3D Magnetic Free Energy and Flaring Activity Using 83 Major Solar Flares

In this Letter, we examine the relationship between 3D magnetic free energy (MFE) and flaring activity using 83 major solar flares (M-class and X-class) in nine solar active regions (ARs). For this, we use 998 nonlinear force-free field extrapolations compiled by the “Institute for Space-Earth Environmental Research Database” at Nagoya University. These ARs produced at least three major flares with distinct rising and peak phases of 3D MFE. For each phase, the solar flare occurrence rate (FOR) is calculated as a ratio of the number of flares to the duration. The major results from this study are summarized as follows. First, there is no clear linear correlation (CC = 0.15) between 3D MFE and flare peak flux. Second, the FOR (3.4 day−1) of the rising phase is a little higher than that (3.1 day−1) of the peak phase, depending on AR. Third, for several flares, there are noticeable decreases in 3D MFE, which correspond to the typical energy of a major flare (about 1032 erg). Fourth, it is interesting to note that there are noticeable enhancements in FORs at several local maxima of 3D MFE, which may be associated with flux emergence and/or shearing motions. Fifth, the flare index rates, which are defined as the summation of flaring activity divided by the duration, of rising and peak phases are 151 day−1 and 314 day−1, respectively. Our results imply that the traditional and simple “storage and release” model does not apply to flare activities, and the random perturbation may be important for triggering flares.


Introduction
Solar flares are sudden and intense energy releases of magnetic fields, which are some of the most significant eruptions in the solar system (Tandberg- Hanssen & Emslie 1988;Benz 2008;Svestka 2012).When a strong flare occurs, it affects telecommunications and satellites around the Earth (Doherty et al. 2004).To gain deeper insights into the physics of solar flares, it is important to estimate the magnitude of magnetic energy encapsulated within magnetic fields (Regnier & Priest 2007).The magnetic energy is a result of the Sun's extremely hot and convective interior, which generates electric currents that, in turn, produce magnetic fields.It is now widely known that magnetic field is a fundamental driver of solar activity such as flares (Barnes & Sturrock 1972;Shibata & Magara 2011).The magnetic energy within a volume V within a magnetic configuration is computed using the formula: The triggering of major solar flares is attributed to two primary mechanisms: (1) flux emergence and cancellation (Parker 1955;Archontis 2008;Choudhary et al. 2013), and (2) shearing motions (Wang et al. 1994(Wang et al. , 2002;;Kusano et al. 2004).These mechanisms are responsible for triggering and releasing magnetic free energy (MFE) during solar eruptions, such as flares.The MFE, denoted as E free in Equation (2), is defined as It is the excess energy stored within a magnetic configuration: total magnetic energy (E tot ) minus the potential energy as its minimum-energy state (E pot ) (Low 1989;Regnier & Priest 2007).There is no free energy in a potential field configuration, which we accept as a minimum-energy state for a given normal magnetic field at the photosphere (Regnier & Priest 2007).Several studies have shown that the value of MFE decreases after major flares, implying that the stored magnetic energy should be released through solar flares (Sun et al. 2012;Tarr et al. 2013).
There have been several studies on the relationship between MFE and solar flaring activity (Sun et al. 2012(Sun et al. , 2015;;Choudhary et al. 2013;Tarr et al. 2013;Veronig & Polanec 2015;Jiang et al. 2016;Liu et al. 2016).Their studies have a couple of limitations: (1) their results are based on 2D magnetic field data, and (2) they did not use a large sample of data for the relationship.Choudhary et al. (2013) examined the relationship between 2D MFE and solar flare, where MFE is calculated using the magnetic virial theorem (Klimchuk et al. 1992).They suggest that the largest flares (X-class) are observed when MFE exceeds 50% of the total magnetic energy.
3D coronal magnetic fields are typically extrapolated from photospheric magnetic fields under the nonlinear force-free assumption.They represent the full complexity of magnetic fields, which are crucial for understanding the dynamic behavior of solar flares, especially in regions with strong magnetic nonpotentiality (Liu et al. 2014).The Institute for Space-Earth Environmental Research (ISEE) Database for Nonlinear Force-Free Field (NLFFF) of Solar Active Regions was developed by the Hinode Science Center, ISEE, Nagoya University (Kusano et al. 2021). 4These data were created using the NLFFF code (Inoue et al. 2013).Based on this database, Kusano et al. (2020) proposed a flare prediction model named κ-scheme.According to this model, if any point on a polarity inversion line (PIL) satisfies the condition of magnetohydrodynamic instability, an X-class flare usually occurs.
In this Letter, we study 3D MFEs and flaring activity using 83 major solar flares from the NLFFF extrapolation data sets of ISEE.Our study is contrasted with the previous studies in the following aspects.First, we use 998 NLFFF extrapolation data from nine flare-productive active regions (ARs).Second, for the first time, we examine the relationship between 3D MFE and flaring activity using a large data set.Third, we examine the relationship in view of two distinct phases of AR evolution: the rising phase and the peak phase.The data and method we use in this study are described in Section 2. The results and discussion are presented in Section 3. Finally, we give a summary and a conclusion of our findings in Section 4.

