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Force-free equilibria containing two vertically arranged magnetic flux ropes of like chirality and current direction are considered as a model for split filaments/prominences and filament-sigmoid systems. Such equilibria are constructed analytically through an extension of the methods developed in Titov & Demoulin and numerically through an evolutionary sequence including shear flows, flux emergence, and flux cancellation in the photospheric boundary. It is demonstrated that the analytical equilibria are stable if an external toroidal (shear) field component exceeding a threshold value is included. If this component decreases sufficiently, then both flux ropes turn unstable for conditions typical of solar active regions, with the lower rope typically becoming unstable first. Either both flux ropes erupt upward, or only the upper rope erupts while the lower rope reconnects with the ambient flux low in the corona and is destroyed. However, for shear field strengths staying somewhat above the threshold value, the configuration also admits evolutions which lead to partial eruptions with only the upper flux rope becoming unstable and the lower one remaining in place. This can be triggered by a transfer of flux and current from the lower to the upper rope, as suggested by the observations of a split filament in Paper I. It can also result from tether-cutting reconnection with the ambient flux at the X-type structure between the flux ropes, which similarly influences their stability properties in opposite ways. This is demonstrated for the numerically constructed equilibrium.
Aims. We study an evolving bipolar active region that exhibits flux cancellation at the internal polarity inversion line, the formation of a soft X-ray sigmoid along the inversion line and a coronal mass ejection. The aim is to investigate the quantity of flux cancellation that is involved in flux rope formation in the time period leading up to the eruption.
Methods. The active region is studied using its extreme ultraviolet and soft X-ray emissions as it evolves from a sheared arcade to flux rope configuration. The evolution of the photospheric magnetic field is described and used to estimate how much flux is reconnected into the flux rope.
Results. About one third of the active region flux cancels at the internal polarity inversion line in the 2.5 days leading up to the eruption. In this period, the coronal structure evolves from a weakly to a highly sheared arcade and then to a sigmoid that crosses the inversion line in the inverse direction. These properties suggest that a flux rope has formed prior to the eruption. The amount of cancellation implies that up to 60% of the active region flux could be in the body of the flux rope. We point out that only part of the cancellation contributes to the flux in the rope if the arcade is only weakly sheared, as in the first part of the evolution. This reduces the estimated flux in the rope to similar to 30% or less of the active region flux. We suggest that the remaining discrepancy between our estimate and the limiting value of similar to 10% of the active region flux, obtained previously by the flux rope insertion method, results from the incomplete coherence of the flux rope, due to nonuniform cancellation along the polarity inversion line. A hot linear feature is observed in the active region which rises as part of the eruption and then likely traces out the field lines close to the axis of the flux rope. The flux cancellation and changing magnetic connections at one end of this feature suggest that the flux rope reaches coherence by reconnection immediately before and early in the impulsive phase of the associated flare. The sigmoid is destroyed in the eruption but reforms quickly, with the amount of cancellation involved being much smaller than in the course of its original formation.
Slow rise and partial eruption of a double-decker filament. I. observations and interpretation
(2012)
We study an active-region dextral filament that was composed of two branches separated in height by about 13 Mm, as inferred from three-dimensional reconstruction by combining SDO and STEREO-B observations. This "double-decker" configuration sustained for days before the upper branch erupted with a GOES-class M1.0 flare on 2010 August 7. Analyzing this evolution, we obtain the following main results. (1) During the hours before the eruption, filament threads within the lower branch were observed to intermittently brighten up, lift upward, and then merge with the upper branch. The merging process contributed magnetic flux and current to the upper branch, resulting in its quasi-static ascent. (2) This transfer might serve as the key mechanism for the upper branch to lose equilibrium by reaching the limiting flux that can be stably held down by the overlying field or by reaching the threshold of the torus instability. (3) The erupting branch first straightened from a reverse S shape that followed the polarity inversion line and then writhed into a forward S shape. This shows a transfer of left-handed helicity in a sequence of writhe-twist-writhe. The fact that the initial writhe is converted into the twist of the flux rope excludes the helical kink instability as the trigger process of the eruption, but supports the occurrence of the instability in the main phase, which is indeed indicated by the very strong writhing motion. (4) A hard X-ray sigmoid, likely of coronal origin, formed in the gap between the two original filament branches in the impulsive phase of the associated flare. This supports a model of transient sigmoids forming in the vertical flare current sheet. (5) Left-handed magnetic helicity is inferred for both branches of the dextral filament. (6) Two types of force-free magnetic configurations are compatible with the data, a double flux rope equilibrium and a single flux rope situated above a loop arcade.
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.