MAGNETIC STRUCTURE PRODUCING X- AND M-CLASS SOLAR FLARES IN SOLAR ACTIVE REGION 11158

We numerically extrapolate the 3D coronal magnetic field assuming it as the NLFFF by using the MHD relaxation method developed by Inoue et al. (2011, 2012a, 2012b), which has already been applied to the solar active region NOAA 10930. This method can numerically solve the following equations:

$$\begin{equation} \frac{\partial {\boldsymbol {v}}}{\partial t} = - ({\boldsymbol {v}}\cdot {\boldsymbol {\nabla }}){\boldsymbol {v}} + \frac{1}{\rho } {\boldsymbol {J}}\times {\boldsymbol {B}} + \nu {\boldsymbol {\nabla }}^{2}{\boldsymbol {v}}, \end{equation} \tag{ 1 }$$

$$\begin{equation} \frac{\partial {\boldsymbol {B}}}{\partial t} = {\boldsymbol {\nabla }}\times ({\boldsymbol {v}}\times {\boldsymbol {B}} - \eta {\boldsymbol {J}}) - {\boldsymbol {\nabla }}\phi , \end{equation} \tag{ 2 }$$

$$\begin{equation} {\boldsymbol {J}} = {\boldsymbol {\nabla }}\times {\boldsymbol {B}}. \end{equation} \tag{ 3 }$$

$$\begin{equation} \frac{\partial \phi }{\partial t} + c^2_{h}{\boldsymbol {\nabla }}\cdot {\boldsymbol {B}} = -\frac{c^2_{h}}{c^2_{p}}\phi , \end{equation} \tag{ 4 }$$

where ${\boldsymbol {B}}$ is the magnetic flux density, ${\boldsymbol {v}}$ is the velocity, ${\boldsymbol {J}}$ is the electric current density, ρ is the pseudodensity, and ϕ is a scalar potential. The pseudodensity is assumed to be proportional to $|{\boldsymbol {B}}|$ in order to ease the relaxation by equalizing the Alfvén speed in space. The last equation (Equation (4)) is used to avoid deviation from $\nabla \cdot {\boldsymbol {B}}=0$ and was introduced by Dedner et al. (2002).

The length, magnetic field, pseudodensity, velocity, time, and electric current density are normalized by L₀ = 184.32 Mm, B₀ = 2500 G, ρ₀ = |B₀|, V_A ≡ B₀/(μ₀ρ₀)^1/2, where μ₀ is the magnetic permeability, whose value corresponds to 4π × 10⁻⁷ H/m, τ_A ≡ L₀/V_A and J₀ = B₀/μ₀L₀, respectively. The non-dimensional viscosity ν is set as a constant (1.0 × 10⁻³), and the non-dimensional resistivity η is given as

$$\begin{equation} \eta = \eta _0 + \eta _1 \frac{|{\boldsymbol {J}}\times {\boldsymbol {B}}||{\boldsymbol {v}}|^2}{\boldsymbol {|B|}}, \end{equation} \tag{ 5 }$$

where η₀ is a constant specific to each case whose values are shown in Table 1, and η₁ = 1.0 × 10⁻³ in non-dimensional units. The second term is introduced to accelerate the relaxation to the force-free field, particularly in the weak field region. The other parameters $c_h^2$ and $c_p^2$ are set to the constants 0.04 and 0.1, respectively.

The velocity field is capped at v_max so that it does not correspond to a large value. Let us define $v^{*} = |{\boldsymbol {v}}|/|{\boldsymbol {v}}_{A}|$ and specify that if the value of v* becomes larger than the value of v_max given in Table 1, the velocity is modified as follows:

$$\begin{equation} {\boldsymbol {v}} \Rightarrow \frac{v_{{\rm max}}}{v^{*}} {\boldsymbol {v}}. \end{equation} \tag{ 6 }$$

