Force-free field modeling of twist and braiding-induced magnetic energy in an active-region corona
aa r X i v : . [ a s t r o - ph . S R ] N ov A ccepted F ebruary
19, 2018 for publication in A p J Preprint typeset using L A TEX style emulateapj v. 5 / / FORCE-FREE FIELD MODELING OF TWIST AND BRAIDING-INDUCED MAGNETIC ENERGY IN ANACTIVE-REGION CORONA
J. K. T halmann and
S. K. T iwari and T. W iegelmann Accepted February 19, 2018 for publication in ApJ
ABSTRACTThe theoretical concept that braided magnetic field lines in the solar corona may dissipate a su ffi cient amountof energy to account for the brightening observed in the active-region corona, has been substantiated by high-resolution observations only recently. From the analysis of coronal images obtained with the High ResolutionCoronal Imager, first observational evidence of the braiding of magnetic field lines was reported by Cirtain et al.(2013) (hereafter CG13). We present nonlinear force-free reconstructions of the associated coronal magneticfield based on vector SDO / HMI magnetograms. We deliver estimates of the free magnetic energy associated toa braided coronal structure. Our model results suggest ( ∼
100 times) more free energy at the braiding site thananalytically estimated by CG13, strengthening the possibility of the active-region corona being heated by fieldline braiding. We were able to assess the coronal free energy appropriately by using vector field measurementsand attribute the lower energy estimate of CG13 to the underestimated (by a factor of 10) azimuthal fieldstrength. We also quantify the increase of the overall twist of a flare-related flux rope which had been claimedby CG13. From our models we find that the overall twist of the flux rope increased by about half a turn withintwelve minutes. Unlike another method, to which we compare our results to, we evaluate the winding of the fluxrope’s constituent field lines around each other purely based on their modeled coronal 3D field line geometry –to our knowledge for the first time.
Subject headings:
Sun: photosphere — Sun: corona — Sun: magnetic fields — Sun: evolution — Sun: activity— Sun: flares INTRODUCTION
The plasma of the solar corona is much hotter ( & K)than that of the photosphere ( ∼ ff erent. Theplasma temperature in the AR corona is 8-20 × Kwhich is by a factor of 4-10 higher than that of the quiet-Sun corona (e. g. Zirker 1993). The most widely be-lieved phenomenon that accounts for the heating of the AR(magnetically-closed) corona is the braiding of (ensembles of)magnetic field lines (which numbers their crossings; see e. g.,Berger & Asgari-Targhi 2009). It results in high temperatureseither by Joule (ohmic) heating of currents induced by en-tangled magnetic field lines (Parker 1972) or by nano-flaresoccurring when neighboring, oppositely directed field linesreconfigure via magnetic reconnection (Parker 1983, 1988;Priest et al. 2002). The latter is supported by theoretical mod-els (e. g. Klimchuk 2006). The former was recently inves-tigated by Bourdin et al. (2013) who compared synthesizedemission from a forward 3D MHD coronal model to actualcoronal images. They were able to show that the field line [email protected] Institute of Physics / IGAM, University of Graz, Universit¨atsplatz 5,8010 Graz, Austria Max Plank Institute for Solar System Research, Max-Planck-Str. 2,37191 Katlenburg-Lindau, Germany braiding delivered an energy input required for the observedheating of the AR-corona by ohmic dissipation.The braiding of magnetic field lines can be caused by ran-dom displacements of where magnetic field lines are line-tiedat a photospheric level (i. e. of their footpoints) or by vor-tical motions of the photospheric plasma, the latter result-ing in the twisting (winding) of field lines about each other.Di ff erent MHD models on magnetic field line braiding havebeen developed and most of them support the former mech-anism (e. g., Gudiksen & Nordlund 2005; Rappazzo et al.2008; van Ballegooijen et al. 2011, and references therein).The observational evidences of these processes have neverbeen very clear, longing for high resolution observations. Thedevelopment of recent space-based instruments, e. g., the So-lar Optical Telescope (SOT) on board Hinode (Kosugi et al.2007; Tsuneta et al. 2008; Suematsu et al. 2008) and the At-mospheric and Imaging Assembly (AIA) on board
