INTERPRETING ERUPTIVE BEHAVIOR IN NOAA AR 11158 VIA THE REGION'S MAGNETIC ENERGY AND RELATIVE-HELICITY BUDGETS

A physically meaningful way to quantify the non-potentiality or, in other words, the eruptive capacity of solar active regions (ARs) is by calculating these regions' non-potential, "free" magnetic energy and, perhaps, the associated (relative) magnetic helicity. The free magnetic energy is the excess energy stored in the magnetic configuration of the AR, above the reference, potential energy corresponding to the AR's magnetic energy in the absence of electric currents. Magnetic helicity, on the other hand, is the corresponding degree of twist, writhe, and linkage of the AR's magnetic field lines (Berger 1999 and references therein). Adopting the formalism of the relative magnetic helicity (Berger 1984), necessary in case the studied magnetic configuration permeates a given boundary (such as the photosphere, for ARs), nonzero (left- or right-handed) total magnetic helicity implies nonzero free energy. On the other hand, zero magnetic helicity, as a signed quantity, may imply a nonzero free magnetic energy, in case equal and opposite amounts of left- and right-handed helicity are simultaneously present in the configuration. The lack of electric currents automatically means a zero free magnetic energy and a zero relative helicity, at the same time meaning the inability of an AR to energetically power solar eruptions, that is, flares and coronal mass ejections (CMEs; e.g., Schrijver 2009 and references therein).

Contrary to free magnetic energy, the possibly decisive role of magnetic helicity in solar eruptions is strongly debated. While free magnetic energy dissipates in non-ideal processes such as magnetic reconnection, helicity is—to a very good approximation—conserved in the course of reconnection. Indeed, helicity dissipation appears proportional to the inverse square of the magnetic Reynolds number (e.g., Freedman & Berger 1993; Berger 1999), and this makes it insignificant for the solar atmosphere given the very large value of the latter. If not transferred due to reconnection via existing magnetic connections to larger scales, therefore (i.e., in remote parts of the global solar magnetic field), then helicity can be bodily removed from ARs in the form of CMEs (Low 1994; DeVore 2000). In smaller amounts, helicity can also escape to the heliosphere via unwinding motions of "open" magnetic field lines in case helical, "closed" field lines transfer their helicity to them via interchange reconnection (e.g., Pariat et al. 2009; Raouafi et al. 2010).

Outstanding questions that might well involve magnetic helicity and its evolution in the solar AR corona include (1) the triggering of solar eruptions and (2) the causal relationship between the two eruption aspects, namely, flares and CMEs (a.k.a. the flare–CME connection). Numerous observational and modeling efforts have long suggested that ARs with large helicity budgets tend to be more eruptive than others (e.g., Canfield et al. 1999; Nindos & Andrews 2004; LaBonte et al. 2007; Nindos 2009; Georgoulis et al. 2009; Smyrli et al. 2010; Tziotziou et al. 2012) with some of these works actually attributing eruption onset to helicity, via the helical kink instability (e.g., Rust & LaBonte 2005; Török & Kliem 2005; Gibson et al. 2006; Kumar et al. 2012). A "large" helicity budget implies a dominant sign of helicity in ARs. On the other hand, there have been counterarguments over helicity's necessity for eruptions in successful eruption models that utilize roughly equal and opposite amounts of left- and right-handed helicity (Phillips et al. 2005; Zuccarello et al. 2009).

A reasonable step ahead would be to simultaneously and self-consistently calculate the budgets of free magnetic energy and relative magnetic helicity in solar ARs. Existing, widely applied techniques to this purpose involve time integration of the relative-helicity and energy injection rates obtained via the Poynting theorem for helicity and energy, respectively, on the photospheric boundary (e.g., Berger & Field 1984; Kusano et al. 2002) or evaluation of the relative-helicity formula in the three-dimensional AR corona (Berger 1984; Finn & Antonsen 1985). Both approaches have nontrivial prerequisites: helicity- and energy-rate calculation require a photospheric flow-velocity field that involves significant uncertainties (e.g., Welsch et al. 2007), together with the field-generating vector potential in the lower boundary (Chae 2001). Volume calculation of helicity and energy, on the other hand, requires an extrapolated three-dimensional magnetic field and its generating vector potential, including the corresponding current-free field and vector potential. Furthermore, nonlinear force-free (NLFF) field extrapolations are model dependent and subject to uncertainties and ambiguities (Schrijver et al. 2006; Metcalf et al. 2008). Besides the computational expense stemming from the current spatial-resolution capabilities of observed solar vector magnetograms that render these calculations (particularly the volume calculation) infeasible for routine use, the additional prerequisites contribute to our overall inability to credibly monitor the magnetic energy and helicity budgets in solar ARs.

With these limitations in mind, Georgoulis & LaBonte (2007) introduced a technique that calculates the snapshot (instantaneous) free-magnetic-energy and relative-magnetic-helicity budgets in solar ARs using only a single vector magnetogram for each calculation. The calculation is self-consistent and strictly adopts the relative-helicity formulation (i.e., a zero relative helicity for a zero free energy). However, this first effort assumed linear force-free fields that are known to be unrealistic for solar ARs. Recently, Georgoulis et al. (2012b) generalized the method to accommodate NLFF fields for the first time. Again, the method does not require any extrapolations, velocities, or vector-potential calculations. Tziotziou et al. (2012) applied this method to 162 vector magnetograms of 42 different ARs to construct the first free-magnetic-energy–relative-magnetic-helicity (EH) diagram of solar ARs. They reported a monotonic relation between free energy and relative helicity, thereby confirming that eruptive ARs tend to accumulate large budgets of both quantities. They further inferred well-defined thresholds of free energy and relative helicity for ARs to enter major (at least M-class) flaring and eruptive territory. These thresholds are 4 × 10³¹ erg and 2 × 10⁴² Mx², respectively. Calculation is straightforward and needs only a small fraction of the computational resources needed for a more conventional calculation of free-energy and helicity budgets.

These developments occurred in excellent timing with the advent of the Solar Dynamics Observatory (SDO) mission (Pesnell et al. 2012) featuring the Helioseismic and Magnetic Imager (HMI) vector magnetograph (Scherrer et al. 2012). Besides the constant-quality, high-spatial-resolution SDO/HMI data, the instrument's distinguishing feature is its unprecedented cadence. This has led to detailed, lengthy time series of AR vector magnetograms acquired within 12 minutes of each other. Such time series are ideal for the NLFF energy/helicity calculation method of Georgoulis et al. (2012b) in an effort to (1) test the method extensively, (2) investigate the statistical robustness of the EH diagram of Tziotziou et al. (2012), and (3) address the aforementioned outstanding questions regarding free energy, helicity, and their combined or distinct role in solar eruptions.

From the thus-far released SDO/HMI data sets, NOAA AR 11158 is undoubtedly the best-studied subject (see Section 3 below). The intensely eruptive AR gave the first X-class flare of solar cycle 24, thus ending the unusually prolonged lull of solar eruptive activity associated with the latest solar minimum. The SDO/HMI time series covered a five-day interval of the evolution of the AR, including the major X2.2 flare and a lengthy series of other eruptive and combined flares.

This work is structured as follows: the SDO/HMI data on NOAA AR 11158 are described in Section 2. A brief overview of existing studies on the AR is presented in Section 3. Section 4 presents a summary of the methodology of Georgoulis et al. (2012b) for the calculation of free magnetic energy and relative magnetic helicity, as well as one of the most credible methods for deriving these budgets from time integration of the respective injection rates. This latter method is used as a reference for comparison with our instantaneous NLFF budget calculation. Section 5 provides our results in detail, while Section 6 summarizes the study, discusses the ramifications of our results, and presents our conclusions.

NOAA AR 11158 emerged in the eastern solar hemisphere, near disk center, on 2011 February 11. During its first disk transit, GOES 1–8 Å X-ray flux detectors registered 56 C-class, 5 M-class, and the first X-class (X2.2) flare of solar cycle 24 (01:44 UT on 2011 February 15) as stemming from this AR. The X-class flare was associated with conspicuous halo-CME and EUV wave events (Schrijver et al. 2011). The AR evolved quickly from a simple dipole to a complex, predominantly quadrupolar configuration with an enhanced and strongly sheared polarity inversion line (PIL). Snapshots of this rapid, dynamical evolution are presented in Figure 1. Extensive descriptions of the AR's configuration and evolution can be found in Schrijver et al. (2011), Jiang et al. (2012), Liu et al. (2012a), and Vemareddy et al. (2012a), while Sun et al. (2012b) provided a detailed description of the long-term evolution of the AR, its magnetic field, and observed shear motions along the main PIL. A discussion of the most important works in the AR is presented in Section 3.