An initial condition is given by the potential field extrapolated using the Fourier method from the normal component on the vector field assuming a periodic condition and an exponential decay in the height direction, whose analytical formula can be found in Equation (13.3.4) in Sturrock (1994). First, we calculate the 3D potential field according to the following equation:

$$\begin{equation} {\boldsymbol {B}} = \Sigma {\boldsymbol {b}}_{(m,n)}^{\ast } {\rm exp}(ik_m x + ik_n y - |k|z), \end{equation} \tag{ 7 }$$

where k_m = 2πm/L_x, k_n = 2πn/L_y, $|k| = \sqrt{k_m^2+k_n^2}$, m = −n_x/2, ..., −1, 1, ..., n_x/2, and n = −n_y/2, ..., −1, 1, ..., n_y/2, respectively. ${\boldsymbol {b}}_{(m,n)}^{\ast }$ consists of three components, $b_{x(m,n)}^{\ast } = -i k_m b_{z(m,n)}^{\ast }/|k|$, $b_{y(m,n)}^{\ast } = -i k_n b_{z(m,n)}^{\ast }/|k|$, and $b_{z(m,n)}^{\ast }$. However, after this method, the vector field is slightly deviated from the original one because ${\boldsymbol {b}}_{(0,0)}^{\ast }$ (averaged component) was not counted in the process of the inverse Fourier transformation. Therefore, as a next step, we calculate the potential field again using the original vector field without removing the ${\boldsymbol {b}}_{(0,0)}^{\ast }$ and the lateral and top boundaries of the 3D magnetic field obtained from Equation (7). The potential field is obtained from the normal component of the magnetic field at all of the boundaries according to the ${\boldsymbol {\nabla }}^2 \Phi =0$ (where ${\boldsymbol {B}}= -{\boldsymbol {\nabla }}\Phi$) after total flux conservation ∫B_ndS = 0 is satisfied in the entire domain, where dS represents the surface element on all boundaries, and the subscript n represents the component normal to the surfaces of the boundaries.

The observed vector field is given as the bottom boundary condition, and the three components of the magnetic field are fixed. However, the observational data conditions cannot perfectly satisfy the induction equation at the end, so the inconsistent boundary condition generates errors related to ${\boldsymbol {\nabla }}\cdot {\boldsymbol {B}}$ near the boundary area. Equation (4) can reduce these errors dramatically, as shown in Table 1 (see $D=\int |{\boldsymbol {\nabla }}\cdot {\boldsymbol {B}}|^2dV$). On the top and side boundaries, the transverse fields are determined by an induction equation (2) where a perfect conductive wall is assumed (i.e., the velocity and electric fields are set to zero), while the normal component is kept with at the initial condition to conserve the total flux. Thus, the side and top boundaries deviate from the real situation; therefore, our analysis is limited to closed loops in the central area.

The Runge–Kutta scheme with fourth-order accuracy for the temporal integral and the central finite difference with second-order accuracy for the spatial derivative are applied as the numerical scheme for this calculation. The numerical domain is covered by 184.32 × 184.32 × 184.32 Mm³, whose area is given by a 128 × 128 × 128 grid. The vector field given as the boundary condition is obtained by 4 × 4 binning from the original vector field 512 × 512. The grid interval corresponds to about 1688 km pixel⁻¹, i.e., about 23; therefore, we focus on the scale of the field lines connecting the sunspots in the central area of the numerical domain.

We analyzed one X-class flare (X2.2) that occurred at around 01:48 UT on 2011 February 15 and M-class flares (M6.6, M1.0, and M1.1) that occurred at around 17:34 UT on February 13, and 01:36 UT and 07:40 UT on February 16, respectively. The GOES X-ray profile is shown in Figure 1(a). The observation of the vector field by HMI/SDO covered this time span, and the HMI team has already provided these data, which are projected and remapped to heliographic coordinates with a spatial resolution of ∼05 and a cadence of 12 minutes (http://jsoc.stanford.edu/jsocwiki/ReleaseNotes). The vector magnetic fields are obtained from the Very Fast Inversion of the Stokes Vector algorithm (Borrero et al. 2011) based on the Milne–Eddington approximation, and the minimum energy method (Metcalf 1994; Leka et al. 2009) was used to solve the 180° ambiguity in the azimuth angle of the magnetic field. The vector fields before each of the four flares are investigated (M6.6, X2.2, M1.0, and M1.1) as shown in Figure 1(b), in which the gray scale represents the distribution of the normal component of the magnetic field (B_z).

