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We clarify a relationship of the dynamics of a solar flare and a growing Coronal Mass Ejection (CME) by investigating the dynamics of magnetic fields during the X2.2-class flare taking place in the solar active region 11158 on 2011 February 15, based on simulation results obtained from Inoue et al. 2014. We found that the strongly twisted lines formed through the tether-cutting reconnection in the twisted lines of a nonlinear force-free field (NLFFF) can break the force balance within the magnetic field, resulting in their launch from the solar surface. We further discover that a large-scale flux tube is formed during the eruption as a result of the tether-cutting reconnection between the eruptive strongly twisted lines and these ambient weakly twisted lines. Then the newly formed large flux tube exceeds the critical height of the torus instability. The tether-cutting reconnection thus plays an important role in the triggering a CME. Furthermore, we found that the tangential fields at the solar surface illustrate different phases in the formation of the flux tube and its ascending phase over the threshold of the torus instability. We will discuss about these dynamics in detail., Comment: 38 pages, 14 figures, accepted for publication in The Astrophysical Journal

We performed a magnetohydrodynamic (MHD) simulation using a nonlinear force-free field (NLFFF) in solar active region 11158 to clarify the dynamics of an X2.2-class solar flare. We found that the NLFFF never shows the drastic dynamics seen in observations, i.e., it is in stable state against the perturbations. On the other hand, the MHD simulation shows that when the strongly twisted lines are formed at close to the neutral line, which are produced via tether-cutting reconnection in the twisted lines of the NLFFF, consequently they erupt away from the solar surface via the complicated reconnection. This result supports the argument that the strongly twisted lines formed in NLFFF via tether-cutting reconnection are responsible for breaking the force balance condition of the magnetic fields in the lower solar corona. In addition to this the dynamical evolution of these field lines reveals that at the initial stage the spatial pattern of the footpoints caused by the reconnection of the twisted lines appropriately maps the distribution of the observed two-ribbon flares. Interestingly, after the flare the reconnected field lines convert into the structure like the post flare loops, which is analogous to EUV image taken by SDO. Eventually, we found that the twisted lines exceed a critical height at which the flux tube becomes unstable to the torus instability. These results illustrate the reliability of our simulation and also provide an important relationship between flare-CME dynamics., Comment: 29 pages, 11 figures, accepted for publication in The Astrophysical Journal

We develop a nonlinear force-free field (NLFFF) extrapolation code based on the magnetohydrodynamic (MHD) relaxation method. We extend the classical MHD relaxation method in two important ways. First, we introduce an algorithm initially proposed by cite{2002JCoPh.175..645D} to effectively clean the numerical errors associated with $nabla cdot vec{B}$. Second, the multi-grid type method is implemented in our NLFFF to perform direct analysis of the high-resolution magnetogram data. As a result of these two implementations, we successfully extrapolated the high resolution force-free field introduced by cite{1990ApJ...352..343L} with better accuracy in a drastically shorter time. We also applied our extrapolation method to the MHD solution obtained from the flux-emergence simulation by cite{2012ApJ...748...53M}. We found that NLFFF extrapolation may be less effective for reproducing areas higher than a half-domain, where some magnetic loops are found in a state of continuous upward expansion. However, an inverse S shaped structure consisting of the sheared and twisted loops formed in the lower region can be captured well through our NLFFF extrapolation method. We further discuss how well these sheared and twisted fields are reconstructed by estimating the magnetic topology and twist quantitatively., Comment: 36 pages, 12 figures, accepted for publication in The Astrophysical Journal

We study the three-dimensional magnetic structure of solar active region 11158, which produced one X-class and several M-class flares on 2011 February 13$-$16. We focus on the magnetic twist in four flare events, M6.6, X2.2, M1.0, and M1.1. The magnetic twist is estimated from the nonlinear force-free field extrapolated from the vector fields obtained from the Helioseismic and Magnetic Imager on board the Solar Dynamic Observatory using magnetohydrodynamic relaxation method developed by \cite{2011ApJ...738..161I}. We found that strongly twisted lines ranging from half-turn to one-turn twist were built up just before the M6.6- and X2.2 flares and disappeared after that. Because most of the twist remaining after these flares was less than half-turn twist, this result suggests that the buildup of magnetic twist over the half-turn twist is a key process in the production of large flares. On the other hand, even though these strong twists were also built up just before the M1.0 and M1.1 flares, most of them remained afterwords. Careful topological analysis before the M1.0 and M1.1 flares shows that the strongly twisted lines were surrounded mostly by the weakly twisted lines formed in accordance with the clockwise motion of the positive sunspot, whose footpoints are rooted in strong magnetic flux regions. These results imply that these weakly twisted lines might suppress the activity of the strongly twisted lines in the last two M-class flares., Comment: 30 pages, 10 figures, The Astrophysical Journal, accepted

We analyze the temporal evolution of the three-dimensional (3D) magnetic structure of the flaring active region (AR) NOAA 10930 by using the nonlinear force-free fields extrapolated from the photospheric vector magnetic fields observed by the Solar Optical Telescope on board {\it Hinode}. This AR consisted mainly of two types of twisted magnetic field lines: One has a strong negative (left-handed) twist due to the counterclockwise motion of the positive sunspot and is rooted in the regions of both polarities in the sunspot at a considerable distance from the polarity inversion line (PIL). In the flare phase, dramatic magnetic reconnection occurs in those negatively twisted lines in which the absolute value of the twist is greater than a half-turn. The other type consists of both positively and negatively twisted field lines formed relatively close to the PIL between two sunspots. A strong CaII image began to brighten in this region of mixed polarity, in which the positively twisted field lines were found to be injected within one day across the pre-existing negatively twisted region, along which strong currents were embedded. Consequently, the central region near the PIL distributed with a mix of differently twisted field lines and the strong currents may play a prominent role in flare onset., Comment: 28 pages, 8 figures, The Astrophysical Journal, accepted