EXTRAPOLATION OF THE SOLAR CORONAL MAGNETIC FIELD FROM SDO/HMI MAGNETOGRAM BY A CESE–MHD–NLFFF CODE

The HMI on board the SDO provides photospheric vector magnetograms with a high resolution both in space and time. It observes the full Sun with a 4 K × 4 K CCD whose spatial sampling is 0.5 arcsec pixel⁻¹. Raw filtergrams are obtained at six different wavelengths and six polarization states in the Fe i 6173 Å absorption line, and are collected and converted into observable quantities (such as Dopplergrams, continuum filtergrams, and line-of-sight and vector magnetograms) on a rapid time cadence. For the vector magnetic data, each set of filtergrams takes 135 s to be completed. To obtain vector magnetograms, Stokes parameters are first derived from filtergrams observed over a 12 minute interval and then inverted through the Very Fast Inversion of the Stokes Vector (Borrero et al. 2011). The 180° azimuthal ambiguity in the transverse field is resolved by an improved version of the "minimum energy" algorithm (Leka et al. 2009). Regions of interest with strong magnetic fields are automatically identified near real time (Turmon et al. 2010). A detailed description on how the vector magnetograms are produced can be found on the Web site http://jsoc.stanford.edu/jsocwiki/VectorPaper.

The magnetogram data we use here is downloaded from http://jsoc.stanford.edu/jsocwiki/VectorPaper, where the HMI vector magnetic field data series hmi.B_720s_e15w1332 are released for several ARs. There are two special formats, i.e., direct cutouts and remapped images. We use the remapped format which is more suitable for modeling in local Cartesian coordinates, since the images are computed with a Lambert cylindrical equal area projection centered on the tracked region. For our test, we select two ARs, AR 11158 and AR 11283, both of which produced X-class flares and were very non-potential. The full resolution of the data is about 05 pixel⁻¹ and we rebin them to 1'' pixel⁻¹ for the NLFFF modeling.

AR 11158 is a well-known target studied in many recent works for different purposes (e.g., Schrijver et al. 2011; Sun et al. 2012b; Liu et al. 2012; Jing et al. 2012), and was selected by Wiegelmann et al. (2012) for a special test on optimizing their extrapolation code with HMI data. This AR has a multipolar and complex structure. It produced several major flares and coronal mass ejections (CMEs) during its disk passage, including the first X-class flare of cycle 24 on 2012 February 15. Figure 2 (the top panel) shows a vector magnetogram of AR 11158 taken at 20:36 UT on 2011 February 14, which will be used for our computation. This AR is well isolated from others and is almost flux-balanced (see Table 1), with the main polarities concentrated in the central field of view (FoV). As can be seen, the field shows a strong shearing along the polarity inversion lines (PILs).

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Table 1. Quality of the Magnetograms

Data	flux	force	torque	Sx	Sy	Sz
AR 11158
Raw map	0.017	0.072	0.073	5.05E-03	6.12E-03	1.05E-03
Preprocessed map	0.019	0.002	0.002	7.83E-05	1.01E-04	8.74E-05
Numerical potential	0.019	0.001	0.002	8.78E-05	8.42E-05	8.74E-05
AR 11283
Raw map	−0.111	0.193	0.143	1.12E-02	1.15E-02	1.94E-03
Preprocessed map	−0.119	0.012	0.017	1.73E-04	1.95E-04	1.79E-04
Numerical potential	−0.119	0.011	0.017	1.98E-04	1.63E-04	1.79E-04

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The other AR, AR 11283, is also very eruptive, generating several X-class flares and CMEs. We select a magnetogram taken at 22:00 UT on September 6, just prior to a major flare at 22:20 UT. The magnetogram is shown in the bottom panel of Figure 2. As an input for extrapolation, this data is not as good as AR 11158's, since the flux is dispersed with some strong polarities almost on the edge of the FoV. In addition, the total flux is not well balanced (see Table 1), which is unfavorable for our extrapolation.

