Webinar on magnetic topology (23/04/20 at 3pm BST)

There will be three talks of ~35mins each plus some question time. The entire webinar will last approximately 2 hours and will take place using Zoom. Please contact one of the organisers (Simon Candelaresi, simon.candelaresi at glasgow.ac.uk; David MacTaggart, david.mactaggart at glasgow.ac.uk) if you have any questions.

The link to the Zoom webinar will be sent to individuals directly.


  • Sauli Lindberg (University of Helsinki): Taylor's conjecture on magnetic helicity conservation

    Woltjer recognised magnetic helicity as a conserved quantity of ideal MHD. He proceeded to formulate Woltjer's variational principle which leads to the prediction of a relaxed state where the magnetic field is force-free. As later noted by Moffatt, ideal MHD also has many local conserved quantities (the subhelicities over magnetically closed subvolumes that move with the fluid). The relaxed state is nevertheless observed to be essentially independent of the local behaviour of the initial state. A way out of the dilemma was conjectured by Taylor in 1974: in the presence of slight resistivity, the subhelicities cease to be conserved but the total magnetic helicity remains an approximate invariant. Berger showed in 1984 (under mild extra assumptions) that magnetic helicity dissipates much slower than magnetic energy. However, a rigorous mathematical proof straight from the MHD equations has been lacking. In mathematical terms, Taylor's conjecture translates into the statement that magnetic helicity is conserved in the ideal (inviscid, non-resistive) limit. I will discuss my recent proof, in collaboration with Daniel Faraco, of the mathematical version of Taylor's conjecture.

  • Long Chen (University of Durham): The topology of resistive magnetic relaxation

    In plasmas, complex braided magnetic structures are known to self-organise into simple configurations. While Parker (1983) has conjectured that reconnection events will ultimately lead to only two flux tubes with opposite helicity, Yeates et al. (2010, 2015) have identified the topological degree as a separate constraint. We trace the evolution of topology using a combination of 3D and 2D simulations. We find that there are two distinct phases: a fast reconnection phase constrained by the topological degree, followed by a diffusion dominated phase with merging of discrete flux tubes. Resistivity and boundary conditions both affect the reconnection events and the resulting topology. Whether the final state could reach Parker’s state depends on various aspects, which I will discuss in this talk.

  • Christopher Berg Smiet (Princeton Plasma Physics Lab): The alternating-hyperbolic sawtooth

    The sawtooth oscillation is an instability in the core of Tokamak fusion reactors where a fast instability mixes the plasma, reducing the temperature and efficiency of fusion reactions. Conventional theory predicts that this crash is caused when the central safety factor q_0 (inverse of the winding number of field lines at the location of the magnetic axis) is 1, but a significant number of observations has measured a crash at q_0 ~ 0.7. We present a new model that explains these observations through a topological change of the field surrounding the magnetic axis. We construct a correspondence between the structure of the field around the magnetic axis and elements of the simple Lie group SL(2, R). The group structure indicates a topological transition of the magnetic surfaces surrounding the axis into an alternating hyperbolic geometry when q_0=2/3, which fits with measurements of q_0=0.7 ±1. We find an unstable ideal 2/3 mode localized on the axis, and the magnetic perturbation associated with this mode induces the topological transition of the axis. At high amplitude this mode can stochastisize a region around the axis, which leads to redistribution of fluxes and a reset of the sawtooth cycle.