Practical Calculation of Magnetic Energy and Relative Magnetic Helicity Budgets in Solar Active Regions Manolis K. Georgoulis Research Center for Astronomy and Applied Mathematics Academy of Athens, Athens, Greece Helicity Thinkshop on Solar Physics Beijing, Oct  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 / 17 Beijing, 12 – 17 Oct Outline 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? 3 / 17 Beijing, 12 – 17 Oct  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 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 ~ |H m | [ Woltjer – Taylor theorem (LFF field state)] Source: SoHO/LASCO

Helicity rates vs. helicity budgets 4 / 17 Beijing, 12 – 17 Oct 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 than the rate,of relative magnetic helicity?

Analysis made possible if vector magnetograms are available 5 / 17 Beijing, 12 – 17 Oct Tic mark separation: 10” NOAA AR 10930, 12/11/06, 13:53 – 15:15 UT Continuum intensity Vertical electric current density Magnetic field vector The main magnetic polarity inversion line in the AR Azimuth disambiguation has been performed using the NPFC method of Georgoulis (2005) Hinode SOT/SP

Calculation of magnetic energy and relative magnetic helicity budgets: I. LFF field approach 6 / 17 Beijing, 12 – 17 Oct Current-free (potential) magnetic energy: Total magnetic energy: Free (non-potential) magnetic energy: Relative magnetic helicity: where: Surface-integral calculation (Georgoulis & LaBonte 2007) NOAA AR  ≈ ± Mm -1

Results of the LFF field approximation 7 / 17 Beijing, 12 – 17 Oct  Two active regions tested: 01/25/00, 19:02 UT NOAA AR 8844 Non-Eruptive NOAA AR 9167 Eruptive 09/15/00, 17:48 UT Force-free parameter Magnetic flux Current-free magnetic energy Total magnetic energy Free magnetic energy Relative magnetic helicity Ratio (eruptive / non-eruptive) 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 8 / 17 Beijing, 12 – 17 Oct Volume-integral energy-helicity calculation : Current-free magnetic energy: Free magnetic energy: Total magnetic energy: Relative magnetic helicity:, where e.g, Longcope & Malanushenko (2008) NLFFF extrapolation for NOAA AR (Wiegelmann 2004)

Is there any better way than volume integrals? What if we knew the photospheric magnetic connectivity? 9 / 17 Beijing, 12 – 17 Oct 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? 10 / 17 Beijing, 12 – 17 Oct 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  i,  j, with vector positions r i, r j 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 magnetic helicity budgets: II. NLFF field approach 11 / 17 Beijing, 12 – 17 Oct Surface-summation energy-helicity calculation: preliminary analysis (Georgoulis et al., 2010) Current-free magnetic energy: Total magnetic energy: Free magnetic energy: Relative magnetic helicity: 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 L fg close is chosen such that free energy is kept to a minimum:  Mutual term of relative L fg is defined following Demoulin et al., (2006):

Summary: NLFF magnetic energy and helicity budget calculation 12 / 17 Beijing, 12 – 17 Oct Volume expressions Surface expressions Current-free magnetic energy: Total magnetic energy: Free magnetic energy: Relative magnetic helicity:, where Relative magnetic helicity: where fluxes  i and alpha-values  i stem from the analysis of magnetic connectivity

Results: preliminary comparison of free magnetic energies 13 / 17 Beijing, 12 – 17 Oct 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 magnetic helicities 14 / 17 Beijing, 12 – 17 Oct 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 regions? 15 / 17 Beijing, 12 – 17 Oct NOAA AR Magnetic energy (erg) 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 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? 16 / 17 Beijing, 12 – 17 Oct E np > 3 x erg H m > 2 x Mx 2 Eruptive regions tend to have large free magnetic energy (> 3 x erg) and relative magnetic helicity (> 2 x Mx2) The “threshold” helicity magnitude shows excellent agreement with the typical CME helicity budgets (DeVore 2000; Georgoulis et al. 2009)

Summary and Conclusions 17 / 17 Beijing, 12 – 17 Oct 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 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 hypothesis  After calculating the NLFF field helicity, we can find the  -value that would give the same helicity for a LFF field:  Then we can use this  -value to calculate a LFF field total energy: 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 extrapolation NLFFF extrapolation for NOAA AR (Wiegelmann 2004) Logarithm of the free magnetic energy as a function of altitude – most of it close to the photosphere (< 20 Mm)