlyapprox- acting.
imately 10times per eddy turnovertime, andhas ampli- We characterize processes by the following quantities.
tude such that b v 1. The Reynolds number The process duration τ is the time between the final
rms rms
∼ ∼
is given by Re=v (L /2π)/ν. state of the process and its initial state (normalized to
rms ⊥
We analyze snapshots dumped at a cadence (∆t)−1 τ ). The length L for a state is the maximum dis-
eddy
fromfoursimulationsshowninTableI.Themainresults tance between any two constituent points (normalized
are from runs with 5123 resolution, with the Re = 1250 to the perpendicular box size L ). This is generalized
⊥
casehavingthehighestcadenceandlongesttimeinterval. for a process as the maximum length, L , among con-
max
Due to computational constraints, ∆t is larger than the stituent states. The instantaneous Ohmic energy dissi-
internal time step in the simulation. The minimum ca- pation rate for a state is = dVηj2, with integration
E
dence required to properly track structures is estimated over the constituent points ofRthe state (normalized to
by requiring that the distance advected by the flow dur- average total energy dissipation rate, 1). This
tot
E ≈
ing ∆t is less than the typical current sheet thickness, is generalized to the total dissipated energy of a pro-
3
cess, E = dt dVηj2, by combining the energy dis- relations, with examples shown in the scatterplots in
sipation ratRes oRf all constituent states (normalized to Fig. 3. We find that Lmax τ, previously inferred
∼
E τ 1). We also consider the peak energy in solar flare observations [45], while τ2 and
tot tot eddy max
≈E ≈ E ∼
dissipationrate, ,whichisthemaximumenergydis- E τ3. These scalings are consistent with the estimate
max
E ∼
sipation rate among constituent states. E = dt dVηj2 τ τV ηj2 τ3, assum-
∼ Emax ∼ max thr ∼
ing thRat vRolume scales as length squared [13, 14] and
Results.— For the following analysis, we choose current densities are near j . From these correlations,
thr
jthr/jrms ≈ 6.8. This threshold is high enough to avoid E ∼Em3/a2x, and hence P(E) is shallower than P(Emax).
percolation of processes through space and time. Pro- Finally, we consider the evolution of processes over
cesses that exist during the initial or final snapshots do their durations, based on the constituent states at
not significantly contribute to the statistical results; we the given times. We focus on the energy dissipation
retain them in distributions for better statistics. rate, (t) for 0 < t < τ. Although this time-series
E
An example process, with duration τ 0.5 and 31 is irregular for any given process, we find that pro-
≈
distinct paths, is shown in Fig. 1. Representative states cesses of all durations exhibit similar average evolu-
are shown (in green) on a subdomain of the simulation tions. In Fig. 4, we show the energy dissipation rate
grid. We alsoshowaschematic ofthe paths andinterac- normalized to the peak, (t)/ , versus normalized
max
E E
tionsintheprocess. Theprocessincludesadivisionafter time, t/τ, averaged for all processes. We find that
the structure is stretched. A large number of paths are (t/τ)/ sin(πt/τ). Minor deviations suggest
max
hE E i ≈
produced during the final stages, as the process decays atemporalasymmetry,quantifiablebythefirstmoment,
toward the threshold. t/τ = τ(t/τ) (t)dt/ τ (t)dt, which is 0.5 for sym-
h iE 0 E 0 E
We now consider the statistical properties of the metric funRctions. We meRasure a very small but consis-
processes from the four simulations in Table I. The tent asymmetry: for Re = 800,1250,1800 , t/τ =
E
{ } h i
mean number of states per snapshot is Nstate = 0.483,0.483,0.476 . Hence a flare event grows slightly
194,288,657,1328 . For fixed cadence of ∆ht−1 =i 32, f{asterthanitdecay}s;thisissimilartoobservedsolarand
{ }
the mean number of processes per eddy turnover time stellarflares,althoughtheasymmetryisamplifiedbythe
is Nproc = 914,1271,4272,11608 , strongly increasing Neupert effect [28, 29, 32, 36, 46].