SDO (Lemen et al. 2012), and their delivery of high-resolutioncoronal images allow us to have a closer look to the mech-anisms heating the coronal plasma. In particular, the data ob-tained from NASA’s recently flown rocket carrying the HighResolution Coronal Imager (Hi-C; see Golub et al. 2006, andCirtain et al. (2013)) with a spatial resolution of ∼ ′′ (6times that of AIA with ∼ ′′ ), have given an unique opportu-nity for a fresh look at the coronal heating mechanism. UsingHi-C data, Cirtain et al. (2013) (hereafter “CG13”) claimedfirst observational evidence of the braiding mechanism to de-liver the amount of energy required to heat the AR corona.However, a direct computation of the free magnetic energystored in the AR loops was not possible due to the lack ofdirect coronal magnetic field measurements. In this work,we close this gap using a nonlinear force-free (NLFF) coro-nal magnetic field model to substantiate the estimated energybudget. Another important aspect of the analysis of CG13 was theseemingly increasing twist of a magnetic structure during therising phase of a small flare (which the 5-minute observationtime of Hi-C covered). The twist of a magnetic structure isdetermined by the winding of the magnetic field lines arounda central axis and is related to its helicity (e. g. Berger 1999).Attempts to estimate the twist of AR magnetic fields havebeen made based on the length of field lines and the force-free parameter where they are line-tied at a photospheric level(e. g., Leamon et al. 2003) but were suspected to underesti-mate an AR’s global twist using a “best-fit” force-free param-eter. Leka et al. (2005) suspected that only when applied tothin flux tubes this method may correctly recover the windingof the flux rope axis (see also, e. g., Inoue et al. 2012). Onan AR scale, however, other guesses for a global value of theforce-free parameter might be appropriate (e. g., Tiwari et al.2009). Here, we try a novel approach to estimate the windingof a flux rope’s constituent field lines in the corona by usingtheir 3D geometry as inferred from the NLFF modeling. Inthis way we aim to verify the overall increase of the twist of aflare-associated structure and to compare the result to the esti-mate of the twist based on the field line length and the averageforce-free parameter at their footpoints. OBSERVATIONS AND MODELING
We first align the
SDO / HMI (Schou et al. 2012) vectormaps (Borrero et al. 2010, with the 180 ◦ -ambiguity of thetransverse field resolved following Metcalf (1994); Leka et al.(2009)) of NOAA AR 11520 on 2012 Jul 11 at 19:00 UTand a co-temporal AIA 19.3 nm coronal image using stan-dard IDL mapping software. In the same way, we align theHi-C observation at 18:55 UT and the AIA 19.3 nm observa-tion at 19:00 UT and select sub-fields which cover the field-of-view of the vector maps. The AIA 19.3 nm image (seeFigure 1a) shows patterns of concentrated strong emission(especially above regions of strong negative polarity; com-pare Figure 1b) on top of weaker emission on larger scalesand weakest emission in the center of the AR where largefilament channels run. The strong emission is found espe-cially around ( x , y ) ≈ (90,80) Mm, outlining a narrow, stronglyemitting magnetic structure. The Heliophysics Events Knowl-edgebase lists an AIA-flare associated to this strongly emit-ting structure and triggered for being registered by the systemin 17.1 nm and 13.1 nm. The small flare started at ∼ ∼ ∼ ′′ , and consequently that of the asso-ciated NLFF model) is clearly below that of the Hi-C data,we should still be able to grossly estimate the coronal energycontent. This is what at best can be done as long as mag-netic field measurements of higher resolution (e. g., from theSOT / Spectro-Polarimeter (SP) with ∼ ′′ in fast-mode oper- \protect http: // / hek / index.html (a)(b) F ig . 1.— (a) AIA 19.3 nm image on 2012 Jul 11 at 19:00 UT, coveringa similar region as the Hi-C 19.3 nm observations presented recently byCG13 (compare their Figure 1). (b)
