Theory and Applications of Helicity

Abstract

Linking number, writhe and twist are three important measures of a curve's geometry. They have been well studied and their de nitions extended to open curves situated between two horizontal planes [1]. However, many applications of these tools involve geometries that have a curved nature to them [2]. For example, the magnetic coronal-loops in the Sun's atmosphere share a spherical boundary (the photosphere). We reformulate these ideas in a spherical geometry, and then explore the oddities of this curved space to show that our new concept is consistent with its older, at counterpart. The second part of this project concerns a series of datasets from plasma experiments at Basic Plasma Science Facility, UCLA, Los Angeles. These experiments involve the creation of ux ropes inside a large (18m) plasma machine. A strong background magnetic eld is applied which ensures that eld lines travel from one end of the cylindrical device to the other. Due to mutual J B forces, the ux ropes twist and tangle about each other. We study three separate datasets: the rst one involving two ux ropes; the second, three ux ropes; the nal two ux ropes. The last experiment is perhaps the most exciting as the plasma velocity has been recorded. This extra data allows us to employ two di erent non-equivalent concepts of magnetic helicity. First, we use the surface ux formulation that makes various ideal assumptions, discarding several terms in Ohm's law. This is compared to helicity calculated by use of winding numbers { a construction without these ideal assumptions. By examining the di erence of these two results, it is shown that we may arrive at a measure of the resistivity present in the system. The plasma investigations described above rely on being able to seed magnetic eld lines across the length of the machine. This is not a simple process. The dataset itself is spatially non-uniform which makes numerical integration to obtain eld lines di cult. Even before integration is considered, a method to interpolate on our data grid of magnetic ux density is needed. This requires further careful considerations. Any interpolator must ensure that the data remains divergence-free; this requirement imposes conditions on the continuity of the derivatives. We have written a code to perform tricubic spline interpolation, and demonstrate that by using a particular method for xing the coe cients, this level of continuity can be achieved.