Asymptotic windings, surface helicity and their applications in plasma physics

TL;DR

This paper explores the mathematical properties of surface helicity in plasma physics, providing rigorous interpretations, linking it to surface topology, and applying these insights to optimize coil designs in fusion devices.

Contribution

It offers a rigorous physical interpretation of surface helicity, relates it to surface topology, and connects it to windings and rotational transform in plasma physics, with applications to coil design.

Findings

Surface helicity is non-trivial only on surfaces with holes.

Surface helicity relates to average poloidal and toroidal windings.

Symmetric toroidal surfaces minimize surface helicity among fixed-area surfaces.

Abstract

In [J. Cantarella, J. Parsley, J. Geom. Phys. 60:1127 (2010)] Cantarella and Parsley introduced the notion of submanifold helicity. In the present paper we investigate properties of surface helicity and in particular answer two open questions posed in the aforementioned work: (i) We give a precise mathematically rigorous physical interpretation of surface helicity in terms of linking of distinct field lines. (ii) We prove that surface helicity is non-trivial if and only if the underlying surface has non-trivial topology (i.e. at least one hole). We then focus on toroidal surfaces which are of relevance in plasma physics and express surface helicity in terms of average poloidal and toroidal windings of the individual field lines of the underlying vector field which enables us to provide a connection between surface helicity and rotational transform. Further, we show how some of our results may be utilised in the context of coil designs for plasma fusion confinement devices in order to obtain coil configurations of particular "simple" shape. Lastly, we consider the problem of optimising surface helicity among toroidal surfaces of fixed area and show that toroidal surfaces admitting a symmetry constitute global minimisers.