ELECTRIC CURRENTS IN FLARE RIBBONS: OBSERVATIONS AND THREE-DIMENSIONAL STANDARD MODEL

The SDO, launched in 2010, has a full solar disk field of view and comprises three instruments. Among them, the HMI (Scherrer et al. 2012; Schou et al. 2012) measures the vector magnetic fields in the Fe i absorption line at 6173 Å. It has a spatial resolution of 1''with a 05 pixel size and a cadence of 45 s for the line-of-sight magnetic field. For the transverse magnetic field, the cadence is reduced to 12 minutes by averaging the four Stokes profiles before the inversion in order to enhance the signal-to-noise ratio. This time cadence still allows us to obtain the magnetograms just before the impulsive phase and after the peak of the flare.

The AIA (Lemen et al. 2012; Boerner et al. 2012) on board SDO has four identical telescopes providing high temporal (cadence of 12 s) and spatial resolution (15 with a 06 pixel size) in different filters, with nine in the EUV wavelengths and one in visible wavelengths. In Figure 1, a southwest portion of the Sun is shown during the impulsive phase of the flare. The wavelength of 335 Å, corresponding to a temperature of 2.5× 10⁶ K (see O'Dwyer et al. 2010), is selected throughout the paper since the flare ribbons are the least saturated in this filter (as it allows us to see coronal structures with high temperatures).

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In order to analyze photospheric structures, observations near the central region of the solar disk are required, so as to avoid the degradation of observations due to projection effects. The 2011 February 15 event, occurring in active region 11158, is a very good candidate because of the strength of the event (X2.2) and for its location near the center disk (Figure 1). It is also a good candidate because such a compact active region flare allows better measurements of the horizontal magnetic field vector in the related sunspots, with the magnetic field being more intense in those regions.

The active region 11158 is part of a complex region involving four magnetic polarities (Figure 2) that evolved over several days, leading to two M-class flares (February 13 & 14) and one X-class flare on February 15 (Schrijver et al. 2011; Gosain 2012; Inoue et al. 2013). The 2011 February 13 M6.6 flare was investigated in details by Liu et al. (2013), who also showed the formation of two flare ribbons and observational evidences of magnetic reconnection via the tether-cutting reconnection model. Although the magnetogram shows a quadrupolar structure for the magnetic polarities, the X2.2 flare occurred in the central bipole region as the flare ribbons extending from [x, y] = [170'', −240''] to [240'', −210''], so over a very limited portion of the Sun (Figure 1). Sun et al. (2012), Wang et al. (2012), and Petrie (2013) found that the horizontal photospheric magnetic field was enhanced and more sheared in between the flare ribbons by about 30% after the X2.2 flare, while the vertical field component had no significant modifications. Note that Petrie (2013) also showed maps of the vertical current density but without any detailed analysis as we perform below. Nonlinear force free field extrapolations complemented the vector magnetograms by allowing the study of the coronal magnetic field evolution (Sun et al. 2012; Inoue et al. 2013; Liu et al. 2013; Dalmasse et al. 2013). In particular, the topology of this active region, along with the presence of a flux rope and QSLs, was investigated by Zhao et al. (2014).

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We downloaded the HMI level-1b IQUV data and inverted them with the Milne–Eddington inversion code UNNOFIT (see Bommier et al. 2007). We selected a region covering AR 11158 and applied a solar rotation compensation to select the same region over 4 hr of observation. We thus treated 20 maps, from 2011 February 15 00:00 UT to 2011 February 15 03:48 UT.

The specificity of UNNOFIT is that a magnetic filling factor is introduced to take into account the unresolved magnetic structures, as a free parameter of the Levenberg–Marquardt algorithm that fits the observed set of profiles with a theoretical one. However, for further application we used only the averaged field, i.e., the product of the field with the magnetic filling factor, as recommended by Bommier et al. (2007). The interest of the method lies in a better determination of the field inclination. We tried to compensate the rather poor spectral HMI resolution by interpolating with a cubic spline function between the six spectral points. However, we found that the inversion results were better, with an apparent better spatial resolution, without the interpolation, i.e., with six spectral points adjusted on the computed theoretical profile of the four Stokes profiles. In the code, the loop on the pixels is parallelized by using the OpenMP method. We had 12 cores running in parallel for the inversion of one map on our computer, and since we were allowed to invert five maps simultaneously, it took one hour of computation to treat one hour of observational data.