Vertical magnetic field component of theHMI vector map at 19:00 UT (black / white represents negative / positive polar-ity). Rectangular boxes outline sub-regions which are used for analysis of atwisted (S1) and a braided (S2) structure. S1 encompasses to a great extendthe connectivity of the south-west part of the AR to which a recorded AIA-flare was associated. S2 outlines the region associated to a braided structure,focused on by CG13 (see their Figure 3). ation) are not available with a high temporal cadence or duringtimes of flare occurrences (the nearest-in-time SOT / SP mea-surement was completed about 1 hour before the start of theflare).HMI vector maps are available at a ∼ ff ects, we transform the magnetic field vectors tothe Heliographic coordinate system, i. e. transform the longi-tudinal and transverse field components to their vertical andhorizontal correspondents (following Gary & Hagyard 1990).The resulting local magnetic field vectors are then prepro-cessed following Wiegelmann et al. (2006) to gain force-freeconsistent boundary conditions (e. g. Aly 1984; Low 1985;Aly 1989) for the NLFF relaxation in the volume above(Wiegelmann & Inhester 2010; Wiegelmann et al. 2012). ALaplace problem for the magnetic scalar potential, matchingthe normal component of the NLFF solution on the volumes’boundary, is solved whose gradient resembles the associatedpotential field solution.For the available nearest-in-time SP vector map around17:54 UT (Skumanich & Lites 1987; Lites et al. 2007) we re-solve the ambiguity of the transverse field using the samemethod as used for the available HMI vector products(Leka et al. 2009). Hereafter, we treat the obtained vectormap in the same way as discussed above for the HMI dataand reconstruct the NLFF field above a flux-balanced vectormap. We find the sub-region which corresponds to the field-of-view of HMI by cross-correlation of the vertical magneticfield component prior to NLFF modeling. Given the di ff erentplate scale of the instruments, this allows us to consider nearlythe same sub-volume in the SP model.From Figure 1a it is evident that high, over-arching coro-nal field lines above AR 11520 do not contribute to the AIAemission pattern in the center of the AR. This makes it dif-ficult to verify the NLFF model solution (by comparison ofmodeled magnetic field lines to coronal loops seen in the AIAimage) since only the (open) field at the edges of the activeregion and some low-lying structures in its center are clearlyseen in the AIA image. Much of the central part of the ac-tive region emission is dominated by low-lying dark filamentchannels. Therefore, we verify the model results by compar-ison of to the strong AIA emission in the south-east of theAR (in Section 3.2). We, however, can indicate the globalquality of the NLFF reconstruction based on the HMI vectormaps in form of the current-weighted (CW) average of sin θ ,where 0 ≤ θ ≤ ◦ is the angle between the vectors of mag-netic field and electric current density (De Rosa et al. 2009).We find h CW sin θ i ≃ h θ i ≃ ◦ . (An entirely force-free field gives h θ i = ◦ .) RESULTS
Magnetic free energy of braided structure
We define “S2” (dashed rectangle in Figure 1), covering thearea around the observed braided structure shown in Figure 3bof CG13. Our NLFF model solutions adhere a spatial reso-lution of ∼ ′′ (when based on HMI data) and ∼ ′′ (whenbasing the modeling on SP data). The braided strands ob-served by the Hi-C instrument were exhibited angular widthsof ∼ ′′ , i. e. are below the resolution limit of our mod-els. Nevertheless, we estimate the free magnetic energy, E free ,of the volume around the observed braided structure but as-sume that the retrieved values represent some lower boundto the real amount of free energy present in the coronal vol-ume. In accordance to the assumption of CG13, we con-sider a sub-volume of ∼ km which should cover the ob-served braided structure and its nearest surrounding (whose“footprint” is outlined as S2 in Figure 1b). From the mod-els based on HMI data (called “HMI model(s)” hereafter), wefind E free ∝ J which is for all evaluated times about 5% ofthe total magnetic energy in this sub-volume (see Table 1).We repeat the order-of-magnitude estimate as described inCG13 by evaluating the free energy within a certain volume V as to be ∝ B φ V / π , where B φ is the azimuthal field of thebraided structure. For the latter, we estimate the magnitude ofthe average magnetic field in a vertical plane perpendicular tothe thought axis of the braided structure. We find B φ ∝ mTand using V = km , our analytically estimated amountof free energy becomes E free ∝ J. CG13, for comparison,
