Manolis K. Georgoulis

Practical Calculation of Magnetic Energy Practical Calculation of Magnetic Energy and Relative Magnetic Helicity and Relative Magnetic Helicity BudgetsBudgets in in

Solar Active RegionsSolar Active Regions

Manolis K. Georgoulis Research Center for Astronomy and Applied Mathematics

Academy of Athens, Athens, Greece

Helicity Thinkshop on Solar PhysicsHelicity Thinkshop on Solar Physics Beijing, 12-17 Oct. 2009

D. Rust, B. LaBonte, A. Pevtsov, A. Nindos, M. Berger, T. Wiegelmann, and a number of NASA research grants

Thanks to: Prof. H. Zhang & the Organizers of this Meeting for kind support

Constraints to coronal evolution placed by magnetic helicity

Helicity rates vs. helicity budgets

Calculation of magnetic energy and relative magnetic helicity budgets

2 / 17Beijing, 12 – 17 Oct. 2009

OutlineOutline

Via the linear force-free (LFF) field approximation

Via the nonlinear force-free (NLFF) field approximation

Volume-integral evaluation using extrapolation results

Surface-summation evaluation using photospheric magnetic connectivity

Preliminary results

Correlations between LFF and NLFF energy and helicity budgets

NLFF field energy and helicity budgets

An energy-helicity criterion for eruptive solar active regions

Conclusions – future prospects

Why should magnetic helicity be important for solar Why should magnetic helicity be important for solar coronal activity?coronal activity?

3 / 17Beijing, 12 – 17 Oct. 2009

Theoretical reasons : Observational reasons :

We can see it (!) and there is increasing evidence of its presence in eruptive active regions and CMEs

from Rust & LaBonte (2005)

Magnetic helicity cannot be dissipated effectively by magnetic reconnection

2M

m

m1/2M

m

m R~H

ΔHor R~

H

ΔH

so it can only be bodily transported (CMEs?)

Unless magnetic helicity is not removed, a magnetic system cannot return to the ground, current-free state

currentfreecurrenttotal EEE

~ |Hm|[ Woltjer – Taylor theorem (LFF field state)]

Source: SoHO/LASCO

Helicity rates vs. helicity budgetsHelicity rates vs. helicity budgets

4 / 17Beijing, 12 – 17 Oct. 2009

Calculations of relative magnetic helicity mainly deal with the helicity injection rate, rather than the helicity budget, in active regions:

However:

The helicity injection rate lacks a reference point

Calculation of the velocity field u is non-unique and highly uncertain

What if we tried calculating the budget, rather What if we tried calculating the budget, rather than the rate,of relative magnetic helicity?than the rate,of relative magnetic helicity?

- B uA [2dt

dH

S

pm

Sd] u BAp

Analysis made possible if vector magnetograms Analysis made possible if vector magnetograms are availableare available

5 / 17Beijing, 12 – 17 Oct. 2009

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 ARAzimuth disambiguation has been performed using the NPFC method of Georgoulis (2005)

Hinode SOT/SP

Calculation of magnetic energy and relative Calculation of magnetic energy and relative magnetic helicity budgets: I. LFF field approachmagnetic helicity budgets: I. LFF field approach

6 / 17Beijing, 12 – 17 Oct. 2009

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

Surface-integral calculation (Georgoulis & LaBonte 2007)

NOAA AR 10030

≈ -0.053 ± 0.011 Mm-1

Results of the LFF field approximationResults of the LFF field approximation

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Two active regions tested:

01/25/00, 19:02 UT

NOAA AR 8844

Non-EruptiveNon-Eruptive

NOAA AR 9167

EruptiveEruptive

09/15/00, 17:48 UT

0123456789

10

Force-freeparameter

Magnetic flux Current-freemagneticenergy

Total energy Free energy Relativemagnetichelicity

Force-freeparameter

Magneticflux

Current-freemagneticenergy

Totalmagneticenergy

Freemagneticenergy

Relativemagnetichelicity

Ra

tio

(e

rup

tiv

e /

no

n-e

rup

tiv

e)

For nearly the same force-free parameter, and a ratio of ~ 3.3 in the magnetic flux, current-free, and total magnetic energy, the respective ratios for the free magnetic energy and relative magnetic helicity are ~9.