Generally, the raw magnetogram cannot be inputted directly into the NLFFF code because the photospheric field is intrinsically non-force-free and violates the force-free assumption. According to the derivation by Molodenskii (1969) and Aly (1989), an ideally force-free magnetogram should fulfill the following conditions:

$$\begin{eqnarray} \int _{S} B_{z} {\ d} x {\ d} y = 0,\quad F_{x} = \int _{S}B_{x}B_{z} {\ d} x {\ d} y &=& 0,\nonumber \\ F_{y} = \int _{S}B_{y}B_{z} {\ d} x {\ d} y = 0,\quad F_{z} = \int _{S}E_{B} {\ d} x {\ d} y &=& 0,\nonumber \\ T_{x} = \int _{S}yE_{B} {\ d} x {\ d} y = 0,\quad T_{y} = \int _{S}xE_{B} {\ d} x {\ d} y &=& 0, \nonumber \\ T_{z} = \int _{S}(yB_{x}B_{z}-xB_{y}B_{z}) {\ d} x {\ d} y &=& 0 \end{eqnarray} \tag{ 3 }$$

where $E_{B} = B_{x}^{2}+B_{y}^{2}-B_{z}^{2}$. These expressions are derived from the volume integrals of the total divergence, magnetic force, and torque of an ideally force-free field

$$\begin{equation} \int _{V} \nabla \cdot \mathbf {B} {\ d} V = 0, \ \ \int _{V} {\mathbf {j}} \times {\mathbf {B}} {\ d} V = {\mathbf {0}},\ \ \int _{V} {\mathbf {r}} \times ({\mathbf {j}} \times {\mathbf {B}}) {\ d} V = {\mathbf {0}} \end{equation} \tag{ 4 }$$

(by using Gauss' divergence theorem, the volume integrals 4 can be transformed to the surface integrals 3). In the case of the coronal field, the surface integrals of Equation (3) are usually restricted within the bottom magnetogram since the contribution from other boundaries can be neglected. To assess the real data with respect to the force-free condition, three parameters are usually computed as (Wiegelmann et al. 2006)

$$\begin{eqnarray} \epsilon _{\rm flux} &=& \frac{\int _{S} B_{z} {\ d} x {\ d} y}{\int _{S} |B_{z}| {\ d} x {\ d} y},\ \ \epsilon _{\rm force} = \frac{|F_{x}|+|F_{y}|+|F_{z}|}{\int _{S} P_{B} {\ d} x {\ d} y},\nonumber \\ \epsilon _{\rm torque} &=& \frac{|T_{x}+|T_{y}|+|T_{z}|}{\int _{S} \sqrt{x^{2}+y^{2}}P_{B} {\ d} x {\ d} y}, \end{eqnarray} \tag{ 5 }$$

where $P_{B} = B_{x}^{2}+B_{y}^{2}+B_{z}^{2}$. Small values of these quantities, e.g., _flux, _force, _torque ≪ 1 indicate a good input for the NLFFF modeling. Table 1 shows that for AR 11158, the force-free condition is satisfied quite well with flux almost balanced and _force, _torque less than 0.1; however, for AR 11283, it is worse. Note that the flux non-balance will have a negative effect on the extrapolation (see Section 4).

Besides being non-force-free, the observed data contains measurement noise which is also unfavorable for practical implementation of extrapolation. To this end, preprocessing of the raw magnetogram has been proposed by Wiegelmann et al. (2006) to remove the force and noise for providing better input for NLFFF modeling. To be consistent with our extrapolation code using a magnetic splitting form, we recently developed a new code for magnetogram preprocessing (C. W. Jiang & X. S. Feng 2013, in preparation), in which the vector magnetogram is also split into two parts for a potential field and a non-potential field; we deal with the two parts separately. Preprocessing of the potential part is simply performed by taking the data sliced at a plane about 400 km above the photosphere² from the 3D potential field which is extrapolated from the observed vertical field. Then the non-potential part is modified and smoothed by an optimization method to fulfill the constraints of total magnetic force-freeness and torque-freeness, which is similar to the method proposed by Wiegelmann et al. (2006). We have paid particular attention to what extent the force needs to be removed and the data can be smoothed. As for practical computation based on numerical discretization, an accurate satisfaction of force-free constraints is apparently not necessary. Also the extent of the smoothing for the data must be carefully determined in order to mimic the expansion of the magnetic field from the photosphere to some specific height. We use the force-free and smooth values calculated from the preprocessed potential-field part as a reference to guide the preprocessing of the non-potential field part, i.e., we require that the target magnetogram has the same force-free and smoothness levels as its potential part. These requirements can well restrict the free parameters, i.e., the weighted factors in the optimization function.

