C ALCULATING S EPARATE M AGNETIC F REE E NERGY E STIMATES FOR A CTIVE R EGIONS P RODUCING M ULTIPLE F LARES : NOAA AR11158 Lucas Tarr & Dana Longcope Department of Physics, Montana State University Abstract It is well known that photospheric flux emergence is an important process for stressing coronal fields and generating magnetic free energy, which may then be released during a flare. The Helioseismic and Magnetic Imager (HMI) onboard the Solar Dynamics Observatory (SDO) captured the entire emergence of NOAA AR 11158. This region emerged as two distinct bipoles, possibly connected underneath the photosphere, yet characterized by different photospheric field evolutions and fluxes. The combined active region complex produced 15 GOES C–class, 2 M– class, and the X2.2 Valentine’s Day Flare during the four days after initial emergence on February 12th, 2011. The M and X class flares are of particular interest because they are nonhomologous, involving different subregions of the active region. We use a Magnetic Charge Topology together with the Minimum Current Corona(MCT/MCC: Longcope, 1996, 2001) model of the coronal field to model field evolution of the complex. Combining this with observations of flare ribbons in the 1600 ˚ A channel of the Atmospheric Imaging Assembly (AIA) onboard SDO, we generate a separate energy estimate for each major flare using their respective unique subsets of stressed magnetic domains. This work is supported under contract SP02H3901R from Lockheed–Martin to MSU. 1. Partitioning and Feature Tracking 2011-02-11 12:10 -650 -600 -550 -500 -450 -400 -300 -250 -200 -150 N6 N5 N3 N2 N1 P1 P3 P5 P8 20 40 60 80 100 Hours -2•10 5 0 2•10 5 P1 N1 N2 P3 N3 P8 N11 N19 N23 N25 N26 N28 N29 P31 P37 P39 P52 P53 N56 P59 P64 M6.6 M2.2 X2.2 2011-02-12 14:10 -450 -400 -350 -300 -250 -200 -300 -250 -200 -150 N19 N16 N11 N3 N2 P1 P3 P8 P21 P24 2011-02-13 17:22 -200 -150 -100 -50 0 50 -300 -250 -200 -150 N47 N44 N37 N35 N29 N28 N26 N25 N19 N3 N2 P1 P3 P31 P37 P39 P41 P44 P49 P52 P59 2011-02-15 01:46 100 150 200 250 300 350 -300 -250 -200 -150 N87 N85 N81 N78 N56 N47 N35 N29 N28 N26 N25 N19 N2 P1 P3 P39 P52 P53 P59 P61 P64 P82 P83 P84 P89 GOES Radiation curve starting at 2011-02-11 00:00 000:000 020:000 040:000 060:000 080:000 100:000 C M X GOES Radiation curve starting at 2011-02-11 00:00 000:000 020:000 040:000 060:000 080:000 100:000 10 -8 10 -7 10 -6 10 -5 10 -4 10 -3 M6.6 M2.2 X2.2 Flux (10 16 Mx) Figure 1: Upper panels: four samples of the partitioned magnetogram time series. Inset: flux over time in regions with at least 4 × 10 20 Mx. Lower: GOES flux ( W/m 2 ), with green lines at the times of the four upper panels. AR11158 North-South Flux Imbalances 0 20 40 60 80 100 Hours since 2011-02-11 08:10 -5•10 5 0 5•10 5 M6.6 M2.2 X2.2 South South Signed North North Signed Total Signed North Neg + South Pos External External Signed Flux (10 16 Mx) Figure 2: Solid: Positive and negative flux in the Northern (blue) and Southern (black) emergence zones; Dashed: the same using signed flux, including all regions (dashed green), and just central regions (dashed red). The flux in each region is distributed among all other regions, thus defining the system’s connectivity. The amount of flux connecting region j to region k at time i according to a potential field configuration is P i j,k . We quantify the flux evolution of each region according to the method of Tarr & Longcope (2012): P i = P 0 + i-1 X j =0 Δ j S + i-1 X j =0 Δ j R ≡ F i + i-1 X j =0 Δ j R. (1) • Matrix equation describing magnetic connectivity of the active region complex • P i , P 0 : Potential field connectivity at time i and initial time 0 • F i : Connectivity of the constrained field given P 0 and an evolving lower boundary ∑ Δ i S • ∑ Δ i R: Available flux for coronal redistribution at time i (difference between constrained and potential field connectivities) 3. Magnetic Charge Topology with MCC -5.0•10 3 0 5.0•10 3 1.0•10 4 1.5•10 4 2.0•10 4 2.5•10 4 3.0•10 4 Domain Fluxes (10^16 Mx) N25 N56 N29 N88 N19 N81 N59 N60 N73 N26 N64 N61 N3 N45 N65 N37 N2 N25 N56 N29 N88 N19 N81 N59 N60 N73 N26 N64 N61 N3 N45 N65 N37 N2 M6.6 M2.2 X2.2 50 60 70 80 90 100 Hours since 2011-02-11 00:00) -2•10 4 -1•10 4 0 1•10 4 2•10 4 3•10 4 4•10 4 Flux difference from potential (10^16 Mx) N56 N25 N2 N29 N88 N19 N59 N60 N73 N81 N26 N64 N61 N3 N45 N65 N37 N56 N25 N2 N29 N88 N19 N59 N60 N73 N81 N26 N64 N61 N3 N45 N65 N37 Figure 3: Top: Elements of Δ i S P 52,* (domains with which P52 emerged). Bottom: Elements of Δ i R P 52,* (flux difference from a potential field configuration