Data and Method
In this study, we use the ISEE database compiled by Kusano et al. (2020), which provides 3D magnetic field data for solar ARs within ±60°from the central meridian of the Sun.We calculate total magnetic energy and potential energy using Equation (1), and the 3D MFE is calculated using Equation (2).For this, we use NLFFF and potential fields, which are provided by the ISEE database.
We present three different ways of estimating the errors of 3D MFE as follows.First, we refer to 3D coronal magnetic extrapolations by Inoue et al. (2013) who compared their results with Low & Louʼs solution (Low & Lou 1990).The mean vector error is from 0.95 to 0.98, and the energy ratio is from 0.99 (for 256 3 grids) to 1.04 (for 64 3 grids).Because we use about 512 3 , the resulting error of the energy ratio is thought to be around 1%. Second, we define the background value as the running median average of 3D MFEs within 36 minutes and the error is thought to be the rms value of the difference between 3D MFE and its background.From this methodology, the calculated average errors for 3D MFE and 2D MFE density are 0.80% and 0.38%, respectively.Third, the error propagation of 3D MFE can be expressed as where . Since B p comes from B l , its error is assumed to be replaced by B l .We use typical errors, 10 Gauss for B l and 100 Gauss for B t .Since the large contribution of MFE comes from strong magnetic fields, B is assumed to be about 3000 Gauss (Hoeksema et al. 2014).As a result, the error is about 7%.In the figures, we display the error bars using the second estimation for convenience.
We select nine major flare-productive ARs with the following conditions.First, we consider ARs where major flares (Mand X-class) occur at least three times during the observing period.Second, we select ARs that have their distinct rising and peak phases (above the 80% of the maximum MFE) of 3D MFE, which serve as indicators of MFE evolution and its impact on flaring activity.In case it is not clear to determine the ending of their peak phases using 3D MFE, we confirm the peak phase of the 3D MFE using 2D MFE density, which is a proxy for total photospheric MFE density (TOTPOT) from the Space-weather HMI Active Region Patches (SHARP) parameter (Bobra et al. 2014;Hoeksema et al. 2014).We select the maximum value of the 3D MFE to determine phases.The peak phase is defined as the period from 80% to the maximum of 3D MFE.The rising phase is defined as the period from the observation start time to 80% of the maximum 3D MFE.The SHARP parameter named total unsigned flux (USFLUX) is used in this work to examine the flux emergence during the flare events.2D MFE density and the total unsigned flux (Φ) are calculated as and,

| | ( ) F = S B dA, 5
z which are described in Table 3 of Bobra et al. (2014), respectively.The details of nine solar ARs are given in Table 1.In total, there were 83 major flares in these ARs.
We consider the solar "flare occurrence rate (FOR)" to estimate flaring activity, and it is calculated by dividing the total number of flares by the total duration of each phase.To sum up the contribution of the flare's strength, we use the flare index (FI): where F C (×10 −6 ), F M (×10 −5 ), and F X (×10 −4 ) are GOES peak fluxes in unit of watts m −2 of C-, M-, and X-class flares for the given AR, respectively.Then, we estimate the solar "flare index rate" (FIR), which is calculated by dividing the FI by the total duration for each phase.

