Context. Helicity is a fundamental property of magnetic fields, conserved in ideal MHD. In flux rope geometry, it consists of twist and writhe helicity. Despite the common occurrence of helical structures in the solar atmosphere, little is known about how their shape relates to the writhe, which fraction of helicity is contained in writhe, and how much helicity is exchanged between twist and writhe when they erupt. Aims. Here we perform a quantitative investigation of these questions relevant for coronal flux ropes. Methods. The decomposition of the writhe of a curve into local and nonlocal components greatly facilitates its computation. We use it to study the relation between writhe and projected S shape of helical curves and to measure writhe and twist in numerical simulations of flux rope instabilities. The results are discussed with regard to filament eruptions and coronal mass ejections (CMEs). Results. (1) We demonstrate that the relation between writhe and projected S shape is not unique in principle, but that the ambiguity does not affect low- lying structures, thus supporting the established empirical rule which associates stable forward (reverse) S shaped structures low in the corona with positive (negative) helicity. (2) Kink-unstable erupting flux ropes are found to transform a far smaller fraction of their twist helicity into writhe helicity than often assumed. (3) Confined flux rope eruptions tend to show stronger writhe at low heights than ejective eruptions (CMEs). This argues against suggestions that the writhing facilitates the rise of the rope through the overlying field. (4) Erupting filaments which are S shaped already before the eruption and keep the sign of their axis writhe (which is expected if field of one chirality dominates the source volume of the eruption), must reverse their S shape in the course of the rise. Implications for the occurrence of the helical kink instability in such events are discussed. (5) The writhe of rising loops can easily be estimated from the angle of rotation about the direction of ascent, once the apex height exceeds the footpoint separation significantly. Conclusions. Writhe can straightforwardly be computed for numerical data and can often be estimated from observations. It is useful in interpreting S shaped coronal structures and in constraining models of eruptions.

The structure of the coronal magnetic field prior to eruptive processes and the conditions for the onset of eruption are important issues that can be addressed through studying the magnetohydrodynamic (MHD) stability and evolution of nonlinear force-free field (NLFFF) models. This paper uses data-constrained NLFFF models of a solar active region (AR) that erupted on 2010 April 8 as initial conditions in MHD simulations. These models, constructed with the techniques of flux rope insertion and magnetofrictional relaxation (MFR), include a stable, an approximately marginally stable, and an unstable configuration. The simulations confirm previous related results of MFR runs, particularly that stable flux rope equilibria represent key features of the observed pre-eruption coronal structure very well, and that there is a limiting value of the axial flux in the rope for the existence of stable NLFFF equilibria. The specific limiting value is located within a tighter range, due to the sharper discrimination between stability and instability by the MHD description. The MHD treatment of the eruptive configuration yields a very good agreement with a number of observed features, like the strongly inclined initial rise path and the close temporal association between the coronal mass ejection and the onset of flare reconnection. Minor differences occur in the velocity of flare ribbon expansion and in the further evolution of the inclination; these can be eliminated through refined simulations. We suggest that the slingshot effect of horizontally bent flux in the source region of eruptions can contribute significantly to the inclination of the rise direction. Finally, we demonstrate that the onset criterion, formulated in terms of a threshold value for the axial flux in the rope, corresponds very well to the threshold of the torus instability in the considered AR.

Two well known phenomena associated with erupting filaments are the transient coronal holes that form on each side of the filament channel and the bright post-event arcade with its expanding double row of footpoints. Here we focus on a frequently overlooked signature of filament eruptions: the spike- or fan-shaped brightenings that appear to mark the far endpoints of the filament. From a sample of non-active-region filament events observed with the Extreme- Ultraviolet Imaging Telescope on the Solar and Heliospheric Observatory, we find that these brightenings usually occur near the outer edges of the transient holes, in contrast to the post-event arcades, which define their inner edges. The endpoints are often multiple and are rooted in and around strong network flux well outside the filament channel, a result that is consistent with the axial field of the filament being much stronger than the photospheric field inside the channel. The extreme ultraviolet brightenings, which are most intense at the time of maximum outward acceleration of the filament, can be used to determine unambiguously the direction of the axial field component from longitudinal magnetograms. Their location near the outer boundary of the transient holes suggests that we are observing the footprints of the current sheet formed at the leading edge of the erupting filament, as distinct from the vertical current sheet behind the filament which is the source of the post-event arcade.

We analyze the evolution of a sigmoidal (S-shaped) active region toward eruption, which includes a coronal mass ejection (CME) but leaves part of the filament in place. The X-ray sigmoid is found to trace out three different magnetic topologies in succession: a highly sheared arcade of coronal loops in its long-lived phase, a bald-patch separatrix surface (BPSS) in the hours before the CME, and the first flare loops in its major transient intensity enhancement. The coronal evolution is driven by photospheric changes which involve the convergence and cancellation of flux elements under the sigmoid and filament. The data yield unambiguous evidence for the existence of a BPSS, and hence a flux rope, in the corona prior to the onset of the CME.