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Institute
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.
Testing magnetofrictional extrapolation with the Titov-Demoulin model of solar active regions
(2010)
We examine the nonlinear magnetofrictional extrapolation scheme using the solar active region model by Titov and Demoulin as test field. This model consists of an arched, line-tied current channel held in force-free equilibrium by the potential field of a bipolar flux distribution in the bottom boundary. A modified version with a parabolic current density profile is employed here. We find that the equilibrium is reconstructed with very high accuracy in a representative range of parameter space, using only the vector field in the bottom boundary as input. Structural features formed in the interface between the flux rope and the surrounding arcade - "hyperbolic flux tube" and "bald patch separatrix surface" - are reliably reproduced, as are the flux rope twist and the energy and helicity of the configuration. This demonstrates that force-free fields containing these basic structural elements of solar active regions can be obtained by extrapolation. The influence of the chosen initial condition on the accuracy of reconstruction is also addressed, confirming that the initial field that best matches the external potential field of the model quite naturally leads to the best reconstruction. Extrapolating the magnetogram of a Titov-Demoulin equilibrium in the unstable range of parameter space yields a sequence of two opposing evolutionary phases, which clearly indicate the unstable nature of the configuration: a partial buildup of the flux rope with rising free energy is followed by destruction of the rope, losing most of the free energy.
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.
Using Big Bear Solar Observatory film data recently digitized at NJIT, we investigate a Moreton wave associated with an X9 flare on 1990 May 24, as well as its interactions with four filaments F1-F4 located close to the flaring region. The interaction yields interesting insight into physical properties of both the wave and the filaments. The first clear Moreton wavefront appears at the flaring-region periphery at approximately the same time as the peak of a microwave burst and the first of two gamma-ray peaks. The wavefront propagates at different speeds ranging from 1500-2600 km s(-1) in different directions, reaching as far as 600 Mm away from the flaring site. Sequential chromospheric brightenings are observed ahead of the Moreton wavefront. A slower diffuse front at 300-600 km s(-1) is observed to trail the fast Moreton wavefront about one minute after the onset. The Moreton wave decelerates to similar to 550 km s(-1) as it sweeps through F1. The wave passage results in F1's oscillation which is featured by similar to 1 mHz signals with coherent Fourier phases over the filament, the activation of F3 and F4 followed by gradual recovery, but no disturbance in F2. Different height and magnetic environment together may account for the distinct responses of the filaments to the wave passage. The wavefront bulges at F4, whose spine is oriented perpendicular to the upcoming wavefront. The deformation of the wavefront is suggested to be due to both the forward inclination of the wavefront and the enhancement of the local Alfven speed within the filament channel.
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.
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.