Results and Discussion
In this work, we analyze the dependence of the GOES X-ray flare flux on 3D MFE, as shown in Figure 1.We show that there is no clear linear correlation between 3D MFE and flare peak flux, where the correlation coefficient is 0.15.3D MFE, at the same time when the GOES X-ray flux is at peak, is calculated by using the linear interpolation method.Our results are in the same line with the previous observations that there is no meaningful correlation between waiting time and flare peak flux (Wheatland 2000;Moon et al. 2001).Our results together with the above ones support that flares are triggered by random perturbations such as flux emergence (Moon et al. 2001).In our data sets, there are 13 X-class flares, in which the X-ray peak flux is above the 10 −4 watts m −2 .It is interesting to note that most X-class flares (11) occurred during the peak phase of 3D MFE as shown in Figure 1.
We define two distinct flaring phases for each AR, based on 3D MFE, and analyze the flare number and duration for each phase.Figure 2 shows the temporal evolution of 3D MFE, 2D MFE density, and total unsigned magnetic flux of NOAA ARs 11429 and 11430 from 2012 March 4 to 11.From our calculations, the maximum value of 3D MFE is 9.16 × 10 32 erg at 22:24 UT on 2012 March 6.The peak phase of the AR 11429 is from 22:00 UT on 2012 March 5 to 09:00 UT on 2012 March 8.There are 12 M-class (vertical green dashed lines in Figure 2) and 3 X-class flares (vertical red dashed lines in Figure 2).A total of 10 major flares occurred during the peak phase, and 3 major flares occurred during the rising phase.And, during the declining phase, there were 2 major flares.
We note a frequently flaring time interval (denoted by the black box) when there were six major flares.This looks like a "local maximum" of 3D MFE, which abruptly increases and then keeps steady or fluctuates in 3D MFE.There are little changes in total unsigned fluxes during the local maximum.During this interval, there was a continuous flux emergence, which seemed to trigger these sequential flares.This interpretation is consistent with recurrent homologous solar eruption observations of this AR (Dhakal et al. 2020).The maximum value of 3D MFE corresponds to about 35% of the total energy, which was 2 hr before two X-class flares during the peak phase of 3D MFE as shown in Figure 2. The step-wise sudden energy loss (about 10 32 erg in 3D MFE) observed during the X-class flares is shown as a black arrow in Figure 2. It is interesting to note that this energy loss corresponds to a typical energy of major flares (Hudson 1991;Emslie et al. 2005;Schrijver 2007;Guo et al. 2008;Fang et al. 2012).Thus, we guess that the sudden energy loss in 3D MFE is mainly transformed into the emitting energy of these X-class flares.
Figure 3   It is also noted that there is a relatively significant flaring activity during a short time interval (indicated by the black box) in AR 12192 (Figure 3(c)).This period is about 3 hr, and four M-class flares occurred during the period.The location of these four M-class flares is the core of the AR 12192, which was found by Chen et al. (2015).Thus, from our results together with the above studies, we think that emerging fluxes and shearing motions are the main contributors to these major flares indicated by the black box in Figure 3.
Figure 4 shows the temporal evolution of 3D MFE, 2D MFE density, and total unsigned magnetic flux of the NOAA AR 12673 (a) and 12975 and 12976 (b).In NOAA AR 12673 (Figure 4(a)), during the rising phase, the 3D MFE remains relatively steady and shows a significant number of flaring activity (nine M-class flares).This activity is associated with a local maximum of 3D MFE, denoted as the black box in Figure 4(a).This period was from 18:00 UT on 2017 September 4 to 07:00 UT on 2017 September 5.During this period, total unsigned flux steadily increases.Thus, we think   Yang et al. (2017) showed that at the PIL between the newly emerging negative fields and the surrounding positive ones, the magnetic fields were highly sheared in AR 12673, resulting in the accumulation of MFE.This interpretation can explain the abrupt accumulation of 3D MFE just before the flares.From 3D MFE and total unsigned flux together with the previous studies, we think that both magnetic shearing motions and emerging fluxes play an important role in triggering and releasing multiple major flare energies during the peak phase of 3D MFE.
In AR 12975 and 12976 (Figure 4(b)), there are six M-class flares during the rising phase, and there are three major flares (two M-class and one X-class) during the peak phase.There was a local maximum of 3D MFE, which is denoted as a black box in Figure 4(b).It is interesting to note that during the rising phase, 3D MFE and total unsigned flux show different trends; the total unsigned flux decreases during this local maximum, while 3D MFE increases.We can assume that, during this period, the contribution of shearing motions and emerging fluxes and/or cancellations, which are identified by a timeseries animation, are responsible for triggering and releasing the flares.
Table 1 describes the FOR values of each AR, total, and when only X-class flares are included, for the rising and peak phases of 3D MFE.It is shown that in several ARs (12192, 12975, and 12976) FOR values during the rising phase are higher than those of the peak phase.In the other ARs, FOR values are higher during the peak phase than those during the rising phase of 3D MFE.It is interesting to note that FOR is not always higher during the peak phase of 3D MFE than during the rising phase.On average, the FOR value of the rising phase (3.42) is even higher than that of the peak phase (3.13).The standard deviation of the rising and peak phase FOR are 1.35 and 1.84, respectively.Since the difference between two average FOR values is not smaller than the standard deviations, this difference is not statistically significant.However, when X-class flares only are considered, it is impressive that the FOR of the peak phase is above 5 times that of the rising phase.
We have considered nine ARs that have more than or equal to three major flares from 2011 to 2022, which covers the nearly full cycle of solar cycle 24 and the first part of solar cycle 25.It is noted that solar activities during this period are relatively weak compared to the previous cycles.If the data are extended to previous solar cycles, we may get more interesting insights into the relationship between MFE and major solar flares depending on the solar cycle and the hemispheric locations of ARs.
We calculated the FIR value of each phase of 3D MFE.From the FIR result, we show that the flare strength during the peak phase (FIR = 314 day −1 ) is significantly higher than that of the rising phase (FIR = 151 day −1 ).From FIR and FOR results, we can say that ARs in the peak phase have a tendency to produce stronger flares.
In this study, we examine 3D MFE and 2D MFE density variations after the M-class and X-class flares occur.We present our results in Table 2.We determine "decrease" or "increase" based on the variations in MFE occurring within 36 minutes before and after the peak time of the flare.To determine the noise level, we define the background value as the median of MFE within ±36 minutes of MFE (consisting of seven data points).Then, we calculate the standard deviation value (σ) using the difference between MFE and its background value.Then, we compare σ with the mean difference of MFE after and before the flare peak time: Mean(MFE(i+1), MFE(i+2), MFE(i+3)) − Mean(MFE(i-3), MFE(i-2), MFE(i-1)), where i is the closest time to the flare peak time.If the mean difference is smaller than σ, it is categorized as a "decrease."Conversely, if it is larger than σ, it is categorized as an "increase."Additionally, if it falls within the negative −σ to positive σ range, it is designated as "not clear."This classification criterion is applied to both 3D MFE and 2D MFE density.With this criterion, we classified 83 major flares into five distinct categories, as shown in Table 2.These categories show different groups.According to the standard idea on flare energy release, 3D MFE should decrease after the flare.However, in case there are significant emerging fluxes during the event, 3D MFE may not decrease.If we only consider X-class flares, about 69% of events show noticeable decreases in 3D MFE.Another interesting point is that for a significant fraction (about one-third) of the events, 3D MFE decreases, while 2D MFE density increases.These cases may be related to many reports on shear increases associated with strong flares (Wang et al. 2002;Liu et al. 2005;Simões et al. 2013).