After the inversion, the 180° remaining azimuth ambiguity was resolved by applying the ME0 code developed by Metcalf, Leka, Barnes, and Crouch (Leka et al. 2009) and available at http://www.cora.nwra.com/AMBIG/. After resolving the ambiguity, the magnetic field vectors were rotated into the local reference frame, where the local vertical axis is the Oz axis.

Figure 2(a) shows the four magnetic polarity regions almost located on the central part of the solar disk. The coordinates are given in the local solar reference (i.e., not in the line-of-sight coordinates). The four polarities started emerging at the end of 2011 February 10, and as they kept evolving, the trailing negative polarity of the west bipolar system moved closer to the positive leading polarity of the east bipolar system. After several flares involving the magnetic reconnection between the two bipolar fields, the two central polarities are strongly linked by magnetic arcades. These two polarities encounter each other with a close "fly-by," leading to a strong shear of the magnetic arcades (see Toriumi et al. 2013). This provides the conditions for further flares to occur, including the studied X2.2 flare, in between these two polarities.

The X-class flare of February 15 took place in the central bipolar region (contoured with a black square in Figure 2(a)). As such, although the whole domain presents a rather complex quadrupolar region, the X-class eruptive flare can be studied only by focusing on the central region. Figure 2(a) shows the polarities of B_z after the peak of the flare, which are not significantly different from before the flare, as shown by Sun et al. (2012) and Petrie (2013).

Figure 2(b) shows the vertical component J_z of the photospheric current at 02:00 UT of AR 11158. The current density J_z is directly derived from the magnetic field components via a centered difference method from the equation $\nabla \times \boldsymbol {B} = \mu _0 \boldsymbol {J}$ written in the local frame. Since the vertical component of the magnetic field does not change much over time, we are interested in the changes of the vertical current density during the eruptive flare, which are directly related with the changes of the horizontal magnetic field. This vertical current is the only one that could be computed from a vector magnetogram observed with one spectral line. It is still the most interesting component for flare studies as those currents are expected to enter in the corona. Knowing the vertical component of the electric current density is also of importance for extrapolating the coronal magnetic field (e.g., Wheatland & Gilchrist 2013).

The J_z map grid has a mesh size of ≈370 km. The map is centered on the central bipole, and the coordinates are the local solar coordinates. The values of |J_z| below the noise level of 20 mA.m⁻² are not shown. This level is estimated by the formula δB_T/(μ₀Δx) (see Gary & Démoulin 1995), where δB_T is the transverse field error and could be of the order of ∼100 G (see, e.g., Wiegelmann & Inhester 2010), and Δx = 05 is the grid size.

In the region of interest (the two central polarities, see the square in Figure 2(a)), we can already point out the strong current density regions near the polarity inversion line (PIL). The negative (positive) density currents shown in green (yellow or red) mostly appear in the negative (positive) polarity. However, mixed densities also appear further away from the PIL, probably because of the presence of the interlocking-comb structure in the penumbra of the sunspots (Thomas & Weiss 2012; Solanki & Montavon 1993), as already seen in other current maps from Venkatakrishnan & Tiwari (2009).

Near the PIL, the positive and negative current density regions align with each other, which is suggestive of opposite current ribbons. Those current densities extend upward or downward while contouring the negative or positive polarities, so that they both have a J-shape. The time evolution of those high current density regions is investigated in Section 3.

We show in Figure 3 a zoom of the photospheric current maps in the central bipole at four different times before (01:36 UT and 01:48 UT) and after (02:00 UT and 02:12 UT) the impulsive phase of the flare. This zoom corresponds to the black square indicated in Figure 2(a), and the times are restricted by the temporal resolution of HMI (12 min). The strong and coherent currents are dominantly direct currents along the magnetic field ($\boldsymbol {J} \times \boldsymbol {B}>0$ with a dominant positive current helicity) as found by Petrie (2013). We also show four squares indicating different regions, such as the hook and the straight part of the J-shaped current ribbons. This allows the reader to quickly pinpoint the different changes occurring during the flare evolution. We refer to H− (H+) for the hook part of the ribbon in the negative (positive) polarity and refer to S− (S+) for the straight part of the ribbon in the negative (positive) polarity.