Zoom In Zoom Out Reset image size

We use the Ca ii H 3968 Å images in these flares for comparison with the extrapolated fields. These images were provided by the Broadband Filter Imager on SOT on board Hinode. The fields of view are 18828 × 11158 with a pixel size of 1728 × 1024 at 17:35:38 UT on February 13, 11158 × 11158 with 1024 × 1024 at 01:50:18 UT on February 15, 18828 × 11158 with 1728 × 1024 at 01:40:39 on February 16, and 11158 × 11158 with 1024 × 1024 at 07:42:13 on February 16. We also used the EUV images at 94 Å at 23:59:28 UT on 2011 February 14 obtained from AIA/SDO, whose view extracted from the full disk image corresponds to the same as that of HMI used in this study.

First, we check the force freeness of the reconstructed field. As an example, Figure 2(a) shows the force-free α map related to the extrapolated field at 00:00 UT on February 15. The vertical and horizontal axes represent the values of the force-free α estimated on opposite footpoints of each field line. Note that these values are estimated in the plane at 1688 km, i.e., 1 grid size in this case, above the photosphere, and the field lines are traced from the region where B_z < −50 G. Although the extrapolated field cannot satisfy the force-free condition perfectly because of the deviation from the force-free state in the photosphere, these dotted points seem to be generally distributed along the green line, as shown in our previous result (Inoue et al. 2012a).

Zoom In Zoom Out Reset image size

Next, we compare the 3D field lines structure with the observational data at the various wavelengths to check its reliability. As an example, Figure 2(b) shows selected 3D magnetic field lines in blue; they are extrapolated from the vector field obtained from HMI/SDO at 00:00 UT on February 15, around 1.5 hr before the X2.2 flare, and plotted over the distribution of the B_z component. These selected field lines start from points on the photosphere with B_z  >  250 G. A strong shear field clearly formed on the PIL between the sunspots located in the center of the plot. This feature is consistent with earlier works (e.g., Sun et al. 2012; Wiegelmann et al. 2012; Jiang & Feng 2013).

The same selected field lines are plotted with blue and gray lines in Figure 2(c) over the distribution of the B_z component, and then the brightest part of the AIA's 94 Å image (I > 1.0 × 10⁵(DN)) is superimposed. The blue lines are arbitrarily chosen from the field lines in Figure 2(b) that are located in the strongly enhanced areas of the AIA image. The inset shows the same image without the field lines revealing high-intensity area in the EUV image is not only enhanced near the central area but also extends in the east–west direction; nevertheless, the blue lines can cover these regions.

Figure 2(d) shows the another set of selected field lines in orange and gray plotted over the B_z distribution with the Ca ii images obtained from FG/Hinode. Only the region with an Ca ii signal greater than 1000(DN) observed at 01:50:18 UT on February 15 is drawn in this plot. The inset also shows the same picture without the field lines. The orange field lines correspond to the strong shear field along the PIL with respect to the gray lines, and their footpoints are rooted in the region where the Ca ii images are strongly enhanced.

This result is consistent with our previous study (Inoue et al. 2011), which confirms the reliability of the extrapolated field as well as the fact that magnetic reconnection is occurred along these strong non-potential field lines because the strong Ca ii illumination is deeply related to magnetic reconnection (Priest & Forbes 2002; Forbes 2000).

