New energy and helicity bounds for knotted and braided magnetic fields

Abstract

In this article we present a review of some of the author's most recent results in topological magnetohydrodynamics (MHD), with an eye to possible applications to astrophysical flows and solar coronal structures. First, we briefly review basic work on magnetic helicity and linking numbers, and fundamental relations with magnetic energy and average crossing numbers of magnetic systems in ideal conditions. In the case of magnetic knots, we focus on the relation between their groundstate energy and topology, discussing the energy spectrum of tight knots in terms of ropelength. We compare this spectrum with the one given by considering the bending energy of such idealized knots, showing that curvature information provides a rather good indicator of magnetic energy contents. For loose knots far from equilibrium we show that inflexional states determine the transition to braid form. New lower bounds for tight knots and braids are then established. We conclude with results on energy-complexity relations for systems in presence of dissipation.

Keywords:

• Magnetic knots
• Links
• Braids
• Energy spectrum
• Ropelength
• Helicity
• Crossing number

Acknowledgments

R.L.R. wishes to thank Francesca Maggioni for her help to produce the plots of figures 6 and 7 based on SONO data.