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The left column of Figure 3 shows the different times of the zoomed-in J_z maps, while the right column shows the maps of the base difference of the direct electric current density. The base reference image is chosen at 01:24 UT, well before the onset of the flare. We only show the evolution of the direct currents that stay direct (i.e., we show the currents both satisfying $J_z^{\rm phot}$(01:24) $B_z^{\rm phot}$(01:24) > 0 and $J_z^{\rm phot} (t) B_z^{\rm phot}(t)>0$).

There is almost no evolution between 01:36 UT and 01:48 UT (Figure 3). However, most of the changes can be seen between 01:48 UT (before the flare peak) and 02:00 UT (after the flare peak). In H−, the hook shape becomes structured at 02:00 UT, while the signal is more randomly oriented before the flare peak. There is a signal increase at the base of the hook, around x = 85 Mm and y = 78 Mm, as it is clearly seen in the base difference (right) figure at 02:00 UT. The hook also becomes more consistent in the H+ region with an increase of the current density around x = 123 Mm, y = 82 Mm. There is also a strong increase of the current density in the straight parts of the ribbons (see the dark red ribbons in the base difference image). In the S− region, there is a broadening of the current ribbon, as J_z increases in the higher part (y ≈ 80 Mm) of the region. The positive current ribbon does not broaden as much, but it elongates toward the lower x region with an increase in the current density. All these changes remain at 02:12 UT when the flare brightness decreases significantly (see the light curve in 335 Å in Figure 5), indicating that the signals we analyze are unlikely to be related to polarization artifacts.

Although the current structures that we see appearing after the flare at 02:00 UT have a coherent spatial organization, polarization artifacts should be seriously considered, especially since these appear at flare ribbon locations. Indeed, during solar flares such effects have often been reported at the chromospheric level (see Hénoux & Karlický 2013 and references therein). The nonthermal particle beams and neutralizing return currents are generally cited as possible sources for an extra polarization signal. However, more recently, Štěpán & Heinzel (2013) showed that this polarization is not due to energetic particles impacting the chromosphere but to a radiation anisotropy near the chromospheric ribbons. Since this polarization effect is mainly seen for chromospheric lines (e.g., Hα, Na D2) in which ribbons are always very bright, the present photospheric data, which do not reveal any obvious signal in the intensity (Stokes I in the Fe i 6173 Å line), should avoid such artifacts. However, since the interest of this paper is to analyze in detail the evolution of the high current density region, we investigate in the following whether current signals are due to coherent, physical changes of the magnetic field.

So as to verify the reality of the current ribbons described above, we show in Figures 4(b) and (c) two zooms of the same region, the hook of the J-shape negative current ribbon (H−), also indicated with the square in Figure 4(a). These two zooms also show the vector map of the horizontal magnetic field superposed on top of the J_z maps and represent the two times taken before (b) and after (c) the flare peak.

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The horizontal field vector $\boldsymbol {B}_{h}$ has well-resolved vectors that smoothly rotate across the current ribbon, all along the straight and curved part of the hook shown in Figure 4. This is a clear indication of a consistent magnetic field signal free of artifacts, as otherwise the $\boldsymbol {B}_{h}$ directions would appear more randomly oriented, especially near the edges of the ribbons during the flare (Figure 4(c)). Also, the rotation of the vector field is a very good indicator of the presence of a current layer. Indeed, the sudden rotation of $\boldsymbol {B}_h$ should be expected in the presence of a high current density region, as is depicted in the Harris sheet model in 2.5D with a guide field along the current and field component across the sheet (Harris 1962).

