Non-potential Field Formation in the X-shaped Quadrupole Magnetic Field Configuration

The SOT (Ichimoto et al. 2008; Shimizu et al. 2008; Suematsu et al. 2008; Tsuneta et al. 2008) on board Hinode has two focal plane instruments, i.e., the Filtergraph (FG) and the SP (Lites et al. 2013). In this study, we used series a of the SP data.

The SP performs spectropolarimetric observations with two magnetically sensitive Fe i lines at 6301.5 Å and 6302.5 Å. We used nine vector magnetic field maps derived from the SP observations. The maps have an effective pixel size of 03 with a field of view (FOV) of 280'' × 130''. The spectral sampling was 21.549 mÅ pixel⁻¹. The SP maps were taken with fast-mapping mode by slit scanning over almost the entire active region every two or three hours. Because we want to investigate the difference between the global coronal 3D magnetic field during flare productive and less productive periods, we used nine SP maps obtained at 10:42–11:35 UT, 16:54–17:45 UT, and 23:19-00:14 UT on 2014 February 1, and 02:20–03:15 UT, 07:12–08:07 UT, 10:20–11:15 UT, 16:00–16:52 UT, and 19:20–20:15 UT on 2014 February 2, and 07:50–08:45 UT on February 3. As shown in Figure 1, the observation period was between 21 hr before flare 1 and 14 hr after flare 3.

For the calibration of the Stokes profiles, we used the Solarsoft routine SP_PREP (Lites & Ichimoto 2013). When we derived physical parameters from the Stokes profiles, we solved the radiative transfer equation (del Toro Iniesta 2007),

$$\begin{eqnarray}&&\displaystyle \frac{d{\boldsymbol{I}}}{d\tau }={\boldsymbol{K}}({\boldsymbol{I}}-{\boldsymbol{S}}),\end{eqnarray} \tag{ 2 }$$

where ${\boldsymbol{I}}\equiv (I,Q,U,V)$ is the Stokes vector, τ is the optical depth, ${\boldsymbol{K}}$ is the propagation matrix, and ${\boldsymbol{S}}$ is the source function. We applied the Milne-Eddington atmosphere (ME) model to the two lines (Fe i doublet at 6301.5 and 6302.5 Å) to derive the physical parameters by a nonlinear least square fitting using the code based on MELANIE (Socas-Navarro 2001). This simultaneous inversion has been shown to provide better results than the use of only a single line (Lites et al. 1994). The 180° ambiguity in the transverse magnetic field direction was solved with the minimum energy ambiguity resolution method (Metcalf 1994; Leka et al. 2009).

The AIA was used to investigate the coronal and chromospheric structures of the flaring region in NOAA 11967. In this study, we use 1600 Å, which contains C iv line and continuum, for investigating the behaviors of the flare ribbons in the chromosphere and 131 Å, which contains Fe vii, xx, and xxiii lines, for the coronal hot plasma in the flaring region. The pixel size was 06 and the temporal cadence was 12s for 131 Å images and 24s for 1600 Å images. The HMI on board SDO provided the full-disk photospheric magnetic field. The HMI measures polarimetric signals ($I\pm Q,I\pm U,I\pm V$) of magnetic sensitive Fe i 6173 Å line at 6 narrow bands (bandwidth 76 mÅ +/− 10 mÅ) in the line in a 135 s sequence and derive the magnetic field vectors, which are averaged over 12 minutes to increase the signal-to-noise ratio.

The Solarsoft routine AIA_PREP was used for the AIA data calibration and the alignment between the AIA and HMI data. We investigated the coronal and chromospheric dynamics by visually inspecting of the EUV(131 Å) and UV (1600 Å) data acquired near the occurrence of flare 1 (08:00UT-09:00UT), flare 2 (09:00UT-10:00UT), and flare 3 (18:00UT-18:30UT).

