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| - II. Non-potential modelling
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| - Eruptions from quiet Sun coronal bright points
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| - Context. Our recent observational study shows that the majority of coronal bright points (CBPs) in the quiet Sun are sources of one or more eruptions during their lifetime. Aims. Here, we investigate the non-potential time-dependent structure of the magnetic field of the CBP regions with special emphasis on the time-evolving magnetic structure at the spatial locations where the eruptions are initiated. Methods. The magnetic structure is evolved in time using a non-linear force-free field (NLFFF) relaxation approach based on a time series of helioseismic and magnetic imager (HMI) longitudinal magnetograms. This results in a continuous time series of NLFFFs. The time series is initiated with a potential field extrapolation based on a magnetogram taken well before the time of the eruptions. This initial field is then evolved in time in response to the observed changes in the magnetic field distribution at the photosphere. The local and global magnetic field structures from the time series of NLFFF field solutions are analysed in the vicinity of the eruption sites at the approximate times of the eruptions. Results. The analysis shows that many of the CBP eruptions reported in a recent publication contain a twisted flux tube located at the sites of eruptions. The presence of flux ropes at these locations provides in many cases a direct link between the magnetic field structure, their eruption, and the observation of mini coronal mass ejections (mini-CMEs). It is found that all repetitive eruptions are homologous. Conclusions. The NLFFF simulations show that twisted magnetic field structures are created at the locations hosting eruptions in CBPs. These twisted structures are produced by footpoint motions imposed by changes in the photospheric magnetic field observations. The true nature of the micro-flares remains unknown. Further 3D data-driven magnetohydrodynamic modelling is required to show how these twisted regions become unstable and erupt.
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