TABLE 1M agnetic energies associated to field line braiding E nl ff E pot E free [ × J]17:54 ⋆ Total, potential and free magnetic energy ( E nl ff , E pot and E free = E nl ff − E pot , respec-tively) of the 3D model fields in the volume above S2. Non-starred values are basedon the HMI models with a resolution of ≈ ′′ and covering ∼ × × , i. e., ∼ × km . ⋆ ) Values are based on the SP model with a resolution of ≈ ′′ and a volume of ∼ × × , i. e., ∼ × km .Based on the findings of a previous statistical analysis, the error of the energy esti-mates can be assumed as ∼
1% for both E pot and E nl ff and as ∼
10% for E free (seeThalmann et al. 2013). used B φ ∝
10 mT, yielding E free ∝ J only. Even when tak-ing into account the statistical error of our model-based freeenergy estimate ( ∼
10% for E free ; see Thalmann et al. 2013),the estimated free energy we find is much larger than thatestimated by CG13 (larger by a factor 10 ). Therefore, thediscrepancy of the energy estimates may be attributed to theunderestimated azimuthal magnetic flux assumed by CG13.We, additionally compare the free energy estimate at18:00 UT with that of a NLFF model based on nearest-in-timeSP vector map at 17:54 UT (called “SP model” hereafter).The order of magnitude agrees ( E free ∝ J), confirming ourfree energy estimates from the HMI models. Moreover, wefind ∼
60% more energy in the SP model. This supports theresults of a recent case study by Thalmann et al. (2013) whichalready indicated that the energy estimates based on modelsusing SP data exceed those of models based on HMI data.
Magnetic field geometry
To investigate the magnetic topology associated to the AIA-flare we select a sub-field (solid outline “S1” in Figure 1)which properly adheres the associated magnetic connection.A negative-polarity island (NPI) is discernible in the north-west of S1 (Figure 2a). The surrounding of the NPI exhibits,besides the regions towards the center of the AR, the high-est values of the vertical current density (red and blue filledcontours). An inspection of the co-temporal emission in thedi ff erent AIA channels (Figure 2b) reveals that the distinct ar-eas of maximum intensity in the various wavelengths are co-spatial. In order to outline brightest structures in the di ff erentwavelength channels consistently, the contours outline the 98percentile for each wavelength channel. This means that thecontours outline the region within which pixels of highest in-tensity are located. Depending on the wavelength channel,either kernels of emission (at 33.5, 21.1, 19.3 and 9.4 nm;presumably outlining substructures) or larger-scale emission(at 30.4, 17.1 and 13.1 nm), surrounding and partly coincid-ing the former. The fact that the strongest emission in all ofthe di ff erent channels is co-spatial indicates a multi-thermalemission which originates from spatially close regions.Now, how does the magnetic field configuration which isassumed to be partly outlined by the observed emission looklike in detail? At 19:00 UT, two minutes before the end of thesmall flare. We assume that the flare-related reconnection in anarrow current sheet somewhere near the strong emission pat-tern was accomplished by 19:00 UT and that the coronal fieldwas close to a nearly force-free post-flare configuration. Asshown in Figure 2c, a bundle of twisted field lines is present in (a) (b) (c)NPIN1N2 NPIN1N2 F ig . 2.— (a) Sub-field S1 of 2D NLFF lower boundary at 19:00 UT. The gray-scale background reflects the vertical magnetic field, B z , (black / white representsnegative / positive polarity). White / black contours are drawn at ±
