As mentioned previously, the efficacy of photospheric magnetic field observations in studies looking for pre-eruptive signatures of solar flare occurrence is limited due to the fact that the eruption occurs higher in the solar atmosphere in the corona. Though vector magnetic field observations are not available for the corona as they are for the photosphere, modeling of the coronal magnetic field provides an effective approach to study the state of the field during solar flare events and potentially find such signatures. Coronal magnetic field models use the photospheric magnetic field measurements as a boundary condition to model the corona. The four main types of coronal models are the Magnetohydrodynamics (MHD) model, the Magnetohydrostatic (MHS) model, the Force-Free models, and the Potential-Field Source-Surface (PFSS) model. Table 1 lists these models in order of most to least complex, as well as some of the key physical effects neglected. In the present study, the coronal magnetic field was modeled using NLFFF extrapolation; thus, the focus of this section will be on Force-Free modeling.

The photospheric vector magnetic field used as a boundary condition for extrapolation was generated following the same procedure as (Yurchyshyn et al., 2022). Specifically, using the same cylindrical equal area (CEA) projection used to produce the hmi.sharp_cea_720s series data (Bobra et al., 2014), HMI magnetic field measurements rebinned to 1 Mm pixel scale were transformed to a local Cartesian coordinate system. A reference point centered on the base of the Cartesian box is used in the transformation, ensuring that the base maps are projected onto their LoS counterparts with minimum distortion (Yurchyshyn et al., 2022). The center of the data cube—i.e., the Cartesian box—was set to the HARP center coordinates, and the size of the box was 400 × 200 × 200 for x, y, z pixels with z in the vertical. The relatively large horizontal extent was to ensure that the extropolated closed-field configurations over the AR would be fully captured. As previously mentioned, NLFFF extrapolation was performed using the weighted optimization code from Fleishman et al. (2017).

This procedure—from HMI data retrieval and coordinate transformation through NLFFF extrapolation—is brought to fruition numerically as part of the GX Simulator package (Nita et al., 2015; Nita et al., 2018). The package is freely available from the SolarSoft IDL library. The GX Simulator was used to create the [400,200,200] box structures containing the extrapolated magnetic field from the HMI field measurements for a time period of 1 h before through 1 h after the flare start time - in this case 2012-03-06 23:00Z to 2012-03-07 01:00Z. Given the 12-min cadence for the HMI dataset, this meant 11 coronal magnetic field extrapolations were generated. A schematic of the NLFFF extrapolated magnetic field for NOAA AR 11429 at 00:00UT on 7 March 2012 (the closest HARP time to the X5.4 flare start time) produced using the GX Simulator is shown in Figure 1.

The SDO Advanced Imaging Assembly (AIA) (Lemen et al., 2012) provides nearly simultaneous full-disk images of the corona and transition region—up to 0.5 solar radii above the solar limb—from 10 different wavelength channels with 1.5 arcsec spatial resolution and 12 s temporal resolution. The AIA 171 Å channel captures emissions from Fe ix ions at temperatures around 0.6 MK, showing the quiet corona and upper transition region. The AIA 211 Å channel captures emissions from Fe xiv ions at temperatures around 2 MK, showing the active region corona. The AIA 1600 Å channel captures emissions from C iv ions at temperatures around 10,000 K, showing the upper photosphere and lower transition region. Areas of flare brightening were determined by the overlap of the brightest areas seen in the SDO/AIA 171 Å, 211 Å, and 1,600 Å imagery. To find such areas, imagery from the three AIA channels—for the same field-of-view (FOV) as that of the magnetogram—were overlaid using the JHelioviewer tool (Müller et al., 2017; see Figure 2). The 1,600 Å imagery was set to the red channel; the 171 Å imagery was set to the green channel; and the 211 Å imagery was set to the blue channel for the overlaid image. Pixels (within the same boxed area shown in Figure 2) that had RGB values greater than or equal to 250—i.e., overlapping bright pixels from the three AIA images—identified the areas of flare brightening. Figure 3 illustrates the evolution of the cumulative area of flare brightening from the start of the X5.4 flare through 1 hour after. The cumulative area is very similar to the cumulative flare ribbon from Kazachenko et al. (2017). The alignment between areas of flare brightening and regions of strong surface magnetic field strength and high twist was compared to test the predictive capability of such regions in identifying flare location.

