David N. Hosking

Imagine stirring up a fluid with very low viscosity and then allowing the turbulent motions to decay away. How would the energy of the flow depend on time? The answer is that energy decays at the “turnover” (roughly, rotation) rate of the largest eddies. As the flow becomes increasingly weak, that rate is constrained by dynamical invariants related to the random momentum carried by large patches of fluid. This enables the energy to be calculated as a function of time.

What about magnetic fields in a plasma? In this case, magnetic structures relax towards local quasi-equilibrium states with random, non-zero magnetic helicity—a measure of the twist and linkage of magnetic field lines. Even when the total magnetic helicity vanishes, large regions contain random excesses of positive or negative helicity, and a statistical measure of these fluctuations remains approximately conserved as the field relaxes. In cases where the energy of the magnetic field dominates over that of plasma flows, magnetic reconnection—the breaking and rejoining of field lines that lets neighbouring structures merge—controls the decay timescale, while helicity conservation constrains how field strength and structure size evolve together (Phys. Rev. X, 2021).

These ideas explain several puzzling features of relaxing chaotic magnetic fields, chief among them their tendency to transfer energy to larger scales, even when their total helicity vanishes. They also have consequences for cosmology: they imply that magnetic fields generated at the electroweak phase transition, a fraction of a nanosecond after the Big Bang, could survive to the present with strengths several orders of magnitude greater than earlier estimates, consistent with magnetic-field constraints inferred from gamma-ray observations of blazars (Nature Commun., 2023).

Magnetically confined plasmas in fusion experiments can sometimes lose their equilibria in dramatic fashion. In tokamaks operating in high-confinement mode, type-I edge-localised modes (ELMs) expel hot plasma from the edge, causing potentially catastrophic heat loads on surrounding components. One explanation for these abrupt eruptions is that the plasma edge is metastable beforehand: stable to small perturbations, but unstable to sufficiently large ones. I have been looking for simplified plasma regimes in which such states and their relaxation can be studied tractably.

Two-dimensional interchanges of flux tubes, in which tubes of magnetic flux swap places much as buoyant fluid rises in convection, turn out to be one. There, we can calculate the available energy of a metastable equilibrium (the energy it could release by rearranging its flux tubes) by combinatorial optimisation, and develop statistical predictions for the state to which it relaxes (J. Plasma Phys., 2025).

A rotating Z-pinch, a column of plasma confined by the magnetic field of a current flowing along it, provides a second example. In that case, a sheared flow can sustain a transport barrier: a layer in which a steep pressure gradient is held stable. Under slow heating, the barrier steepens until a stable flux-tube equilibrium disappears in a saddle-node bifurcation, triggering an eruption. The transport barrier then rebuilds, and the cycle repeats (arXiv, 2026).

Notes from the lectures I gave at the 2023 NSF/APS-DPP Summer School on Plasma Physics for Astrophysicists, on the fundamentals of fluid dynamics and on MHD relaxation.