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For our analysis we use cutout vector magnetic field data taken with the HMI on board SDO. HMI is a full disk (4096 × 4096) filtergraph with a 05 pixel size that samples the Fe i 617.3 nm photospheric line through a 76 mÅ filter at six wavelength positions along the line, thus covering a range of λ₀  ±  17.5 pm. The first of HMI's two CCD cameras records filtergrams at these six wavelength positions in two polarization states and uses them to obtain Dopplergrams and line-of-sight (LOS) magnetograms at a cadence of 45 s. The second camera acquires six polarization states at these six wavelength positions every 135 s and uses them to compute the four Stokes parameters (I, Q, U, and V) necessary to derive the vector magnetic field. This is accomplished by a Milne–Eddington-based inversion approach (Borrero et al. 2011). Averaged 720 s filtergrams are used for the derivation of the Stokes parameters in order to enhance the signal-to-noise ratio and suppress five-minute p-mode oscillations.

We use 600 vector magnetograms of NOAA AR 11158 covering a five-day period (2011 February 12–16) with a 12 minute cadence. The photospheric area covered by these magnetograms is 650 × 600 pixels, or 325'' × 300'' on the image (observer's) plane. Figure 1 provides a sequence of snapshots and a detailed vector magnetogram of the region. The vector magnetogram data were acquired by SDO's Joint Science Operations Center (JSOC). Although the released vector magnetograms were already treated for the azimuthal 180° ambiguity, for reasons explained in Georgoulis (2012) they were re-processed using the non-potential field calculation (NPFC) of Georgoulis (2005), as revised in Metcalf et al. (2006). For our analysis we use the heliographic components of the magnetic field vector on the heliographic plane, derived with the de-projection equations of Gary & Hagyard (1990). As typical single-value uncertainties for the LOS and transverse field components and for the azimuth angle (δB_l, δB_tr, and δϕ, respectively), we use the mean value of the JSOC-provided respective uncertainties of all pixels with an LOS field B_l lying within 30% of the magnetogram's maximum/minimum B_l value. This guarantees the exclusion of numerous quiet-Sun pixels within the field of view (FOV), with B_l values close to the derived errors, that would contribute with unrealistically high uncertainties for the AR magnetic field and azimuth angle. The mean values of the derived errors are δB_l ∼ 32.5 G, δB_tr ∼ 38.2 G, and δϕ ∼ 29.

Figure 2 shows the evolution of the unsigned (total) magnetic flux (thick black curve) and the corresponding positive- and negative-polarity fluxes (red and blue curves, respectively) over the observing interval. The thick orange curve denotes the unsigned magnetic flux participating in the magnetic connectivity matrix of the AR (see Section 4.1 for a definition) which is necessary for our calculations. Note that the AR was approximately flux balanced, with the positive-polarity flux showing a small (∼5% on average) excess over its negative-polarity counterpart. The dynamical evolution of the AR kicks in after decimal day 12.8 (∼19:00 UT on 2011 February 12) and is signaled by the abrupt increase of all flux budgets, particularly the unsigned connected flux (orange curve). Subsequent increases and decreases of this flux budget reflect the physical evolution in the AR, from flux emergence and enhancement of the connectivity matrix to flaring and eruptions (see Section 5.3 for a detailed description). During the observing interval the AR released 1 X-class, 3 M-class, and 25 C-class flares.

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NOAA AR 11158 has already been the subject of numerous studies, using a variety of methods and observations, and also including the same HMI data set used for the present study. Schrijver et al. (2011) presented the first detailed work, supported by MHD modeling, on several aspects of the X2.2 flare of 2011 February 15 such as the study of expanding loops above the PIL, the observed EIT wave event and the accompanying halo CME. The evolution of magnetic field and magnetic energy in NOAA AR 11158 were discussed in detail by Sun et al. (2012b) with the use of NLFF field extrapolations. More recently, Sun et al. (2012a) focused on the M2.2 flare triggered on 2011 February 14 and used NLFF field extrapolations to study non-radial eruptions modulated by the local magnetic field geometry and the existence of a coronal null point.

Several studies focused on horizontal fields, shear motions along the PIL, Lorentz-force calculations, and peculiar flows (proper motions) in NOAA AR 11158. An NLFF field computation of the M6.6 flare of February 13 (Liu et al. 2012a) showed a 28% increase of the mean horizontal field near the magnetic PIL and a downward collapse of a strong horizontal current system above the AR's photosphere after the flare. Wang et al. (2012) reported a similar result for the X2.2 flare, showing a rapid, irreversible enhancement (30%) of the horizontal magnetic field at the PIL and an increase of both inclination and shear of the photospheric field as a response to the flare. This agrees with the results of Sun et al. (2012b) who used an NLFF field extrapolation to show an enhancement of the horizontal field that becomes both more inclined and more parallel to the PIL. Gosain (2012) also found that the field became more horizontal after the flare close to the PIL. Jiang et al. (2012) detected a clockwise rotation in the sunspot, 20 hr prior to the X-class flare that stopped 1 hr after the flare and suggested that these motions were associated with the buildup of helicity in the region and the shearing of the main neutral line. Abrupt changes during the X2.2 flare in the vertical electric current, the Lorentz force vector and the field vector, that became stronger and more horizontal, were also described by Petrie (2012); these changes were mainly concentrated near the PIL where the shear increased. Alvarado-Gómez et al. (2012) used HMI Dopplergram data to identify, through standard methods of local helioseismology, seismic sources in the X2.2 flare event. They correlated these sources with the HMI magnetic field changes and estimated the work done by the Lorentz force integrated over the entire AR. Liu et al. (2012b) studied photospheric flows derived from HMI vector fields and sub-photospheric flows derived by time–distance helioseismology from HMI Dopplergrams to conclude that horizontal flows associated with flux emergence, including apparent shear motion along the PIL, do not extend deeply into the subsurface. Horizontal proper motions of a few 100 m s⁻¹ along the main neutral line of the X2.2 flare were measured by Beauregard et al. (2012) using a local correlation tracking (LCT) method on HMI continuum images and longitudinal magnetograms. Maurya et al. (2012) further reported magnetic and Doppler transients and locations of spectral line reversals during the flare's impulsive phase that do not correspond to real changes of the photospheric magnetic and velocity field.

Concerning helicity budgets in NOAA AR 11158, Vemareddy et al. (2012a) studied the helicity injection (through shuffling) and its behavior and found that changes of the helicity flux signal due to injection of negative helicity in regions of dominant positive helicity are co-spatial and co-temporal with flaring/eruptive sites. Jing et al. (2012), using the same HMI vector magnetogram data set as our study, calculated the average current helicity as a function of altitude and relative helicity with an NLFF coronal field extrapolation method. Nindos et al. (2012), in a study of the role of the overlying background field in solar eruptions, also calculated from the same HMI vector magnetogram data the helicity flux and budget in NOAA AR 11158. Electric current, fractional current helicity (i.e., helicity due to vertical electric currents), photospheric free energy, and angular shear were also recently derived by Song et al. (2013) in an attempt to quantify the non-potentiality of NOAA AR 11158. Vemareddy et al. (2012b), on the other hand, studied two sub-regions of the AR with rotating sunspots and showed that proper and rotational motions play a key role in enhancing the magnetic non-potentiality of the AR by injecting helicity and twisting the magnetic fields, thus increasing the free energy.

Recently, Liu & Schuck (2012), in a comprehensive study of magnetic helicity and energy in NOAA AR 11158, studied the region's evolution by deriving and integrating helicity injection from both shuffling and emergence. They concluded that helicity is mainly contributed by shuffling while the energy buildup is mainly attributed to emergence.

