Twist and writhe measure basic geometric properties of a ribbon or tube. While these measures have applications in molecular biology, materials science, fluid mechanics and astrophysics, they are under-utilized because they are often considered difficult to compute. In addition, many applications involve curves with endpoints (open curves); but for these curves the definition of writhe can be ambiguous. This paper provides simple expressions for the writhe of closed curves, and provides a new definition of writhe for open curves. The open curve definition is especially appropriate when the curve is anchored at endpoints on a plane or stretches between two parallel planes. This definition can be especially useful for magnetic flux tubes in the solar atmosphere, and for isotropic rods with ends fixed to a plane.

We reconsider the topological interpretation of magnetic helicity for magnetic fields in open domains, and relate this to the relative helicity. Specifically, our domains stretch between two parallel planes, and each of these ends may be magnetically open. It is demonstrated that, while the magnetic helicity is gaugedependent, its value in any gauge may be physically interpreted as the average winding number among all pairs of field lines with respect to some orthonormal frame field. In fact, the choice of gauge is equivalent to the choice of reference field in the relative helicity, meaning that the magnetic helicity is no less physically meaningful. We prove that a particular gauge always measures the winding with respect to a fixed frame, and propose that this is normally the best choice. For periodic fields, this choice is equivalent to measuring relative helicity with respect to a potential reference field. But for aperiodic fields, we show that the potential field can be twisted. We prove by construction that there always exists a possible untwisted reference field.

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Magnetic helicity flux gives information about the topology of a magnetic field passing through a boundary. In solar physics applications, this boundary is the photosphere and magnetic helicity flux has become an important quantity in analysing magnetic fields emerging into the solar atmosphere. In this work we investigate the evolution of magnetic helicity flux in magnetohydrodynamic (MHD) simulations of solar flux emergence. We consider emerging magnetic fields with different topologies and investigate how the magnetic helicity flux patterns corresponds to the dynamics of emergence. To investigate how the helicity input is connected to the emergence process, we consider two forms of the helicity flux. The first is the standard form giving topological information weighted by magnetic flux. The second form represents the net winding and can be interpreted as the standard helicity flux less the magnetic flux. Both quantities provide important and distinct information about the structure of the emerging field and these quantities differ significantly for mixed sign helicity fields. A novel aspect of this study is that we account for the varying morphology of the photosphere due to the motion of the dense plasma lifted into the chromosphere. Our results will prove useful for the interpretation of magnetic helicity flux maps in solar observations. †

A measure of the writhing of a curve is introduced and is used to extend the Cȃlugȃreanu decomposition for closed curves, as well as the polar decomposition for curves bound between planes. The new writhe measure is also shown to be able to assess changes in linking due to belt-trick and knotting type deformations, and further its utility is illustrated on examples taken from elastic rod parameter-continuation studies. Finally C++ and Mathematica codes are made available and shown to be faster than existing algorithms for the numerical computation of the writhe.

We introduce a technique for evaluating the changing connectivity of a vector field whose integral curves (field lines) form tangled tubular bundles. Applications of such fields include magnetic flux ropes, relativistic plasma jets, stirred two-dimensional fluids, superfluid vortices, and polymer networks. The technique is based on maps of the field line winding-the average entanglement of a given field line with all other field lines. Previously this had been developed for divergence-free vector fields. By extending some previous theoretical results, we show how it can be applied to any vector field that forms a tubular bundle. We demonstrate the efficacy of this technique on data from laboratory plasma experiments with two interacting magnetic flux ropes. Performed in the UCLA Large Plasma Device, the plasma's magnetic field structure is too complex to identify a single dominant current sheet as an expected site of magnetic reconnection. Previously, this complex structure had restricted the ability to analyze the evolving magnetic connectivity, but this is no such restriction to our method. We demonstrate that the plasma establishes a periodically oscillating cycle of magnetic field structure variation which, while triggered by an ideal instability, is dominated by magnetic reconnection. This reconnection leads to periodically varying coherence of a merged central flux rope, a conclusion supported by analysis of the writhing structure of the magnetic field.

Aims. Sigmoidal structures in the solar corona are commonly associated with magnetic flux ropes whose magnetic field lines are twisted about a mutual axis. Their dynamical evolution is well studied, with sufficient twisting leading to large-scale rotation (writhing) and vertical expansion, possibly leading to ejection. Here, we investigate the behaviour of flux ropes whose field lines have more complex entangled/braided configurations. Our hypothesis is that this internal structure will inhibit the large-scale morphological changes. Additionally, we investigate the influence of the background field within which the rope is embedded. Methods. A technique for generating tubular magnetic fields with arbitrary axial geometry and internal structure, introduced in part I of this study, provides the initial conditions for resistive-MHD simulations. The tubular fields are embedded in a linear force-free background, and we consider various internal structures for the tubular field, including both twisted and braided topologies. These embedded flux ropes are then evolved using a 3D MHD code. Results. Firstly, in a background where twisted flux ropes evolve through the expected non-linear writhing and vertical expansion, we find that flux ropes with sufficiently braided/entangled interiors show no such large-scale changes. Secondly, embedding a twisted flux rope in a background field with a sigmoidal inversion line leads to eventual reversal of the large-scale rotation. Thirdly, in some cases a braided flux rope splits due to reconnection into two twisted flux ropes of opposing chirality -a phenomenon previously observed in cylindrical configurations. Conclusions. Sufficiently complex entanglement of the magnetic field lines within a flux rope can suppress large-scale morphological changes of its axis, with magnetic energy reduced instead through reconnection and expansion. The structure of the background magnetic field can significantly affect the changing morphology of a flux rope.

Magnetic helicity is a measure of the entanglement of magnetic field lines used to characterize the complexity of solar active region (AR) magnetic fields. Previous attempts to use helicity-based indicators to predict solar eruptive/flaring events have shown promise but not been universally successful. Here we investigate the use of a quantity associated with the magnetic helicity, the magnetic winding, as a means to predict flaring activity. This quantity represents the fundamental entanglement of magnetic field lines and is independent of the magnetic field strength. We use vector magnetogram data derived from the Helioseismic Magnetic Imager (HMI) to calculate the evolution and distribution of the magnetic winding flux associated with five different ARs, three of them with little flaring activity/nonflaring (AR 11318, AR 12119, AR 12285) and two highly active with X-class flares (AR 11158, AR 12673). We decompose these quantities into “current-carrying” and “potential” parts. It is shown that the ARs that show flaring/eruptive activity have significant contributions to the winding input from the current-carrying part of the field. A significant and rapid input of current-carrying winding is found to be a precursor of flaring/eruptive activity, and, in conjunction with the helicity, sharp inputs of both quantities are found to precede individual flaring events by several hours. This suggests that the emergence/submergence of topologically complex current-carrying field is an important element for the ignition of AR flaring.