Summary and Conclusion
In this study, we have examined the relationship between 3D MFE and flaring activity using 83 major flares.For this, we have used about 1000 NLFFF extrapolations.The major results from this study are as follows.First, we analyze the relationship between the flare peak flux with 3D MFE and find that there is no clear linear correlation between them, which does not support the traditional "storage and release" model (Rosner & Vaiana 1978).Second, for the first time, we have examined the flaring activities depending on the phases of 3D MFE: rising and peak (defined as when MFE is above 80% of its maximum value) phases.It is found that the flaring rates of the rising phase (3.42) are a little higher than those of the peak phase (3.13), which is different from a common idea on flare occurrence.Of course, when only X-class flares are considered, the flaring rates of the peak phase are noticeably higher than those of the rising phase.Third, it is interesting to note that multiple flares occurred during the short time intervals, which are characterized by local maxima of 3D MFE.During these intervals, there are continuous flux emergencies and/or shearing motions, which are thought to be mainly responsible for triggering and releasing major flares.Fourth, we have examined the variations of 3D MFE and 2D MFE density during the events.It is noted that there are different groups showing different patterns.When X-class flares only are considered, most (about 69%) of the events show noticeable decreases in 3D MFEs, supporting the conventional idea that the stored magnetic energies are released in the form of flares.
According to the "storage and release" model of Rosner & Vaiana (1978), the flare flux should be positively correlated with the stored energy in sheared magnetic fields.However, we can not see any noticeable correlation between them (see Figure 1).One interesting trend is that there is a higher probability of producing stronger flares in the peak phase of 3D MFE than in the rising phase.Thus, our results can be interpreted in view of the logistic avalanche model of flares that solar flares could potentially emerge from an avalanche-like process within a self-organized critical system (see Figure 1 of Aschwanden et al. 1998).In this system, we expect several observations of flares that are consistent with our findings.First, there are many different scales of flares for a given 3D MFE.Second, we cannot see a clear correlation between 3D MFE and flare peak flux.Third, there is a trend to have stronger flares when 3D MFE is high.
shows the temporal evolution of 3D MFE, 2D MFE density, and total unsigned magnetic flux of NOAA AR 11158 (a), 11514 and 11515 (b), and 12192 (c).In NOAA AR 11158 (Figure 3(a)), during the rising phase, only one M-class flare was observed.There is one M-class and one X-class flare during the peak phase of 3D MFE.All three major flares that occurred in AR 11158 show a sudden decrease of 3D MFE after the events.It seems that the first two major flares occurred during the continuous flux emergence, which is in the same argument of Toriumi et al. (2014), who insisted that AR 11158 is more likely to be created from a single split emerging flux.NOAA AR 11515, near AR 11514, is one of the strongest ARs in the solar cycle 24.In the case of NOAA AR 11514 and 11515 (Figure 3(b)), there occurred a large number of M-class flares during both the rising and peak phases.We note two sequential local maxima (indicated by the black box) of 3D MFE, which is characterized by relatively significant flaring activity.During this period, eight M-class flares occurred for about 12 hr.It is noted that magnetic fluxes increase and then decrease, while 3D MFE steadily decreases and then rises.This fact may imply that emerging fluxes and flux cancellations play an important role in triggering and releasing multiple flare energy, supporting the result of Louis et al. (2014).
Figure 3(c) shows three parameters of NOAA AR 12192, which is one of the largest and large energy stored ARs in solar cycle 24 (Chen et al. 2015; Sun et al. 2015; Bamba et al. 2017; Jain et al. 2017).The maximum value of 3D MFE is 1.76 × 10 33 erg in this AR 12192.During the rising phase, there are three M-class and one X-class flare.Six M-class and three X-class flares occurred during the peak phase of 3D MFE.There were noticeable decreases in 3D MFE and emerging fluxes during the two X-class flares, indicated by the black arrows in Figure 3(c).Chen et al. (2015) observed the apparent shearing motions of emerged photospheric fluxes in the AR 12192 core, which play an important role in triggering the X-class flares.

