Magnetic Helicity and the Calabi Invariant

In this thesis we investigate the relationship between the Calabi invariant (Calabi, 2015) and magnetic helicity. We establish under which conditions both measures agree and importantly under which geometric conditions we can apply the Calabi invariant. We demonstrate how the Calabi invariant can be used to refine the techniques available in understanding the winding of magnetic fields tied to astrophysical plasmas. After an introduction to the literature on magnetic helicity, asymptotic winding and the Calabi invariant itself, we establish in Chapter 6 the first results of this thesis. We derive an alternative expression for the Calabi invariant in terms of the pullback of a symplectic potential and readily apply this expression to calculate the helicity of a magnetic field in a cylindrical domain. Next we adapt the Calabi invariant to relative field line helicity and in addition provide an extension of the Calabi invariant to fields with non-vanishing boundary contributions. Examples in cylindrical and spherical coordinates are provided. Following from there, in Chapter 7 we apply the Calabi invariant to a magnetic field modelled on an iterated braid (Wilmot-Smith et al., 2008, 2015). Poincar'e plots are provided to illustrate that the Calabi invariant can allow us to consider the small-scale behaviour of magnetic fields whose overall classical helicity vanishes. In this process we construct a gauge invariant Asymptotic Field Line Helicity for space-filling sets of field lines. A connection between the asymptotic field line helicity and the Calabi invariant is demonstrated, along with a connection to the Winding Number (Prior and Yeates, 2014), Schwartzman Cycles (Schwartzman, 1957) and Fathi's Lemma (Fathi, 1980). In Chapter 8 we note that the constructions provided in Chapters 6 and 7 require the existence of a (global) cross-section to the flow. In general the existence of this cross-section for a generic flow in arbitrary domain is difficult to demonstrate. However an existing result on intrinsically harmonic forms (Calabi, 1969) is applied to a magnetic field in a cylindrical domain in an attempt to establish existence criteria. Moreover the case of right-handed vector fields (Ghys, 2009) is discussed, and the existence of a bounding field line for the cross-section is proposed. Finally we end the thesis with a short review of the material covered, the results given and some comments on how to develop this work moving forward.