Temporal Behaviors of Magnetic Helicity Injections by Self and Mutual Sunspot Rotations

Magnetic helicity is a physical parameter used to quantify the complexity of magnetic fields, providing an indication of the energy state in the coronal magnetic structure. We investigate the temporal evolution of magnetic helicity and its relationship to the occurrence of a variety of flares in the solar active region NOAA 12297, which was well observed using the Solar Dynamics Observatory/Helioseismic and Magnetic Imager in 2015 March. The active region produced many M-class flares and an X-class flare in two distinctive areas, both of which had a similar magnetic evolution, i.e., the opposite polarity of an emerging flux developed beside a preexisting sunspot, but exhibited flares with different magnitudes and frequencies. We derived the spatiotemporal evolution of the magnetic helicity injections and evaluated how spinning and braiding helicity injections evolved with time in the two areas. In one area, we observed a remarkable evolution, in which a negative spinning helicity injection in the preexisting sunspot increased in a positive helicity system, followed by the occurrence of the X-class flare. The negative helicity injection was clearly caused by the flux emergence that developed along the outer edge of the preexisting sunspot. The other area showed positive braiding helicity injections, with spinning helicity injections fluctuating concurrently with flux emergence, changing their signs several times, i.e., variable energy, and helicity input. The observed temporal behaviors of the helicity injections may explain different types of flare occurrences in the regions.


Introduction
A solar flare is an explosive phenomenon, which releases a large amount of energy in a short timescale in the solar outer atmosphere.Free energy stored in the magnetic structure is released via magnetic reconnection to energy in various forms (e.g., thermal and nonthermal).The free energy is developed in the complex three-dimensional structure of the coronal magnetic field, which is morphologically visible as bright non-potential structures in extreme ultraviolet (EUV) and X-ray wavelengths.An X-ray sigmoid (Rust & Kumar 1996;McKenzie & Canfield 2008), which is an S-or inverse S-shaped bright structure seen in soft X-rays is a remarkable coronal feature that implies a complex and twisted topology.Its appearance has been recognized as a precursor to the occurrence of a large flare or a coronal mass ejection (CME; Sterling & Hudson 1997;Canfield et al. 1999;Kawabata et al. 2018).The shear may be relaxed during flares, evolving to current-free (i.e., potential) magnetic fields (Sakurai et al. 1992).
Magnetic helicity is a physical quantity that describes the twist and linking of magnetic field lines and is essential for the evaluation of the energy state in the corona.To monitor the temporal evolution of free energy stored as the complexity of the magnetic field, the magnetic helicity may be used as a proxy for the free energy stored in the coronal magnetic field.This measure is well conserved in the corona because the flux frozen-in condition is maintained there and the magnetic helicity has an inverse-cascading property (Frisch et al. 1975;Pouquet et al. 1976).However, the magnetic helicity in the corona cannot be estimated directly because measurements of the coronal magnetic field are not available.Alternatively, the injection of the magnetic helicity into the corona is estimated using the measurements of the magnetic field and velocity in the photosphere (Chae et al. 2001;Kusano et al. 2002;Démoulin & Berger 2003).Accumulating the injected helicity allows us to track the storage of free energy in the corona.
CMEs are another manifestation of the explosive activities in the corona.Continuous helicity injection of the same sign leads to the overaccumulation of magnetic helicity in the corona, which has to be eliminated by CMEs (Zhang & Low 2005).Both the buildup of magnetic helicity and the sufficiently weak background (i.e., confining) magnetic field are preferable conditions for eruptive CMEs (Vemareddy 2017(Vemareddy , 2019)).
In addition to the buildup of free energy, some triggering process is required to ignite solar flares.In this regard, solar flares may be more easily triggered when positive and negative helicities cancel each other.Kusano et al. (2003) proposed a flare trigger model in which the annihilation of magnetic fields with opposite magnetic helicities produces solar flares.This model is supported by several observations that implied a role of magnetic helicity cancellation in triggering flares (e.g., Park et al. 2010Park et al. , 2013;;Vemareddy et al. 2012a).In this sense, for further understanding of the flaring mechanisms, it is essential to reveal the relationship between temporal, spatial, and sign variations of magnetic helicity injection and flare productivities.Since solar flares are not necessarily accompanied by CMEs and vice versa (Sun et al. 2015), the role of the cancellation of magnetic helicity in triggering CMEs is an open question.Observationally, Park et al. (2012) statistically studied the temporal evolution of the accumulation of magnetic helicity in active regions and found that faster CMEs tend to be produced in active regions where the sign of magnetic helicity injection changes in their evolutions.On the other hand, Vemareddy (2015) reported that the accumulation of monotonic helicity produced strong eruptions but the sign reversal of the accumulation of helicity resulted in no eruption.Elucidating the relationship between magnetic helicity and the eruptivities is still an important task from the perspective of space-weather forecasting.
We need to distinguish the dominant processes of helicity injection for a better understanding of their variabilities.In this regard, rotating sunspots are one of the critical apparent photospheric features that suggest the injection of magnetic helicity into the corona (Ravindra et al. 2011;Vemareddy et al. 2012a).Evershed (1909Evershed ( , 1910) ) first reported spectral observations for sunspot rotation and Stenflo (1969) and Barnes & Sturrock (1972) showed that the rotation can build up sufficient free energy to produce solar flares.Sunspot rotations have been used theoretically to increase free energy in the corona toward developing flares and eruptions (Amari et al. 1996;Tokman & Bellan 2002;Török & Kliem 2003), and observationally studied in relation to flare productivity (Régnier & Canfield 2006;Zhang et al. 2008).Yan et al. (2008) statistically studied the relationship between flare productivity and rotating sunspots, classifying the magnetic pairs of sunspots into six groups, based on the features of their self and mutual rotations.They concluded that there are favorable configurations of sunspots to produce larger flares, such as sunspot rotations whose directions are opposite to solar differential rotation.
The sign of the magnetic helicity in active regions exhibits weak hemispheric preference, independent of the solar cycles (Pevtsov et al. 1995).Negative and positive helicity signs tend to be generated predominantly in the northern and southern hemispheres, respectively.This sign preference, i.e., the hemispheric rule, is attributed to the dynamo mechanism, including the Coriolis force (Holder et al. 2004), Σ-effect (Longcope et al. 1998), and the differential rotation of the Sun (Démoulin et al. 2002).
This study presents a case study on the temporal evolution of magnetic helicity injection in an active region and its influence on the occurrence of a variety of flares.In this study, we particularly focus on the role of self and mutual sunspot rotations on the variabilities of helicity injections.The active region studied here is NOAA 12297, which was well observed on the solar disk using the Solar Dynamics Observatory (SDO) in 2015 March.The region was flare productive for several days, producing many M-class flares and an X-class flare in two distinctive areas.The two areas have a similar magnetic configuration; however, they produce flares with different magnitudes and frequencies.
This work presents a comparison of the two areas based on helicity injection.The temporal evolution of magnetic helicity injection in each area allows us to track how the coronal energy state changes with time.We also identify photospheric motions of the magnetic fields responsible for large helicity injections, which can be compared to the occurrence of solar flares.The paper is organized as follows: Section 2 presents the observations and Section 3 describes the analysis methods; Section 4 presents the results from the analysis; finally, we discuss the results in Section 5 and summarize the study in Section 6.

