Magnetic Helicity and Free Magnetic Energy: Their Relation and Predictive Power in Eruptive Solar Magnetic Configurations

Manolis K. Georgoulis1 & Thomas Wiegelmann2

2008 Fall AGU Meeting

MKG gratefully acknowledges partial support from NASA’s STEREO Heliophysics Guest Investigator Program NNX08AJ10G

San Francisco, CA, December 15 – 19, 2008

1. JHU/APL, 11100 Johns Hopkins Road, Laurel, MD 20726, USA

2. Max-Planck Institute for Solar System Research, Max-Planck, Strasse 2 37191, Katlenburg-Lindau, Germany

SummarySummary

Solar and space weather forecasting seek metrics with predictive power that can help mitigate the adverse effects of solar eruptions and their consequences. These metrics often rely on a reduced, simplified physical background but this does not mean that space weather itself should necessarily trade well-founded physics for operational convenience. Solar eruptions are triggered in tangled, helical, solar magnetic fields stressed well beyond their minimum-energy state. We aim to precisely calculate the two physical quantities thought to be responsible for eruptions, namely, the relative magnetic helicity and the free magnetic energy of the source active regions. Using the nonlinear force-free field (NLFFF) approximation we first relate the magnetic free energy to the relative helicity and then we proceed to distinguish eruptive from non-eruptive active regions purely from their free energy and helicity values. Results with sufficient statistics, pending the availability of high-quality solar vector magnetograms, can establish “all-clear” free-energy and helicity thresholds. Moreover, the free-energy / helicity buildup in solar active regions can be viewed as a course through a sequence of metastable states with eruptions occurring spontaneously when the system reaches local or global minimal stability. This allows the careful use of the self-organized criticality (SOC) concept to describe the dynamics of the system and leads to important conclusions about how far we can go when predicting solar eruptions. For questions, suggestions, or comments, please contact:

Manolis Georgoulis, (240)-228-5508, manolis.georgoulis@jhuapl.edu

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Analysis outlineAnalysis outline

I. Vector magnetograms (VMGs) of solar active regions are used. The analysis entails the following steps:

Azimuth disambiguation of VMGs

Calculation of magnetic energy and relative magnetic helicity in three different ways:

Linear force-free field (LFFF) calculation – a first step

Nonlinear force-free field (NLFFF) calculation, using two different methods:

NLFF field extrapolation and volume evaluation

NLFF magnetic connectivity matrix calculation and surface evaluation

III. Results:

Correlations between LFFF and NLFFF energy and helicity budgets

Magnetic free energy and helicity budgets

Energy – helicity correlations and their predictive power

II. Application to VMGs from twenty two (22) solar active regions

IV. The course toward minimal stability – a possible SOC manifestation?

V. Summary, conclusions, and future work

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Azimuth disambiguationAzimuth disambiguation

We use the Non-Potential Magnetic Field Calculation (NPFC) method of Georgoulis (2005).

The NPFC is an iterative method that calculates self-consistently both the current-free (potential) and the current-carrying (non-potential) magnetic field components that best reproduce the observed field vector

Tic mark separation: 10”

NOAA AR 10930, 12/11/06, 13:53 – 15:15 UT

Continuum intensity

Vertical electric current densityMagnetic field vector

The main magnetic polarity inversion line in

the AR4 / 14

Linear force-free field energy / helicity calculationLinear force-free field energy / helicity calculation

Surface calculation on the plane S of the observations using the linearized analysis of Georgoulis & LaBonte (2007):

Current-free (potential) magnetic energy:

dy'dx' z0,','A0,','B π8

1E g

S

pp yxyx

Total magnetic energy:

p22

t E λFα1E

Free (non-potential) magnetic energy:

p22

np EλFαE

Relative magnetic helicity:

p2

m E λ α F π8H

where:

x y

ml

x y

ml

n

l

n

m mluu

n

l

n

m mluu

uub

uub

1 1

2/1222,

1 1

2/3222,

/

/

2

1F

Sample -value calculation on a VMG from NOAA AR 10930

Flux-weighted average and uncertainties (highlighted)

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Nonlinear force-free field energy / helicity calculationNonlinear force-free field energy / helicity calculation

