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A parametric study of erupting flux rope rotation modeling the "Cartwheel CME" on 9 April 2008
(2012)
The rotation of erupting filaments in the solar corona is addressed through a parametric simulation study of unstable, rotating flux ropes in bipolar force-free initial equilibrium. The Lorentz force due to the external shear-field component and the relaxation of tension in the twisted field are the major contributors to the rotation in this model, while reconnection with the ambient field is of minor importance, due to the field's simple structure. In the low-beta corona, the rotation is not guided by the changing orientation of the vertical field component's polarity inversion line with height. The model yields strong initial rotations which saturate in the corona and differ qualitatively from the profile of rotation vs. height obtained in a recent simulation of an eruption without preexisting flux rope. Both major mechanisms writhe the flux rope axis, converting part of the initial twist helicity, and produce rotation profiles which, to a large part, are very similar within a range of shear-twist combinations. A difference lies in the tendency of twist-driven rotation to saturate at lower heights than shear-driven rotation. For parameters characteristic of the source regions of erupting filaments and coronal mass ejections, the shear field is found to be the dominant origin of rotations in the corona and to be required if the rotation reaches angles of order 90 degrees and higher; it dominates even if the twist exceeds the threshold of the helical kink instability. The contributions by shear and twist to the total rotation can be disentangled in the analysis of observations if the rotation and rise profiles are simultaneously compared with model calculations. The resulting twist estimate allows one to judge whether the helical kink instability occurred. This is demonstrated for the erupting prominence in the "Cartwheel CME" on 9 April 2008, which has shown a rotation of a parts per thousand aEuro parts per thousand 115(a similar to) up to a height of 1.5 R (aS (TM)) above the photosphere. Out of a range of initial equilibria which include strongly kink-unstable (twist I broken vertical bar=5 pi), weakly kink-unstable (I broken vertical bar=3.5 pi), and kink-stable (I broken vertical bar=2.5 pi) configurations, only the evolution of the weakly kink-unstable flux rope matches the observations in their entirety.
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.
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.