CALCULATING SEPARATE MAGNETIC FREE ENERGY ESTIMATES FORACTIVE REGIONS PRODUCING MULTIPLE FLARES: NOAA AR11158

Lucas Tarr & Dana LongcopeDepartment of Physics, Montana State University

AbstractIt is well known that photospheric flux emergence is an important process for stressing coronalfields and generating magnetic free energy, which may then be released during a flare. TheHelioseismic 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 fieldevolutions 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 February12th, 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 togetherwith the Minimum Current Corona(MCT/MCC: Longcope, 1996, 2001) model of the coronalfield to model field evolution of the complex. Combining this with observations of flare ribbonsin the 1600 A channel of the Atmospheric Imaging Assembly (AIA) onboard SDO, we generatea separate energy estimate for each major flare using their respective unique subsets of stressedmagnetic domains. This work is supported under contract SP02H3901R from Lockheed–Martinto MSU.

1. Partitioning and Feature Tracking2011-02-11 12:10

-650 -600 -550 -500 -450 -400

-300

-250

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-150

N6

N5N3

N2N1 P1

P3P5

P820 40 60 80 100Hours

-2•105

0

2•105

P1N1N2P3N3P8N11N19N23N25N26N28N29P31P37P39P52P53N56P59P64

M6.6 M2.2 X2.2

2011-02-12 14:10

-450 -400 -350 -300 -250 -200

-300

-250

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-150

N19

N16

N11

N3

N2P1

P3

P8

P21

P24

2011-02-13 17:22

-200 -150 -100 -50 0 50

-300

-250

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-150

N47

N44

N37

N35

N29

N28

N26

N25

N19N3

N2

P1

P3

P31

P37

P39

P41

P44

P49

P52

P59

2011-02-15 01:46

100 150 200 250 300 350

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-150

N87

N85

N81

N78

N56

N47N35

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

000:000 020:000 040:000 060:000 080:000 100:00010-8

10-7

10-6

10-5

10-4

10-3

M6.6 M2.2 X2.2

Flu

x(1016M

x)

Figure 1: Upper panels: four samples of the partitioned magnetogram time series. Inset: flux over time in regions with at least4× 1020 Mx. Lower: GOES flux ( W/m2), with green lines at the times of the four upper panels.

AR11158 North-South Flux Imbalances

0 20 40 60 80 100Hours since 2011-02-11 08:10

-5•105

0

5•105

M6.6 M2.2 X2.2SouthSouth SignedNorthNorth SignedTotal SignedNorth Neg + South PosExternalExternal Signed

Flux(101

6Mx)

Figure 2: Solid: Positive and negative flux in theNorthern (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’sconnectivity. The amount of flux connecting region j to region k at time i according to a potentialfield configuration is Pij,k. We quantify the flux evolution of each region according to the methodof Tarr & Longcope (2012):

Pi = P0 +

i−1∑j=0

∆jS +

i−1∑j=0

∆jR ≡ Fi +

i−1∑j=0

∆jR. (1)

•Matrix equation describing magnetic connectivity of the active region complex

• Pi, P0: Potential field connectivity at time i and initial time 0

• Fi: Connectivity of the constrained field given P0 and an evolving lower boundary∑

∆iS•∑

∆iR: Available flux for coronal redistribution at time i (difference between constrained andpotential field connectivities)

3. Magnetic Charge Topology with MCC

-5.0•103

0

5.0•103

1.0•104

1.5•104

2.0•104

2.5•104

3.0•104

Dom

ain

Flu

xes

(10^

16 M

x)

N25

N56

N29

N88N19N81N59

N60

N73 N26

N64

N61N3 N45N65N37 N2

N25N56N29N88N19N81N59N60N73N26N64N61N3N45N65N37N2

M6.6 M2.2 X2.2

50 60 70 80 90 100Hours since 2011-02-11 00:00)

-2•104

-1•104

0

1•104

2•104

3•104

4•104

Flu

x di

ffere

nce

from

pot

entia

l (10

^16

Mx) N56

N25

N2

N29

N88N19N59

N60

N73 N81N26N64

N61N3 N45 N65N37

N56N25N2N29N88N19N59N60N73N81N26N64N61N3N45N65N37

Figure 3: Top: Elements of ∆iSP52,∗ (domains withwhich P52 emerged). Bottom: Elements of ∆iRP52,∗(flux difference from a potential field configuration forP52’s domains).

Separators at 2011-02-13 17:22

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P1

P3

P31

P37

P39

P41

P44

P49

P52

P57P59

N2

N3N19

N25

N26

N28

N29

N35

N37

N38

N42

N43

N44

N45

N47

N51

Figure 4: Topological skeleton and mask overlaid onthe magnetogram at the time of the GOES M6.6 flare.Separators shown in color.

An example of elements from (1), ∆jS and ∆jR, is shown in Figure 3. Knowing the differencebetween 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 freemagnetic energy, ∆WMCC due to that current:

Flux difference in domains D linked by separator σ:

ψ(cr)iσ = −

∑D

i−1∑j=0

∆jRD = IL4π

ln

(eI∗

|I|

)(2)

Free magnetic energy: ∆WMCC = 14π

∫ Ψ

Ψpotl

IdΨ = LI2

32π2ln(√eI∗|I|

). (3)

AIA 1600 2011-02-13T17:46:17.12

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P1

P3

P31

P37

P39

P41

P44

P49

P52

P57P59

N2

N3

N19

N25

N26

N28

N29

N35

N37

N38

N42

N43

N44

N45

N47

N51B01

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 1600A image, in log scaling, during theM6.6 flare, with overlaid skeleton. ±75 G contours ofthe LOS magnetogram shown in yellow and blue. Thickblue lines are separators involved in the flare (attachedto red–boxed nullpoints), and green dashed lines are allother separators.

While the total free energy at any time is givenby the sum of Eq. (3) over all separators, wemust remember that

not all separators, and therefore not allstressed domains, are involved in every flare.

We therefore approximate involved domains bynoting that:

• Flare ribbons (observed in AIA 1600A) arewell approximated by spine fieldlines of thepotential field topology

• Separators which relax are those connectingtwo nulls along highlighted spines

•Domains bounded by these separators mustexchange flux to drive a flare

AIA 1600 2011-02-14T17:31:05.12

50 100 150 200 250

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-150

P1

P3

P39P44

P52

P53P59

P61

P64

P73P76P80

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

B12A13

A14 B15

A16

B17

A18

B19

A20

A21

A22

B23

A24

B25B26

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

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-150

P1

P3

P39P44

P52

P53

P59

P61

P64

P82

P83

P84

P88

P89

P92

P95

N2

N19

N25

N26

N28

N29

N35N47

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).

ContactLucas Tarr email: ltarr@physics.montana.eduPhD Candidate address: Montana State University

Department of PhysicsBozeman, 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 emergedseparately and were later smashed together

• Emergence of Eastern destablizing bipoleP52/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 losingflux. Some domains are associated with multiple separators, so to self consistently relax a set ofseparators we employ the following algorithm:

1. For each separator, determine the reconnection direction that minimizes total energy. Onlyrearrage 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.

ReferencesLongcope, 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