Interestingly, these two figures also show the evolution of the horizontal component of the magnetic field during the flare. In particular, the changes of the vector field characteristics are readily seen in the coherent ribbon structure that is formed at the base of the hook: both the direction and intensity of the vectors are changing coherently. These results agree with the quantification of irreversible changes during the flare in the photospheric horizontal magnetic field quantified in Sun et al. (2012) and Wang et al. (2012). We conclude that the electric density changes are not mere artifacts due to polarization effects.

Since the current density significantly changes before and after the peak of the flare in the different regions H+, H−, S+, and S−, we study in the following the evolution of the electric current intensity I_c in these different regions. I_c is readily calculated by the integration of the current density over the different surfaces considered (I_c = $\mathop {\int \!\!\! \int } J_z dS$, where S is the surface contained in the four regions of interest).

We report in Figure 5 the evolution of the electric current integrated from the direct current densities ($J_z^{\rm phot} (t) B_z^{\rm phot}(t)>0$, shown in red) and the return current densities ($J_z^{\rm phot} (t) B_z^{\rm phot}(t)<0$, shown in blue). We have also added the light curve 335 Å AIA filter (green curve), showing the relative light intensity calculated as I_rel = I_335Å(t)/I_335Å(0UT) − 1.

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In all the different regions, the signal for the return currents (i.e., $\boldsymbol {J} \times \boldsymbol {B} <0$) is significantly low compared with that of the direct currents. Furthermore, there is no perceptible evolution of these return currents before, during, and after the flare peak. On the contrary, the direct electric currents (in red) are significantly changing during the impulsive phase of the flare. Indeed, all panels show an increase in the current intensity in the same time range as the light emission increase within the limits of the 12 minute temporal resolution of HMI and of the fluctuations before and after the flare. Then, I_c(direct) increases by a factor of 1.4 in H− during the flare and by about 1.3 in H+. In the regions S− and S+, I_c(direct) increases by 2 and 1.2, respectively. Moreover, during the decaying phase of the flare, the currents evolve slowly (Figure 5) when the flare ribbons have strongly weakened in intensity.

This is the first quantification of the electric current evolution in current ribbons during an eruptive flare. In summary, we found three main characteristics: first, the evolution of the coherent electric ribbons (Figure 3); second, the decrease of the curvature radius of the horizontal magnetic field $\boldsymbol {B}_h$ across the current ribbons (Figure 4); and finally, the slow evolution of the electric current after the sudden increase during the flare impulsive phase (Figure 5). All these further confirm that the current evolution observed is not caused by a modification of the light polarization due to energetic particles or enhanced radiation.

Predictions of the evolution of the flare, and especially the flare loops and the flux rope, have been given in a series of papers (see Aulanier et al. 2012; Janvier et al. 2013). In the following section, we propose to look at the results obtained via a 3D simulation of an eruptive flare in light of the present observations of the flare and electric ribbons.

The evolution of the flare ribbons during the 2011 February 15 event is analyzed with the SDO/AIA instrument. We choose the 335 Å filter since it gives a good compromise between the saturation level and the possibility to clearly see the whole ribbons for a rather long period (i.e., the flare loops saturate in 335 Å later after the formation of the ribbons). The comparison with other filters can be seen in Movie 3 of Schrijver et al. (2011). We show in Figure 6(a) and (d) the flare ribbons before and during the impulsive phase of the flare. Similar changes as pointed out for the evolution of the current density ribbons can be readily seen: the hooks for both flare ribbons become more consistent at 01:53:03 UT, while the emission intensity increases. The hooks also expand and then become rounder (see, e.g., the flare ribbon of the positive polarity, (x, y) ≈ (240'', −210''), where the hook surrounds a dimming region, indicating the presence of a flux rope). During the flare peak, the straight parts of the ribbons also broaden (the ribbon in the negative polarity extends toward higher y values, and inversely for the ribbon in the positive polarity). Moreover, the ribbon in the positive polarity also elongates (the end of the straight part extends from x ≈ 200'' to 190'').

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All those different changes are comparable to that of the current density ribbons as studied in Section 3. Figures 6(b) and (e) show the two J_z maps obtained from HMI (similar to Figure 3 but in the observing frame) superposed on top of the AIA co-aligned images. The brown arrows point to the different changes observed in both the J_z ribbons and the flare ribbons. These two figures demonstrate the similar nature of the current ribbons and the flare emission ribbons.