Observations
In this study, we used the data obtained by the Atmospheric Imaging Assembly (AIA; Lemen et al. 2012) and the Helioseismic Magnetic Imager (HMI; Scherrer et al. 2012;Schou et al. 2012) on board SDO (Pesnell et al. 2012).AIA records full-disk images of the Sun in seven EUV wavelength channels, two UV wavelength channels, and one visible wavelength channel.The pixel size is 0 6, and the cadence is 12 and 24 s for the EUV and UV channels, respectively.We primarily used the series of UV data of 1600 Å to track chromospheric brightening occurring associated with large flares.
The HMI instrument obtains full-disk filtergrams with a spatial sampling of 0 5 pixel −1 for the six narrow bands (76 mÅ width) tuned for the magnetically sensitive photospheric Fe I line at 6173 Å to measure the Stokes vector.The HMI science data processing pipeline was used to calibrate the Stokes vector.The vector magnetic field was derived with the Very Fast Inversion of the Stokes Vector algorithm (Borrero et al. 2011), based on the Milne-Eddington atmospheric model.The 180°ambiguity of the azimuthal component is solved using the minimum energy method (Metcalf 1994;Metcalf et al. 1995;Leka et al. 2009).In this study, we mainly analyzed the Spaceweather HMI Active Region Patch cylindrical equal area (CEA) data set.This data set has a 12 minute cadence and the magnetic field vector is projected onto the local solar coordinate using Lambert CEA projection (Calabretta & Greisen 2002;Sun 2013) from the direct cutout of the fulldisk HMI image in the pixel coordinate reference system.

Methods
In this section, we describe the methods employed to analyze the magnetic field observation.We estimate the physical parameters associated with the non-potentiality of the magnetic field described below.Note that the running average for an hour of observations is used for the parameters.

Magnetic Helicity
Magnetic helicity is a measure used to describe the complexity of a magnetic field.In the solar magnetic field case, we typically use the relative magnetic helicity, where A is the vector potential of the observed magnetic field, B P is the reference potential magnetic field, and A P is its vector potential (Berger & Field 1984;Finn & Antonsen 1985).
If we assume that helicity in a solar active region changes only through the photosphere (i.e., we neglect the dissipation of helicity in the corona and the release into the interplanetary space with CMEs), the helicity injection is formulated using the time derivative of the relative helicity, where S p is the area of an active region in the photosphere and the subscripts "n" and "t" denote the normal and transverse components, respectively.To calculate this quantity, both the photospheric magnetic field and the photospheric velocity field are required; previous studies employed some optical flow methods to derive the velocity field (e.g., local correlation tracking, November & Simon 1988).Démoulin & Berger (2003) suggested that the velocity field, which can be obtained with an optical flow method, is the footpoint velocity field u: and with the footpoint velocity, the helicity injection rate can be written as which can be computed with parameters that are obtainable from observations.To derive u, we applied the Differential Affine Velocity Estimator algorithm (Schuck 2006) for the HMI line-of-sight magnetic field maps.We used an apodizing window of 10 pixels (∼5″) and neglected small magnetic fields of less than 50 G.The window size is similar to that employed by Ravindra et al. (2011) for their local correlation tracking computation, i.e., 4″.It is noteworthy that the smaller apodizing window results in a larger amplitude of velocity and helicity flux, but helicity flux averaged for 1 hr is less sensitive to the different window sizes (Chae et al. 2004;Vemareddy et al. 2012b).
Because the integrand in Equation (4) is not gauge invariant (only the surface-integrated value dH R /dt is gauge invariant), mapping the integrand as the helicity flux density produces spurious signals of fake polarity.As an alternative approach to derive the helicity flux density, Pariat et al. (2005) defined "G θ ," formulated as denoting the relative rotation of x to x¢. n is a unit vector normal to S p , i.e., (0, 0, 1) in the Cartesian coordinate system.Equations ( 5) and (6) indicate that magnetic helicity injection at the position x is the summation of the relative rotation of x to x¢, weighted by the vertical magnetic field at both positions, B n (x) and x B n ( ) ¢ ¢ .The surface integration of G θ coincides with the helicity injection at the surface: 3.2.Spinning and Braiding Helicity Injections Longcope et al. (2007) suggested that the origin of the magnetic helicity injection in the photosphere can be divided into two contributing phenomena, i.e., spinning and braiding motions in the set of unipolar regions, which correspond to self and mutual rotations, respectively.The total helicity injection rate is decomposed of the spinning and braiding helicity  8) is formulated as follows: is the spinning helicity injection by a self-rotation of a given magnetic patch "a," and is the braiding helicity injection by a mutual rotation between magnetic patches a and b.