10 mT. White / black arrows indicate the magnitude and orientation of the horizontal fieldoriginating from negative / positive polarity regions where B z >
10 mT. A negative polarity island (NPI) is visible as a chain of black closed contours at the north-west of S1. Red / blue filled contours resemble the vertical current density j z of ± − . (b) AIA 33.5 nm image covering S1 at 19:00 UT. Contours outlinethe 98 percentile of the maximum intensity in the 33.5 nm (blue), 30.4 nm (red), 21.1 nm (pink), 19.3 nm (brown), 17.1 nm (yellow), 13.1 nm (cyan) and 9.4 nm(dark green) wavelength channel. (c)
Selected field lines calculated from the NLFF magnetic field model above S1 at 19:00 UT. The background shows thenearest-in-time Hi-C 19.3 nm observation at 18:55 UT. The field lines are color-coded according to the absolute current density. The view is along the vertical(in negative z − ) direction. (a) (b) ←← F ig . 3.— Selected field lines at (a) (b) x − ) direction. The vertical extension of the box is ∼
12 Mm. The bottom layer reflects the vertical magnetic field component of the NLFFlower boundary. the reconstructed 3D NLFF field. (Only field lines are shownwhich start close to that part of the NPI where the positivevertical electric current is strongest; compare Figure 2a.) Thefield lines seemingly make up a flux rope which is more com-pact where it emerges from the lower boundary (in the pos-itive polarity region, bordering the NPI in the north-west ofS1) and more extended towards where it re-enters the area ofnegative polarity in the north-east of S1 (near N1). Compari-son to the co-temporal coronal image at 19.3 nm (Figure 2c)shows that the reconstructed field structure does not perfectlyoverlap the coronal emission pattern but does resemble it rea-sonably well.Within the flux rope, the strongest values of absolute cur-rent density (Figure 2c) are found at the center and bot-tom of the tightly twisted parts. When viewed from above,these locations of strong currents well coincide with placesof strongest coronal emission, despite a small spatial devi-ation of a few Mm. This suggests that the observed AIAemission represents dissipated electromagnetic energy whichcould well be induced by magnetic reconnection in strongelectric current concentrations in the twisted flux rope or Jouleheating by ohmic dissipation. The repeated brightening in thiscoronal area observed by CG13, however, supports the for-mer.
Temporal evolution of the twist
The field lines of Figure 2c are again shown in Figure 3b butwhen viewed from the side (along the negative x -direction). Additionally, we display the corresponding field line geome-try at 18:48 UT in Figure 3a which originate within the sameregion of strong positive vertical electric current density asthose at 19:00 UT. Comparison of the field line geometries 12-minutes apart suggests a reconfiguration of the magnetic field(given the above choice of regarded field lines). While no fieldlines connect to the negative polarity (N2) in the south-east ofS1 at 18:48 UT, some do so at 19:00 UT. This reconfigura-tion might be caused by magnetic reconnection related to baldbatches present at the boundaries of the NPI. Unfortunately, arelated investigation is out of the scope of this study.From a visual inspection of Figure 2c, the bundle of fieldlines warps ∼ ∼ (a)(b) (c) (d) F ig . 4.— Sub-set of all considered field lines, ending in the strongest polarity regions of N1 (white rectangle) on the NLFF lower boundary at (a) (b) at 19:00 UT. The view is along the vertical (negative z − ) direction. Projected number of turns of each field line pair estimated along a common thought axisat (c) (d) overall twist (representedby the black solid line in Figure 4c for 18:48 UT and 4d for19:00 UT). The term overall is also to account for the fact thatour method does not distinguish between the twist of the fluxtube axis itself and the winding of the field lines with respectto that twisted axis. The changing steepness of the distribu-tions of the estimated overall twist immediately suggests anincrease from ∼ ∼ ∼ ′′ and allow the fieldlines to originate only from areas of strong vertical electriccurrent), to be able to show clearly represented graphs in Fig-ure 4c and 4d, where we display the winding of every possiblycombination of pairs of field lines ( ∼ ∼ with a finer spatial sampling of 0.25 ′′ and byallowing the field lines to originate anywhere near the NPI,yielding ∼