Finally, a study of loops in NOAA AR 11158 was carried out by Aschwanden et al. (2011), who applied an automated analysis of SDO/AIA image data sets, to detect 570 loop segments, shortly before the X-class flare, at temperatures covering the range of 10^5.7–10⁷ K. Gosain (2012) used SDO/AIA observations to also study the evolution of coronal loops during the X2.2 flare which show three distinct dynamical phases; a slow rise phase prior to the flare, a collapse phase during the flare, and an oscillation phase following the flare-driven implosion.

In this section, we present (1) our method for calculating the instantaneous free-magnetic-energy and relative-magnetic-helicity budgets in an AR (Section 4.1), and (2) the derivation of these budgets by integrating the respective energy/helicity fluxes inferred via the photospheric velocity field (Section 4.2). We emphasize that the main analysis method is the one described in Section 4.1; integrated helicity and energy fluxes are used as a reference for comparison with our results.

4.1. NLFF Energy and Helicity Budgets from Single Vector Magnetograms

Recently, Georgoulis et al. (2012b) proposed a new NLFF method to derive the (instantaneous) free-magnetic-energy and relative-magnetic-helicity budgets using a single photospheric or chromospheric vector magnetogram of the studied AR. This method provides unique results, contrary to model-dependent NLFF field extrapolation methods. In return, it requires a unique magnetic-connectivity matrix, i.e., a matrix containing the flux committed to connections between positive- and negative-polarity flux partitions. This matrix is calculated using a simulated annealing method (Georgoulis & Rust 2007; Georgoulis et al. 2012b) that guarantees connections between opposite-polarity flux partitions while globally minimizing the corresponding connection lengths. The connectivity matrix defines a collection of N magnetic connections, treated as slender force-free flux tubes with known footpoints, flux contents, and variable force-free parameters.

For this collection of flux tubes, the free magnetic energy E_c is the sum of a self term $E_{c_{\rm self}}$, expressing the internal twist and writhe of each tube, and a mutual term $E_{c_{\rm mut}}$, due to interactions between different flux tubes. As such, it is given by

$$\begin{eqnarray} E_c & = & E_{c_{\rm self}} + E_{c_{\rm mut}} \nonumber \\ & = & A d^2 \sum _{l=1}^N \alpha _l^2 \Phi _l^{2 \delta } + \frac{1}{8 \pi } \sum _{l=1}^N \sum _{m=1, l \ne m}^N \alpha _l \mathcal {L}_{lm}^{\rm arch} \Phi _l \Phi _m, \qquad \end{eqnarray} \tag{ 1 }$$

where A and δ are known fitting constants, d is the pixel size of the magnetogram, and Φ_l and α_l are the respective flux and force-free parameters of flux tube l. $\mathcal {L}_{lm}^{\rm arch}$ is the mutual-helicity factor of two arch-like flux tubes that do not wind around each other's axes. Inference of this factor, first introduced by Démoulin et al. (2006), is discussed in detail by Georgoulis et al. (2012b) for all possible cases of intersecting/non-intersecting flux tubes pairs and flux-tube pairs with a "matching" footpoint, per the reduced spatial resolution of the flux-partitioning map. Note that the free energy of Equation (1) is to be taken as a lower limit of the actual energy as the unknown winding factor around flux tubes, which would require knowledge of the three-dimensional magnetic field, has been set to zero. Previous works (Georgoulis et al. 2012b; Tziotziou et al. 2012) have shown that this lower limit is realistic.

Likewise, the respective relative magnetic helicity H_m is the sum of a self $H_{m_{\rm self}}$ and a mutual $H_{m_{\rm mut}}$ term,

$$\begin{eqnarray} H_m & = & H_{m_{\rm self}} + H_{m_{\rm mut}} \nonumber \\ & = & 8 \pi d^2 A \sum _{l=1}^N \alpha _l \Phi _l ^{2 \delta } + \sum _{l=1}^N \sum _{m=1,l \ne m}^N \mathcal {L}_{lm}^{\rm arch} \Phi _l \Phi _m. \qquad \end{eqnarray} \tag{ 2 }$$

We refer the reader to Georgoulis et al. (2012b) for a detailed description of the method and the derivation of uncertainties for all terms of Equations (1) and (2).

4.2. NLFF Energy and Helicity Budgets by Integration of Respective Fluxes

A method to infer the (total) energy and relative-helicity budgets in an AR at time t is by integrating the respective fluxes from initiation t₀ of the AR to time t. The energy and helicity budgets in this case are given by

$$\begin{equation} E \equiv E(t) = \int _{t_0}^t (dE/dt) dt \end{equation} \tag{ 3 }$$

and

$$\begin{equation} H_m \equiv H_m(t) = \int _{t_0}^t (dH_m/dt) dt, \end{equation} \tag{ 4 }$$

respectively, where E(t₀) = H_m(t₀) = 0. The helicity injection rate (Berger & Field 1984) depends on (1) the photospheric velocity vector, whose inference involves significant uncertainties (e.g., Welsch et al. 2007), and (2) the vector potential A_p of the potential magnetic field B_p exhibiting the same normal-field condition as the observed field B.

In the simplest case, photospheric velocities can be inferred from a sequence of magnetograms through LCT (November & Simon 1988) techniques which, however, are highly uncertain due to the ambiguities in image motion. Moreover, LCT techniques tend to underestimate the amount of helicity injected by the shear term (Démoulin & Berger 2003) and are inconsistent with the magnetic induction equation, which governs the evolution of the photospheric magnetic fields (Schuck 2005).

Several alternative methods have been proposed during the past decade (see Welsch et al. 2007 and references therein) for the evaluation of plasma velocities. Here we use the Differential Affine Velocity Estimator for Vector Magnetograms (DAVE4VM) introduced by Schuck (2008), which is a generalization of the Differential Affine Velocity Estimator (DAVE; Schuck 2005, 2006) method. DAVE4VM estimates the plasma velocity $\boldsymbol{\upsilon }$ using the normal component of the ideal induction equation

$$\begin{equation} \partial _t{B_z} + \nabla _h \cdot (B_z \boldsymbol{\upsilon }_h - \upsilon _z\mathbf {B}_h) = 0, \end{equation} \tag{ 5 }$$

where B is the magnetic field vector, z denotes the normal (vertical) axis, and h denotes x- and y-components on the tangential (horizontal) plane. We apply this method to our time series of vector magnetograms on the heliographic plane, using a window size of 21 pixels in DAVE4VM. Since the coordinate system for the affine velocity profile is not aligned with the magnetic field (velocities not necessarily orthogonal to B), the calculated velocity $\boldsymbol{\upsilon }$ has to be further corrected by removing the field-aligned plasma flow $\boldsymbol{\upsilon }_\parallel$ and keeping only the perpendicular (⊥) component that plays a role in the induction equation. The corrected cross-field velocity is given by

$$\begin{equation} \boldsymbol{\upsilon }_\perp = \boldsymbol{\upsilon } - \frac{(\boldsymbol{\upsilon } \cdot \mathbf {B})\mathbf {B}}{B^2}. \end{equation} \tag{ 6 }$$

The normal $\upsilon _{\perp n}$ and tangential $\upsilon _{\perp t}$ components of this velocity $\boldsymbol{\upsilon }_\perp$ are then used to calculate the helicity flux dH/dt|_S across the plane S of the magnetogram. As Berger & Field (1984) have shown, this is derived by

$$\begin{equation} \frac{dH}{dt}\left |\right. _S = 2 \int _S(\mathbf {A}_p \cdot \mathbf {B}_h) \upsilon _{\perp n} dS - 2 \int _S(\mathbf {A}_p \cdot \boldsymbol{\upsilon }_{\perp t}) B_z dS, \end{equation} \tag{ 7 }$$

where the first term roughly corresponds to the helicity flux through emergence and the second term to helicity flux through photospheric shuffling. The vector potential A_p is calculated by means of a fast Fourier transform method, as implemented by Chae (2001). We refer the reader to Liu & Schuck (2012) for a discussion concerning issues related to the computation and interpretation of helicity fluxes related to tangential and vertical flows.