The evolution of the current density can also be studied by means of a numerical simulation. In the following, we present this evolution resulting from a 3D MHD simulation of an eruptive flare. This β = 0 simulation performed with the OHM code reproduces the evolution of an initially torus-unstable flux rope (Aulanier et al. 2010, 2012). The setups of these simulations were inspired by those of the pioneering 3D simulations from Amari et al. (2003a, 2003b) of the flux cancellation model. As the flux rope expands, regions of high current densities are created, where magnetic reconnection takes place. The evolution of reconnected field lines and the topology of the magnetic field (including QSLs) as well as the 3D intrinsic reconnection process have been analyzed in details in Aulanier et al. (2012) and Janvier et al. (2013). In particular, Figure 1 in Janvier et al. (2013) shows the formation of a coronal current layer and its thinning process as the flux rope continues to expand. It also shows the direct connection with the QSLs, and the broadening of the cusp underneath the coronal current structure can be directly linked with the footprints of the photospheric currents.

In Figure 7, we have represented the main features of this 3D model. The grey areas represent the coronal current layer and its extension to the photosphere. More accurately, we only represent here the locations of high electric current densities. These locations are also locations of QSLs, where the magnetic field connectivities are the mostly distorted (see Section 1). The footprints of QSLs and current layer are J-shaped, as can also be seen in images from the simulation in Figures 6(c) and (f).³ The hooks surround the legs of the flux rope, and four magnetic field lines have been represented, prior to reconnection.

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The photospheric currents ribbons separate as time goes by (see pink arrows in Figure 7), similar to the flare ribbons. Indeed, Figure 6 in Aulanier et al. (2012) shows that reconnected field lines, thus flare loops, are anchored in high photospheric current density regions. As flare loops are known to be rooted close to flare ribbons, the evolution of the flare ribbons is expected to trace the evolution of current ribbons, as observed.

The photospheric currents are shown in Figures 6(c) and (f) in the early and later stages of the numerical simulation. The contour plots in cyan and pink trace the negative and positive vertical magnetic fields, respectively. Both the magnetic polarities and the current ribbon positions are comparable to the present flare observations, as follows. The outward motion of the current ribbons, as expected in the flare ribbons evolution, is reproduced. These are pointed out with the yellow arrows. These shapes and displacements of the flare ribbons were also recovered by Inoue et al. (2014) with MHD simulations starting from an NLFFF extrapolation of the same active region, as follows (see their Figure 7 and Figure 6 of this paper).

These shapes and displacements of the flare ribbons were also recovered by Inoue et al. (2014) with MHD simulations starting from an NLFFF extrapolation of the same active region (compare their Figure 7 with our Figure 6).

Furthermore, the current ribbons elongate with an increase of their current density in the straight parts, and the hooks expand while becoming rounder. This is due to the building process of the flux rope: as magnetic reconnection takes place, part of the reconnected magnetic field lines further build up the envelope of the flux rope. This is also shown in Figure 17 of Dudík et al. (2014), where the hooks of the current ribbons surround the newly reconnected magnetic field lines and therefore expand as the flux rope becomes bigger. The 3D structure of the current layer is also shown for a numerical simulation in Figure 11 of Kliem et al. (2013), especially the hook part that contours the flux rope legs.

We also point out the existence of return currents contouring the hooks of the electric current ribbons: the positive current hook in white in the positive polarity is contoured with black (i.e., return current), while the black hook in the negative polarity, indicating direct negative current, is surrounded by white signatures of return currents. This feature also appears in the observations of the photospheric currents (Figures 6(d) and (e)), where both direct current hooks (yellow and green in the positive and negative polarities, respectively) are surrounded by opposite color patches. We conclude that all the observed characteristics present in the J_z maps (Section 3.3) are predicted by the 3D numerical model.