The following regions were identified and used as masks to generate separate data volumes for magnetic field parameter calculations.

1) Active Region—AR

The OR was defined as areas within the same boxed portion as before, with the observed surface (or photospheric) |B| ≥ 1000 G and |T _w|≥ 1.5 turn, where T _w is the twist number of individual field lines calculated using the code developed by Liu et al. (2016). The magnetic field strength threshold is the same as that used in Kusano et al. (2020). The threshold for |T _w| is a common threshold used and considered appropriate to define moderate to high twist. The purpose of the OR was to isolate the portions of the magnetic field involved with the flaring process, as well as investigate if the observed photospheric measurements of field strength and twist can be used to identify the location of flaring within an AR. Figure 5 suggests that the OR aligns quite well with the initial areas of brightening seen in AIA imagery.

The 00:00UT AR and OR—i.e., the AR and OR at the time closest to when the flare was triggered—were used to determine magnetic field line seed locations—i.e., the locations of the photospheric footprints of the field lines. From these seed locations, the field lines within the [400,200,200] box were traced using linear interpolation, as part of the GX Simulator package (Nita et al., 2015; Nita et al., 2018). The coordinates of the field lines identify the volume elements used to calculate the magnetic field parameters—i.e., the data volumes. Data volumes were generated (using the specified seed locations) for each of the 11 extrapolated magnetic fields throughout the time period around the solar flare event. Figure 6, 7 shows the 00:00UT OR data volume.

The magnetic field parameters calculated from the data volumes include four Space weather HMI Active Region Patch (SHARP) parameters—total unsigned flux (ϕ), total unsigned vertical current ( J z Total ) , total unsigned current helicity ( H c Total ) , and absolute value of the net current helicity (Hc_Abs)—as well as the total vertical magnetic field energy ( E B Z ) , total vertical field gradient ( ∇ B z Total ) , and total unsigned magnetic twist flux (τ _Total). The description, units, and formulas for the seven magnetic field parameters are listed in Table 2. The summations for the parameters occur over the individual volume elements within the different data volumes, with the exception of ϕ which is the surface flux. The SHARP parameters investigated were selected due to the potential predictive capability of these parameters found by Bobra and Couvidat (2015). The inclusion of the parameter τ is motivated by the predictive capability of the parameter found by Kusano et al. (2020).

To effectively compare the solar flare events and changes relative to flare occurrence, all parameters were normalized to the value at the flare start time. Time series of every parameter for each region (AR and OR) were plotted using the normalized values. Additionally, the error in each parameter was estimated using the same error analysis used for the total-based SHARP parameters (Bobra et al., 2014).

While α and τ provide a quantification of the overall twist calculated from the field values, it is important to examine the actual twist structure and how it evolves during the solar flare event. To accomplish this, twist maps—like that of Liu et al. (2016) and Yurchyshyn et al. (2022)—were calculated at vertical planes across and along the densest area of the OR and flare brightening, as well as in an area of weak magnetic field for reference (see Figure 8). For this flare event the area aligned well with the PIL, implying the eruption occurred directly over the PIL as opposed to the sliding eruption seen by Yurchyshyn et al. (2022). A highly twisted structure—likely a MFR—is evident in Figure 9 (shown by dark-blue areas of high twist) over the inversion line. The quantification and evolution of the twist structure was analyzed using the same sign-singularity measure (SSM) as that of Yurchyshyn et al. (2022). The SSM is calculated by: S S M r = ∑ L i r | μ i r | , (4) where, μ i r = ∫ L i r U x , y d x d y ∫ L i R U x , y d x d y . (5) U (x, y) is the studied signed parameter (in this case T _w ), and L _i (r) ⊂ L _i (R) represents a unique hierarchy of disjoint squares of size r, which cover the whole square L of size R that bounds the area of interest (Yurchyshyn et al., 2022).