Likewise, the magnetic-energy injection rate (i.e., the Poynting flux) dE/dt|_S, as Kusano et al. (2002) have shown, is given by

$$\begin{equation} \frac{dE}{dt}\left |\right. _S = \frac{1}{4\pi } \int _S B^{2}_h \upsilon _{\perp n} dS - \frac{1}{4\pi } \int _S(\mathbf {B}_h \cdot \boldsymbol{\upsilon }_{\perp t}) B_z dS, \end{equation} \tag{ 8 }$$

where the first term roughly corresponds to the energy flux through emergence and the second term to energy flux through shuffling.

5.1. Helicity and Energy Budgets in NOAA AR 11158

We first apply the method described in Section 4.1 to the 600 SDO/HMI magnetograms of our sample. Figures 3 and 4 show, respectively, the relative-magnetic-helicity and the free-magnetic-energy budgets in the AR as functions of time. Totals (panels (b)) and self terms (panels (c)) are also shown. Given the small amplitudes of the self terms for energy and helicity, the corresponding mutual terms are indistinguishable from the totals and are hence not shown here. Figure 4 also shows the potential and total magnetic energy and their evolution. Uncertainties are shown by gray bars in all plots. For the total and mutual energy/helicity terms these uncertainties are relatively small, while they are relatively large for the self terms.

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All relative-helicity and free-energy terms show a relatively smooth, progressive evolution, with some scatter around the corresponding 72 minute running averages. Scatter increases after decimal day 13.5 when the AR starts building significant budgets of magnetic helicity and free energy powered by continuous flux emergence (Figure 2). An important finding is that the region builds both significant relative magnetic helicity and free magnetic energy, reaching peaks of ∼2 × 10⁴³ Mx² and ∼5.5 × 10³² erg, respectively. These budgets are large enough to power several eruptive flares which, in fact, is what the AR did over the five-day observation interval (Section 2). The peak times of all these flares are marked in Figures 3 and 4. Eruptive flares can be assessed fairly easily by using frequency–time radio spectra such as those of Figures 3(a) and 4(a), obtained by the WAVES instrument on board the Wind mission (Bougeret et al. 1995). Type-II radio activity in particular, corresponding to hours-long frequency drift of the radio emission, is a tell-tale signature of shock-fronted CMEs (see, e.g., Hillan et al. 2012 and references therein). Much faster (shorter) frequency drifts (Type-III bursts) typify magnetic-reconnection episodes in the solar atmosphere. Association of registered GOES flares with observed CMEs is based on (1) correspondence of the peak time of the flare with the onset time of the Type-II radio burst, and (2) available CME catalogs, such as the LASCO, CACTUS, and SEEDS. This is because neither GOES nor Wind/WAVES have spatial resolution. While the flare–CME connection is discussed in Section 5.6 in more detail, from Figures 3(a) and 4(a) we discern that at least the M6.6 flare of February 13, the M2.2 flare of February 14, the X2.2 flare of February 15, and the M1.6 flare of February 16 were eruptive.

NOAA AR 11158, an intensely flaring/eruptive AR shows both a significant budget and a dominant sense of magnetic helicity, positive (right handed) in this case, as already suggested by Tziotziou et al. (2012) for numerous ARs. Notably, as well, no major flare (⩾M1.0) occurs before thresholds of ∼4 × 10³¹ erg and ∼2 × 10⁴² Mx² in free magnetic energy and relative magnetic helicity, respectively, are exceeded, as concluded by Tziotziou et al. (2012). This point is further discussed in Section 5.4.

Comparison of our results (Figure 4) with the respective results by Sun et al. (2012b) shows that although the corresponding potential, total, and free-energy evolution share some qualitative resemblance before the occurrence of the X-class flare, there are notable differences in their evolution afterward: the magnetic energy derived by Sun et al. (2012b) continuously decreases after the flare while in our case free energy increases for almost a day thereafter. Furthermore, there are quantitative differences within the entire observing interval with the maximum free magnetic energy derived by our NLFF method being almost twice that derived by the NLFF field extrapolation (note that our free energy is a lower limit!). Similar differences also exist between our magnetic free energy and the respective results by Nindos et al. (2012), derived from NLFF field extrapolations of 16 Hinode SOT/SP vector magnetograms, that do not differ substantially from the results by Sun et al. (2012b). Such discrepancies are already noted and discussed by Georgoulis et al. (2012b). Finally, the relative-helicity evolution derived by Jing et al. (2012) using an NLFF field extrapolation differs from our result both quantitatively and quantitatively, as it is ∼50% lower.

5.2. Comparison with Helicity/Energy Budgets Derived from Integration of Helicity/Energy Fluxes

Figure 5 shows the emergence and shuffling terms of helicity (Figure 5(a)) and energy (Figure 5(b)) fluxes, as well as their totals, derived by Equations (3) and (4). Emergence and shuffling terms of energy/helicity are always in phase, while changes in the evolution of both beyond the influence of flux emergence mainly reflect shear motions along the PIL, separation motions of connected polarities, and sunspot rotation (Liu & Schuck 2012). Shuffling dominates over emergence in helicity injection rates, while emergence dominates over shuffling in energy injection rates. Moreover, significant changes in the energy flux seem to occur immediately after the X-class flare (early on day 15) while there seems to be a ∼24 hr delay in the helicity flux response. As discussed in Section 4.2, magnetic-energy and helicity budgets at a given time are derived here by integrating the respective fluxes from the start of observation to the time of interest. These budgets should be good approximations of the actual budgets because the start of the observing interval corresponds practically to the birth (t₀) of the AR.

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Integrating the helicity injection rates (Figure 5(a)), we find a total accumulated helicity ∼2.4 × 10⁴³ Mx², with almost two-thirds of it due to shuffling. This is in agreement with the results of Liu & Schuck (2012). In terms of energy, we find a total accumulated value ∼9 × 10³² erg, with ∼77% of it due to emergence; this is slightly different than the percentages reported by Liu & Schuck (2012). We also note that both the shuffling helicity flux profile and the corresponding accumulated helicity are in fairly good agreement with the corresponding calculations of Vemareddy et al. (2012a) who used DAVE instead of DAVE4VM to derive the velocity flow field in the AR. However, Nindos et al. (2012), who also used DAVE for deriving the helicity flux, find an accumulated helicity which is almost 50% lower than our accumulated helicity due to shuffling.

The helicity and energy flux curves derived by Liu & Schuck (2012) show a qualitative agreement but significant quantitative differences with our helicity/energy-flux results. Differences stem mostly from a difference in the FOV since these authors used a significantly smaller FOV that tightly encloses the AR. This difference in the FOV also explains the difference between our unsigned flux (Figure 2) and the corresponding flux derived by Liu & Schuck (2012) and Sun et al. (2012b) who used the same (albeit Lambert de-projected) magnetograms. Use of a ∼200–300 G threshold in our flux derivation, which is equivalent to restricting the FOV very close to the AR, results in a curve that is identical to theirs (not shown here). Further small discrepancies in the helicity flux could result from (1) uncertainties in the velocity calculations, (2) differences in the azimuth disambiguation methods, and (3) differences in the de-projection of the magnetograms; as already mentioned, Liu & Schuck (2012) used the Lambert (cylindrical equal area) de-projection method while we use the heliographic de-projection (Gary & Hagyard 1990).

Comparing the flux-integrated energy and helicity budgets from Section 4.2 with our respective NLFF budgets from Section 4.1 we find a fairly close agreement between the peaks, particularly for the relative helicity (∼2 × 10⁴³ Mx² (instantaneous) versus ∼2.4 × 10⁴³ Mx² (flux-integrated)) and less so for the total magnetic energy (∼16 × 10³² erg (instantaneous) versus ∼9 × 10³² erg (flux-integrated)). Comparing the respective temporal profiles is not straightforward, however, and could even be misleading: in our instantaneous NLFF budgets, activity-related changes reflected on the connectivity matrix incur a direct impact on the budget values. Flux-integrated budgets, on the other hand, will not show an activity-related change unless the photospheric velocity field, primarily, is impacted by this change. This explains the smoothness of the flux-integrated budgets in Figure 5 that is contrasted to the numerous transient variations of the (72 minute averaged) instantaneous budgets. Nonetheless, a point-to-point comparison between the integrated helicity/energy fluxes and the respective NLFF budgets yields significant correlations, at least up to the end of February 15, just before our instantaneous budgets started decreasing. We find linear (Pearson) and rank-order (Spearman) coefficients of the other 0.73 and 0.79, respectively, for relative magnetic helicity, and 0.83 and 0.84, respectively, for total magnetic energy.