Volume calculation via magnetic field extrapolation We use the NLFFF optimization method of Wiegelmann (2004)

Potential magnetic energy:

Total magnetic energy:

dVB π8

1E

V

2NLFFt

Free magnetic energy:

dVB π8

1E

V

2pp

ptnp EEE

Relative magnetic helicity:

, where

dy'dx',0y',x'B'rr

z'rr

z'rr

z'rr

π2

1rA z

S22g

dVBAH NLFFF

V

gm

is a selected gauge expression for the vector potential proposed by Longcope & Malanushenko (2008) – see also DeVore (2000)

NLFFF extrapolation for NOAA AR 10930

Logarithm of free magnetic energy density vs. height for a 2D cross-section of the extrapolated volume

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Nonlinear force-free field energy / helicity calculationNonlinear force-free field energy / helicity calculation

Surface calculation using magnetic field extrapolation and the subsequent magnetic connectivity matrix Based on our preliminary analysis (Georgoulis, 2009, in preparation)

Potential magnetic energy:

dy'dx' z0,','A0,','B π8

1E g

S

pp yxyx

Total magnetic energy:

jicloseij

n

1i

n

ji1,ji

2δi

n

1ii

2t ΦΦLαΦαAλE

Free magnetic energy:

ptnp EEE

Relative magnetic helicity:

ji

n

1i

n

ji1,jij

2δi

n

1ii

2m ΦΦLΦαAλ π8H

where the VMG has been flux-partitioned into n partitions with fluxes i and alpha-values i, i={1,…,n}

Moreover:

A and are known fitting constants

Mutual term of free energy Lfgclose is chosen

such that the free energy is kept to a minimum:

intersectnot do g and f ; 0

intersect g and f ; / Lclose

fgff

Mutual term of relative Lfg is defined in accordance with the analysis of Demoulin et al., (2006):

intersectnot do g f, ; LL

intersect gf, and ; L

intersect g f, and ; L

L

gfgf

)((f)gf

)((f)arch

gf

fgarcharch

garch

g

lengthlength

lengthlength

Connectivity matrix NOAA AR 10930

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Results: comparison of free magnetic energiesResults: comparison of free magnetic energies

We studied VMGs from 22 active regions. Of those, we test the results on 9 active regions for which the NLFFF extrapolation worked satisfactorily without preprocessing (see Wiegelmann et al., [2006] for more information)

NLFFF extrapolation

NLFFF surface calculation

LFFF calculation

These results are preliminary. No attempt has been made to improve the calculations using, for example, preprocessing

The NLFFF free energies calculated using the connectivity matrix are highly correlated to those calculated by the NLFF field extrapolation

A good correlation also exists between the LFFF and the NLFFF free energies from extrapolation, although not as good as the first shown.

This stresses the compromise brought by the LFFF approximation. 8 / 14

Results: comparison of relative magnetic helicitiesResults: comparison of relative magnetic helicities

Comparison for the same 9 active regions

NLFFF extrapolation

NLFFF surface calculation

LFFF calculation

Helicity magnitudes The NLFFF relative helicities

calculated using the connectivity matrix are reasonably close to those calculated by the NLFFF extrapolations

The correlation between the NLFFF helicities is generally better than that between the NLFFF and the LFF helicities, as expected

These results are preliminary. No attempt has been made to improve the calculations using, for example, preprocessing

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Free magnetic energy vs. relative magnetic helicityFree magnetic energy vs. relative magnetic helicityA notable feature in the energy and helicity of eruptive regions A notable feature in the energy and helicity of eruptive regions

NOAA AR

Mag

neti

c energ

y (

erg

)

8210

9026

9165

1003010930

10953

8210

90269165

1003010930

10953

Potential energy

Free energy

Now we focus on the NLFFF energy / helicity calculations of the entire sample of 22 regions.

Of these active regions, 6 were flaring and eruptive, in general (NOAA ARs 8210, 9026, 9165, 10030, 10930, and 10953)

All of the eruptive regions, but NOAA AR 9026, have significant free magnetic energy accompanied by a relative helicity with magnitude > 2 x 1042 Mx2. Non-eruptive active regions do not show this behavior.