500 field line pairs to be considered). Addition-ally, we vary the area around the NPI from which field lineshave to connect to N1 as well as in- and decrease the spatialsampling of field line footpoints. This allows us to estimatethe uncertainty of our overall twist estimate in terms of fieldline selection. This analysis yields an overall (median) twist e T = ± e T = ± P ( | x i − e x | ) / N , where x i are the elements of a sample, e x its median and N is the numberof elements). This means that, on overall, the field line con-figuration acquires more twist in the course of the AIA flare(about half a turn within 12 minutes). This not only quantita-tively confirms what was suspected from the visual inspectionof the NLFF model field line configurations (Figure 3), it alsoverifies what was suspected by CG13 through a pure visualanalysis of a 5-minute sequence of coronal images, namelythat the twist in the flare-related structure increased with time.A corresponding analysis of the magnetic field configurationat 19:12 UT reveals an ongoing but less rapid increase of theoverall twist.For comparison, we also estimate the twist of the individualfield lines using a method as often found in literature (e. g.,Leamon et al. 2003; Inoue et al. 2012). For each field lineof the considered subset we calculate T α = α L / π (where α is the mean value of the force-free parameter α = µ j z / B z at both footpoints and L is the arc-length of the field line).Here, we find a median of e T α = ± e T α = ± ∼ α based methods may underestimatethe twist of larger-scale structures within ARs. The increaseof the overall twist of the 3D magnetic structure is naturallyrelated to a correspondent change in the underlying magneticfield. The median value of the force-free parameter of allanalyzed field lines increases from e α = ± − at18:48 UT to 0.6 ± − at 19:00 UT.To judge the influence of spatial resolution on our geomet-rical twist estimate, we additionally compare the overall twistof the configuration in the SP model at 17:54 UT with thatof the nearest-in-time HMI model (at 18:00 UT). The result-ing overall twist of e T = ± ∼ ′′ ) and e T = ± ∼ ′′ ) agrees within the error ranges (which representagain the mean absolute deviation from the median). We finda similar median value of the force-free parameter from both,the higher-resolution SP ( e α = ± − ) and lower-resolution HMI ( e α = ± − ) model. SUMMARY AND DISCUSSION
Just recently, through the observation of the coronal plasmawith unprecedented spatial resolution, new insights on theprocesses heating the solar corona were gained. Most plau-sibly, such processes involve the reconfiguration of the mag-netic field at coronal heights since the associated magnetic en-ergy outclasses the kinetic, thermal and gravitational energy(e. g. Forbes 2000). Following an analytical expression tonumber the free magnetic energy associated to a spatially re-solved bundle of braided coronal loops (given by Cirtain et al.2013, hereafter “CG13”) an amount of ∝ J of free mag-netic energy was suspected. Furthermore, they estimated thatabout 0.1% of the stored energy was converted into the ob-served radiation. Moreover, they interpreted the analyzed se-quence of coronal images as to depict an increase of the twistof a magnetic structure during the rising phase of a small flare.In this study, we aimed to verify the findings of CG13by compensating the deficiency of direct observations of thecoronal magnetic field by reconstructing the associated ARcoronal nonlinear force-free field. We used measurements ofthe photospheric field vector by
SDO / HMI and