Concluding, we note in passing that helicity- and energy-flux calculations and associated budgets "inherit" the uncertainties of the inferred photospheric velocity field, including caveats or shortcomings in its calculation. We repeated the analysis of Section 4.2 using a classical LCT method applied to successive sub-sequences of five consecutive B_z-images. We found that the LCT-calculated emergence helicity budget is roughly an order of magnitude smaller than that obtained using DAVE4VM-inferred velocities (Figure 5(a)).

5.3. Helicity and Energy Evolution in NOAA AR 11158

5.3.1. Comparison of Helicity/Energy Budgets with Unsigned Connected Flux

As already discussed in Section 2 and elsewhere, NOAA AR 11158 exhibited a very rapid and dynamical evolution, with continuous flux emergence (Figures 1 and 2) and a rapidly evolving connected flux (Figure 2). Connected-flux changes reflect changes in the connectivity matrix that arise from flux emergence, flux cancellation in small-scale reconnection events and/or large-scale flaring/erupting activity. Figures 3 and 4 suggest a good agreement between the unsigned connected flux and the AR's relative-helicity/free-magnetic-energy budgets with significant linear and rank-order correlation coefficients, equal to 0.93 and 0.89 for helicity and 0.88 (both correlation coefficients) for free magnetic energy. It is, however, the potential energy that shows a remarkable correlation with the unsigned connected flux with both correlation coefficients ∼0.99. This suggests that the unsigned flux evolution correlates well with the global magnetic field evolution in the AR while small increases/decreases indicate local and/or larger-scale changes in the non-potentiality of the magnetic field configuration resulting, at times, in significant changes in the relative-helicity and/or free-magnetic-energy budget. Point taken, we will now further discuss the dynamical evolution of NOAA AR 11158 by examining the evolution of magnetic helicity and free magnetic energy in more detail.

5.3.2. Helicity Evolution in NOAA AR 11158

Figure 3 shows the evolution of relative magnetic helicity in NOAA AR 11158. In the pre-eruption phase of the AR, particularly during the first day (2011 February 12), the relative-helicity budget is very low. It reflects the initially simple β-configuration (Figure 1) of the AR, with a magnetic field configuration apparently close to a potential configuration. However, the AR evolved rapidly, within a couple of days, into a more complex βγδ-configuration (Figure 1). Consequently, the helicity budget increased drastically within less than a day starting around the end of February 12, a period that coincides with the first cluster of two C-class flares and an eruptive M6.6 flare (see Figures 3, 4, and 11). Several hours after the M-class flare, late on day 13, a significant decrease of helicity occurs, co-temporally with a small increase of the connected flux, which is not associated with any flaring/eruptive activity. This helicity decrease seems also to be associated with a delayed by 2 hr increase of the free magnetic energy (Figure 4) and probably suggests a large-scale re-organization of the AR. Thereafter, during the early hours of day 14, coinciding with a second cluster of four non-eruptive C-class flares (among them a C8.3 and a C6.6), helicity increases, but this is followed by a very significant, simultaneous decrease of helicity, free energy, and connected magnetic flux shortly before midday (∼14.4). Figure 6 indicates an intense restructuring of the magnetic field configuration accompanied by an increase of the fragmentation in the AR. The former is reflected on the decrease of the connected flux Φ_conn while the latter is judged by the abrupt increase of the flux-tube number. "Open" flux is further increased, implying that previously connected flux within the AR now closes beyond the AR's limits.

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Helicity increases again, during the last half of day 14, when a swarm of six C-class flares (the largest ones being C9.4, C7 and an eruptive C6.6; Figure 11) and an eruptive M2.2 flare occur. An overall increasing helicity trend, starting around decimal day 14.5, interrupted by short, simultaneous decreases of helicity, free energy, and connected flux associated with eruptions (see further discussion in Section 11), could well be related to the reported rotation of the main sunspots (Jiang et al. 2012; Vemareddy et al. 2012a). Such a rotation twists and/or shears nearby magnetic field lines, adding further helicity and free/total energy to the region. Recently, Vemareddy et al. (2012b) showed clear correspondence, in a sub-region of NOAA AR 11158, between the rotational profile of the sunspot and the respective shear and helicity flux. This latest cluster of flares is followed in the early hours of day 15 by the large eruptive X2.2 flare, the first X-class flare of solar cycle 24. The peak of the helicity budget (∼2 × 10⁴³ Mx²) is first reached three hours after the X-class flare. The persistence of sunspot rotation after the X-class flare (Jiang et al. 2012) could explain this further increase of helicity.

It is worth discussing further that during the aforementioned 1.5 day interval before the first helicity peak, despite the occurrence of eruptive flares that removed helicity from the AR, helicity kept on increasing. This behavior seems to be the end result of a continuous, uneven competition between new helicity injection in the region, as Figure 5 also suggests, and helicity removal through repetitive eruptive activity with the former overpowering the latter. There are, as expected, occasional decreases of helicity during this interval associated with eruptive flares, which will be later discussed in Section 5.6, but the magnetograms (Figure 1) attest to a very rapid and intense dynamical evolution of the AR, with continuous emergence of new magnetic flux. As a result, both the unsigned flux and the unsigned connected flux keep increasing throughout the interval of study (Figure 2). The latter generally follows the aforementioned helicity evolution, increasing rapidly between decimal days 12.7 to 15 and then fluctuating around a mean value of 2.4 × 10²² Mx.

After its first peak, the helicity budget shows a considerable decrease that immediately follows the termination of the sunspot rotation reported by Jiang et al. (2012), and probably indicates another re-organization of the AR's magnetic connectivity, after a large sequence of flaring/eruptive activity. This re-organization also relates to the decrease of the unsigned connected flux and the simultaneous increase of free magnetic energy. Figure 7 provides a glimpse of this re-organization with two connectivity matrices before and after the X-class flare, separated by an hour, that clearly show connectivity changes mainly between the AR's eastern sunspot complex, where the flare occurred, and the AR's central complex of sunspots. We also remind the reader that an EIT wave event (Schrijver et al. 2011) accompanied the X-class flare, which could affect the AR and its vicinity. Late on February 15, the helicity budget increases again through two non-eruptive C-class flares and reaches a second, similar peak at decimal day 15.84, at which time a C6.6 non-eruptive flare occurs. Thereafter, a substantial, 30% helicity decrease occurs within 8 hr, accompanied only by two small non-eruptive C-class flares. This remarkable helicity decrease also coincides with a considerable decrease of free magnetic energy and, as Figure 8 shows, a decrease of the connected flux. At the same time the number of flux tubes that comprise the connectivity matrices increases, indicating a rapid fragmentation of the AR (more magnetic partitions, defining a larger number of flux tubes but with less total connected flux), while the open flux also increases indicating a large-scale re-organization of the magnetic configuration. It is the cumulative effect of both processes that leads to these profound helicity and energy decreases while no eruptive activity takes place.

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During the last 17 hr of 2011 February 16 seven non-eruptive C-class flares (among them a C9.9 and a C7.7) and an eruptive M1.6 flare occur while the helicity fluctuates around a value of ∼1.5 × 10⁴³ Mx². This well-defined mean helicity, despite a swarm of (non-eruptive) flares, relates to the absence of major eruptions during this interval—only the eruptive M1.6 flare shows a clear associated helicity decrease (see Section 5.6 and Figure 11).