{ }
with Re. The most complex processes have 103 con-
∼
stituent paths. We find a consistent asymmetry in the Conclusions.— In this Letter, we investigate the com-
interactions: therearemoredivisionsthanmergers,with binedtemporalandspatialintermittencyofenergydissi-
a ratio Nmer/Ndiv = 0.84,0.78,0.80,0.82 . pationinnumericalsimulationsofMHDturbulence. The
{ }
We show in Fig. 2 the probability distributions for conclusions are robust with respect to cadence, resolu-
dissipated energy E and for peak energy dissipation tion, and threshold. We find that a significant fraction
rate . The distribution for dissipated energy, P(E), of energy dissipation occurs in current sheets that are
max
E
hasapower-lawtailwithanindexnear 1.75 0.1,which temporally organized in intense, long-lived flare events
− ±
isclosetothe analogousobservationsfortotalenergyre- withdurationsthatmayspanseverallargeeddyturnover
leased in solar flares [29, 44]. The power law extends times. The process duration scales proportionally to its
across three orders of magnitude in E, from E 10−5 maximum length. The energy dissipated in these in-
≈
up to about E 10−2. For smaller E, the distribution tense processes is distributed as a power law with in-
≈
is shallower and apparently non-universal, likely due to dexnear 1.75,implyingthedominanceoflarge,intense
−
dissipation-range effects and threshold effects. With in- flare events. This can be compared to the spatial struc-
creasingRe, the power law extends to smaller E, consis- tures at fixed times, which have a distribution of energy
tent with the longer inertial range. The distribution for dissipation rates closer to the critical index of 2, sug-
−
peak energy dissipation rate, P( ), exhibits a power gesting that structures of all intensities instantaneously
max
E
law with index close to 2.0 0.1 from 10−4 contribute equally to the overall energy dissipation rate
max
− ± E ≈
to 10−2. Similar indices are observed in distri- [14].
max
E ≈
butions for peak hard X-ray flux in solar flares [e.g., 44] We find that the distributions and scalings are in-
and for energy dissipation rates of states [14]. sensitive to Re, suggesting universality. This is consis-
E
The distribution for process durations τ is also shown tentwiththe factthatthe structuresarecoherentacross
in Fig. 2. The durations extend to well above an eddy inertial-range scales, with only their thickness being set
turnover time, sometimes comparable to the analyzed by the dissipation mechanism. The dependence of the
time interval. The distribution from τ 0.2 to τ 8 statistics on Re has likely saturatedatthe relatively low
≈ ≈
can be fit to a power law with index near 3.2 0.2, Re consideredhere, making the results relevantfor large
− ±
somewhatsteeperthantheindicesrangingbetween 2.2 Re in space and astrophysicalturbulence.
−
and 3.0 for solar flare durations [23, 32, 44], although We find asymmetry in the divisions and mergers of
−
close to the index 3.4 for rise times [28]. current sheets, as well as in the evolution of the energy
−
The process characteristics are related by strong cor- dissipation rate of a process. This temporal asymmetry
4
FIG.1. Statesin atypicalprocess with duration τ ≈0.5, shown in green on apieceof thesimulation lattice (with dimensions
0.10L⊥×0.14L⊥×0.90L⊥,withoutaccountingforelongation ofthelatticevertically). Alsoshownisaschematicofpathsand
interactions in theprocess, with red lines marking the times corresponding to theshown states.
FIG. 2. The probability distributionsfor dissipated energy E,peak energy dissipation rateEmax, and duration τ for Re=800
(red), Re=1250 (blue), and Re=1800 (green).
can possibly be linked to the direct cascade of energy
from large to small scales, giving large structures a ten-
dencytodivideintosmallerstructures. Infuturestudies,
∗
[email protected]
temporal asymmetry may be a useful diagnostic for cur-
[1] W.-C. Mu¨ller, D. Biskamp, and R. Grappin, Physical
rent sheet instabilities, including the tearing instability Review E, 67, 066302 (2003).
[47–50]. [2] C. Chen, A. Mallet, T. Yousef, A. Schekochihin, and
The present work lays out the foundation for a com- T. Horbury,Monthly Notices of the Royal Astronomical
Society, 415, 3219 (2011).
prehensive statistical analysis of dissipative processes in
[3] M. Wan, K. T. Osman, W. H. Matthaeus, and
MHD turbulence, to appear in a future paper [51]. The
S.Oughton,TheAstrophysicalJournal,744,171(2012).
methodology can be applied to many other systems, in-
[4] S. Servidio, W. Matthaeus, M. Shay, P. Cassak, and
cluding hydrodynamic turbulence [52], line-tied MHD P. Dmitruk, Physical Review Letters, 102, 115003
[15, 53], kinetic plasma turbulence [54, 55], avalanching (2009).
systems [56], and other complex dynamical systems. [5] S.Servidio,W.Matthaeus,M.Shay,P.Dmitruk,P.Cas-
sak, andM.Wan,PhysicsofPlasmas,17,032315(2010).