3.2.1. Comparing Magnetic Twist with Ca ii Intensity and Distribution

We estimate the magnetic twist from the 3D extrapolated field lines and compare it with the Ca ii images obtained from SOT/Hinode in detail. The magnetic twist is an important proxy for determining the stability of the magnetic configuration (Kruskal & Kulsrud 1958; Hood & Priest 1979; Török et al. 2004; Inoue & Kusano 2006) as well as a convenient one for clarifying its topology. The twist index is defined as

$$\begin{equation} T_n = \int \frac{dT_n}{dl}dl = \int \frac{J_{||}}{4 \pi B_{||}}dl = \frac{1}{4\pi } \int \alpha dl, \end{equation} \tag{ 8 }$$

where the line integral ∫dl is taken along each magnetic field line, and the force-free α can be obtained from $\alpha = {\boldsymbol {J}} \cdot {\boldsymbol {B}}/|{\boldsymbol {B}}|^2$ for each field line. The physical significance is that it represents the degree of twist of a magnetic field line corresponding to the measurement of the magnetic helicity generated by the current parallel to a field line (Berger & Field 1984; Berger & Prior 2006; Török et al. 2010; Inoue et al. 2012b). Note that following the work of Berger & Prior (2006) we applied this definition only to closed loops connecting the sunspots, as shown in Figure 3.

Zoom In Zoom Out Reset image size

The red lines in the upper panels in Figures 3 represent the contours of half-turn twists (T_n = 0.5), i.e., the regions surrounded by the red contours are dominated by strongly twisted lines (T_n > 0.5), with the Ca ii images around the peak time of the GOES X-ray flux in each flare. Hereafter, "strongly twisted lines" will refer to field lines having more than a half-turn twist (T_n  >  0.5). The white lines represent the contours of |B_z| = 625 G. The M6.6 and X2.2 flares on February 13 and February 15, respectively, reveal that the locations of strongly twisted regions surrounded by the red lines correspond to enhanced regions of strong Ca ii illumination as shown in Figure 2(d). In particular, the profile of the strongly twisted region (T_n > 0.5) at 00:00 UT on February 15 is similar to that of the Ca ii image, indicating that the dramatic magnetic reconnection occurred in the strongly twisted magnetic lines having around or more than a half-turn twist, as shown in Inoue et al. (2011, 2012b). In addition, because the strongly twisted regions on February 15 seem to be larger than that of February 13, this change in the size of the strongly twisted area might be related to the occurrence conditions of X or M-class flares.

On the other hand, the enhancement of the Ca ii image in the M1.0 flare on February 16 appears outside the strongly twisted regions. Therefore, the strongly twisted lines do not seem to be related to this flare despite the growth in the size of the twisted regions due to the clockwise motion of the positive sunspot. However, in the M1.1 flare, the same regions with strongly twisted lines coincided well with the areas of enhanced Ca ii, whose values are smaller than that in the M6.6 or X2.2 flares. At this time, the strongly twisted regions surrounded by the red contours also seem to grow continuously.

The lower panels in Figures 3 show the selected field lines in orange and blue corresponding to magnetic twist of more and less than a half-turn twist, plotted over the upper panels. The footpoints of these field lines lie in the regions where the Ca ii image is strongly enhanced, i.e., these field lines connect the two-ribbon flare across the PIL except that in M1.0 flare and weakly twisted lines in M1.1 flare. According to these results, the magnetic field lines on February 15 are obviously elongated compared with those on February 13 and 16, meaning that stronger shear is formed just before the X2.2 flare compared to the M flares.

3.2.2. Temporal Evolution of the Magnetic Twist during the Major Flares

Figure 4 shows the temporal evolution of the magnetic twist. At 16:00 UT on February 13, the strongly twisted regions in the M6.6 flare, which have 0.5 <T_n < 1.0, are already built up by the clockwise motion of the positive sunspot. Most of the strongly twisted lines disappear from the data after this flare. At 00:00 UT on February 15, before the X2.2 flare, the strongly twisted regions are built up again and seem to dominate larger areas than before the previous M6.6 flare. In contrast to the previous flare, some parts of the strongly twisted regions remained even after the X2.2 flare. Nevertheless, the store-and-release scenario of magnetic helicity appeared clearly during the large flares. As a common feature before each flare, we have never found a strongest twisted region having a value greater than a one-turn twist (T_n > 1.0). This means that the magnetic configuration is stable against the ideal MHD instability; however, even these magnetic configurations of less than one-turn twist can generate large flares.