Mean Angular Rotation
From the spinning helicity injection, we obtain the rotational rate of the whole body of a spot.The mean angular rotation (Longcope et al. 2007) is defined as where Φ a denotes the total magnetic flux of the magnetic patch a:

Average Force-free α av
To evaluate the complexity of the horizontal component of the magnetic field, we calculate α av , which is an average of the force-free α values in a spot.This parameter is the index of the twist of the magnetic field in a sunspot and is related to the current helicity.This value can be described as the ratio of the vertical electric current to the vertical magnetic field.There are some methods to define α av (Hagino & Sakurai 2004).In this study, we employ the following definition:

Overall Evolution of the Active Region
The active region NOAA 12297 appeared from the east limb at the beginning of 2015 March.Our study focuses on the period from 12 UT on March 10 to 10 UT onMarch 13.The region was located close to the disk center, i.e., at S16E13 at 00:30 UT on 2015 March 11. Figure 1 shows the temporal evolution of the magnetic flux distribution in the CEA coordinate.The region was relatively complex; however, it was primarily composed of a static large sunspot (P1) and an emerging flux region (P2b and N2a/b).A pair of small patches (P2a and N3) were preexisting fluxes located northwest of N2.The emerging flux region successively developed between P1 and N3.Starting from the pair of P2b and N2a, the emerging flux region gradually developed with time; a negative magnetic patch was elongated, which is labeled as N2b.Note that P2 and N2 are determined as groups of unipolar patches because they are complicatedly developed with separation and coalescence.In determining each magnetic patch, we set the threshold of 1000 G and neglect magnetic elements smaller than 30 pixels.When we mention N2 (P2), it refers to both N2a and N2b (P2a and P2b).We determined two subregions indicated with orange rectangles in the left-top panel of Figure 1.We call the eastern region, including the large spot P1 and one footpoint N2 of the emerging flux region, Region 1.The western region, i.e., Region 2, has the preexisting small spot N3 and the other footpoint P2 of the emerging flux region.Figure 2 shows the temporal change of the absolute magnetic flux in four magnetic patches, i.e., P1 (red), P2 (magenta), N2 (blue), and N3 (cyan).The vertical solid and dashed lines indicate the time when the flares and CMEs of interest occurred, as explained in the following paragraph.We should note that neglecting a vertical magnetic field of less than 1000 G resulted in an underestimation of the total magnetic flux, particularly in the emerging patches P2 and N2.We investigated how magnetic fluxes and parameters that will be derived in the following sections may be different when a threshold of 500 G is used.Our results showed that magnetic fluxes and helicity injections are underestimated in P2 and N2 by a factor of 1.5, and in P1 and N3 by a factor of 1.2.However, these underestimations do not affect our discussions and conclusions qualitatively.Therefore, we will maintain a threshold of 1000 G for the vertical magnetic field.The large positive spot P1 and small negative spot N3 had an almost constant flux during this period, whereas P2 and N2 exhibited a flux increase from 16 UT on March 10 until 4 UT on March 12, indicating a flux emergence.The magnitude of the emerging flux was (1.5-3.0)× 10 21 Mx.
Figure 3 shows the time profiles of the AIA 1600 Å intensity, integrated in some regions of interest; the top panel displays the total intensity of the entire active region, the middle is that of Region 1 and the bottom is that of Region 2. Different frequencies and magnitudes of flares are clearly observed; Region 1 produced the largest X-class flare in the active region, with only two M-class flares, whereas Region 2 displayed smaller M-class (M3.2-M1.0)flares.Note that the M1.4 flare at 12:09 UT on March 12 was located far from the major concentrations of magnetic flux; thus, it is not denoted in Figure 3.The vertical dashed lines in Figure 3 also show when CMEs occurred.The X-ray classifications of flares associated with CMEs are shown in red.We identified the CMEs originating from this active region by surveying the CME catalog3 (Gopalswamy et al. 2009), neglecting poor and very poor events.We also carefully inspected the images obtained with AIA and the C2 field of view (FOV) of the Large Angle and Spectrometric COronagraph (LASCO; Brueckner et al. 1995) on board the Solar and Heliospheric Observatory (SoHO; Domingo et al. 1995) and identified from which region the CMEs were ejected.Finally, five CMEs originating from Regions 1 and 2 were identified.Just one CME was expelled from Region 1, which accompanies the X2.1 flare, whereas four of five CMEs occurred in Region 2. Half of them follow the C-class flares clarified with parenthesis in the bottom panel of Figure 3.All the CMEs occurred during the emergence of P2 and N2 (see Figure 1).We found that most of the M-flares were not accompanied by CMEs.
Both regions had a similar development of the polarity inversion line, although the regions produced flares with different magnitudes and frequencies.The large spot P1 in Region 1 and small spot N3 in Region 2 were preexisting.The emerging flux activity initiated the development of N2 in Region 1 and P2 in Region 2 moving along the outer boundary of each spot.This movement can be considered as a counterclockwise mutual rotation of opposite-polarity flux, with respect to the preexisting flux.Note that the magnetic flux of P1 was at least one order of magnitude larger than that of N3.

