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### Practical Calculation of Magnetic Energy and Relative Magnetic Helicity Budgets in Solar Active Regions

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

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

Manolis K. Georgoulis

Research Center for Astronomy and Applied Mathematics

Academy of Athens, Athens, Greece

Beijing, 12-17 Oct. 2009

Helicity Thinkshop on Solar Physics

2 / 17

Beijing, 12 – 17 Oct. 2009

- Constraints to coronal evolution placed by magnetic helicity
- Helicity rates vs. helicity budgets
- Calculation of magnetic energy and relative magnetic helicity budgets

- 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 coronal activity?

from Rust & LaBonte (2005)

- Magnetic helicity cannot be dissipated effectively by magnetic reconnection

Source: SoHO/LASCO

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

3 / 17

Beijing, 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

~ |Hm|

[ Woltjer – Taylor theorem (LFF field state)]

Helicity rates vs. helicity budgets activity?

However:

- The helicity injection rate lacks a reference point
- Calculation of the velocity field u is non-unique and highly uncertain

4 / 17

Beijing, 12 – 17 Oct. 2009

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

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

Analysis made possible if vector magnetograms are available activity?

Continuum intensity

Vertical electric current density

Magnetic field vector

5 / 17

Beijing, 12 – 17 Oct. 2009

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

Hinode SOT/SP

Tic mark separation: 10”

The main magnetic polarity inversion line in the AR

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

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

- Current-free (potential) magnetic energy:

- Total magnetic energy:

- Free (non-potential) magnetic energy:

- Relative magnetic helicity:

where:

6 / 17

Beijing, 12 – 17 Oct. 2009

- Surface-integral calculation (Georgoulis & LaBonte 2007)

a≈ -0.053 ± 0.011 Mm-1

NOAA AR 10030

Results of the LFF field approximation helicity budgets: I. LFF field approach

NOAA AR 9167

09/15/00, 17:48 UT

Ratio (eruptive / non-eruptive)

Eruptive

Total

magnetic

energy

Force-free

parameter

Magnetic

flux

Current-free

magnetic

energy

Free

magnetic

energy

Relative

magnetic

helicity

NOAA AR 8844

01/25/00, 19:02 UT

Non-Eruptive

7 / 17

Beijing, 12 – 17 Oct. 2009

- Two active regions tested:

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 calculation, however?

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

- Current-free magnetic energy:

- Total magnetic energy:

- Free magnetic energy:

- Relative magnetic helicity:

, where

e.g, Longcope & Malanushenko (2008)

8 / 17

Beijing, 12 – 17 Oct. 2009

Volume-integral energy-helicity calculation :

NLFFF extrapolation for NOAA AR 10930 (Wiegelmann 2004)

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

9 / 17

Beijing, 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

- Define the connectivity matrices

Which magnetic connectivity? knew the photospheric magnetic connectivity?

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Beijing, 12 – 17 Oct. 2009

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

Discretized view of the photospheric magnetic flux

- An alternative connectivity can result in the minimum possible total connection length in the magnetogram
- To achieve this, we minimize the functional

between any two opposite-polarity fluxes Fi, Fj, with vector positions ri, rj

Convergence of the annealing

- We perform the minimization using the simulated annealing method

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

- Current-free magnetic energy:

- Total magnetic energy:

- A and d are known fitting constants
- Mutual term of free energy Lfgclose is chosen such that free energy is kept to a minimum:

- Free magnetic energy:

- Relative magnetic helicity:

- Mutual term of relative Lfg is defined following Demoulin et al., (2006):

where the VMG has been flux-partitioned into n partitions with fluxes Fi and alpha-values ai

11 / 17

Beijing, 12 – 17 Oct. 2009

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

Summary: NLFF magnetic energy and helicity budget calculation

- Current-free magnetic energy:

- Current-free magnetic energy:

- Total magnetic energy:

- Total magnetic energy:

- Free magnetic energy:

- Free magnetic energy:

- Relative magnetic helicity:

- Relative magnetic helicity:

, where

where fluxes Fi and alpha-values aistem from the analysis of magnetic connectivity

12 / 17

Beijing, 12 – 17 Oct. 2009

Surface expressions

Volume expressions

Results: preliminary comparison of free magnetic energies calculation

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

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Beijing, 12 – 17 Oct. 2009

Limited sample of 9 active regions:

Results: preliminary comparison of relative magnetic helicities

NLFF volume calculation

NLFF surface calculation

LFF calculation

14 / 17

Beijing, 12 – 17 Oct. 2009

Limited sample of 9 active regions:

- 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

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

Potential energy

10030

Free energy

10930

9026

9165

8210

10953

8210

9026

10953

9165

10930

10030

15 / 17

Beijing, 12 – 17 Oct. 2009

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)

Magnetic energy (erg)

WHERE ARE THESE SIX ERUPTIVE REGIONS?

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

NOAA AR

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

An “energy-helicity” eruptive criterion? helicities

Enp > 3 x 1031 erg

Hm > 2 x 1042 Mx2

16 / 17

Beijing, 12 – 17 Oct. 2009

- 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 Conclusions helicities

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Beijing, 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 helicities

Basic mutual helicity configurations helicities

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 hypothesis helicities

LFF energy estimate

NLFF volume integral

NLFF surface integral

Min “Taylor” energy

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

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

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

Cross-section of a NLFF field extrapolation helicities

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

NLFFF extrapolation for NOAA AR 10930 (Wiegelmann 2004)

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