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
-200
-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
-200
-150
N19
N16
N11
N3
N2P1
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
N19N3
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
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
-200 -150 -100 -50 0 50
-300
-250
-200
-150
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
-150 -100 -50 0 50
-300
-250
-200
-150
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
-300
-250
-200
-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
-200
-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: [email protected] 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