Hinode / SOT-SP (with a plate scale of ∼ ′′ and ∼ ′′ , respectively)and employed the associated, static nonlinear force-free equi-librium solution in the 3D model volume above. We firstchecked the free energy within a volume of ∼ km con-taining the braided structure and estimated an amount of ∝ J, i. e. ∼ times more than what the analytical esti-mate of CG13 delivered. We were able to attribute this di ff er-ence to the underestimated strength of the azimuthal magneticfield of the observed braided structure. This firstly, highlightsthe importance of the analysis of the coronal magnetic fieldenergy based on vector magnetic field measurements and / orforce-free model techniques and secondly, allows us to con-clude that even more free magnetic energy is available forheating the AR corona than what was suspected from the ob-servational analysis of the AR corona.We furthermore investigated a magnetic flux rope whichhas been associated to a small AIA-flare. Strongest abso-lute current density was found in those parts of the flux ropewhich were most tightly twisted. We were able to associatethe highly twisted parts of the flux rope to highest coronalemission in all AIA wavelength channels. We interpret thisspatial overlap by the field-aligned currents being dissipatedin the course of magnetic reconnection or ohmic dissipationand yielding the radiative losses which are observed in formof coronal emission signatures.We also looked at the temporal evolution of the overall twistof the flux rope by directly incorporating the modeled 3D ge-ometry of the constituent magnetic field lines. To our knowl-edge this is the first time that the shape of modeled field lineshas been used to quantify the average twist of a flux rope.For each pair of a subset of field lines making up the twistedflux rope we estimated their footpoint-to-footpoint winding.The median number of turns of all possible combinations offield line pairs allowed us to derive the increase of the over- all twist of the flux rope (from about 1.0 to 1.7 turns within ∼
12 minutes). Additionally, we used a method as commonlyused in literature to calculate the AR-twist which involves theforce-free parameter at the line-tied field line footpoints andthe field line length. Here we found similar result, namely thatthe average winding of the flux rope increased by about halfa turn in the course of the small flare. This allowed us to con-firm the assumption of Leka et al. (2005), namely the abilityof the latter method to adequately recover the winding of thinflux tubes (which we assume our field lines are) but to slightlyunderestimate the twist of larger-scale structures. For investi-gating the e ff ect of spacial resolution, we also employed theoverall twist of the structure at around one hour before theflare based on a higher-resolution SP model and a nearest-in-time ( ∼ ffi cient for the heating of the solar corona,(2) were capable to associate the localized strong coronalemission and the strong localized field-aligned currents in atwisted flux rope and (3) presented a novel approach to quan-tify the temporal evolution of the overall twist of a flare-related structure. Our investigation, on the one hand, sup-ports the conclusion drawn by CG13: the free magnetic en-ergy stored in the low-lying coronal loops (braided magneticfield lines) of an AR is su ffi cient to heat the AR corona byradiating the heat which is delivered by small-scale flares.On the other hand, our work underlines the great potential offorce-free coronal field models to partially explain observa-tional emission signatures by the associated modeled coronalmagnetic field structure, which at present time is not routinelyaccessible via direct measurements.We thank the anonymous referee for careful considerationon this manuscript and useful comments. J. K. T., acknowl-edges support from Austrian Science Fund (FWF): P25383-N27 and DFG grant WI 3211 / SDO data are courtesy of the NASA / SDO
AIA and HMI science teams.
Hinode is a Japanese missiondeveloped and launched by ISAS / JAXA, with NAOJ as do-mestic partner and NASA and STFC (UK) as internationalpartners. It is operated by these agencies in co-operationwith ESA and NSC (Norway).
Hinode
SP Inversions wereconducted at NCAR under the framework of the Commu-nity Spectro-polarimetric Analysis Center. We acknowledgethe Hi-C instrument team for making the flight data publiclyavailable. MSFC / NASA led the mission and partners includethe SAO in Cambridge (MA); LMSAL in Palo Alto (CA);UCLAN in Lancashire (UK); and the LPI RAS in Moscow.