5.3.3. Free Magnetic Energy Evolution in NOAA AR 11158

Figure 4 shows the evolution of the free-magnetic-energy budget in NOAA AR 11158. Free energy generally follows the relative-helicity evolution, but with some notable differences. During the first four days of observations, the free magnetic energy shows a similar increase as helicity and finally peaks less than a day after the X-class flare, at the same time (decimal day 15.84). The peak energy is ∼6 × 10³² erg and is followed by a rapid ∼50% decrease. Differences between the free energy and the relative helicity reflect the different physics of energy and helicity removal: free energy is released (dissipated) in the course of any (eruptive or confined) flare, whereas helicity is bodily removed in the course of eruptions only (excluding smaller amounts that escape by the unwinding of "open" field lines in case of interchange reconnection with helical "closed" lines). Indeed, there are occasional decreases of free magnetic energy that correspond to the energy contents of associated non-eruptive flares when helicity does not show any significant changes. A much more significant increase in free magnetic energy than helicity also occurs, immediately after the X-class flare, with an enhanced energy flux during that period, as also Figure 5 indicates. After its peak, free energy decreases with a faster (∼50%) rate than helicity, which only drops by 30%. The initial stages of this considerable decrease, as discussed in Section 5.3.2, are due to fragmentation and large-scale magnetic field re-organization while later this decrease is the result of intense, but non-eruptive, flaring activity, as Figure 4 shows. Small-amplitude enhancements of free energy during this last day of observations coincide with similar enhancements of the energy injection rate.

Following the studied five-day observing interval, there are still sufficient budgets of both free magnetic energy and relative helicity left in NOAA AR 11158 to further power flares and eruptions. Indeed, as the GOES event catalog indicates, two more M-class flares (M6.6 and M1.0) and 31 C-class flares occurred before the AR finally crossed the western solar limb, concluding its first disk passage. When the AR reappeared in the eastern limb (as NOAA AR 11171) it was a decaying AR with no sunspots, having two well-separated, disperse opposite polarity regions and a long Hα filament present along the PIL. The AR concluded its impressive flaring activity, according to the GOES event catalogue, with three more flares (M1.5, C4.5, and C6.4), occurring within a couple of days after its reappearance.

In summary, our magnetic-energy and helicity calculation can provide a plausible interpretation of the evolution in NOAA AR 11158, in qualitative agreement with independent observational and modeling/data-analysis studies. The AR builds rapidly large budgets of right-handed helicity and free energy sufficient to power several confined and eruptive flares. The overall dynamical evolution of the AR is a combination of flux emergence injecting free energy and helicity and flares/eruptions that dissipate or remove parts of these budgets. In some cases we also witness "global" (i.e., AR-wide) re-organizations of the region's magnetic configuration. This usually follows swarms of flaring activity that affects free-energy and relative-helicity budgets. However, reorganizations occur also in the absence of flares and this may be due to fragmentation, in case flux emergence ceases, or possibly due to large-scale readjustments of the AR within the global solar magnetic field.

5.4. Energy–Helicity Diagram of NOAA AR 11158

Recently, Tziotziou et al. (2012) introduced the "energy–helicity" (EH) diagram of solar ARs, demonstrating a monotonic correlation between the NLFF free magnetic energy and relative magnetic helicity, as well as respective thresholds of ∼4 × 10³¹ erg and ∼2 × 10⁴² Mx² for an AR to host major (M-class and higher), generally eruptive, flares. A plausible question is whether this diagram can be reproduced by NOAA AR 11158 alone.

Figure 9 shows the compiled EH diagram for NOAA AR 11158. Magnetograms corresponding to the first 20 hr of February 12 (blue crosses), when the AR has not yet built considerable budgets of magnetic helicity and free energy (Figures 3 and 4), show low, scattered values for both quantities. For this period, there also seems to exist an exponential increase of relative helicity with respect to free energy. If this behavior is real, considering of course the significant uncertainties, it would indicate that ARs tend to accumulate magnetic helicity at higher rates than free magnetic energy. However, this behavior might also be an artifact of our NLFF method that gives rise to exactly zero free energies for exactly potential fields; this is not completely guaranteed for helicity, particularly in case of existing flux imbalances (Georgoulis et al. 2012b). In conclusion, it is rather difficult to disentangle between these two different possibilities.

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Once the AR has accumulated significant helicity and energy budgets (red crosses) we find that the relative-helicity and free-magnetic-energy budgets follow quite well the derived least-squares best fit and the least-squares best logarithmic fit (Equations (3) and (4), respectively, of Tziotziou et al. 2012). This nearly monotonic dependence holds for a large range of helicities and energies. Furthermore, as Figures 3 and 4 suggest, no major flare occurs before both the helicity and the energy thresholds are crossed. In fact, not even C-class flares occur in the AR before these thresholds are exceeded. Once exceeded, however, a major (M-class) flare occurs within hours.

The single-AR EH diagram of Figure 9, relying solely on magnetograms from NOAA AR 11158, strongly attests to the validity of the EH diagram of Tziotziou et al. (2012). It further adds support to the conclusion that both magnetic free energy and helicity are important for ARs, particularly flaring ones.

5.5. Amplitudes and Timing of Mutual- and Self-helicity/Energy Terms

As shown in Figures 3 and 4 and discussed in Section 5.1, the mutual terms of the relative helicity and free energy in NOAA AR 11158 are much larger than the respective self terms. This is not a new result; Régnier et al. (2005) found from NLFF field extrapolations that the mutual helicity in NOAA AR 8210 accounts for ∼95% of the total helicity budget. In NOAA AR 11158, mutual helicity accounts for even more of this budget: mutual-helicity terms are typically ∼10³ times larger than self-helicity terms, thus accounting for ∼99.9% of the total helicity budget. Mutual terms in the free magnetic energy typically account for even more (∼99.93%) of the respective budget. We find, therefore, that mutual terms caused by the interaction of different NLFF flux tubes overwhelmingly dominate the twist and writhe contributions of these flux tubes in the budgets of magnetic helicity and energy. Let us clarify at this point that the true values of the mutual-to-self helicity and energy ratios can be determined only in case of fully resolved magnetic configurations. For N discrete flux tubes (Equations (1) and (2)) we have N self- and N²−N mutual terms. Changing N due to increasing or decreasing, but always imperfect, spatial resolution, will modify the number of terms and hence the values of the ratios. In the extreme case of N = 1, meaning that the observing instrument resolves a single flux tube, there will be no mutual terms. The above point taken, it appears that for N > 1 or N ≫ 1 independent works invariably find a clear dominance of mutual over self terms. This should be a real effect.

Another interesting result occurs when one studies the relative timing between mutual and self terms of helicity and energy. This is exemplified in Figure 10, where we have normalized mutual and self terms of helicity and energy by their respective maxima. We find, then, that (1) there is a hysteresis in the buildup of self terms of helicity and energy with respect to these quantities' mutual terms, and (2) the buildup rate of self terms of helicity and energy is generally lower than that of the respective mutual terms, with the self term of the free energy building up at a slightly lower rate than the self term of the relative helicity. Rough, qualitative features of the mutual-term time series are reproduced by self-term time series for both quantities, but at a delay ranging between several to ∼24 hr. We also find remarkable coincidence in the times of peaks of mutual (helicity and energy) terms, as well as in the respective peak times of self (helicity and energy) terms. Mutual terms peak ∼12 hr earlier than self terms.

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To our knowledge, this is the first time that this result, obviously enabled by the unparalleled cadence of the SDO/HMI magnetogram data, is reported. We consider this an important finding because it attests to a possible conversion of mutual-to-self helicity and energy. More specifically, mutual interactions between an ensemble of flux tubes translate to twist and writhe of individual flux tubes of the ensemble. We cannot think of any way to achieve this, other than magnetic reconnection. Indeed, magnetic helicity is roughly conserved in the course of reconnection forcing interacting pre-reconnection flux tubes to become more helical after reconnection because they interact less than before. Naturally, this reconnection will force interaction changes to the other flux tubes of the ensemble, as well. If, however, a sequence of reconnection episodes occur in order to relax a given magnetic configuration, then the trend appears such that increasingly helical post-reconnection flux tubes are formed. The same essentially happens with free energy (more twisted structures), as well, but under the fact that energy always dissipates in the course of reconnection. These energy losses may be the reason for the slightly lower buildup rate of free energy in Figure 10.

The latter result certainly warrants further investigation. Georgoulis (2011) suggested that the small-scale flickering of numerous magnetic reconnection events occurring routinely along AR PILs effectively transforms shear-induced mutual helicity into self helicity. The relation between PIL formation and the origin of shear has been discussed extensively by Georgoulis et al. (2012a). As a result, a new picture of pre-eruption AR evolution appears to emerge, with typically eruptive ARs characterized by intense PILs building increasingly helical magnetic configurations along these PILs. This argument is in favor of the formation of pre-eruption flux ropes in ARs. Very recently, Patsourakos et al. (2013) reported direct evidence of pre-eruption flux-rope formation via a confined flare, which is obviously a magnetic reconnection episode.

