In this paper, we present a multi-wavelength analysis of two X-class solar eruptive flares of classes X2.2 and X9.3 that occurred in the sigmoidal active region NOAA 12673 on 2017 September 6, by combining observations of Atmospheric Imaging Assembly and Helioseismic Magnetic Imager instruments on board the Solar Dynamics Observatory. On the day of the reported activity, the photospheric structure of the active region displayed a very complex network of δ-sunspots that gave rise to the formation of a coronal sigmoid observed in the hot EUV channels. Both X-class flares initiated from the core of the sigmoid sequentially within an interval of ∼3 hours and progressed as a single "sigmoid-to-arcade" event. Differential emission measure analysis reveals strong heating of plasma at the core of the active region right from the pre-flare phase which further intensified and spatially expanded during each event. The identification of a pre-existing magnetic null by non-force-free-field modeling of the coronal magnetic fields at the location of early flare brightenings and remote faint ribbon-like structures during the pre-flare phase, which were magnetically connected with the core region, provide support for the breakout model of solar eruption. The magnetic extrapolations also reveal flux rope structures prior to both flares which are subsequently supported by the observations of the eruption of hot EUV channels. The second X-class flare diverged from the standard flare scenario in the evolution of two sets of flare ribbons, that are spatially well separated, providing firm evidence of magnetic reconnections at two coronal heights.

In this paper, we present a systematic study of the force-free field equation for simple axisymmetric configurations in spherical geometry and apply it to the solar active regions. The condition of separability of solutions in the radial and angular variables leads to two classes of solutions: linear and nonlinear force-free fields. We have studied these linear solutions and extended the nonlinear solutions for the radial power law index to the irreducible rational form n = p/q, which is allowed for all cases of odd p and cases of q > p for even p, where the poloidal flux ψ ∝ 1/r n and field B ∝ 1/r n+2 ). We apply these solutions to simulate photospheric vector magnetograms obtained using the spectropolarimeter on board Hinode. The effectiveness of our search strategy is first demonstrated on test inputs of dipolar, axisymmetric, and non axisymmetric linear force-free fields. Using the best-fit to these magnetograms, we build three-dimensional axisymmetric field configurations and calculate the energy and relative helicity with two independent methods, which are in agreement. We have analyzed five magnetograms for AR 10930 spanning a period of three days during which two X-class flares occurred which allowed us to find the free energy and relative helicity of the active region before and after the flare; our analysis indicates a peak in these quantities before the flare events which is consistent with the results mentioned in literature. We also analyzed single-polarity regions AR 10923 and 10933, which showed very good fits with potential fields. This method can provide useful reconstruction of the nonlinear force-free (NLFF) fields as well as reasonably good input fields for other numerical techniques.

Magnetohydrodynamics of the solar corona is simulated numerically. The simulation is initialized with an extrapolated non-force-free magnetic field using the vector magnetogram of the active region (AR) NOAA 12192 obtained on the solar photosphere. Particularly, we focus on the magnetic reconnections occurring close to a magnetic null-point that resulted in appearance of circular chromospheric flare ribbons on October 24, 2014 around 21:21 UT, after peak of an X3.1 flare. The extrapolated field lines show the presence of the threedimensional (3D) null near one of the polarity inversion lines-where the flare was observed. In the subsequent numerical simulation, we find magnetic reconnections occurring near the null point, where the magnetic field lines from the fan-plane of the 3D null form a X-type configuration with underlying arcade field lines. The footpoints of the dome-shaped field lines, inherent to the 3D null, show high gradients of the squashing factor. We find slipping reconnections at these quasi-separatrix layers, which are co-located with the post-flare circular brightening observed at the chromospheric heights. This demonstrates the viability of arXiv:1805.00635v1 [astro-ph.SR] 2 May 2018 the initial non-force-free field along with the dynamics it initiates. Moreover, the initial field and its simulated evolution is found to be devoid of any flux rope, which is in congruence with the confined nature of the flare.

The magnetohydrodynamics of active region NOAA 11283 is simulated using an initial non-force-free magnetic field extrapolated from its photospheric vector magnetogram. We focus on the magnetic reconnections at a magnetic null point that participated in the X2.1 flare on 2011 September 6 around 22:21 UT (SOL2011-09-06T22:21X2.1) followed by the appearance of circular flare ribbons and coronal dimmings. The initial magnetic field from extrapolation displays a three-dimensional (3D) null topology overlying a sheared arcade. Prior to the flare, magnetic loops rise due to the initial Lorentz force, and reconnect at the 3D null, leading to expansion and loss of confined plasma that produce the observed pre-flare coronal dimmings. Further, the simulated dynamics documents the transfer of twist from the arcade to the overlying loops through reconnections, developing a flux rope. The nonparallel field lines comprising the rope and lower-lying arcades form an X-type geometry. Importantly, the simultaneous reconnections at the 3D null and the X-type geometry can explain the observed circular and parallel flare ribbons. Reconnections at the 3D null transform closed inner spine field lines into open field lines of the outer spine. The footpoints of these open field lines correspond to a ring-shaped coronal dimming region, tracing the dome. Further, the flux rope bifurcates because of these reconnections, which also results in the generation of open magnetic field lines. The plasma loss along the open field lines can potentially explain the observed coronal dimming.