Manolis K. Georgoulis 1 & Thomas Wiegelmann 2

Magnetic Helicity and Free Magnetic Energy: Their Relation and Predictive Power in Eruptive Solar Magnetic Configurations Manolis K. Georgoulis1 & Thomas Wiegelmann2 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 MKG gratefully acknowledges partial support from NASA’s STEREO Heliophysics Guest Investigator Program NNX08AJ10G 2008 Fall AGU Meeting San Francisco, CA, December 15 – 19, 2008

Summary 2 / 14 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

Analysis outline 3 / 14 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 II. Application to VMGs from twenty two (22) solar active regions III. Results: • Correlations between LFFF and NLFFF energy and helicity budgets • Magnetic free energy and helicity budgets • Energy – helicity correlations and their predictive power IV. The course toward minimal stability – a possible SOC manifestation? V. Summary, conclusions, and future work

Azimuth disambiguation Continuum intensity Vertical electric current density Magnetic field vector 4 / 14 • 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 NOAA AR 10930, 12/11/06, 13:53 – 15:15 UT Tic mark separation: 10” The main magnetic polarity inversion line in the AR

Linear force-free field energy / helicity calculation Flux-weighted average a and uncertainties (highlighted) Sample a-value calculation on a VMG from NOAA AR 10930 5 / 14 • Surface calculation on the plane S of the observations using the linearized analysis of Georgoulis & LaBonte (2007): • Current-free (potential) magnetic energy: • Total magnetic energy: • Free (non-potential) magnetic energy: • Relative magnetic helicity: where:

Nonlinear force-free field energy / helicity calculation 6 / 14 • Volume calculation via magnetic field extrapolation Logarithm of free magnetic energy density vs. height for a 2D cross-section of the extrapolated volume • We use the NLFFF optimization method of Wiegelmann (2004) • Potential magnetic energy: • Total magnetic energy: NLFFF extrapolation for NOAA AR 10930 • Free magnetic energy: • Relative magnetic helicity: , where is a selected gauge expression for the vector potential proposed by Longcope & Malanushenko (2008) – see also DeVore (2000)

Nonlinear force-free field energy / helicity calculation • Moreover: • A and d are known fitting constants • Mutual term of free energy Lfgclose is chosen such that the free energy is kept to a minimum: • Mutual term of relative Lfg is defined in accordance with the analysis of Demoulin et al., (2006): 7 / 14 • Surface calculation using magnetic field extrapolation and the subsequent magnetic connectivity matrix Connectivity matrix NOAA AR 10930 • Based on our preliminary analysis (Georgoulis, 2009, in preparation) • Potential magnetic energy: • Total magnetic energy: • Free magnetic energy: • Relative magnetic helicity: where the VMG has been flux-partitioned into n partitions with fluxes Fi and alpha-values ai, i={1,…,n}

Results: comparison of free magnetic energies NLFFF extrapolation NLFFF surface calculation LFFF calculation 8 / 14 • 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) • 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. These results are preliminary. No attempt has been made to improve the calculations using, for example, preprocessing

Results: comparison of relative magnetic helicities NLFFF extrapolation NLFFF surface calculation LFFF calculation Helicity magnitudes 9 / 14 • Comparison for the same 9 active regions • 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

Free magnetic energy vs. relative magnetic helicity A notable feature in the energy and helicity of eruptive regions Potential energy 10030 10930 Free energy 9026 Magnetic energy (erg) 9165 8210 10953 8210 NOAA AR 9026 10953 9165 10930 10030 10 / 14 • 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

The predictive power of the free energy – helicity correlation 11 / 14 • 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 finding

Other potentially interesting correlations 12 / 14 • 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

Helicity buildup and subsequent eruptions in active regions: could SOC play a role here? 13 / 14 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. NOAA AR 9165 NOAA AR 9026 Overall, a possible SOC behavior may play a significant role in this problem • 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.

Summary, conclusions, and future work 14 / 14 • 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, 2006

Manolis K. Georgoulis 1 & Thomas Wiegelmann 2