For the HMI data, there is a data product, called Space-weather HMI Active Region Patches (SHARP), which provides vector magnetic field data on a 12 minute cadence and automatically identifies and track's active regions (Bobra et al. 2014). We used these vector field data to increase the narrow FOV of Hinode/SP when extrapolating the coronal magnetic field by NLFFF modeling (Figure 3). When we created the boundary data for the modeling, we converted the magnetic field vector map in the line-of-sight coordinates to that in the local frame coordinates and resized the HMI pixel size to the SP pixel size with a cubic polynomial interpolation using 16 neighboring points. The resized HMI data were co-aligned to the SP data by applying a cross-correlation technique. One of the co-aligned HMI data is shown in Figure 3(a). To obtain wider FOV data with higher polarimetric accuracy, the co-aligned HMI data were added to around the SP map. The 2 × 2 binned data of the combined SP/HMI maps, one of which is shown in Figure 3(b), were used as the boundary condition for the NLFFF modeling. The pixel size is 06 × 06. The magenta, red, and blue squares in Figure 3(b) shows the FOV of the Hinode/SOT SP, Figures 9 and 11, and Figures 13 and 15, respectively.

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Several methods are used to solve the boundary value problem and obtain a force-free field (Wiegelmann & Sakurai 2012); here, we used the magnetohydrodynamics (MHD) relaxation method. This method was first developed by Mikic & McClymont (1994) and used zero-β time-dependent MHD codes to achieve stationary equilibrium. Their calculation begins with a potential field calculated from the normal components of the magnetic field at the photosphere, and they control the transverse electric field while maintaining B_z stable; eventually, they obtain a force-free state.

We used the code developed by Inoue et al. (2014) and also see in Inoue (2016), which improved Mikic & McClymont (1994). To achieve a force-free state, we solved the following zero-β MHD equations in the Cartesian coordinates,

$$\begin{eqnarray}&&\displaystyle \frac{\partial {\boldsymbol{v}}}{\partial t}=-({\boldsymbol{v}}\cdot {\boldsymbol{\nabla }}){\boldsymbol{v}}+\displaystyle \frac{1}{\rho }\,{\boldsymbol{j}}\times {\boldsymbol{B}}+\nu {{\rm{\nabla }}}^{2}{\boldsymbol{v}},\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial {\boldsymbol{B}}}{\partial t}={\rm{\nabla }}\times ({\boldsymbol{v}}\times {\boldsymbol{B}}-\eta \,{\boldsymbol{j}})-{\rm{\nabla }}\phi ,\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&{\boldsymbol{j}}={\rm{\nabla }}\times {\boldsymbol{B}},\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial \phi }{\partial t}+{c}_{h}^{2}{\rm{\nabla }}\cdot {\boldsymbol{B}}=-\displaystyle \frac{{c}_{h}^{2}}{{c}_{p}^{2}}\phi ,\end{eqnarray} \tag{ 6 }$$

where ρ is the pseudo density, which is assumed to be proportional to $| {\boldsymbol{B}}|$, ϕ is the convenient potential for ${\rm{\nabla }}\cdot {\boldsymbol{B}}$ cleaning and ν is the viscosity, which was set to be a constant ($1.0\times {10}^{-3}$). The length, magnetic field, velocity, and time were normalized by ${L}_{0}=157$ Mm and ${B}_{0}=4000$ G, ${V}_{{\rm{A}}}\equiv {B}_{0}/{(4\pi {\rho }_{0})}^{1/2}$, and ${\tau }_{{\rm{A}}}\equiv {L}_{0}/{V}_{{\rm{A}}}$, where V_A is the Alfvén velocity. Equations (3)–(6) are the equation of motion, the induction equation, the Ampère's law, and ${\rm{\nabla }}\cdot {\boldsymbol{B}}$ cleaning scheme introduced by Dedner et al. (2002), respectively. The parameters c_p² and c_h² are the advection and diffusion coefficients, respectively, which were fixed at 0.1 and 0.04, correspondingly. The non-dimensional resistivity η is given by

$$\begin{eqnarray}&&\eta ={\eta }_{0}+{\eta }_{1}\displaystyle \frac{| {\boldsymbol{j}}\times {\boldsymbol{B}}| | {\boldsymbol{v}}{| }^{2}}{| {\boldsymbol{B}}{| }^{2}},\end{eqnarray} \tag{ 7 }$$

where ${\eta }_{0}$ and ${\eta }_{1}$ are fixed at $5.0\times {10}^{-5}$ and $1.0\times {10}^{-3}$ in non-dimensional units.