The fact that similar changes are found for the evolution of the current ribbons in the simulation is quite remarkable since the simulation is not a data driven simulation but on the contrary has theoretical boundary conditions solely designed to model a nonsymmetric bipolar magnetic configuration as typically observed in mostly bipolar active regions. The boundary driving is also generic of the shear concentration around PILs (Schmieder et al. 1996) and of the progressive dispersion observed in active regions. Such driving was only used to build the stressed magnetic configuration that contains mostly a magnetic sheared arcade with a moderately twisted flux rope inside. The photospheric driving was stopped during the eruption, which is driven only by the magnetic forces growing after the torus instability and loss of equilibrium occurred. Magnetic reconnection is occurring during all the evolution, mostly at separatrices and QSLs.

To conclude, the 3D MHD simulation only incorporated a simple bipolar magnetic configuration with few generic processes typically observed in the evolution of active regions. The simulations for the torus instability conditions of a flux rope were also performed and the results published (Aulanier et al. 2010) well before the presently studied flare occurred on the Sun. This is a clear indication that the key ingredients for an eruption are incorporated in the MHD simulation.

The results presented above are also striking considering that at the photospheric level, there is a finite plasma β, while the MHD simulation is realized in the zero β limit. However, the electric currents shown in Figure 6(c) and (f) are only the traces at the lower boundary from electric currents present in the coronal volume. They are present along a thin complex volume that follows the QSL system. The photospheric currents therefore have a coronal origin, and they are simply mapped down to the lower boundary along the coronal field lines.

The flare ribbons shown in Figure 6(a) and (d) are formed at the bottom of the corona, and they are directly comparable to the location of reconnected field lines in the simulation, i.e., to the current ribbons. The close correspondence between the flare and current ribbons in the observed flare show that the sharp transition from the corona to the photosphere through the extremely complex chromospheric layer is not able to stop the coronal currents from entering into the photosphere. This is consistent with the horizontal magnetic field around the photospheric inversion line that is getting stronger during the flare, while the vertical field component is not significantly changed (Sun et al. 2012; Petrie 2013).

The present study of the photospheric currents during an eruptive flare event shows an increase in the current density in specific locations that define the current ribbons. The quantification of the current evolution shows in particular that the hook and the straight part of the J-shaped current ribbons are subject to an increase during the impulsive phase of the eruptive flare. A thorough analysis of the vertical component of the currents with J_z maps, in relation to the vector map of the photospheric horizontal magnetic field (Section 3.2), indicates that this increase cannot be attributed to polarization artifacts either due to accelerated particles or to anisotropic radiation.

Also, numerical MHD simulations reproducing the evolution of an eruptive flare indicate that photospheric currents further appear in locations where there was no current before (see, e.g., the differences for the central straight part of the current ribbons in Figures 6(c) and (f)).

Basic theoretical arguments, however, go against these findings. Firstly, during an eruption, because of expansion, the length of the magnetic field lines increases while their end-to-end twist is conserved. This reduces (not increases) the current densities along the flux tube. Secondly, during and after the eruption, the magnetic field is classically expected to evolve toward a more and more potential state, while the magnetic helicity is mostly removed by the coronal mass ejection (i.e., the flux rope) via the reconnection of coronal magnetic field lines.

Indeed, in the 3D standard model, the total magnetic energy decreases and the initial flux rope, which is further built during the reconnection process, is ejected while simple arcades (the flare loops) are formed (Aulanier et al. 2012). Then, the magnetic field globally relaxes to a more potential field.

Extended currents associated with the preeruptive or the erupting flux rope are relatively weak in the present observation. They are actually difficult to observe: as shown in Figure 3 (and also in Figure 8 of Petrie 2013), the base of the flux rope legs that are anchored in the region surrounded by the hook have very weak current densities, with a level similar to the noise level (see white and grainy regions inside the hooks in Figures 3, 4). Such relatively weak current densities at the footpoints of the twisted flux rope may be explained as follows. In nearly cylindrical twisted loops, the current density is nearly inversely proportional to the structure length for a given end-to-end twist. It is therefore expected that for a flux rope running along the PIL of an AR, the extended currents at the photospheric footprints should be relatively weaker than those of equally sheared and shorter loops running across the PIL. These differences are well seen in numerical simulations that incorporate a long twisted flux rope and short-sheared loops (see, e.g., Figures 9 and 10 of Aulanier et al. (2005a) for active-region loops and Figure 6 from this paper, where extended currents at the flare loops footpoints are present between the current ribbons).