for P52’s domains). Separators at 2011-02-13 17:22 -200 -150 -100 -50 0 50 -300 -250 -200 -150 P1 P3 P31 P37 P39 P41 P44 P49 P52 P57 P59 N2 N3 N19 N25 N26 N28 N29 N35 N37 N38 N42 N43 N44 N45 N47 N51 Figure 4: Topological skeleton and mask overlaid on the magnetogram at the time of the GOES M6.6 flare. Separators shown in color. An example of elements from (1), Δ j S and Δ j R, is shown in Figure 3. Knowing the difference between the constrained and potential fields at every time i, we employ the method of Longcope & Magara (2004) to calculate the minimum current flowing along each separator, I , and the free magnetic energy, ΔW MCC due to that current: Flux difference in domains D linked by separator σ : ψ (cr )i σ = - X D i-1 X j =0 Δ j R D = IL 4π ln eI * |I | ! (2) Free magnetic energy: ΔW MCC = 1 4π Z Ψ Ψ potl IdΨ= LI 2 32π 2 ln √ eI * |I | . (3) AIA 1600 2011-02-13T17:46:17.12 -150 -100 -50 0 50 -300 -250 -200 -150 P1 P3 P31 P37 P39 P41 P44 P49 P52 P57 P59 N2 N3 N19 N25 N26 N28 N29 N35 N37 N38 N42 N43 N44 N45 N47 N51 B01 A02 A03 A04 A05 A06 A07 B08 A09 B10 A11 A12 B13 B14 B15 A16 A17 A18 B19 A20 B21 B22 B23 A25 A26 Figure 5: AIA 1600 ˚ A image, in log scaling, during the M6.6 flare, with overlaid skeleton. ±75 G contours of the LOS magnetogram shown in yellow and blue. Thick blue lines are separators involved in the flare (attached to red–boxed nullpoints), and green dashed lines are all other separators. While the total free energy at any time is given by the sum of Eq. (3) over all separators, we must remember that not all separators, and therefore not all stressed domains, are involved in every flare. We therefore approximate involved domains by noting that: • Flare ribbons (observed in AIA 1600 ˚ A) are well approximated by spine fieldlines of the potential field topology • Separators which relax are those connecting two nulls along highlighted spines • Domains bounded by these separators must exchange flux to drive a flare AIA 1600 2011-02-14T17:31:05.12 50 100 150 200 250 -300 -250 -200 -150 P1 P3 P39 P44 P52 P53 P59 P61 P64 P73 P76 P80 P81 P86 P87 N2 N19 N25 N26 N28 N29 N35 N37 N47 N56 N60 N61 N64 N65 N73 N78 N82 A01 B02 B03 B04 A05 A06 B07 A08 B09 A10 A11 B12 A13 A14 B15 A16 B17 A18 B19 A20 A21 A22 B23 A24 B25 B26 A27 B28 B29 B30 A33 Figure 6: Same as Figure 5, during the M2.2 flare. AIA 1600 2011-02-15T02:01:05.12 100 150 200 250 300 350 -300 -250 -200 -150 P1 P3 P39 P44 P52 P53 P59 P61 P64 P82 P83 P84 P88 P89 P92 P95 N2 N19 N25 N26 N28 N29 N35 N47 N56 N81 N83 N85 N87 N88 A02 B03 A04 A05 A06 A07 B08 B09 A10 A11 B12 B13 A14 A15 A16 A17 B18 A19 B20 A21 A22 B23 B24 B25 B26 B27 B28 B29 B30 B31 A33 Figure 7: Same as Figure 5, during the X2.2 flare. • M6.6: most regions, reconnecting flux into low, core–region loops and higher loops from SE to NW. • M2.2: Eastern regions, reconnecting the central regions with the newly emerged bipole (P52/N56) in the SE. Also involves newly arisen coronal null. • X2.2: all regions, with reconnection through the coronal null (170”E, 225”N; spine sources N25/N2). Contact Lucas Tarr email: [email protected] PhD Candidate address: Montana State University Department of Physics Bozeman, Mt 59715 phone: 971.533.0469 2. Quantifying Flux Change Notable features of Figure 1 and Figure 2: • Multiple sites of simultaneous emergence (North and South) • Multiple phases of emergence (t=0hr, 35hr, 70hr) • Strongly sheared central PIL has little ini- tially connected flux: eg., N26–P3 emerged separately and were later smashed together • Emergence of Eastern destablizing bipole P52/N56 prior to M2.2 Flare 4. Relaxing a subset of stressed domains This work will be completed (very shortly!) in a forthcoming paper: Each separator relaxes by exchanging flux between four domains, two gaining flux, two losing flux. Some domains are associated with multiple separators, so to self consistently relax a set of separators we employ the following algorithm: 1. For each separator, determine the reconnection direction that minimizes total energy. Only rearrage coronal flux in that direction. 2. While there is still flux to be rearranged: (a) Propose one small reconnection across each separator (b) Calculate the total energy change for each small event (c) Accept the reconnection generating the largest drop in free energy Please see http:\\solar.physics.montana.edu/tarrl/ for movies, papers, preprints, and ongoing work. References Longcope, D. 1996, Sol. Phys, 169, 91 Longcope, D., & Magara, T. 2004, ApJ, 608, 1106 Longcope, D. W. 2001, Physics of Plasmas, 8, 5277 Tarr, L., & Longcope, D. 2012, ApJ, 749, 64