The results of preprocessed data are also given in Table 1 together with results of its numerical potential-field part. For both magnetograms, the preprocessing reduces the parameters _force and _torque by more than one order of magnitude, making the residual force around the level of numerical error (i.e., the parameters are close to those of the numerical potential field). The parameters S_x, S_y, S_z in the table measure the smoothness of the components B_m (m = x, y, z), which are defined by (see C. W. Jiang & X. S. Feng 2013, in preparation)

$$\begin{equation} S_{m} = \sum _{\rm p}[(\Delta B_{m})^{2}]/\sum _{\rm p}[(\overline{\Delta } B_{m})^{2}] \end{equation} \tag{ 6 }$$

where the summation ∑_p is over all the pixels of the magnetogram, and Δ is a typical five-point two-dimensional-Laplace operator, i.e., for pixel (i, j)

$$\begin{eqnarray} \Delta B_{i,j} &\equiv& B_{i+1,j}+B_{i-1,j}+B_{i,j+1}+B_{i,j-1}-4B_{i,j},\nonumber \\ \overline{\Delta }B_{i,j} &\equiv& B_{i+1,j}+B_{i-1,j}+B_{i,j+1}+B_{i,j-1}+4B_{i,j}. \end{eqnarray} \tag{ 7 }$$

As shown, the smoothness of the preprocessed data is very close to those of their potential parts, and is consistent among three components, which is unlike the raw data with very different smoothness for different components. Smoothing of the data can be clearly seen by comparing the raw and preprocessed magnetograms as shown in Figures 3 and 4, and especially in the J_z map which shows that the random noise is suppressed effectively (but not entirely).

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Note that the constraints of Equation (3) are only necessary conditions, but not sufficient for an ideally force-free magnetogram, meaning that the magnetogram may still contain force even when these conditions are satisfied. However, we can say that the preprocessed magnetogram is more suitable for NLFFF modeling than the raw data, but it is not a completely consistent boundary condition.

The observed coronal loops in X-ray and EUV images give us a proxy of the magnetic field line geometry and are thus a good constraint for the magnetic field model besides the vector magnetograms (e.g., Aschwanden et al. 2012; Malanushenko et al. 2012). In Figure 5 we compare the extrapolated field with coronal loops observed by SDO/AIA in the wavelength of 171 Å, in which the loops are the most visible compared to other channels. The field lines are traced from the photosphere at locations selected roughly according to the footpoints of the visible bright loops, and are rendered with different colors for clarity. The viewing angle of the field lines is aligned with the AIA image. We plot the field lines and the AIA image both alone and overlaid for better inspection of the result. As shown from an overview of the images, most of the extrapolated field lines closely resemble the observed loops. At the central region the field lines are strongly sheared along the major PIL and slightly twisted (as compared with the potential solution), indicating the existence of strong electric currents along the field lines. In the left panel of Figure 6, we plot an image of vertical integral of the current density in the extrapolation volume, and the strong-current regions are outlined by the boxes (labeled as A, B, and C). Note that the currents are strongly localized within the central region B and a smaller region C. The magnetic structures of the strong-current regions are shown in the right panel of Figure 6. As can be seen, the twist of the field lines in region C is much stronger than that in region B. The results support observational studies that show that a filament related to an X-class flare and CME exists in the core region B (Sun et al. 2012b), and there are small eruptions in region C due to flux emergence (Sun et al. 2012a).

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Nevertheless, we note that the misalignments between the modeling field lines and the observations are also obvious, especially for the large loops near the northwest boundary of the FoV. There are several reasons for the misalignments: first, a local Cartesian coordinate system is not adequate to include the large loops which obviously require spherical geometry; second, the FoV of magnetogram may not be large enough to properly characterize the entire relevant current system, which again requires the curvature of the Sun's surface to be taken into account; third, it is not easy to precisely locate the photospheric footpoints of different loops that spread apart distinctly in the corona but are rooted very closely in the photosphere; fourth, the coronal field may be rather dynamic, e.g., it expands or oscillates due to eruptions, which makes the static extrapolation fail. We know that this visual comparison between the model result and observation is very preliminary, and further critical comparison is required, for example, with the 3D loops reconstructed with multi-points observation (DeRosa et al. 2009; Aschwanden 2011).

Routinely, we check the quality of the numerical result by computing several metrics. The force-freeness of the extrapolation data is usually measured by a current-weighted sine metric (CWsin) defined by

$$\begin{equation} {\rm CWsin} \equiv \frac{\int _{V}{J}\sigma dV}{\int _{V}{J} dV}; \ \ \sigma = \frac{{ {\mathbf {J}} }\times { {\mathbf {B}} }}{{{JB}}}, \end{equation} \tag{ 8 }$$

where B = |B|, J = |J| and V is the computational volume. We also compute a current-square-weighted sine metric (C²Wsin) similarly defined by

$$\begin{equation} {\rm C^{2}Wsin} \equiv \frac{\int _{V}J^{2}\sigma dV}{\int _{V}J^{2} dV}, \end{equation} \tag{ 9 }$$

with more weight on the strong-current regions. The divergence-freeness is measured by 〈|f_i|〉

$$\begin{equation} \langle |f_{i}|\rangle = \frac{1}{V}\int _{V}\frac{\nabla \cdot \mathbf {B}}{6B/\Delta x} dV. \end{equation} \tag{ 10 }$$

