Flux Ropes as Singularities of the Vector Potential

Abstract

A flux rope is a domain of concentration of the magnetic field $\textbf{B}$. Insofar as $\textbf{B}$ outside such a domain is considered as vanishingly small, a flux rope can be described as the core of a singularity of the outer vector potential $\textbf{A}$, whose topological invariant is the magnetic flux through the rope. By 'topological' it is meant that $\oint_C\textbf{A}\cdot\mathrm d\textbf{s}$ measures along any loop $C$ surrounding the flux rope the same constant flux $\Phi$. The electric current intensity is another invariant of the theory, but non-topological. We show that, in this theoretical framework, the linear force-free field (LFFF) Lundquist model and the non-linear (NLFFF) Gold-Hoyle model of a flux rope exhibit stable solutions distributed over quantized strata of increasing energies (an infinite number of strata in the first case, only one stratum in the second case); each stratum is made of a continuous set of stable states. The lowest LFFF stratum and the unique NLFFF stratum come numerically close one to the other, and match with a reasonable accuracy the data collected by spacecrafts travelling across magnetic clouds. The other LFFF strata do not match these data at all. It is not possible at this stage to claim which model fits better the magnetic cloud data. We also analyze in some detail the merging of tubes belonging to the same stratum, with conservation of the magnetic helicity, and the transition of a tube from one stratum to another one, which does not conserve magnetic helicity.