The velocity field was adjusted as below to avoid increasing when the value of v* becomes larger than the value of ${v}_{\max }$,

$$\begin{eqnarray}&&{\boldsymbol{v}}\to \displaystyle \frac{{v}_{\max }}{{v}^{* }}{\boldsymbol{v}},\end{eqnarray} \tag{ 8 }$$

where ${v}^{* }=| {\boldsymbol{v}}| /| {{\boldsymbol{v}}}_{{\rm{A}}}|$, where ${v}_{\max }=0.01$.

The initial condition was the potential field calculated from the normal component of the observed magnetic field and the calculation used Green's function method (Schmidt 1964; Sakurai 1982). The magnetic field at the top and side boundaries was maintained at the initial state (or potential field) and the normal component of the magnetic field at the bottom boundary was also fixed. We varied the transverse component at the bottom boundary ${{\boldsymbol{B}}}_{\mathrm{BC}}$ as follows,

$$\begin{eqnarray}&&{{\boldsymbol{B}}}_{\mathrm{BC}}=\gamma {{\boldsymbol{B}}}_{\mathrm{obs}}+(1-\gamma ){{\boldsymbol{B}}}_{\mathrm{pot}},\end{eqnarray} \tag{ 9 }$$

where ${{\boldsymbol{B}}}_{\mathrm{obs}}$ and ${{\boldsymbol{B}}}_{\mathrm{pot}}$ are the transverse component of the observational and potential fields, respectively. We increased $\gamma =\gamma +d\gamma$ when $\int | {\boldsymbol{j}}\times {\boldsymbol{B}}{| }^{2}{dV}$ was reduced below a critical value. In this study, we set $d\gamma =0.1$ and when γ became equal to 1, ${{\boldsymbol{B}}}_{\mathrm{BC}}$ was consistent with the observed field.

Spatial derivatives were calculated by integrating the second-order central differences and temporal derivatives were integrated by the Runge–Kutta–Gill method to fourth-order accuracy. We used the combined SP and HMI data at the boundary condition on the bottom, which is shown in Figure 3(b). The numerical domain was set to (0, 0, 0) < (x, y, z) < (1.5, 1.0, 0.5), which was resolved by 540 × 360 × 180 nodes. The relaxation for each map was computed using 144 processors of the JAXA Supercomputer System generation 2 (JSS2) and the computation time was approximately 5 hr. To visualize the field lines, we used the UCAR's Vapor 3D visualization package (www.vapor.ucar.edu).

In this study, we attempted to detect the formation of a flare-productive 3D magnetic field structure, i.e., the magnetic field configuration, in which the magnetic reconnection is likely to occur. A necessary and sufficient condition for reconnection in 3D is the existence of a thin current layer, in which the ideal MHD breaks down. Priest & Démoulin (1995) proposed a model of reconnection in the absence of null points. They suggested that the current layer is formed in quasi-separatrix layers (QSLs), in which the gradient of field-line linkage is steep but continuous. To define QSLs, the norm N has been proposed (Priest & Démoulin 1995; Demoulin et al. 1996), which is,

$$\begin{eqnarray}&&N=\sqrt{\left[{\left(\displaystyle \frac{\partial X}{\partial x}\right)}^{2}+{\left(\displaystyle \frac{\partial X}{\partial y}\right)}^{2}+{\left(\displaystyle \frac{\partial Y}{\partial x}\right)}^{2}+{\left(\displaystyle \frac{\partial Y}{\partial y}\right)}^{2}\right]},\end{eqnarray} \tag{ 10 }$$