The authors would like to thank Jean Carlos Perez
[6] A. Greco, W. Matthaeus, S. Servidio, and P. Dmitruk,
for his support in conducting the numerical simula- Physical Review E, 80, 046401 (2009).
tions. This work was supported by NASA grant No.
[7] V.Zhdankin,S.Boldyrev,J.Mason, andJ.Perez,Phys-
NNX11AE12G, US DOE award DESC0003888, and the ical Review Letters, 108, 175004 (2012).
NSF Center for Magnetic Self-Organization in Lab- [8] J. Jim´enez, A. A. Wray, P. G. Saffman, and R. S. Ro-
oratory and Astrophysical Plasmas at U. Wisconsin- gallo, Journal of Fluid Mechanics, 255, 65 (1993).
Madison. [9] F. Moisy and J. Jim´enez, Journal of Fluid Mechanics,
5
FIG. 3. Scatterplots of maximum length Lmax, peak energy dissipation rate Emax, and dissipated energy E versus the process
duration τ for Re=1250.
maninasab, G. Witzel, R. Sch¨odel, M. Garc´ıa-Mar´ın,
L.Meyer,G.Bower,etal.,AstronomyandAstrophysics,
500, 935 (2009).
[20] J. Albert, E. Aliu, H. Anderhub, P. Antoranz, A. Ar-
mada, C. Baixeras, J. Barrio, H. Bartko, D. Bastieri,
J. Becker, et al., The Astrophysical Journal, 669, 862
(2007).
[21] M. Tavani, A. Bulgarelli, V. Vittorini, A. Pellizzoni,
E.Striani,P.Caraveo,M.Weisskopf,A.Tennant,G.Pu-
cella, A.Trois, et al.,Science, 331, 736 (2011).
[22] A. Abdo, M. Ackermann,M. Ajello, A. Allafort, L. Bal-
dini, J. Ballet, G. Barbiellini, D. Bastieri, K. Bechtol,
R. Bellazzini, et al., Science, 331, 739 (2011).
[23] N. B. Crosby, M. J. Aschwanden, and B. R. Dennis,
Solar Physics, 143, 275 (1993).
[24] T. Shimizu, Publications of the Astronomical Society of
Japan, 47, 251 (1995).
FIG. 4. The evolution of energy dissipation rate relative to
[25] G. Boffetta, V. Carbone, P. Giuliani, P. Veltri, and
the peak, E(t/τ)/Emax, averaged across all processes of all A. Vulpiani,Physical review letters, 83, 4662 (1999).
durations, for Re = 800 (red), Re = 1250 (blue), and Re = [26] C.ParnellandP.Jupp,TheAstrophysicalJournal,529,
1800 (green). This is well fit by a sine function (black), with
554 (2000).
small asymmetric deviations.
[27] I. Hannah, S. Christe, S. Krucker, G. Hurford, H. Hud-
son, andR.P.Lin,TheAstrophysicalJournal,677,704
(2008).
[28] S. Christe, I. Hannah, S. Krucker, J. McTiernan, and
513, 111 (2004). R. P. Lin, TheAstrophysical Journal, 677, 1385 (2008).
[10] T. Leung, N. Swaminathan, and P. Davidson, Journal [29] M. J. Aschwanden, T. TARBELL, R. W. Nightingale,
of Fluid Mechanics, 710, 453 (2012). C.J.Schrijver,C.C.Kankelborg,P.Martens,H.P.War-
[11] S.L.Wilkin,C.F.Barenghi, andA.Shukurov,Physical ren, et al., The Astrophysical Journal, 535, 1047E`1065
review letters, 99, 134501 (2007). (2000).
[12] V.M.Uritsky,A.Pouquet,D.Rosenberg,P.D.Mininni, [30] V. M. Uritsky, M. Paczuski, J. M. Davila, and S. I.
and E. F. Donovan, Physical Review E, 82, 056326 Jones, Physical review letters, 99, 025001 (2007).
(2010). [31] V.M.Uritsky,J.M.Davila,L.Ofman, andA.J.Coyner,
[13] V. Zhdankin, D. Uzdensky, J. Perez, and S. Boldyrev, The Astrophysical Journal, 769, 62 (2013).