Region 1
The top row of Figure 4 displays three snapshots of the vertical magnetic field of Region 1.The cyan arrows present the horizontal velocity field derived using the DAVE method.As can be seen in panels (a)-(c), an emerging flux patch N2a was located north of the large spot P1, and it exhibited a proper  The bottom row of Figure 4 displays magnetic helicity maps G θ corresponding to the panels in the top row.The large spot P1 exhibited positive helicity injection in (d).However, (e) exhibits negative helicity injection in P1; the image in panel (e) was obtained at a time after the emergence of N2b started, and the negative sign is opposite to the preferred sign of the global magnetic helicity in the southern hemisphere.In panel (f), P1 shows positive helicity injection again.The sign of G θ in the emerging flux N2 showed mixed distributions; however, positive helicity injection was predominant.
From 8 UT on March 11 to 0 UT on March 12, the negative spinning helicity injection decreased in P1 and the positive braiding helicity injection between P1 and N2 increased.This implies that positive helicity started competing with negative helicity.The two subsequent M-class flares occurred after the period, suggesting both positive and negative helicity contributed to the occurrence of the flares.Before the emergence of N2b (around 4 UT on March 11), we can find the dark features with both increasing and decreasing gradients, corresponding to clockwise and counterclockwise rotations, respectively.After the emergence of N2b, a few umbral spirals located in 45°-150°(clarified with green curves in Figure 7(b)) rotated counterclockwise at high speed.The rotational speed of these dark features is 3-4 deg hr −1 and these values are consistent with the mean angular rotation plotted in Figure 6.Even after the emergence of N2b finished at 3 UT on March 12, the dark features in 150°-200°show slow counterclockwise rotations that were driven by the mutual rotation of N2b with respect to P1.

Region 2
The top row in Figure 8 shows the temporal evolution of photospheric magnetic fields in Region 2 from March 10-12.This region primarily consisted of two magnetic elements: positive P2 and a preexisting negative sunspot N3.The P2 patch is the leading end of the emerging flux and its counterpart is N2 in Region 1.During the emergence of P2, it was intensely stretched in the northwestern direction.The leading edge of P2b moved fast and increased in size, whereas the trailing edge was almost stationary and remained south of N2.This formed a long polarity inversion line between P2 and N3, where most of the flares occurred.The elongating motion of P2 corresponds to a counterclockwise mutual rotation between P2 and N3.
The bottom row in Figure 8 displays the magnetic helicity injection density G θ maps.We notice that the leading part of P2b injected positive magnetic helicity.Negative injections were observed predominantly at the trailing part of P2b, which came into contact with N3.No strong opposite injections were observed inside P2a and N3, which is different from the large spot P1 in Region 1.
Figure 9 shows the temporal evolution of spinning helicity injections by P2 (red pluses) and N3 (blue triangles), and braiding helicity injection by the combination of them (black squares).The vertical solid and dashed lines indicate M-class flares and CMEs in Region 2, respectively.We note that P2 consists of several positive patches undergoing complex development such as separation and coalescence.
The spinning helicity injection by N3 was significantly smaller than the other injections, owing to its small magnetic flux (see Figure 2).That of P2 and braiding helicity injection between P2 and N3 were comparable and predominant in Region 2. The emerging flux P2 provided significant spinning helicity injections, which were variable over 3 days.A negative injection was observed after the emergence of P2 began at 21  UT on March 10 (panel (d) in Figure 8).The injection then became positive after the first flare and peaked between the second and third flares.After the third flare, the spinning helicity injection became negative, followed by four M-class flares.However, the braiding helicity injection, which is a combination of P2 and N3, was almost positive from the beginning of the P2 emergence and gradual decrease with fluctuations until the end of the analysis period.From a total of seven M-flares occurring in Region 2, four occurred with a negative spinning helicity injection in P2 and a positive braiding helicity injection, implying the role of helicity annihilation in triggering these flares.
The temporal evolution of the mean angular rotation and force-free alpha α av for small spot N3 is shown in Figure 10.The maximum rotational rate of N3 was −8.6 deg hr −1 , which is significantly faster than that of P1, owing to the small magnetic flux in N3 (see Equation ( 11)).The average force-free alpha α av was always positive, as in P1 of Region 1, following the hemispheric preference of the current helicity.

Discussion
Two flare-productive regions, Regions 1 and 2, were examined from the viewpoints of the magnetic helicity injection and its evolution.The regions have a similar magnetic configuration but produced flares of different magnitudes and frequencies; the largest X-class flare was in Region 1, whereas most of the M-class flares were observed in Region 2. The origins of the different flare occurrences are discussed in the following sections.