Zoom In Zoom Out Reset image size

The twist profiles at 00:00 UT and 07:00 UT on February 16 are 40–90 minutes before the M1.0 and M1.1 flares, respectively. The strongly twisted regions of more than a half-turn twist (T_n = 0.5) are built up again owing to the continuous clockwise twist and shear motions of the positive sunspot. In contrast to the previous large-scale flares (M6.6 and X2.2), most parts of the strongly twisted regions remained even after these flares. According to the result in Figure 3(c), because the M1.0 flare occurred outside the strongly twisted region, the strongly twisted region does not seem to be deeply related to this flare. In other words, these strongly twisted lines remain in the stable state despite their continuing growth. On the other hand, in the M1.1 flare, even stronger twisted regions of more than one turn appear and develop. Nevertheless, all the areas dominated by the strongly twisted lines are weakly illuminated by the Ca ii image, as shown in Figure 3(d), and the most strongly twisted regions remained after this flare, as seen in Figure 4. Therefore, this result implies that the strongly twisted lines do not seem to be relaxed enough to become untwisted lines. These results differ from the common feature of the previous large flares. These topics will be discussed later.

In addition to the above analyses, we show the temporal evolution of the ratio of a fragment of the magnetic flux dominated by values of more than a half-turn twist to the total flux in Figure 5, whose formula is written by

$$\begin{equation} F = \frac{\int _{T_n>0.5} B_z dS}{\int B_z dS}, \end{equation} \tag{ 9 }$$

where the dS is a surface element. These fluxes are estimated to integrate only those B_z in positive polarities in the same view as in Figure 4. From this result, the M6.6 and X2.2 flares on February 13 and 15 also clearly exhibit the store-and-release process of the magnetic helicity during the solar flare, and the flux ratio value in the X2.2 event is about three times larger than that of the M6.6 flare. On the other hand, in the last two flares (M1.0 and M1.1 flares), we can see again that the continuous shear and twist motions of the positive sunspot after the large flares regenerate the strongly twisted lines; consequently, a flux comparable to that before the previous large flare is regained. Jing et al. (2012) also indicated that the relative magnetic helicity or helicity injection retain their increasing trend after X2.2 flares. Nevertheless, the magnetic flux dominated by the strongly twisted lines is mostly unchanged during the M1.0 and M1.1 flares.

Zoom In Zoom Out Reset image size

3.2.3. Decrement of the Magnetic Twist during the Major Flares

We investigate in more detail the decrement of the magnetic twist during each flare event to determine how much magnetic twist is needed to cause large flares. Figure 6 shows the distribution map for the values of the magnetic twist (vertical axis) versus the B_z component (horizontal axis). We clearly see that most dotted points in the upper area, above the horizontal dotted line, disappear after each flare; thus, the density of the distribution of less than a half-turn twist seems to be partially enhanced. This result suggests that the strongly twisted lines (in excess of a half-turn twist) relax into less twisted lines. Because many dotted points are, even after each flare, widely distributed in the twist range T_n < 0.5, the buildup of twisted field lines having more than a half-turn twist is an important process for generating the large flares. Moreover, the dotted points above the horizontal dashed line before the X2.2 flare are obviously greater in number and composed of a more twisted region than those before the M6.6 flare, as shown in Figure 5. On the other hand, Figure 7 shows the distribution maps before and after the M1.0 and M1.1 flares. In this case, most of the dotted points remained in the upper region (values greater than a half-turn twist) even after the flare and these distributions do not seem to change dramatically during these flares, as we expected from the above results.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Figure 8 shows the selected 3D field lines before each flare. The strongly twisted lines (orange) lie on the neutral line in each case. Although some weakly twisted lines (blue) in addition to the strongly twisted lines also form the shear structure at 16:00 UT on February 13, in the last two flares (M1.0 and M1.1 flares), some field lines whose footpoints are rooted in regions of strong magnetic field seem to surround the strongly twisted lines. These relationships between the strongly (T_n > 0.5) and weakly (T_n < 0.5) twisted field lines, especially those surrounding the strongly twisted lines, might imply some influences on the dynamics of each flare.