We care about different energy contents, i.e., the total energy E_tot, the potential energy E_pot and the free energy E_free

$$\begin{equation} E_{\rm tot} = \int _{V} \frac{B^{2}}{8\pi } dV,\ \ E_{\rm pot} = \int _{V} \frac{B_{\rm pot}^{2}}{8\pi } dV,\ \ E_{\rm free} = E_{\rm tot}-E_{\rm pot}, \end{equation} \tag{ 11 }$$

where B_pot is the potential field strength. The results of the metrics are given in Table 2 for extrapolations from both the raw and preprocessed magnetograms. We compute the metrics for four different regions including the full extrapolation box and the subregions A, B, and C as outlined in Figure 6.

For the full region, our results of the current-weighted sine is ∼0.3, which means that the mean misalignment angle between the magnetic field and current is about 17°. This value is much larger than those from our previous benchmark tests using ideal or synthetic magnetograms (which are ∼0.1 (6°) or smaller; see Jiang & Feng 2012), but is comparable to previously reported results by other NLFFF codes on real magnetograms (e.g., the average CWsin by various NLFFF codes applied to AR 10930 (Schrijver et al. 2008) and AR 10953 (DeRosa et al. 2009) are 0.36 and 0.28, respectively). With such a large misalignment angle, the result seems to be far away from an exactly force-free solution which has a zero misalignment angle. However, it should be noted that the metric CWsin may not be a good monitor for numerical solutions, which unavoidably have random numerical errors because of limited resolution. As a simple example, CWsin is close to 1 even for a potential field solution computed by Green's function method or other numerical realization. This is because the numerical difference, used for computing the current J = ∇ × B from B, gives small but finite currents whose directions are randomly from 0° to 180°; thus an average of the full volume should give a misalignment angle of ∼90°. The noise in the observation data, mainly in the weak field regions, is also a major source for random numerical errors. In these regions, the actual magnetic elements are probably smaller than the observed or numerical pixel size, and the field directions generally exhibit a random pattern on the image. To reduce such errors in computing the metric, we can either increase the weight of the current (e.g., use C²Wsin) or compute CWsin within the strong-current subregions only. As is expected, the misalignment angle decreases significantly by measuring in this way. For the full region, C²Wsin are only half that of CWsin. For the subregions, CWsin is also only half or less for the full region, reaching the level of those from the benchmark tests (Jiang & Feng 2012). In particular, the misalignment angle is only about 4° in subregion B, showing that the force-free assumption is modeled very well.

Regarding the energy contents, it is interesting to note that the free energy of subregion A exceeds that of the full region, meaning that the free energy content in the full volume excluding subregion A is negative. This is, however, not surprising as we know that any sub-volume energy content of the non-potential field may be lower than the potential energy (e.g., Mackay et al. 2011; Jiang et al. 2012a). Also the measurement error may result in this negative free energy since it is very small compared to the total free energy. Regardless of which case is true, it can be clearly seen that the spatial distribution of free energy is largely co-spatial with that of the current, since the subregion A contains most of the currents of the entire volume. This confirms that the free energy in the corona is actually stored by the current-carrying field (where non-potentiality is strong), but not necessarily in the magnetic flux concentrations.

Finally, we compare the results extrapolated from the raw and preprocessed magnetograms. Inspecting the force-freeness and divergence-freeness metrics shows that the improvement by preprocessing is negligible. This is because the raw data already satisfies the boundary force-free conditions well. Due to the smoothing, the result for the preprocessed data gives slightly lower energy content than those for the raw data.

Figure 7 compares the AIA 171 Å loops with the reconstructed field in the same way as Figure 5. For this AR, the NLFFF model appears to perform only slightly better than the potential model (there are some loops even produced by the NLFFF model that are even worse than the potential model near the northeast boundary of the FoV). The clearest misalignment with the observation is the large closed loop pointed by the arrow in the AIA image. The potential and force-free models failed to recover this group of loops and give open field lines instead. This, however, is not unexpected since the inputted magnetogram has flux unbalanced by −10%. So there must be field lines from the negative polarity opening in the FoV. The reason for this flux unbalance may be that the positive flux in the east is rather dispersed (much more than the negative polarity), and thus properly be underestimated by the observation.