Difference in Helicity Injections
Regions 1 and 2 exhibited different behaviors in the evolution of the helicity injections.A remarkable behavior   was observed in Region 1-the temporal behavior of the negative spinning helicity by P1 was closely related to the occurrence of an X-class flare.It increased toward the occurrence of the X2.1 flare, followed by a gradual decrease.The injection rate was predominant in Region 1, with a maximum value of approximately −4 × 10 37 Mx 2 s −1 .
Because magnetic helicity is related to the free energy of the system, the annihilation of positive and negative magnetic helicity results in a decrease in the total free energy that the system can contain.This means that the annihilation effectively releases excessive magnetic energy in a solar flare (Kusano et al. 2003).This explains why the X-class flare occurred around the peak of the negative helicity injection in P1, and subsequent to that, there was no significant flaring activity, with the exception of two M-class flares.The observed decrease of the negative spinning helicity and increase of positive braiding helicity may be considered as an increase in free energy, resulting in the occurrence of M-flares in Region 1.
Before the X2.1 flare, the α av value started to increase, resulting in an angular rotation in the large spot P1.A drag was formed by the fast emergence of the opposite-polarity N2b, influencing the northern portion of P1.The force may enhance a local twist, which develops a helical motion, observed in a CME accompanied by the X2.1 flare.
Region 2 is characterized by a comparable contribution of the spinning helicity injection by the emerging sunspot group P2, and the braiding helicity injection by P2 and N3.Owing to the continuous counterclockwise mutual rotation, positive braiding helicity was injected during the period of analysis.However, the plot of the spinning helicity injection in P2 displays its variable development.The spinning helicity injection was negative for a while after the emergence of P2, and it sharply changed to positive, followed by two M-class flares.Before the M3.2 flare, it became negative again.Helicity injections in Region 2 exhibit a fluctuated profile.These complex evolutions may be related to the recurrent occurrences of seven M-class flares.
It may be noteworthy that six of the 10 flares in Regions 1 and 2 occurred when there was a spinning helicity injection with a sign opposite to that of the braiding helicity injection.Additionally, for the M1.2 flare in Region 1 at 4 UT on March 13 , the negative spinning helicity in P1 had been injected until 1 UT, against the positive braiding helicity injection.In Region 2, a negative spinning helicity by P2 was injected until just before the occurrence of the M1.8 flare with positive braiding helicity.We can consider that the coexistence of positive and negative helicity injections is a preferable condition for the occurrence of both X-class and recurrent M-class flares.
In both Regions 1 and 2, the braiding helicity injections were typically positive.This reflects the contributions of the macroscopic counterclockwise rotation of the patches, not the small-scale flows inside each patch.

Net Helicity in Subregions
In the top panel of Figure 11, the sum of the spinning and braiding helicity injections (see Figure 5) is compared with the net helicity in Region 1. Net helicity is formulated as the surface integration of the helicity flux density G θ in the FOVs shown in Figure 4.The temporal evolutions of these helicity parameters show the same tendency.This means that the magnetic helicity in this region is dominated by the strong magnetic patches P1 and N2, and the contributions by weak magnetic regions are negligible.
On the other hand, these helicity parameters in Region 2 are compared in the bottom panel in Figure 11.In this region, the net helicity injection keeps increasing until 1 UT on March 12 and reaches 6.9 × 10 37 Mx 2 s −1 .The total G θ is much different from the sum of the spinning and braiding helicity injections.For the purpose of determining the magnetic patches P2 and N3, we employ the large threshold value of a magnetic field of 1000 G and neglect weaker magnetic fluxes surrounding these patches, causing the underestimation of spinning and braiding helicity injections.However, as we mentioned in Section 4.1, since setting the threshold value to 500 G changes the helicity injection rate by a factor of 1.5, neglecting the peripheral weak fields might not be the primary reason for the discrepancy.We consider that the positive scattered magnetic fluxes outside P2 and N3 (e.g., in the southern part of Region 2), which include stronger fluxes than 1000 G, make a considerable contribution to the net helicity and is the main reason for the large difference.These neglected fluxes contribute to G θ , i.e., the net helicity, not only in the scattered flux region but also in P2 and N3 due to their relative rotations, whereas the complicated behaviors of spinning and braiding helicity injection produced the recurrent solar flares, the accumulation of positive magnetic helicity due to the scattered magnetic fluxes might be important for the occurrences of frequent CMEs in Region 2.