Zoom In Zoom Out Reset image size

We investigate in detail the relationship between these strongly and weakly twisted lines before each flare by using the connectivity map in the upper panels in Figure 9, which focuses on specific field lines for which both footpoints are rooted in the regions of the strong magnetic field (B_z  >  500 G and B_z < −500 G). Consequently, both footpoints of the selected field lines are rooted in the regions shown in white in both polarities, where the white lines correspond to the contour at |B_z| = 500 G.

Zoom In Zoom Out Reset image size

These field lines are shown in the middle panels in Figure 9. According to these results, at 16:00 UT on February 13, all the selected closed field lines form a similar shear structure, which are strongly related to the M6.6 flares because the Ca ii image is strongly illuminating in those footpoints in Figure 3. Afterward, these twisted lines are deformed due to the subsequently strong sheared and twisted motion of the sunspot. Consequently, at 00:00 UT on February 15, the weakly twisted lines (blue) seem to partially cover the strongly twisted lines (orange). In fact, the X2.2 flare can be considered to be induced by the strongly twisted lines over the half-turn twist from Figure 3. Eventually, on February 16, most of the strongly twisted lines were covered by weakly twisted lines. These results clearly show that the weakly twisted lines, which constituted the core field on February 13, play a role in the overlying field lines surrounding the strongly twisted lines. Therefore, as time passes, the strongly twisted lines seem to fall into an unfavorable condition to escape from the lower coronal region.

In the lower panels in Figure 9, we show a side view of the 3D field lines before each flare in order to more understand the overview profiles of the field lines qualitatively. We clearly see at 00:00 UT on February 15 that the part of the weakly twisted field lines in blue extends to a higher position compared to those on February 13. On the other hand, although these extended loops still remain at 00:00 UT and 07:00 UT on February 16, they seem to become a bit more compact than earlier. This might be related to the report by Sun et al. (2012), which indicated that this active region decreased in volume after the X2.2 flare.

Figure 10 shows the flux ratio of the weakly twisted lines connecting both polarity regions in white in Figure 9 to the magnetic flux related to the entire closed field, which is composed of the strongly and weakly twisted lines connecting same areas. All of the flux values are estimated to integrate only those B_z in the positive polarities before the flares on February 15 and 16, and the flux ratio formula is defined as

$$\begin{equation} F_{r} = \frac{\int _{0<T_n<0.5} B_z dS}{\int B_z dS}, \end{equation} \tag{ 10 }$$

where dS is a surface element. Note that because we are interested in the deformed weakly twisted lines due to subsequently strong twisted and sheared motion of the sunspot after the M6.6 flare, the value on February 13 is not plotted in Figure 10.

Zoom In Zoom Out Reset image size

This result shows us that this flux ratio remains 62%–67% during the three flares. Therefore, these results reveal no large variations in the magnetic flux. Before the X2.2 flare on February 15, even though the weakly twisted lines whose footpoints are both at |B_z|  >  500 G had already deformed, they covered the strongly twisted fields only in part. This surrounding condition might allow the escape of the strongly twisted field from the lower coronal region. On the other hand, the magnetic field configuration is changed by the strong shear and twist motions of the sunspot, thus the weakly twisted field lines covered the entire strongly twisted field region on February 16. Furthermore, along with the results in Figure 10, because the "total" magnetic energy accumulated in the root of the weakly twisted lines is superior to that of the strongly twisted field lines, these probably have the potential to confine the dynamics of the strongly twisted field in the M1.0 and M1.1 flares. For example, this confinement indicates the suppression of the exhaustion at the magnetic reconnection site and consequent rapid energy release.