5.6. Helicity and Energy Contents of Major Eruptive Flares and the Flare–CME Connection

An admittedly untenable pursuit of relevant analyses of the past was to detect discrete, significant (i.e., beyond uncertainties) changes in free magnetic energy and relative magnetic helicity that could be related to eruptions on a case-by-case basis. Obviously this was attempted under the assumption that eruptions have energy and helicity contents that are significant fractions of the respective budgets possessed by the host AR. In this respect, Metcalf et al. (2005) were unable to determine a discrete decrease of the free magnetic energy in NOAA AR 10486 as a result of an X10 flare in the region, by applying the virial theorem to Imaging Vector Magnetograph (IVM) data. Using vector magnetograms from the Spectropolarimeter (SP) of the Solar Optical Telescope (SOT) on board Hinode with a cadence of a few hours, on the other hand, Georgoulis et al. (2012b) were able to estimate a change in helicity and energy of the order 5 × 10⁴² Mx² and 2 × 10³² erg, respectively, apparently related to an X3.4 flare triggered in NOAA AR 10930. The calculation was made as a proof of concept for the NLFF helicity and energy calculation of Section 4.1. However, given the crude cadence of the SOT/SP data, that result was handled with care.

We consider the high-cadence SDO/HMI data of this study ideal for such an investigation. Using NLFF field extrapolations, Sun et al. (2012b) detected a decrease of ∼3.4 × 10³¹ erg after the X2.2 flare of 2011 February 15 in NOAA AR 11158. In this study, we attempt to detect changes in both free magnetic energy and relative magnetic helicity. Moreover, we do not restrict the analysis to the X-class flare alone; instead, we study the helicity and free-energy evolution in the course of the four largest eruptive flares in NOAA AR 11158 during the five-day observing interval. The results are shown in Figure 11 and are overplotted on co-temporal Wind/WAVES frequency–time spectra for reference. All shown curves are 72 minute averages. Given the small uncertainties in helicity and free energy (Figures 3 and 4), all notable changes that last much longer than the HMI cadence of 12 minutes should be considered significant and stemming purely from changes in the calculated connectivity matrices.

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Figure 11 leads us to three major conclusions: first, in at least three of the four eruptive flares studied (Figures 11(a), (c), and (d)) there is a significant decrease in the magnitudes of the AR's relative helicity and free energy. Second, and perhaps more importantly, these decreases start occurring well before the onset of the flare and the projected launch time of the associated CME. Third, decreases are not instantaneous but transient, lasting for 2–3 hr. Flares and CMEs occur toward the end (15–30 minutes before the end) of these dips. After the end of the dips, the relative-helicity and free-energy budgets follow the underlying, general trends of these budgets for the period of interest.

The properties of the relative-helicity and free-energy decreases, including properties of the four eruptive flares, are summarized in Table 1. The peak amplitude of the helicity and energy decreases in Figure 11 is viewed as the respective helicity and energy contents of the associated eruption. In this respect, the eruption related to the X2.2 flare (Figure 11(a)) has an energy content of ∼8.4 × 10³¹ erg, a factor of <3 larger than the estimate of Sun et al. (2012b). The helicity budget of the respective CME is ∼2.6 × 10⁴² Mx², in excellent agreement with estimated typical helicity contents of CMEs (DeVore 2000; Georgoulis et al. 2009). The other three eruptions are associated with energy releases of the order 10³¹ erg to several times this budget. For the eruptions related to the M2.2 and M1.6 flares (Figures 11(c) and (d), respectively) the helicity content of the CMEs stays close to 2–3 × 10⁴² Mx². For the eruption associated to the M6.6 flare (Figure 11(b)), however, it is an order of magnitude smaller ∼3 × 10⁴¹ Mx². We are unclear on why this CME is so much weaker but this is corroborated by the facts that (1) the CME shock appears rather weak in Wind/WAVES spectra (at least weaker than the others in Figures 11(a), (c), and (d)), and (2) the LASCO CME catalog has registered this CME as "Poor" and "Partial Halo."

Table 1. Temporal Properties of the Calculated Decrease in Relative Magnetic Helicity and Free Magnetic Energy in NOAA AR 11158 for the Four Largest Eruptive Flares that Occurred in the AR during the Observing Interval

No	Date	Decrease Properties			Flare Properties		ΔHm	ΔEc
		Start	Peak	End	Onset	Class	(Mx2)	(erg)
1	2011 Feb 15	23:22a	00:13	02:00	01:44	X2.2	2.6 × 1042	8.4 × 1031
2	2011 Feb 13	16:25	16:50	18:10	17:28	M6.6	2.9 × 1041	1 × 1031
3	2011 Feb 14	15:35	16:25	17:50	17:20	M2.2	2.8 × 1042	4.9 × 1031
4	2011 Feb 16	11:13	12:35	13:44	14:19	M1.6	2 × 1042	6.5 × 1031

Notes. The estimated eruption-related peak decreases ΔH_m and ΔE_c are also provided. All times are universal times. Helicity and energy values have been obtained by the 72 minute average time series of the respective quantities (Figures 3 and 4, respectively). ^aOn 2011 February 14.

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In an attempt to explain why we observe these decreases in the AR's relative helicity and free energy prior to major eruptive flares, we construct in Figure 12 the respective time series for the total connected flux Φ_conn (i.e., the flux participating in the connectivity matrix), the total "open" flux Φ_open (i.e., the flux closing beyond the bounds of the AR), and the number of flux tubes participating in the connectivity matrix. Figure 12 shows that co-temporal with helicity/energy decreases are slight decreases in the connected flux Φ_conn for all four eruptive flares. The most significant of these decreases pertains to the X2.2 flare and associated eruption (∼6%; Figures 11(a) and 12(a)) while the least significant pertains to the M6.6 flare with the weakest CME (∼0.7%; Figures 11(b) and 12(b)). These decreases do not seem to be associated with respective increases in the "open" flux Φ_open (except, perhaps, a very slight increase in Figure 12(c)); this would mean that previously "closed" connections, i.e., connections with both ends within the AR, now close beyond it. Instead, these decreases imply that the normal field component B_z that determines the connected flux becomes somewhat weaker prior to the eruptions. This is in line with multiple recent works that advocate for (Fisher et al. 2012; Hudson et al. 2012) or report (Petrie 2012; Sun et al. 2012b) enhancements in the horizontal magnetic field and slighter decreases in the normal field component. Those arguments and findings mainly refer to the photospheric vicinity of the eruption but, as we can see from Figure 12, they are also detectable over the entire AR. However, our results do not support an irreversible or permanent decrease of the connected flux; in this respect, our results run counter to those works advocating irreversibility, at least in terms of a B_z-decrease. Point taken, note that Sun et al. (2012b) did not find an unambiguously irreversible B_z-decrease in the case of the X2.2 flare in NOAA AR 11158.

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Thanks to the unprecedented cadence of the SDO/HMI magnetogram data, our results help clarify an additional crucial point: so far the above eruption-related flux changes were confirmed by inspecting the pre- and post-eruption configurations. Here we deduce that flux-decreases occur mostly prior to eruptions, with the eruptions themselves occurring either at the last stages of these decreases or shortly thereafter. This is sound evidence toward resolving the elusive flare–CME connection: from our results, the unstable magnetic structure that will evolve to become a CME starts ascending clearly before the flare and the a posteriori projected CME launch time. To our understanding, this is the only possibility that can interpret the connected-flux decreases as changes in the Lorentz force, but also as contraction/collapse of pre-eruption loops as the CME progenitor increases in size and ascends faster (Vourlidas et al. 2012). The flare is a result of the drastic perturbation caused by this ascension. Note that this interpretation has appeared also in earlier works (Zhang & Dere 2006) but only now has this picture been clearly supported by independent observational, modeling, and theoretical works.

The present study includes only a limited sample of eruptive flares. In follow-up works, we will further test our findings with larger flare samples and aim to compare results between eruptive and confined flares.