where (X, Y) is another footpoint of the field line connecting from a selected footpoint at (x, y). N is the norm of the displacement gradient tensor and is evaluated only at the boundary. They claimed that QSLs exist at the locations of high N. N is non-dimensional and independent of the coordinate system. However, the norm has a problem in defining QSLs. If we calculate the norm at different footpoints of the same field line, the values are generally different, making the determination of the QSLs ambiguous. To solve this problem, a new definition of QSLs called squashing factor Q was proposed by Titov (1999) and improved in later studies (Titov et al. 2002; Titov 2007; Pariat & Démoulin 2012). The squashing factor Q is defined as

$$\begin{eqnarray}&&Q=\displaystyle \frac{{N}^{2}}{{\rm{\Delta }}},\end{eqnarray} \tag{ 11 }$$

where the Jacobian matrix D of the field-line mapping and its determinant Δ are

$$\begin{eqnarray}D=\left(\begin{array}{cc}\partial X/\partial x & \partial X/\partial y\\ \partial Y/\partial x & \partial Y/\partial y\end{array}\right)=\left(\begin{array}{cc}a & b\\ c & d\end{array}\right),\end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray}&&{\rm{\Delta }}={ad}-{bc}.\end{eqnarray} \tag{ 13 }$$

The squashing factor Q is defined by the product of the norms ${N}_{+}$ and ${N}_{-}$, which are calculated from each footpoints of the same field line. There is a relation between ${N}_{+}$ and ${N}_{-}$,

$$\begin{eqnarray}&&{N}_{+}=N,{N}_{-}=\displaystyle \frac{N}{{\rm{\Delta }}}.\end{eqnarray} \tag{ 14 }$$

Therefore, Q values calculated from the footpoints of the same field line are equal. Titov et al. (2002) showed that the determinant of the Jacobian matrix (Δ) is related to the ratio of the normal component of the magnetic field at the boundary,

$$\begin{eqnarray}&&| {\rm{\Delta }}| =\displaystyle \frac{| {B}_{z,+}| }{| {B}_{z,-}| }.\end{eqnarray} \tag{ 15 }$$

Although the squashing factor Q is originally defined only at the boundary, Pariat & Démoulin (2012) extended it to a 3D domain assuming that the squashing factor is invariant along a field line.

By using Equations (11) and (15), we calculated the squashing factor in a 3D domain using the results of the force-free modeling. We integrated the field lines using the fourth-order Runge–Kutta method and chose the step size by adaptive step-size control.

The EUV 131 Å images obtained by SDO/AIA for the three flares are shown in the top left of Figures 4–6, which provide the spatial distribution of hot coronal plasmas immediately after the main phase of each flare. The time plots of the GOES X-ray flux (bottom right) are shown on the bottom right of Figures 4–6.

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When the flares occurred, flare ribbons appeared in the chromosphere. The AIA 1600 Å images were used to identify the flare ribbons. The upper right panel of Figures 4–6 show an AIA 1600 Å image in the initial phase of each flare. When flare 1 occurred, four flare ribbons appeared in response to the appearance of hot coronal plasmas. In Figure 4, we can identify the localized kernels of the flare ribbons located at two sunspots (P1, N1) and bright features located in the plage region (P2, N2). After the main phase, the P1-N2 and P2-N1 loops appeared with the pre-existing P1-N1 loops for all the three events in AIA 131 Å. The significant changes in the connectivity of EUV bright structures near the time of the main phase suggest the magnetic reconnection in the coronal magnetic structures formed above the quadrupole magnetic polarity distribution. Figures 5 and 6 show flare ribbons similar to those of flare 1 in 1600 Å, which indicates that they are characterized as homologous flares. Both flare 2 and flare 3 show four flare ribbons identified in the main phase.

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We performed NLFFF modeling using the method described in the previous section. First, we examined the consistency of the NLFFF modeling by comparing the footpoint connectivity of the field lines with the location of the flare ribbons observed by AIA 1600 Å. The force-free field modeling revealed four sets of connectivities in the magnetic field lines that originated in the four flare ribbons. One of the results of NLFFF modeling is shown in Figure 7; it is based on the SP data obtained between 07:12 and 08:07 UT on 2014 February 2, approximately one hour before the occurrence of flare 1. The background map shows the vertical component of the magnetic field. The field lines are drawn from randomly selected points in each region. The blue, green, red, and yellow lines describe the magnetic field lines for the P1-N1, P1-N2, P2-N2, and P2-N1, respectively. The gray arrow in the upper panel shows the direction of view in the bottom panel.