In the previous subsection, we discuss the magnetic structure inducing the X- and M-class solar flares in terms of the temporal evolution and topologies of the twisted field lines formed in the lower corona. However, the overlying field lines surrounding these twisted lines also play an important role in determining whether or not CMEs are produced even if the twist value in the twisted line is more than a one-turn twist (Guo et al. 2010). Therefore, we need to discuss the overlying field lines quantitatively. The decay index is one of quantitative indexes of the overlying fields by estimating how rapidly the strength of the magnetic field decreases with height, which is given by the following equation:

$$\begin{equation*} n(z) = - \frac{z}{|{\boldsymbol {B}}|}\frac{\partial |{\boldsymbol {B}}|}{\partial z}. \end{equation*}$$

The decay index is often referred to an estimation of a criterion for the torus instability whose threshold lies in the range of 1.1 ⩽n_c ⩽ 2, which was introduced by Kliem & Török (2006) and Démoulin & Aulanier (2010). An eruption of the flux rope was numerically confirmed by Török & Kliem (2007), Fan (2010), and Aulanier et al. (2010).

In this study, as mentioned in Section 2, we can discuss only the closed field lines in the central area within a height of around few dozens of Mm due to the problem associated with lateral boundary conditions. Because of this, the decay index in the overlying field lines higher than few dozens of Mm is strongly affected by these boundary conditions. Furthermore because the decay index cannot be applied to the non-potential field structure retrieved with our mode within a height of around few dozens of Mm, it is difficult to discuss it in the framework of our study. However, Nindos et al. (2012) already calculated the decay index related to the NLFFF (Wiegelmann et al. 2012) in AR11158 in the long-term period of February 11–16. They showed the long-term temporal evolution of the decay index at each height and concluded that the onset of eruptions does not depend critically on the long-term evolution of the decay index of the background field before CMEs. In other words, this result might support that the temporal evolution of the twisted field lines play an important role in controlling the flares and CMEs in this active region.

In fact, partial halo and halo CMEs were observed to be associated with the M6.6 and X2.2 flares, however, we were not able to observe the CMEs just after M1.0 and M1.1 flares (http://cdaw.gsfc.nasa.gov/CMElist/). Only one CME that originated from AR11158 was observed after the X2.2-class flare occurred. However, the flare which is the source of this CME occurred in the edge of this active region. Therefore these observations might support that flux rope eruption is confined by the weak twisted line plotted in blue in Figure 9 during the M1.0 and M1.1 flares.

In this study, the strongly twisted field lines before the M6.6 and X2.2 flares distribute in 0.5 < T_n < 1.0, which implies that the magnetic configuration is stable against the MHD instability. Although these magnetic twists obtained from our study seem to be weak to induce the large flares, this is because the NLFFF approximation omits the necessary physics in a dynamic process. i.e., missing the tether-cutting process. Some authors also have supported tether-cutting reconnection as a feasible process causing eruption in this active region as described in Section 1. Because of conservation of the magnetic helicity, tether-cutting reconnection in the strongly twisted lines generates longer and more strongly twisted field lines just before an eruption. For example, Amari et al. (2010, 2011) showed the strongly twisted flux tube along the PIL through the tether-cutting reconnection in the twisted field lines due to the flux cancellation process or converging flows on the photosphere. Eventually, this flux tube is successfully launched from the lower corona. More recently, Kusano et al. (2012) indicated two types of emerging flux that produce the long, strongly twisted field lines in the pre-existing coronal magnetic field through tether-cutting-like reconnection, where the initial preexisting field is assumed to form uniformly sheared arcades in the linear force-free approximation. They reported that the angle between the emerging flux and the pre-existing shear field lines is important. Thus, tether-cutting process would be feasible of producing large flares even the accumulated twist in the solar active region is less than a one-turn twist.