Rui Liu¹, Bernhard Kliem², Viacheslav S. Titov³, Jun Chen¹, Yuming Wang¹, Haimin Wang^4,5, Chang Liu^4,5, Yan Xu^4,5, & Thomas Wiegelmann^6
1 CAS Key Laboratory of Geospace Environment, Department of Geophysics and Planetary Sciences, University of Science and Technology of China, Hefei 230026, China
² Institute of Physics and Astronomy, University of Potsdam, 14476 Potsdam, Germany
³ Predictive Science Inc., 9990 Mesa Rim Road, Suite 170, San Diego, CA 92121, USA
⁴ Space Weather Research Laboratory, New Jersey Institute of Technology, University Heights, Newark, NJ 07102-1982, USA
⁵ Big Bear Solar Observatory, New Jersey Institute of Technology, 40386 North Shore Lane, Big Bear City, CA 92314-9672, USA
⁶ Max-Planck-Institut für Sonnensystemforschung, Justus-von-Liebig-Weg 3, 37077 Göttingen, Germany

The magnetic flux rope (MFR) is a key structure of fundamental importance in various models of solar eruptions. The helical kink instability of MFRs has received a lot of attention, whereby magnetic twist (winding of magnetic field lines around the rope axis) is abruptly converted to magnetic writhe (winding of the axis itself). The instability is triggered when the twist exceeds 1.25 turns with photospheric line tying included¹.

A twist number T_w can be employed to quantify how many turns two infinitesimally close field lines wind about each other²:

This must be done with caution, however, due to the subtle yet significant differences between T_w and the twist number of a curve about an axis in a general geometry, referred to here as T_g (ref. 3). T_w is a close approximation of T_g only in the vicinity of the magnetic axis and if the MFR possesses certain degree of coherence, i.e., cylindrical symmetry. The calculation of T_g is often complicated by the determination of the axis. In contrast T_w can be straightforwardly computed without resorting to the geometry of an MFR; moreover, it has a local extremum at the axis if the MFR possesses some degree of cylindrical symmetry. Thus, T_w provides a convenient means to characterize and quantify magnetic configurations.

Figure 1 | a) GOES 1–8 Å light curve during 2013 August 10-12; b) Free magnetic energy derived from the NLFFF. Also shown is the peak |T_w| (red) within the MFR’s cross section in the cutting plane C_A, scaled by the right y-axes. Vertical dotted lines indicate the peak times of the flares studied, whose GOES classes are annotated in a), where confined/eruptive flares are marked by grey/black arrows, respectively. Panels c) and d) shows a snapshot of AR 11817 on 2013 August 11. White (black) colors refer to positive (negative) B_z, saturated ±800 G. Red (blue) arrows represent the tangential field component originating from negative (positive) B_z. The rectangle in (c) indicates the FOV of photospheric T_w and Q maps in Fig. 2. In panel d), the map of B_z is superimposed by the velocity field averaged over 9 hours from 12:10-21:10 UT on August 11. This time interval corresponds to an enhanced injection of helicity and Poynting fluxes. In Panel (d) green arrows indicate tangential velocities (v_⊥t), and orange contours refer to normal velocities (v_⊥n) at 0.05 and 0.08 km s⁻¹ (upflows). Only those vectors (either field or velocity) at the pixels with |B_z| > 150G are plotted.

Applying this idea to NOAA Active Region 11817 during 2013 August 10–12 (Fig. 1), we studied a series of C-and-above flares (marked by arrows in Fig. 1(a)) and found that they are all associated with an MFR located along the major polarity inversion line, where shearing and converging photospheric flows are present (see Fig. 1(d)). With the nonlinear force-free field modeling based on HMI vector magnetograms, we identified the MFR through mapping squashing factor Q (ref. 4) and twist number T_w⁵. Taking the C2.1 flare on August 11 for example, the footpoints of the MFR can be seen straightforwardly on the photospheric maps of T_w and Q (Fig. 2(a) and (b)) as two conjugate compact regions with enhanced T_w of the same sign and enclosed by high Q lines. The MFR’s cross section is displayed by vertical maps of T_w (Fig. 2(e) and (h)), enclosed similarly by high Q lines.