APPENDIXESTIMATION OF FIELD LINE TWIST
To estimate the overall twist of the flare-related flux rope, we consider each pair of all considered field lines separately (red andblack solid line in Figure 5a). The average footpoint position of each field line pair is connected by a thought “principal axis” (PA)of a thought thin flux tube thought to be composed of the two field lines. Planes normal to the PA (“axis-normal (AN) planes”) (a) (b) (c) F ig . 5.— (a) A pair of randomly selected field lines from the subset of field lines shown in Figure 4b at 19:00 UT. A principal axis (PA) is defined by the meanstart and end location of the footpoint locations of the field line pair. Dashed lines mark planes perpendicular to and along the PA (called “axis-normal” (AN)planes). (b)
Intersections of the field lines with the AN planes, projected into the a common plane. Colored diamonds correspond to projected intersections ofthe field lines with the AN planes equally colored in (a). The small figure in (b) shows the progression of the angle calculated between each pair of field lineintersections with respect to the first measured (“zero”) angle. (c)
Resulting winding number of the field line pair (gray dotted curve) as estimated from theprojected angles in (b). The black dashed line shows the median value of the gradient, corresponding to the average winding of the particular field line pair. are defined and used for further analysis (in this particular case 19 planes; dashed colored lines in Figure 5a). We determine thelocations in 3D space where both of the field lines intersect these AN planes and project them into a common plane (coloreddiamonds in Figure 5b, the color accords to the respective AN plane in Figure 5a which a field line intersects). This allows usto calculate the angle spanned between each of the pair of intersections (diamonds linked by dotted lines of same color; smallpanel in Figure 5b; each angle is color-coded according to the dotted lines connecting pairs of intersections). The spanned anglenaturally is in the range − π ≤ θ ≤ π and we take the angle measured from the first intersection as reference (zero) angle. Thuswe are able to count how often two field lines are winding around each other along the thought common axis. For the examplefield line pair discussed here about 1.5 turns (gray dotted line in Figure 5c). The average gradient of the winding curve of eachfield line pair delivers the average twist of the thought thin flux tube they define (black dashed line). The median winding of allpossible pairs of field lines which connect the NPI and N1 in Figure 4a finally delivers the overall twist of the entire flux tube. APPLICABILITY AND UNCERTAINTY OF THE METHOD
The ability of our method to reasonably recover the twist of a flux rope depends also on its thickness. The thinner it is (i. e. thesmaller its cross section compared to, e. g., its length or curvature radius) the better our method is expected to work. Theseare flux ropes whose constituent field lines are everywhere close in space, i. e. where the common thought flux rope axis andconsequently the PA well represent each of its constituent field lines. However, there might be pairs of field lines whose line-tiedends are close by each other on one end of the field lines but whose footpoints on the other end of the field lines are located faraway from each other (like in case of, e. g., an expanding flux tube). The present analysis compensates partly for this for that weconsidered only field lines which connect the NPI and N1, i. e. by selection the field lines’ footpoints should be relatively closein space. We find that the mean flux tube diameters ( d ; determined by the mean relative footpoint positions of a constituent pairof field lines) are by a factor 10 – 10 smaller than the mean flux tube length ( l ; defined by the mean arc-length of the field linepair). About 90% (10%) of the considered field line pairs exhibit d / l ≤ / or spatialsampling). Here, we also note the influence of the number of AN planes used for the analysis (19 in the presented case). We findthat the results for the overall twist are almost identical when using ∼
10 or ∼ AN planes (or anything in between) and thatthe uncertainty conforms with that given in Section 3.3, namely ∝ ff erent portions ofthe considered field lines in space (Figure 5b) are a ffl icted with a greater or lesser uncertainty. To test our results of the overalltwist of the entire ensemble of considered field lines, we tilt the AN planes with respect to the PA-normal direction until they arealmost parallel to the PA (i. e., we tilt them up to ± ◦ with respect to the PA-normal direction). The arising uncertainty for theoverall twist conforms with the uncertainty ranges discussed above. REFERENCESAly, J. J. 1984, ApJ, 283, 349—. 1989, Sol. Phys., 120, 19Aschwanden, M. J. 2004, Physics of the Solar Corona. An Introduction(Praxis Publishing Ltd)Berger, M. A. 1999, Plasma Phys. Control. Fusion, 41, B167 Berger, M. A., & Asgari-Targhi, M. 2009, ApJ, 705, 347Borrero, J. M., Tomczyk, S., Kubo, M., et al. 2010, Sol. Phys., 273, 267Bourdin, P.-A., Bingert, S., & Peter, H. 2013, ArXiv e-printsCirtain, J. W., Golub, L., Winebarger, A. R., et al. 2013, Nature, 493, 501De Rosa, M. L., Schrijver, C. J., Barnes, G., et al. 2009, ApJ, 696, 1780
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