Occurrences of CMEs
NOAA 12297 produced nine M-class flares and one X-class flare, while five CMEs were observed by the SoHO/LASCO coronagraph.For the comparison among the scenario of helicity buildup and occurrences of flares and CMEs, we display helicity injection (red) and accumulation (black) in the whole active region shown in Figure 12.The accumulated helicity amounted to ∼1.03 × 10 43 Mx 2 during the 3 days.
As a whole, this active region shows the accumulation of monotonic helicity.However, the total helicity injection rate was nearly zero in 12-21 UT on March 11, and was slightly negative after 16 UT when the X2.1 flare occurred.This is because the negative helicity injection by P1 was comparable to the positive injection in the rest of the active region.In this sense, the subregion Region 1 is similar to the group characterized by sign reversal of helicity injection in Park et al. (2012).In Region 1, the frequency of CMEs is scarce and just one CME occurred following the X2.1 flare.We consider the sign reversal of the accumulation of helicity to be important in triggering a flare but might not have a direct role in causing CMEs.The eruptive CME in Region 1 was caused by the X-class flare possibly triggered by the annihilation of opposite magnetic helicity.On the other hand, positive magnetic helicity is injected into Region 2 in total.Therefore, this subregion is similar to the group characterized by monotonically increasing helicity in Park et al. (2012).This may lead to overaccumulation of magnetic helicity, resulting in more frequent occurrences of CMEs.
All the identified CMEs occurred in the flux emergence period of the P2-N2 pair (see Figure 2).In Region 2, we could not find a clear relationship between helicity cancellation and CMEs.Therefore, we speculate that the accumulations of twist owing to flux emergence P2 resulted in the frequent occurrence of CME accompanied by the flares in Region 2.Moreover, the bottom panel in Figure 11 shows that the positive scattered magnetic fluxes in Region 2, which are neglected when determining the magnetic patches P2 and N3, might bring about a large amount of positive helicity into Region 2 during this period, implying that these neglected magnetic fields play a great role in the occurrence of the CME, whereas, the strong background magnetic field in P1 might prevent frequent eruptions of coronal mass in Region 1.We had no CMEs after the end of the flux emergence (∼ 4 UT on March 12), although magnetic helicity monotonically accumulated and five M-class flares occurred.We consider this is because the net helicity injection rate in Region 2 turned into a decrease after the flux emergence period.

Accuracy of Helicity Flux Density
In Figures 5 and 9, we display the maps of magnetic helicity density flux based on the photospheric flow field, i.e., G θ (see Equation ( 5)).Great attention should be paid to an interpretation of the spatial distribution of G θ because helicity flux density is essentially gauge dependent and the surfaceintegrated helicity flux dH R /dt meets the gauge invariance.A more precise formulation for the spatial distribution of helicity flux is the connectivity-based flux density G Φ (Pariat et al. 2005;Dalmasse et al. 2014;Vemareddy & Démoulin 2017).Helicity flux coming into a closed elementary magnetic field through positive and negative footpoints x a  , respectively, is formulated as where f + and f − = 1 − f + are the fractions of the magnetic helicity injected into the elementary flux through the positive and negative footpoints (see Equation (29) in Pariat et al. 2005).Nevertheless, G θ is based on the relative rotation of the photospheric magnetic elements, i.e., dθ/dt (see Equation ( 6)); therefore, the proxy tells us a quantitative contribution of sunspot rotations to coronal non-potentialities.Considering that the computation of G Φ is a computationally consuming task, we believe the G θ is still an excellent approximation for helicity flux density.See the Appendix for the influence of only considering the connectivity between the two main magnetic patches included in each subregion.

Similar Examples
The results of this study can be compared with previous studies on NOAA 10930, which produced an X3.4 flare on 2006 December 13.This active region constituted a large negative sunspot and a small positive sunspot (Kubo et al. 2007).In the middle of its evolution, the size and magnetic flux of the small sunspot increased and it moved clockwise toward the large sunspot.The large sunspot subsequently changed the sign of the magnetic helicity injection, from negative to positive (Park et al. 2010).The main sunspot also changed its rotational direction several times (Ravindra et al. 2011).The clockwise direction is the same as the direction of mutual rotation between a big and small spot.The maximum speed of the rotation was approximately 1°hr −1 .The rotational rate is approximately half of that of the reversed rotation that we obtained for Region 1 in NOAA 12297.Magara & Tsuneta (2008) identified that NOAA 10930 exhibited saturation in the time profile of the accumulation of magnetic helicity.An X3.4 flare was detected after these features.This comparison implies that the properties of reversals of helicity sign and sunspot rotations observed in NOAA 10930 and 12297 are significant for the triggering of large solar flares.Note that the NOAA 10930 was located in the southern hemisphere, but negative helicity was predominant there (Su et al. 2009).
Another comparable example is NOAA 11158, which appeared on the solar disk in 2011 February in the southern hemisphere and produced a large number of flares and CMEs.Similar to the case of NOAA 12297, this active region had a complex evolution where several emerging magnetic fluxes collided with each other.In this active region, Figure 12.Temporal evolution of the accumulation of magnetic helicity(black) and total helicity injection rate (red) in the entire active region.The vertical lines are the same as those in Figure 2. Vemareddy et al. (2012a) examined two distinct domains containing a group of positive and negative sunspots.In one domain ("R1" in their nomenclature), which produced most of the CMEs in this active region, counterclockwise self-rotation of the main sunspot injected intense negative helicity flux into globally positive helicity system (Vemareddy et al. 2012b), similar to Region 1 in our case.Another region ("R2") produced most of the flares, including the X-class, and was dominated by the shearing motion of a sunspot with respect to another static sunspot.They reported intermittent opposite helicity injections into R2, which might drive the successive flares.This evolution is similar to that in Region 2 in NOAA 12297.The evolutions of magnetic helicity injections dH/dt in R1 and R2 are similar to these of the net helicity in Regions 1 and 2 in NOAA 12297, respectively (see Figure 11).There are differences between NOAA 12297 and 11158 regarding their evolutions and activities.For instance, the largest flare was produced in Region 1 for NOAA 12297 but in R2 for 11158, and just one CME associated with the X2.1 flare was expelled from Region 1 in NOAA 12297.Nevertheless, we can speculate that the two active regions have similar evolutions of sunspot rotations leading to similar energetic activities.

