# UCL Discovery

## The theory and applications of writhing

Prior, C.B.; (2010) The theory and applications of writhing. Doctoral thesis , UCL (University College London).

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## Abstract

Writhe measures the extent to which a curve is kinked and coiled about itself in space. It has generally been expressed as a double integral. This measure can be interpreted as the average number of signed crossings seen by each viewer, over all possible viewpoints of the curve. This simple geometrical interpretation is used to describe the established properties of the writhe, as applied to closed spacecurves. These descriptions differ from previous work as they do not require the construction of an artificial ribbon structure. A major feature of this thesis concerns the evaluation of the writhe along a preferred direction. A directional measure termed the polar writhe will be developed which can be applied to generic curves (open or closed) . This single integral expression is shown to be equivalent to the double integral writhe measure for closed curves. However for open curves the two measures are shown to differ. Further, it is shown that the polar writhe has distinct advantages when analysing curves with a strong directional bias. The thesis then discusses in detail the properties of both the writhe and the polar writhe measures for open curves. The use of artificial closures for both measures is examined. In the case of the writhe a new closure is defined that allows the evaluation of the writhe using single integral expression via the theorems of Fuller. This closure is unique in that it can be applied to open curves whose end points are in general position. A simple expression for calculating the non-local polar writhe is derived which generalises a closed curve expression defined in (Berger Prior J. Phys. A: Math. Gen. 39, 8321-8348, (2006)). A quantitative study on the effect of the choice of evaluation direction of the polar writhe is conducted. The polar writhe formulation is applied to a simple linear force-free magnetic field model where the field lines form loops above a boundary plane. Loops with a sufficient amount of kinking are generally seen to form S or inverse S (Z) shaped structures. Such field lines structures are commonly observed in the Sun’s corona. A popular measure of the field line morphology is the magnetic helicity. We use the polar writhe, the correct form for the writhe helicity in the coronal region, to challenge some popular assumptions of the field. Firstly, the writhe of field lines of significant aspect ratio (the apex height divided by the foot point width) can often have the opposite sign to that assumed in a recent review paper by Green et al (Solar Phys., 365-391, (2007)). Secondly, we demonstrate the possibility of field lines forming apparent Z shaped structures which are in fact constructed from a pair of S shapes and have a writhe sign expected of an S shaped structure. Such field lines could be misinterpreted without full knowledge of the line’s three dimensional structure. Thirdly, we show that much of the interesting morphological behaviour occurs for field lines located next to separatrices.

Type: Thesis (Doctoral) The theory and applications of writhing An open access version is available from UCL Discovery English UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics http://discovery.ucl.ac.uk/id/eprint/19502