Figure 2 | MFR before the C2.1 flare at 19:27 UT on 2013 August 11. (a) logQ above 1 (white) and saturated at 5 (black). (b) T_w saturated at ±1.8 and blended with logQ. In (a) and (b) a rectangle encloses one of the MFR’s footpoint regions. This rectangular region is enlarged and redisplayed in the upper left corner. (c–f) Current density j_x, logQ, T_w, and decay index in the cutting plane C_A (denoted in (b)). (g–h) logQ and T_w in the cutting plane C_B (denoted in (b)). Representative field lines of the MFR (green), BPSS (orange) and QSL (magenta) are shown in both cutting planes. The crosses indicate where the field line threads the cutting plane. The cross section of the MFR in C_A is identified by clicking on its high Q boundary, which is replotted as black dots in (c).

The field line with the extremum |T_w| proves to be a reliable proxy of the rope axis (black curves in Fig. 3(a)). The key to identifying and precisely locating the axis of the MFR is the vertical twist map (Fig. 2(e) and (h)). We computed twist maps in many vertical cut planes and traced a field line from the peak-|T_w| point in each map. It was found that all such field lines traced in the range x = [74.5, 107.3] Mm coincide within the limits of numerical accuracy. This includes nearly the whole axis of the MFR, except for a very short section (Δx < 2.5 Mm) at its east end. Placing a cut plane perpendicular to the axis one can see that the in-plane field vectors display a rotational pattern centered on the intersection point and that the current density normal to the plane is enhanced in this area, consistent with the existence of an MFR (Fig. 3(b)).

Figure 3 | A double-decker MFR configuration. a) Twisted field lines of the double-decker MFR in the NLFFF on 2013 August 11 at 19:10 UT. b) Normal current density and in-plane field vectors in a vertical cut plane oriented perpendicularly to the MFR axis (77 deg to the x axis) at (x, y) = (83.9, 54.9) Mm. c) Isosurface of T_w = −1 viewed from the same perspective as Panel (a). Physical units of the coordinates are Mm in (a) and (b).

The vertical twist map in Fig. 2(e) reveals the existence of a second, smaller MFR under the main MFR. This is corroborated by enhanced current density and a rotational pattern of the in-plane field components in this area in a vertical cut (Fig. 2(b)). The field line plot (Fig. 3(a)) and the isosurface of T_w = −1 (Fig. 3(c)) directly display a double-decker configuration of two left-handed flux ropes. Such a configuration was suggested previously as a candidate for partial eruptions⁶.

The MFR is close to the threshold of the helical kink instability, as gauged by both T_w and T_g as well as the global Kruskal-Shafranov criterion in our series of NLFFF models. The high-resolution NST observations of the C2.1 flare on August 11 also indicate the onset of this instability. The analysis of the decay index excludes the torus instability as the eruption onset mechanism for all the flares studied, since the MFR is located well below the height where the decay index reaches its threshold value of 1.5 (e.g., Fig. 2(f)). Hence, the helical kink instability is identified as the prime candidate onset mechanism for the considered flares.

The MFR’s peak |T_w| temporarily increases within half an hour before each flare while it decreases after the flare peak for both confined and eruptive flares (Fig. 1(b)). This pre-flare increase in |T_w| has little effect on the AR’s free magnetic energy or any other parameters derived for the whole region, due to its moderate amount and the MFR’s relatively small volume, while its decrease after flares is clearly associated with the stepwise decrease in the whole region’s free magnetic energy due to the flare. Thus, we suggest that T_w may serve as a useful parameter in forewarning the onset of eruption.

References

[1] Hood, A. W. & Priest, E. R., 1981, GApFD, 17, 297
[2] Berger, M. A. & Prior, C. 2006, JPhA, 39, 8321
[3] Liu, R., Kliem, B., Titov, V. S., et al., 2016, ApJ, 818, 148
[4] Titov, V. S., Hornig, G., & Démoulin, P., 2002, JGR, 107, 1164
[5] The code developed by R. Liu and J. Chen is available online.
[6] Liu, R., Kliem, B., Török, T., et al., 2012, ApJ, 756, 59