Summary
We studied the magnetic helicity injected by self and mutual sunspot rotations in NOAA 12297 to understand the relationship between flares and magnetic field evolution.This active region has two flare-productive regions labeled Region 1 and 2 that produced X-and M-flares in different manners.Region 1 produced an X2.1 flare and Region 2 generated seven M-class flares.We focused on the similar configuration of their magnetic field evolutions.Both regions contained the footpoint of an emerging flux, rotating counterclockwise to a spot.To analyze this active region, we derived the magnetic helicity flux density G θ , spinning and braiding helicity injections, the rotational rate of sunspots, and the average force-free alpha α av .Measuring the spinning and braiding helicity injections allowed us to quantify the effects of the complicated sunspot motions on solar flares.
1.In Region 1, we identified the change in the sign of the spinning helicity injection, from positive to negative, in a large sunspot (P1) before the X-class flare occurred.This opposite helicity injection started with the fast emergence of a part of emerging fluxes (N2b).2. In Region 2, the positive braiding helicity injection between a small sunspot (N3) and emerging fluxes (P2) occurred with the spinning helicity injection of the emerging flux, which changed its sign several times.We consider that this variable contribution of helicity induced the frequent and recurrent M-class flares.3. The majority of the flares occurred in Regions 1 and 2 when positive and negative helicity coexisted.The annihilation of opposite magnetic helicity may be significant to the triggering of flares.
Our results showed that the type of helicity injection is significant when determining the frequency and magnitude of flares.Future work may focus on a statistical survey of the spinning and braiding helicity injections in various flareproductive active regions, and numerical MHD simulations of the coronal response to the rotational reversal of sunspots, which may more clearly reveal the influence of helicity annihilation on flare triggers.
The HMI and AIA data used in this study were courtesy of NASA/SDO and the AIA/HMI science team.This work was supported by MEXT/JSPS grant Nos.JP15H05814 and JP15H05750.T.H. is supported by the Leading Graduate Course for Frontiers of Mathematical Sciences and Physics, The University of Tokyo, MEXT.

Appendix Influence of Unconsidered Magnetic Connectivities
In this study, we separated an active region into distinct subregions (Regions 1 and 2) and evaluated the spinning and braiding helicity injection rate there just by focusing on self and mutual sunspot rotations and by considering only the connectivities between two main magnetic patches in each subregion.Here, we discuss how significantly the magnetic connectivities ignored in the evaluation may influence the helicity injection rate.The magnetic patches N2 and P2 included in Regions 1 and 2, respectively, are the two footpoints of an emerging magnetic flux that appeared at the center of the active region and thus the two subregions should be magnetically connected.In order to estimate the influence of the connectivities neglected in this study, we compared the following two helicity parameters.One (black triangles in Figure 13) is the helicity injection in the whole active region.The other (red diamonds in Figure 13) is the sum of the following amounts: 1.The spinning and braiding helicity injection considered in this study.2. The sum of G θ in the region outside the patches we determined (i.e., outside P1, P2, N2, and N3).
The difference between the two helicity parameters corresponds to the helicity injection due to the relative rotations between P1 (P2, N2, N3) and the magnetic fluxes other than N2 (N3, P1, P2), respectively (in other words, due to the connectivities we did not consider).
Figure 13 shows that the two parameters evolve with similarity in short-term variations but the difference in the injection rate is ∼2.24 × 10 37 Mx 2 s −1 on average, indicating how significant the neglected connectivities are in the results.The difference was small before 16 UT on March 11 when the X-class flare occurred in Region 1.After the flare, the difference became larger, especially the largest at 10:30 UT on March 12.Both parameters have decreasing trends toward 16 UT on March 11, and start increasing after then.Additionally, both the helicity parameters synchronously have some peaks.Thus, we conclude that a drawback of our study that we did not consider the connectivities between Regions 1 and 2 does not have a strong impact on our discussion and conclusions.Spinning and braiding helicity injections are useful to infer the behavior of the magnetic complexity due to sunspot rotations qualitatively.For further quantitative discussion, a more complex but accurate formulation of helicity injection, such as G Φ as discussed in Section 5.4, should be employed.

Figure 1 .
Figure 1.Temporal evolution of the vertical magnetic field (B z ) in NOAA 12297.The maps are displayed using the CEA coordinates.Magnetic patches of our interest are clarified with red and blue contours that highlight |B z | larger than 1000 G.The preexisting main magnetic patches are labeled as follows: the largest sunspot is P1, a small negative spot in the northwest is N3, and the positive region associated with N3 is P2a.An emerging flux pair, newly developed between March 10 and 12, is labeled as P2b and N2a, and a negative elongated patch that emerged from 10 UT on March 11 is labeled as N2b.Two distinctive areas focused on in this study are shown with orange rectangles; Region 1 includes P1 and N2, and Region 2 includes P2 and N3.Cyan arrows are approximately directed at the disk center.

Figure 2 .
Figure 2. Temporal change of the absolute magnetic flux in a large positive spot P1 (red triangles), an emerging flux P2 (magenta triangles) and N2 (blue diamonds), and a negative small spot N3 (cyan diamonds).The threshold of 1000 G in B z is used to obtain the absolute magnetic flux in each patch.The vertical solid and dashed lines indicate the flares and CMEs of interest, respectively.