It is worth noticing that extended patches of strong current densities have been clearly observed from earlier ground-based magnetograms (e.g., Hagyard 1988, Hofmann et al. 1988, and de La Beaujardiere et al. 1993). But such reports do not contradict our present observation of barely detectable current densities at the base of the flux rope, obtained with a more recent spaceborne instrument. On the one hand, old and recent observed extended current patches can simply be associated with relatively short sheared loops (as explained above). On the other hand, they may also be related to current layers associated with flare ribbons, when narrow current density patches are located along separatrices or quasi-separatrix layers footprints (see, e.g., Mandrini et al. 1995 and Démoulin et al. 1997). The patches appear extended because of poor spatial resolution, which is seven times lower in these measurements than in the HMI data. Note that the inversion technique was also improved in between, refining the measurements of the vertical current density that appear localized in the present study.

Interestingly, our present interpretation of the relatively weak magnitude of flux rope currents, and the difficulty of measuring them, raises questions about the possibility of recovering preeruptive flux ropes in nonlinear force-free field extrapolations on a regular basis.

The picture of the current evolution presented above leaves out the evolution of localized coronal current layers that are formed during the building phase and even more during the flux rope expansion. In an evolving magnetic field, current sheets form on separatrices and even more generally on QSLs (Démoulin et al. 1996). The formation of these currents is intrinsic to the evolution of a magnetic field with a complex topology, as the photospheric line-tying and the drastic change of the field line shape impose a different magnetic field evolution on each side of a QSL. Then, current layers associated with separatrices and QSLs build up in eruption models from the earliest 2.5D models to the most recent 3D ones.

In models of eruptive flares, a current sheet builds up below the flux rope (e.g., Lin & Forbes 2000 and references therein). In 2.5D, this current sheet extends around the initial magnetic X-point and stays in the corona even in a spherical geometry, because of the supposed invariance in one direction (e.g., Lin et al. 1998). However, within a 3D model, this current sheet extends all the way down to the photosphere (e.g., Titov & Démoulin 1999). This is generalized to the buildup of a current layer at the hyperbolic flux tube (HFT), which is the core of QSLs (Titov et al. 2002).

During an evolution, slowly driven by boundary conditions, the current layer formed at the QSL becomes thinner with time. At some point of this evolution, the current layer becomes thin enough for reconnection to take place (Aulanier et al. 2006). Reconnection can take place even earlier, when the driver is an ideal instability: the evolution of the current thinning then occurs much faster, on the Alfvén time scale. This was shown in Figure 4 of Aulanier et al. (2012), where the current layer below the erupting flux rope becomes rapidly thinner and reconnection occurs, forming the flare loops and further building the flux rope.

In such a case, the driver is the torus instability, and the strength of this ideal instability implies that the coronal current layer collapses and triggers an efficient magnetic energy release via reconnection. This also explains the impulsive nature of an eruptive flare and the sudden increase in localized regions of the current density.

To conclude, there are two complementary, and not contradictory, evolutions of the electric currents during an eruptive flare. First, as expected by the general intuition, models of eruptive flares have shown that extended currents decrease as the magnetic configuration evolves toward a more potential state. These extended currents are less localized than the coronal current layer, and their photospheric signatures should be seen at footpoints of the flux rope magnetic field lines (inside the region surrounded by the hooks, where the flux rope legs are; see schematic in Figure 7). However, the evolution of these currents are difficult to observe because of their low intensities but could be computed via numerical simulations. Second, a flare also involves magnetic reconnection taking place in current layers. The associated flare driver further builds a thin coronal region where the current density increases. Contrary to the extended currents, these localized currents have high intensities, making them easier to observe. The predictions from the 3D standard model of the footprints of these regions, i.e., the current ribbons, have proven to be correct, as shown with the present analysis of the February 15 eruptive flare event.