Figure 2 .
Figure 2. The temporal evolution of 3D MFE, 2D MFE density, and total unsigned flux of AR 11429 and AR 11430 from 2012 March 4 to 11. Top: the blue dots represent 3D MFE, and the blue solid line between the dots is the linear interpolation value of 3D MFE.The dark gray solid line represents 2D MFE density.The yellow solid line indicates total unsigned flux.We define two flare phases based on 3D MFE: the rising and peak phases.The peak phase is highlighted as a light gray box.The green and red vertical dashed lines represent the peak time of M-class and X-class flares, respectively.The black box indicates the "local maximum."Bottom: three SDO/HMI SHARP magnetograms of rising, peak, and declining phases, respectively.

Figure 3 .
Figure 3.The temporal evolution of 3D MFE, 2D MFE density, and total unsigned flux of AR 11158 (a), AR 11514 and 11515 (b), and AR 12192 (c).The figure descriptions are identical to those in Figure 2.

Figure 4 .
Figure 4.The temporal evolution of 3D MFE, 2D MFE density, and total unsigned flux of AR 12673 (a) and AR 12975 and 12976 (b).The figure descriptions are identical to those in Figure 2.

Table 1
The Flare Occurrence Rate (FOR) during the Rising and Peak Phases of 3D

Table 2
Variations of 3D MFE and 2D MFE Density during the Major Solar Flares Event Number (Percentage)