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Because flare ribbons result from transient heating at the chromospheric footpoints of the magnetic field lines associated with magnetic reconnection, the field lines from one flare ribbon could connect to another flare ribbon. By comparing the two locations, we can confirm the reliability of the NLFFF modeling (Inoue et al. 2011). In Figure 8, the flare ribbons of flare 1 observed by AIA 1600 are shown by the green contours on the vertical magnetic field map. The light blue filled area shows the footpoint of the magnetic field lines connecting from the magenta region. Similarly, the red filled area shows the footpoint of the magnetic field lines connecting from the yellow region. The top panel shows a result of potential field modeling, whereas the bottom panel displays that of NLFFF modeling. Clearly, in potential field modeling, the field lines extending from the flare ribbons (magenta, yellow) do not connect to the flare ribbon located in the negative sunspot (N1). On the other hand, in NLFFF modeling, the connectivity of field lines seems more consistent with the location and shape of the observed flare ribbon in the N1 region, though the red footpoints in the N1 region are slightly deviated from the location of the flare ribbon in the N1 region. This result suggests that the NLFFF modeling performed in this study can reproduce the connectivity of the field lines very well.

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We analyzed the temporal evolution of the NLFFF based on the time series of the SP map. The energy for producing solar flares, i.e., the free energy, can be calculated from the results of the NLFFF modeling and is defined as

$$\begin{eqnarray}&&{E}_{\mathrm{free}}=\int \displaystyle \frac{{B}_{\mathrm{nlfff}}^{2}-{B}_{\mathrm{potential}}^{2}}{8\pi }{dV}.\end{eqnarray} \tag{ 16 }$$

Figure 9 shows the temporal evolution of the free energy density distribution, which is averaged along the vertical direction (z-axis) at each position. The FOV shown in Figure 9 is indicated by the red rectangle in Figure 3. In the region between N1 and P2, which is enclosed in the yellow rectangle, the free energy was increased between 10:42UT and 16:54UT on 2014 February 1. The free energy integrated in the yellow rectangle at 16:54UT on 2014 February 1 was $4.7\times {10}^{31}$ erg, which is sufficient to produce an M-class flare. Subsequently, the free energy exhibited only a slight increase (∼50% at most), though we cannot confirm whether these changes were more significant than the error bar. Possible sources of errors include the error from the observations, the inversion of the spectropolarimetric data, the spatial resolution of the boundary data, and the NLFFF calculation method. Although the evaluation of such errors is important, it is difficult to consider all possible errors. This paper does not provide the uncertainty in the free energy. Because the uncertainty should be evaluated, the evaluation of the errors will be reported in our subsequent paper. A decrease was also observed between 02:20UT and 07:12UT on 2014 February 2, as shown by the white rectangle in Figure 9. In summary, the present data show that the largest part of the free energy required for the M-class flares was stored more than 10 hr before the occurrence of the first M-class flare. The stored free energy was localized around the P2-N1 region and it was not distributed in the entire quadrupole field structure.

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Figure 10 shows maps of the absolute values of the electric current distribution at 10:42UT and 16:54UT on 2014 February 1, which were averaged along the z-axis at each position. The FOV is the same as that of Figure 9. The free energy results from the magnetic fields generated by the electric currents. The electric currents at a given point can create free energy everywhere and it is straightforward to interpret the maps of electric current as the difference from the potential configuration. However, the electric current in Figure 10 was also concentrated in the area inside the yellow rectangle simultaneously with the free energy in Figure 9. Therefore, in this case, we can deduce that the free energy density maps of Figure 9 also contain the information of the distribution of the non-potential field configuration.