The magnetic-free-energy and relative-magnetic-helicity budgets in NOAA AR 11158 have been studied using high-cadence, constant-quality, space-based vector magnetograms acquired by the HMI magnetograph on board SDO. Budgets have been derived using the novel NLFF method introduced by Georgoulis et al. (2012b) that requires single photospheric (or chromospheric) vector magnetograms. The constructed time series of budgets for free energy and relative helicity are comprised of values independent from each other. For reference, we use energy and helicity budgets obtained by time integration of energy and helicity injection rates, respectively, inferred using the DAVE4VM photospheric flow field. Successive points in the budget time series are related to each other in this case. The only alternative way to infer magnetic energy and helicity budgets is by (NLFF, preferably) magnetic field extrapolations, relying also on single vector magnetograms and hence providing independent budgets in the time series. However, both extrapolations and energy/helicity fluxes require additional nontrivial information in order to yield energy and helicity budgets (see the Introduction). Much of this information, particularly the NLFF extrapolated and the velocity fields, is strongly model dependent.⁴ Therefore, if we can show that our method reliably calculates energy and helicity budgets at a given AR without requiring information other than this AR's vector magnetogram, then this or refined future techniques of this kind might become methods of choice for the task at hand.

Keeping in mind that the rate-integrated helicity and energy budgets do not share the same physical meaning with our instantaneous budgets, we find a qualitative correspondence between the two (Section 5.2). Indeed, the peak of the integrated helicity in the region is ∼2.5 × 10⁴³ Mx² while our instantaneous budget shows a peak ∼2 × 10⁴³ Mx². The total energy budget from our method, on the other hand, peaks at ∼1.6 × 10³³ erg, while the peak of the flux-integrated budget is lower, ∼9 × 10³² erg. The morphologies of the two time series are different due to their different physical origins and meaning. This being said, our NLFF field energy and helicity budgets depend more sensitively on the AR's eruptive and non-eruptive evolution. We conclude, therefore, that NLFF methods such as the one implemented here are indeed important for decoding the dynamical evolution of ARs from high-cadence vector magnetogram data.

Stepping on this central conclusion, our analysis has led to a meaningful interpretation of the evolution and the eruptive activity of NOAA AR 11158. Besides free magnetic energy, analysis provides further evidence that magnetic helicity is, indeed, an important ingredient of the overall evolution in solar ARs.

The attained budgets of free magnetic energy and relative helicity in NOAA AR 11158 imply clearly that the AR builds up both significant amounts of energy and dominant (in this case positive (right-handed)) helicity before entering major flaring and eruptive territory. Our method only shows the increases in the energy/helicity budgets without providing a rigorous physical investigation of why this happens except, of course, the increasing size and complexity of the magnetic connectivity matrix that is fueled by continuous flux emergence in the AR (Figure 2). Previous observational works that reported shearing motions along the AR's PIL and rotating sunspots (Section 3) complement the picture and our results. The AR hosts major eruptions when both previously inferred free-energy and relative-helicity thresholds, 4 × 10³¹ erg and 2 × 10⁴² Mx², respectively, are exceeded (Tziotziou et al. 2012). In addition, the constructed—from the 600 magnetograms of this AR alone—EH diagram clearly follows and further validates the monotonic correlation between the two quantities found by Tziotziou et al. (2012) from the study of 42 ARs (Section 5.4). Notably, the construction of the original EH diagram did not include SDO/HMI magnetograms.

Superposed on the overall increasing tendency of free-energy and relative-helicity budgets throughout the five-day observing interval are prolonged, significant decreases of these budgets that last for several hours. We have identified two different reasons for these dips: first, a "large-scale," AR-wide re-organization of the magnetic configuration and, second, confined/eruptive flaring activity in the AR.

An AR-wide re-structuring that results in energy/helicity decreases manifests itself via (1) a decrease of the connected flux Φ_conn participating in the connectivity matrix, accompanied by an increase of the respective "open" magnetic flux Φ_open, or (2) an increase of the number of active connections (thought as slender flux tubes) accompanied by a decrease in Φ_conn (Figures 6 and 8). In the first case, flux closing within the AR up until a certain time now closes beyond its limits. As "open" flux is not currently included in our energy/helicity calculations, this results in a decrease of the calculated budgets. In the second case, the AR fragments to more partitions at smaller Φ_conn leading to the same net result.

An eruption in the AR, on the other hand, will result in a temporarily smaller Φ_conn, apparently due to the eruption-related weakening of the normal-field component (Section 5.6). This has been attributed to the action of the Lorentz force in the eruption area and is supported by observations. However, the new result in our case is that the weakening occurs before the observed onset of the eruption, i.e., the flare onset and/or the projected CME launch time. We interpret this finding as solid evidence that the CME progenitor starts ascending well before the flare. This sheds light on the elusive flare–CME connection and interprets flares as consequences of the CME–progenitor destabilization, rather than as triggers of this destabilization. This behavior is clearly seen in three out of four major eruptive flares the AR hosted within the observing interval. In the remaining eruptive flare the AR exhibited qualitatively the same behavior but to a much smaller extent. The most conspicuous decrease of Φ_conn was ∼6% and was associated with the X2.2 flare of 2011 February 15, giving rise to a helicity content of the CME of the order of 2.6 × 10⁴² Mx² and an energy content for the eruption as a whole of the order of 8.4 × 10³¹ erg. If similar results are confirmed, and we plan to carry out follow-up studies of this type, we envision a routine calculation of eruption energetics and helicity in the future. At the moment, nevertheless, it appears safe to conclude that ARs possess much higher free-energy and helicity budgets than those of the largest eruptions that they may ever host. When this is no longer valid, the AR in question has already entered the final decay phase that will result in its demise.

Another finding of this study was that eruption-related energy and helicity decreases are not irreversible or permanent but transient and restored within several hours by the general trend of the budgets in the AR. The interpretation of this effect, especially in view of works that show permanent decreases of B_z and/or increases of the horizontal field, remains to be investigated.

A further important finding that corroborates the credibility of our calculations is the AR's evolution during the last day of the observing interval. Activity in the AR on 2011 February 16 was characterized by repeated but confined flaring activity, lacking conspicuous CMEs. Monitoring the free magnetic energy of the AR (Figure 4), we notice a nearly monotonic decrease by ∼3 × 10³² erg within less than a day, mostly due to energy dissipation in the course of repeated flaring. The relative helicity, however, fluctuates around a well-defined mean value of ∼1.5 × 10⁴³ Mx² (Figure 3) due to its rough conservation in the course of the confined flares. Such different behavior stemming from the core properties of magnetic free energy and relative magnetic helicity accounts for nonlinearities in the EH diagram of solar ARs.

Concluding, we underline the new, emerging picture of the pre-eruption evolution in ARs that our results have enabled and previous studies have envisioned: this picture relies on the timing between mutual and self terms of free energy and relative helicity. We find a significant (much larger than the observational cadence) hysteresis of self-term development in comparison to mutual terms that are developed earlier and at higher buildup rates (Section 5.5). We consider this evidence of a mutual-to-self conversion of energy and helicity, rather than an independent, out-of-phase buildup of both. Moreover, the higher buildup rate of self-helicity compared to the respective rate of the self-free-energy, in view also of the rough conservation of helicity in an isolated, non-eruptive magnetic structure points to a simple (and single) interpretation of this conversion: magnetic reconnection. The most profound reconnection area of an AR is its PIL(s); the stronger the PIL, the more intense the flickering of magnetic reconnection along it. Reconnection along intense PILs is due to the action of shear that adds stresses to the already complex magnetic configuration, while shear is caused by Lorentz force that acts along the PIL and only there (Georgoulis et al. 2012a). As hinted by Georgoulis (2011) and shown here more convincingly, this logical, hierarchical sequence of dynamical evolution results in increasingly helical magnetic structures as PIL evolves. Stated otherwise, this may well be independent evidence in favor of pre-eruption helical flux-rope formation that was hypothesized, theorized, or observationally reported by multiple previous works. We plan to further elucidate this result in future follow-up studies.

We are grateful to P.W. Schuck for valuable, extensive discussions pertaining to DAVE4VM and its application to this data set. The data are used courtesy of NASA/SDO and the HMI science team. Y.L. was supported by NASA contract NAS5-02139 (HMI) to Stanford University. This work was supported from the EU's Seventh Framework Programme under grant agreement No. PIRG07-GA-2010-268245.