Astrophys.J., 771, 124 (2013). [32] A.Veronig,M.Temmer,A.Hanslmeier,W.Otruba, and
[14] V. Zhdankin, S. Boldyrev, J. Perez, and S. Tobias, As- M. Messerotti, Astronomy and Astrophysics, 382, 1070
trophys.J., 795, 127 (2014). (2002).
[15] M. Wan, A. F. Rappazzo, W. H. Matthaeus, S. Ser- [33] M. Audard, M. Gu¨del, and E. F. Guinan, The Astro-
vidio, andS.Oughton,TheAstrophysicalJournal,797, physical Journal Letters, 513, L53 (1999).
63 (2014). [34] R. Pallavicini, G. Tagliaferri, and L. Stella, Astronomy
[16] G. Momferratos, P. Lesaffre, E. Falgarone, and G. P. and Astrophysics, 228, 403 (1990).
des Forˆets, Monthly Notices of the Royal Astronomical [35] M. Gu¨del, M. Audard, V. L. Kashyap, J. J. Drake,
Society,443, 86 (2014). and E. F. Guinan, The Astrophysical Journal, 582, 423
[17] V. Carbone, L. Sorriso-Valvo, E. Martines, V. Antoni, (2003).
and P. Veltri, Physical Review E, 62, R49 (2000). [36] A.Telleschi,M.Gu¨del,K.Briggs,M.Audard,J.-U.Ness,
[18] T.DiMatteo,A.Celotti, andA.C.Fabian,MonthlyNo- and S. L. Skinner, The Astrophysical Journal, 622, 653
ticesoftheRoyalAstronomicalSociety,304,809(1999). (2005).
[19] A.Eckart,F.Baganoff,M.Morris,D.Kunneriath,M.Za-
6
[37] E. Parker, The Astrophysical Journal, 330, 474 (1988). Physics of Plasmas, 16, 112102 (2009).
[38] H.Hudson,Solar Physics, 133, 357 (1991). [48] D.Uzdensky,N.Loureiro, andA.Schekochihin,Physical
[39] Given the distribution for energy dissipation, P(E) ∼ Review Letters, 105, 235002 (2010).
E−α,thecriticalindexisderivedbynotingthatthetotal [49] Y.-M. HuangandA. Bhattacharjee, Physical review let-
energy dissipation, Etot ∝ REEmmianxEP(E)dE, scales with ters, 109, 265002 (2012).
thelowerboundEminifα>2andwiththeupperbound [50] N. F. Loureiro, R. Samtaney, A. A. Schekochihin, and
Emax if α<2. D. A.Uzdensky,Physics of Plasmas, 19, 042303 (2012).
[40] M. K. Georgoulis, Solar Physics, 228, 5 (2005). [51] V. Zhdankin, D. A. Uzdensky, S. Boldyrev, and J. C.
[41] D. Biskamp, Magnetohydrodynamic turbulence (Cam- Perez, In preparation.
bridge UnivPr, 2003). [52] J.McWilliams,J.Weiss, andI.Yavneh,JournalofFluid
[42] J.Perez,J.Mason,F.Cattaneo, andS.Boldyrev,Phys- Mechanics, 401, 1 (1999).
ical Review X, 2, 041005 (2012). [53] A.RappazzoandM.Velli,PhysicalReviewE,83,065401
[43] Two points are considered spatially connected if sepa- (2011).
rated by fewer than two grid spacings. [54] E. Leonardis, S. Chapman, W. Daughton, V. Royter-
[44] K.R.Bromund,J.M.McTiernan, andS.R.Kane,The shteyn, and H. Karimabadi, Physical Review Letters,
Astrophysical Journal, 455, 733 (1995). 110, 205002 (2013).
[45] M. J. Aschwanden, R. A. Stern, and M. Gu¨del, The [55] H. Karimabadi, V. Roytershteyn, M. Wan,
Astrophysical Journal, 672, 659 (2008). W. Matthaeus, W. Daughton, P. Wu, M. Shay,
[46] M.Temmer,A.Veronig,A.Hanslmeier,W.Otruba, and B. Loring, J. Borovsky, E. Leonardis, et al., Physics of
M. Messerotti, Astronomy and Astrophysics, 375, 1049 Plasmas (1994-present), 20, 012303 (2013).
(2001). [56] M.Bregman andM.Gedalin,PhysicaA:Statistical Me-
[47] A.Bhattacharjee,Y.-M.Huang,H.Yang, andB.Rogers, chanics and its Applications, 387, 2328 (2008).