How realistic is the LFF field How realistic is the LFF field calculation, however?calculation, however?

Calculation of magnetic energy and relative Calculation of magnetic energy and relative magnetic helicity budgets: II. NLFF field approachmagnetic helicity budgets: II. NLFF field approach

8 / 17Beijing, 12 – 17 Oct. 2009

Volume-integral energy-helicity calculation :

Current-free magnetic energy:

dVB π8

1E

V

2pp

Free magnetic energy:

ptnp EEE


dVB π8

1E

V

2NLFFt

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

z'rr

z'rr

z'rr

π2

1rA z

S22g

Relative magnetic helicity:

, wheredVBAH NLFFF

V

gm

e.g, Longcope & Malanushenko (2008)

NLFFF extrapolation for NOAA AR 10930 (Wiegelmann 2004)

Is there any better way than volume integrals? What if Is there any better way than volume integrals? What if we knew the photospheric magnetic connectivity?we knew the photospheric magnetic connectivity?

9 / 17Beijing, 12 – 17 Oct. 2009

Start from the normal magnetic field

Partition the magnetic flux into a sequence of discrete concentrations

Identify the flux-weighted centroids for each partition

ijij L,Φ

Define the connectivity matrices

pnp2p1

2n2221

1n1211

Φ....ΦΦ

................

Φ....ΦΦ

Φ....ΦΦ

M

pnp2p1

2n2221

1n1211

L....LL

................

L....LL

L....LL

L

Which magnetic connectivity?Which magnetic connectivity?

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An alternative connectivity can result in the minimum possible total connection length in the magnetogram

To achieve this, we minimize the functional

ji

ji

ji

ji

ΦΦ

ΦΦ

rr

rr

between any two opposite-polarity fluxes i, j, with vector positions ri, rj

We perform the minimization using the simulated annealing method

Discretized view of the photospheric magnetic flux

Any connectivity (potential, non-potential) can be used with or without flux partitioning

Convergence of the annealing

Calculation of magnetic energy and relative Calculation of magnetic energy and relative magnetic helicity budgets: II. NLFF field approachmagnetic helicity budgets: II. NLFF field approach

11 / 17Beijing, 12 – 17 Oct. 2009

Surface-summation energy-helicity calculation: preliminary analysis (Georgoulis et al., 2010)


dy'dx' z0,','A0,','B π8

1E g

S

pp yxyx


jicloseij

n

1i

n

ji1,ji

2δi

n

1i

2i

2pt ΦΦLαΦαAλEE


ptnp EEE


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

A and are known fitting constants

Mutual term of free energy Lfgclose is

chosen such that 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 following Demoulin et al., (2006):

intersectnot do g f, ; LL

intersect gf, and ; LL

intersect g f, and ; LL

L

gfgf

)((f)closefggf

)((f)closefg

arch

gf

fgarcharch

garch

g

lengthlength

lengthlength

Summary: NLFF magnetic energy and helicity Summary: NLFF magnetic energy and helicity budget calculationbudget calculation

12 / 17Beijing, 12 – 17 Oct. 2009

Volume expressionsVolume expressions Surface expressionsSurface expressions


dVB π8

1E

V

2pp


dy'dx' z0,','A0,','B π8

1E g

S

pp yxyx


dVB π8

1E

V

2NLFFt


jicloseij

n

1i

n

ji1,ji

2δi

n

1i

2i

2pt ΦΦLαΦαAλEE


ptnp EEE


ptnp EEE

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

z'rr

z'rr

z'rr

π2

1rA z

S22g


, wheredVBAH NLFFF

V

gm


ji

n

1i

n

ji1,jij

2δi

n

1ii

2m ΦΦLΦαAλ π8H

where fluxes i and alpha-values i

stem from the analysis of magnetic connectivity

Results: preliminary comparison of free magnetic Results: preliminary comparison of free magnetic energiesenergies