The used magnetograms were taken before major eruptions

Therefore, an envisioned reliable and fast calculation of the free magnetic energy and relative magnetic helicity budgets is a physical and effective way of distinguishing eruptive from non-eruptive active regions

This calculation may lead to enhanced predictive capability of solar eruptions

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The predictive power of the free energy – helicity correlationThe predictive power of the free energy – helicity correlation

The correlation between the free magnetic energy and the relative magnetic helicity is very high

In the NLFFF approximation this means that free energy is associated with helicity of a predominant sense – not nearly equal and opposite amounts of helicity

An “eruptive” zone appears to exist for Enp > ~ 3x1031 erg and |Hm| > 2 x 1042 Mx2

Only better statistics can establish this potentially important finding11 / 14

Other potentially interesting correlations Other potentially interesting correlations

Upper left: Relative magnetic helicity vs. normalized (with respect to total) free magnetic energy

Upper right: Normalized (with respect to flux squared) relative magnetic helicity vs. free magnetic energy

Lower left: Normalized relative magnetic helicity vs. normalized free magnetic energy

None of these plots shows such a good correlation as the helicity vs. energy plot. Therefore, large absolute magnitudes of free energy and relative helicity appear as important eruption drivers

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Helicity buildup and subsequent eruptions in active regions: Helicity buildup and subsequent eruptions in active regions: could SOC play a role here?could SOC play a role here?

Eruptive active regions embark on a helicity buildup phase accompanying the flux emergence phase. Eruptions may well occur during this buildup. What determines the maximum helicity an active region can accumulate?

Is the completion of the energy / helicity buildup phase related to some marginal stability exhibited by a given system parameter? If so, what is this parameter?

If this parameter can be observed or calculated, then one can detect the kickoff of the marginal stability (possibly SOC) phase. Marginal system stability is a requirement for SOC state to occur. The other requirement is the existence of a threshold-driven instability, which is believed to be the case for solar eruptions.

NOAA AR 9026

NOAA AR 9165

The helicity curves at the left have been calculated by means of integration of the helicity injection rates. Rates have been calculated using SoHO /MDI data and the LCT velocity flow. Overall, a Overall, a possible SOC possible SOC behavior may behavior may play a significant play a significant role in this role in this problemproblem

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Summary, conclusions, and future work Summary, conclusions, and future work

Current Analysis

• We test three different methods of energy / helicity calculation; two of them utilize the NLFFF and one uses the LFFF approximation.

• The two NLFFF methods reasonably agree in their estimates of the free magnetic energy and relative magnetic helicity, with the former correlation being stronger than the latter.

• The LFFF approximation typically gives larger energies and helicities

• Applying one of the NLFFF methods in a relatively sizeable sample of 22 active regions we find that (1) the free magnetic energy, (2) the magnitude of the relative helicity, and (3) the combination of the two appear to hold significant predictive power.

• Only eruptive active regions have free magnetic energy > 3 x 1031 erg and relative magnetic helicity magnitude > 2 x 1042 Mx2.

Future prospects

• A reliable calculation of the magnetic energy and relative magnetic helicity budgets in solar active regions should be a continual pursuit.

• Given the necessary buildup of free energy and helicity in active regions for eruptions to occur, the magnetic topology may be evolving through a series of metastable states during buildup. The system may eventually experience a state of local or global marginal stability with respect to certain parameter(s).

• If this(ese) parameter(s) are found, we will have achieved a fundamental physical understanding of solar active regions. The SOC concept may be particularly useful to this purpose.

• The detection of the state of marginal stability, rather than individual morphological or other “precursors” or “proxies” may be our best chance to probabilistically predict solar eruptions.

Further reading:

• Demoulin, P., Pariat, E., & Berger, M. A., SoPh, 233, 3, 2006

• DeVore, C. R., ApJ , 539, 944, 2000

• Georgoulis, M. K., ApJ, 629, L69, 2005

• Georgoulis, M. K. & LaBonte, B. J., ApJ, 671, 1034, 2007

• Longcope, D. W. & Malanushenko, ApJ, 674, 1130, 2008

• Wiegelmann, T., SoPh, 219, 87, 2004

• Wiegelmann, T., Inhester, B., & Sakurai, T., SoPh, 233, 215, 200614 / 14