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Near the major polarity the structure of the loops is very complex and the extrapolated field shows highly sheared and twisted structures, indicating a significant non-potentiality there. Actually this was the site of the flare and filament eruption. The distribution of the vertically integrated current shows a strong concentration of current in this region, as denoted by A in Figure 8. In the same figure, we show the local field structure and the observations from different wavelengths of much higher temperature than 171 Å. The magnetic field exhibits a multi-flux rope configuration. The most remarkable structure is a sigmoid, i.e., the S-shaped loop in the AIA 94 Å and 335 Å images. The sigmoid can be seen most clearly in the AIA 94 Å wavelength (6.3 MK) with rather thin but enhanced shape, and is also well shaped in the soft X-ray image taken by Hinode/XRT. In the fourth panel of the figure the field lines are plotted overlying on the AIA 94 Å image. It demonstrates that our extrapolation has recovered the sigmoid rather precisely, at least in morphology (see the precise alignment of the field lines with the shape of the sigmoid). The distribution of the current also roughly resembles the shape of the sigmoid, suggesting that the enhancement of EUV and X-ray emission associated with the sigmoid is made by the strong field-aligned current via Joule heating of the plasma. This sigmoid is located between the major positive and negative polarities and the currents reside mostly in the northeast part, as shown by the current distribution. The twist of the sigmoid field lines is not strong as modeled in other cases such as Roussev et al. (2012) or Savcheva et al. (2012), and this sigmoid is composed of a single flux rope, which is also different from their results with two flux ropes or double-J shaped current pattern. The observation and modeling suggest that there seems to be another flux rope overlying the sigmoid, and the flare and CME may result from the eruptions of these flux ropes, which is left for future study.

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Similarly, we compute the metrics of force-freeness and divergence-freeness for both the full region and the subregion and the results are given in Table 3. By comparing the results using the raw and preprocessed data, we find that evidently the preprocessed result is closer to force free, especially of the full region for which the raw data gives CWsin ∼ 0.4 (24°) while the preprocessed data gives CWsin ∼ 0.3 (17°). Thus for this AR the preprocessing indeed improves the extrapolation greatly. Also the divergence is reduced by the preprocessing. It is noticeable that the total energy content is doubled by the preprocessing, reaching 10³² erg. However, even this improvement of the free energy is likely to underestimate the actual value, considering that a X-class flare and CME erupted immediately (Feng et al. 2013). Still, the current is strongly localized and the free energy is concentrated within the strong-current region, i.e., subregion A, which occupies only less than one percent of the full volume, but contains most of the free energy.

It is important to monitor the relaxation process in order to study whether the iteration converges, since there is no theory to guarantee this. Here we study the convergence process of the computations by temporal evolution of several monitors, including the residual of the field between two successive iterations

$$\begin{equation} {\rm res}^{n}({\mathbf {B}}) = \sqrt{\frac{1}{3}\sum _{\delta = x,y,z} \frac{\sum _{i}\left(B_{i\delta }^{n}-B_{i\delta }^{n-1}\right)^{2}}{\sum _{i}\left(B_{i\delta }^ {n}\right)^{2}}} \end{equation} \tag{ 12 }$$

(where n denotes the iteration step), the metric CWsin, and the total energy content. We record the residual by every ten steps and compute CWsin and total energy by every ten τ_A. The results for extrapolation of both ARs are plotted in Figure 9. As can be seen, the system converges smoothly and fast. During the first 10 τ_A, the residual continues to increase because the transverse field is inputted at the bottom continuously, which drives the system away from the initial potential field. After this driving process, the residual drops immediately, indicating a fast relaxation of the system. With about 40 τ_A (nearly 10000 iterations), the residual is already reduced to ∼10⁻⁵, and all the metrics and energy almost stagnate afterward. Thus the computations can actually be terminated once the residual is below 10⁻⁵, which is consistent with our previous studies for benchmark cases (Jiang & Feng 2012; Jiang et al. 2012b). It is also noteworthy that the convergence process is rather smooth, without any obvious oscillation or abrupt variation of the residual or the metrics, so the iteration is "safe." This is a good feature of our code over other iteration codes for extrapolation, e.g., the Valori et al.'s (2007) magnetofrictional code or the Wheatland's (2006) Grad-Rubin-like code, which usually show strong oscillatory in the iteration or even fail to converge occasionally (Schrijver et al. 2008; DeRosa et al. 2009).

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