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Limited sample of 9 active regions:

NLFF volume calculation

NLFF surface calculation

LFF calculation

Connectivity matrix has been calculated from line-tracing of a NLFF field extrapolation

Very good agreement between NLFF volume / surface expressions

Acceptable agreement between LFF and NLFF expressions

Results: preliminary comparison of relative Results: preliminary comparison of relative magnetic helicitiesmagnetic helicities

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NLFF volume calculation

NLFF surface calculation

LFF calculation

Connectivity matrix has been calculated from line-tracing of a NLFF field extrapolation

Reasonable agreement between NLFF volume / surface expressions

Fair to poor agreement between NLFF and LFF expressions

Limited sample of 9 active regions:

A quiz: can you identify the eruptive active A quiz: can you identify the eruptive active regions?regions?

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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 (NOAA ARs 8210, 9026, 9165, 10030, 10930, and 10953)

WHERE ARE THESE SIX WHERE ARE THESE SIX ERUPTIVE REGIONS?ERUPTIVE REGIONS?

In terms of free magnetic energy, the eruptive regions have a noticeable fraction of their total energy being available for release

In terms of relative magnetic helicity, the eruptive regions have clearly larger magnitudes than the non-eruptive ones

An “energy-helicity” eruptive criterion?An “energy-helicity” eruptive criterion?

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Enp > 3 x 1031 erg

Hm > 2 x 1042 Mx2

Eruptive regions tend to have large free magnetic energy (> 3 x 1031 erg) and relative magnetic helicity (> 2 x 1042 Mx2)

The “threshold” helicity magnitude shows excellent agreement with the typical CME helicity budgets (DeVore 2000; Georgoulis et al. 2009)

Summary and ConclusionsSummary and Conclusions

17 / 17Beijing, 12 – 17 Oct. 2009

Adopting that magnetic helicity is an important physical quantity in the solar atmosphere, we attempt a calculation of the relative magnetic helicity and energy budgets from single vector magnetograms of solar active regions

Calculation of the relative helicity budget does not require knowledge of the velocity field and hence avoids its shortcomings. Plus, it provides more information than simply calculating helicity injection rates.

Energy-helicity budget calculation for a LFF field has been achieved. We presented here a more general NLFF field calculation that appears to be working satisfactorily.

For a dataset of 22 active-region vector magnetograms it appears that the 6 eruptive active regions show larger free magnetic energy and larger magnitude of relative magnetic helicity.

An eruptive criterion for an active region may be defined here – there is important physics in the “energy-helicity” diagram for a statistically significant sample

FUTURE PROSPECTS: verify calculations and results, increase the sample of active regions, test different connectivity solutions, detailed uncertainty analysis, etc. etc.

BACKUP SLIDES

Basic mutual helicity configurationsBasic mutual helicity configurations

From Demoulin et al. (2006)

To be consistent with a minimum free magnetic energy, we assume that all the possible configurations collapse to that of picture (a).

Testing the Taylor hypothesisTesting the Taylor hypothesis

After calculating the NLFF field helicity, we can find the -value that would give the same helicity for a LFF field:

P

m2T E F

H

λ π8

1α

Then we can use this -value to calculate a LFF field total energy:

p22

Ttotal E λFα1E(T)

per the Woltjer-Taylor theorem, this energy should be the minimum possible

NLFF surface integral

Min “Taylor” energy

LFF energy estimate

NLFF volume integral

Cross-section of a NLFF field extrapolationCross-section of a NLFF field extrapolation

NLFFF extrapolation for NOAA AR 10930 (Wiegelmann 2004)

Logarithm of the free magnetic energy as a function of altitude – most of it close to the photosphere (< 20 Mm)

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Manolis K. Georgoulis