Abstract
Braided vector fields on spatial subdomains which are homeomorphic to the cylinder play a crucial role in applications such as solar and plasma physics, relativistic astrophysics, fluid and vortex dynamics, elasticity, and bio-elasticity. Often the vector field’s topology—the entanglement of its field lines—is non-trivial, and can play a significant role in the vector field’s evolution. We present a complete topological characterisation of such vector fields (up to isotopy) using a quantity called field line winding. This measures the entanglement of each field line with all other field lines of the vector field, and may be defined for an arbitrary tubular subdomain by prescribing a minimally distorted coordinate system. We propose how to define such coordinates, and prove that the resulting field line winding distribution uniquely classifies the topology of a braided vector field. The field line winding is similar to the field line helicity considered previously for magnetic (solenoidal) fields, but is a more fundamental measure of the field line topology because it does not conflate linking information with field strength.
Highlights
The entanglement of vector field integral curves in tubular subdomains homeomorphic to the cylinder has long been of wide interest
As well as extending the class of domains and vector fields considered, we prove that the field line winding measure uniquely determines the field line mapping and whether or not the two vector fields can be linked by an ideal evolution
Our goal in this paper is to describe the field-line topology of a braided vector field v in M
Summary
The entanglement of vector field integral curves (field lines) in tubular subdomains homeomorphic to the cylinder has long been of wide interest. C B Prior and A R Yeates bundles of magnetic field lines—known as magnetic flux ropes—are potentially important for the generation of large scale magnetic fields (Childress and Gilbert 1995, Moffatt and Proctor 1985, Nordlund et al 1992, Bao and Yang 2010) In stellar atmospheres such as the Sun’s corona, twisted and braided magnetic fields play a crucial role in dynamic phenomena such as coronal heating, jet formation, and coronal mass ejections (Rust and Kumar 1996, Török and Kliem 2005, Wilmot-Smith 2015, Prior and MacTaggart 2016, Yeates and Hornig 2016). Prior and Yeates (2014) showed that the H admits an analogous interpretation for so-called ‘braided’ magnetic fields, where magnetic field lines connect between two planar boundaries rather than being tangent In this case, H is gauge dependent, but the authors showed that there is a particular gauge, the ‘winding gauge’, in which H is the equal to the average winding number between all pairs of field line curves.
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