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Figure 11(a) shows the temporal evolution of the QSLs on a photospheric vertical magnetic field. The FOV is the same as that of Figure 9. The red contours indicate the location of the QSLs ($\mathrm{log}Q\gt 1$) in the lower chromosphere (∼400 km above the formation height of the Fe i line). To compare the location between the QSLs and flare ribbons, we have plotted the QSLs in the lower atmosphere. The QSLs in the N1 region that are enclosed by the yellow square show clear temporal changes. New QSLs (QSL1 and QSL2, indicated by the yellow arrows in Figure 11(a)) appeared by 23:19 UT on 2014 February 1, gradually became longer, and were eventually connected to each other by 07:20 UT, one hour before the occurrence of flare 1. The flare ribbons observed by AIA 1600 Å, shown by green contours in the 07:12 UT frame, are located along QSL1 and QSL2. Note that the good correspondence between QSLs and flare ribbons has been reported in previous studies (Savcheva et al. 2015; Inoue et al. 2016). After flares 1 and 2, the QSLs in the N1 region do not show significant temporal evolution. Even after flare 3, the QSLs remained in the N1 region. For comparison, Figure 11(b) shows the location of the QSLs at 07:12 UT on 2014 February 2, which was derived from the potential field calculation. The QSLs in N1 cannot be observed in the potential field.

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Figure 12 shows the temporal evolution of magnetic field lines P1-N1 (blue lines) and P2-N1 (yellow lines) near the formation time of the new QSLs in N1. At 10:42 UT and 16:54 UT on 2014 February 1, the positive footpoints of the P1-N1 and P2-N2 lines are continuously distributed and aligned with each other. At 23:19 UT on February 1 and 02:20 UT on 2014 February 2, the positive footpoints of the P2-N1 lines are clearly separated from those of the P1-N1 lines, creating a QSL between them.

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Figure 13 shows the temporal evolution of the photospheric magnetic field distribution around the P2-N1 region. The FOV is indicated by the blue rectangle in Figure 3. The blue contours represent the positive-polarity patches with the magnetic flux density of 300 G. Figure 14 shows the temporal evolution of the vertical positive magnetic flux (red solid line) and the surface area (blue solid line) measured with the blue contours of Figure 13. The vertical magnetic field flux increases by a factor of two and the surface area shows a similar increase. This increase indicates an emergence of magnetic flux in the P2-N1 region.

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Although the free energy clearly increased between 10:42 UT and 16:54 UT on 2014 February 1, the vertical magnetic fluxes in Figures 13 and 14 show a slight increase during that time. To discuss the origin of free energy, we present Figure 15, which shows the absolute value of the horizontal component of the magnetic flux density $(=\sqrt{({B}_{x}^{2}+{B}_{y}^{2})})$ in the P2-N1 region in the photosphere at 10:42 UT and 16:54 UT on 2014 February 1. The FOV is indicated by the blue rectangle in Figure 3, which is the same as that of Figure 13. An increase of the horizontal component can be seen in the black closed curve, which corresponds to the location where free energy increased.

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The P2 region contains two groups of positive flux, P2a and P2b, which are shown by the yellow and orange circles in Figure 13. In the N1 region, an elongated negative-polarity field (N1a), enclosed by the white circle, developed between 23:19 UT on 2014 February 1 and 07:12 UT on 2014 February 2. The N1a patch is a negative-polarity counterpart of the emerging flux with the positive-polarity P2b patch. Figure 16 shows the temporal evolution of the magnetic field lines derived from NLFFF modeling in the area containing N1a, P2a, and P2b. The magnetic field lines are plotted on the vertical magnetic field map, and the P2a-N1 and P2b-N1 field lines are drawn in yellow and orange, respectively. The P2b lines were initially connected to N1a in the period between 23:19 UT on 2014 February 1 and 02:20 UT on 2014 February 2. However, the field lines connecting from the N1a region changed from P2b (orange lines) to P2a (yellow lines) at 07:12 UT on 2014 February 2. This transition may require the magnetic reconnection between P2a-N1 lines and P2b-N1 lines.

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