As mentioned in the previous section, the area within the AR with observed surface |B| ≥ 1000 G and |T _w|≥ 1.5 turn—i.e., the OR—aligns well with the area of initial brightening in SDO/AIA 171 Å, 211 Å, and 1,600 Å imagery during the onset of flaring (Figure 5). During the first couple minutes following flare initiation (from 00:02UT through 00:06UT), 45% of the bright pixels from the AIA imagery are encapsulated by or within 2 pixels of the OR; 67% are encapsulated or within 5 pixels; and 78% are encapsulated or within 10 pixels. As the X-ray flux quickly increases during the eruptive phase and pixels in the AIA imagery become saturated from the eruption, the area of brightening significantly outgrows the OR. However, the alignment throughout the pre-flare phase and beginning of the eruptive phase suggests that measurements of the photospheric magnetic field strength along with the twist calculated from the field measurements has the potential to provide a good indication of the flare location within an AR if triggered.

Figure 10 shows the vertical twist maps at each timestep from 1 hour before through 1 hour after the X5.4 solar flare, as well as the X-ray flux measured by GOES during this time window. It is evident from the figure that a MFR - defined as areas with |T _w|≥ 1 - was present in the pre-eruptive magnetic field configuration, decayed slightly between the start and max of the flare, then decayed significantly following the peak and release of the associated CME. As the MFR decays following the eruption, twist higher in the corona is enhanced. Figure 11 shows the sign-singularity measure curves calculated from the vertical twist maps, as well as the time series of the total sign-singularity measure, which was calculated as the area under the different curves. The time series provides a quantification of the twist structure evolution visualized in Figure 10. The initial decrease in the total sign-singularity measure indicates the MFR becoming more coherent right up until the eruption, at which point an expected increase in the measure is seen as a result of the decaying MFR and overall fragmentation of the twist structure (Yurchyshyn et al., 2022).

Figure 12 shows the temporal evolution of the seven investigated magnetic field parameters calculated from the extrapolated magnetic field. As mentioned in Section 3.4, for each parameter the values have been normalized to the value at the HARP data time nearest the start time of the solar flare—in this case 2012-03-07 00:00UT.

The AR experiences a fairly consistent increase of about 2% in ϕ, with a peak flux just before the start of the flare. Conversely, the OR experiences a decrease of about 8% in ϕ, with a minimum just before the start of the flare and a fairly consistent increase throughout the duration of the event. The AR also experiences a peak in E B z just before the start of the flare, followed by a rapid decrease of 6% just after the start. The OR experiences a maximum in E B z 24 min before the flare start, then an overall decrease of approximately 28% through the maxtime of the flare. The peak in ϕ and E B z for the OR 24 min prior to the start of the flare is likely the result of field lines in close proximity to the MFR—that initially were part of the strapping field—being stretched and sheared until magnetic reconnection occurred, causing them to wrap around the axis of the MFR. As all these field lines become part of the MFR, winding along the full length of it, the once more vertically oriented loop structures become elongated in the horizontal. The rest of the strapping field—with footprints further from the axis of the MFR—continue to be stretched in the vertical, hence the steady increase in the parameters for the AR up to a maximum just before the flare start. Once the instability of the magnetic field reaches the point to trigger the eruption, energy is lost and the field near the point of eruption begins to relax to a less stressed state. The AR field directly over the decaying MFR is no longer stretched as extensively, and the OR field re-configures to more simple loop structures.

The OR experiences a 20%–25% reduction in ∇ B z Total , J z Total , H c Total , and H c Abs , and a 50% reduction in τ _Total after the eruption, with post-flare values remaining lower than pre-flare values. The AR experiences only a 8%–12% reduction in ∇ B z Total , J z Total , H c Total , and H c Abs , and a 20% reduction in τ _Total. Furthermore, throughout the remaining duration of the flare event, the AR sees an increase in all the of the parameters to values around or just above the pre-flare values. The increase to pre-flare values seen for AR but not the OR suggests that while the portions of the magnetic field involved in the flaring process (OR) relax to a less stressed state following the onset of the solar flare, the eruptive motions can enhance the complexity of the magnetic field over other sections of the AR further from the point of eruption, consistent with the findings of Verma (2018) and Garland et al. (2022).