Figure 3 .
Figure 3. AIA 1600 Å light curves in the entire active region (top), in Region 1 (middle), and in Region 2 (bottom).The vertical dashed lines represent when the CMEs were identified in the SoHO/LASCO C2 FOV.X-ray classification of each flare larger than the C-class is also shown in red if the flare is associated with a CME and in black if not associated with a CME.Two C-class flares associated with the CMEs are shown in red with parenthesis.
shows the temporal change of the magnetic helicity injections.Red triangles indicate spinning helicity injection by P1, whereas blue plus signs spinning helicity injection by N2.Black squares denote braiding helicity injection by the combination of P1 and N2.Vertical lines indicate flares and a CME occurring in Region 1.On March 10, the spinning helicity by P1 was small and positive.From 21 UT on March 10, it changed its sign to negative and became larger rapidly until 17 UT on March 11.The X2.1 flare occurred at 16 UT on March 11.This temporally corresponds to the peak of the negative helicity injection in P1.Subsequently, the spinning helicity injection in P1 gradually approached zero and became positive after 0 UT on March 13.The spinning helicity injection by P1 dominated Region 1 in the period between 12 UT and 20 UT on March 11 (4 hr before and after the X-class flare), during which the spinning injection by P1 was more than five times larger than the other injections;

Figure 4 .
Figure 4. (Top row) The vertical magnetic field B z of Region 1 in the CEA coordinate system shown in the grayscale background.The cyan arrows represent the horizontal velocity field.Magnetic patches (P1, N2a, and N2b) are labeled again.The scale of the horizontal velocity field, corresponding to 200 m s −1 is denoted by a green line at the top-left corner of each panel.(Bottom row) Magnetic helicity injection density G θ of Region 1. Red and blue contours display the patches, defined as P1 and N2, respectively.

Figure 5 .
Figure 5. Temporal evolution of spinning helicity injection by P1 (red triangles) and N2 (blue plus signs), and braiding helicity injection by P1 and N2 (black squares).The vertical solid and dashed lines indicate the occurrence of the three flares and a CME in Region 1, respectively.

Figure 6
Figure6shows the mean angular rotation and average forcefree alpha α av of the large spot P1.Until 21 UT on March 10, P1 exhibited slow clockwise rotation (i.e., negative mean angular rotation).The average rotational rate is −0.3 deg hr −1 .The rotation was reversed to counterclockwise at 0 UT on March 11 and was accelerated until the occurrence of the X-class flare.This change was associated with the new emergence of the N2b elongated patch.The counterclockwise rotation of P1 was the same as the direction of the mutual rotation between P1 and N2.The maximum value of the mean angular rotation was approximately 2.2 deg hr −1 .After the flare, the rotation speed of P1 decelerated and changed back to clockwise on March 13.However, the α av value gradually increased from an early period on March 11 and increased rapidly from 11 UT.This temporal behavior has a delay and is different from that of the mean angular rotation.Even after the X-class flare, α av continued to increase.This evolution of α av indicates that positive current helicity was injected into P1 during the emergence of N2b, contrary to what occurred in the magnetic helicity injection.In order to double check the rotation of P1, we created an angle-time plot (Figure7(b)) by straightening the HMI continuum signals on the red circle displayed in Figure7(a), and tracking the dark features of umbral spirals.The circle has a radius of 21 pixels (∼10 5) and its center corresponds to the center of gravity of the umbra.Before the emergence of N2b (around 4 UT on March 11), we can find the dark features with both increasing and decreasing gradients, corresponding to clockwise and counterclockwise rotations, respectively.After the emergence of N2b, a few umbral spirals located in 45°-150°(clarified with green curves in Figure7(b)) rotated counterclockwise at high speed.The rotational speed of these dark features is 3-4 deg hr −1 and these values are consistent with the mean angular rotation plotted in Figure6.Even after the emergence of N2b finished at 3 UT on March 12, the dark features in 150°-200°show slow counterclockwise rotations that were driven by the mutual rotation of N2b with respect to P1.

Figure 6 .
Figure 6.Temporal evolution of mean angular rotation and average force-free alpha α av of the large spot P1.The vertical lines are the same as in Figure 5.

Figure 7 .
Figure 7. (Top) The HMI continuum map focusing on P1.The red circle of the radius of 21 pixels (∼10 5) represents the angle-time plot of P1.The west direction is set to 0°and the angle increases counterclockwise.(Bottom) The angle-time plot.The green curves are overplotted where north parts of the umbra rotated counterclockwise at high speed.

Figure 8 .
Figure 8. (Top row) The vertical magnetic field B z of Region 2 in the CEA coordinate system shown in the grayscale background.The cyan arrows represent the horizontal velocity field.Magnetic patches (P2a, P2b, and N3) are labeled again.The scale of the horizontal velocity field corresponding to 200 m s −1 is indicated by the green line at the top-left corner of each panel.(Bottom row) Magnetic helicity injection density G θ of Region 2. Red and blue contours display the patches defined as the sunspot group of P2 and N3, respectively.

Figure 9 .
Figure 9. Temporal evolution of the spinning helicity injection by N3 (blue triangles) and P2 (red plus signs), and the braiding helicity injection by N3 and P2 (black squares).The vertical solid and dashed lines indicate the occurrence of the seven M-class flares and CMEs in Region 2, respectively.

Figure 10 .
Figure 10.Temporal evolution of the mean angular rotation and α av of the preexisting spot N3.The vertical lines are the same as those in Figure 9.

Figure 11 .
Figure 11.Comparison between the sum of the spinning and braiding helicities (black triangles) and the net helicity (red diamonds) in Region 1 (top) and 2 (bottom).