On the existence, stability and relaxation of discontinuous magnetohydrodynamic states

Abstract

In the continuum limit, plasmas are governed by magnetohydrodynamic (MHD) models - nonlinear dynamical systems that admit a wealth of spatially and temporally complex phenomena. Under many circumstances, MHD phenomena are strongly dictated by the structure of equilibrium magnetic fields. In the absence of dissipation (e.g. resistivity), smooth pressure profiles are only guaranteed for configurations with continuously nested flux surfaces, the existence of which is guaranteed with axisymmetry. This thesis explores whether equilibria of the Multi-Region Relaxed MHD (MRxMHD) model, which produces MHD equilibria with discontinuous, stepped pressure profiles, are physically and dynamically accessible.

The accessibility of two characteristic features of the MRxMHD model are examined; (i) the `ideal' interfaces, across which pressure discontinuities of the MRxMHD model are supported, and (ii) volume-localised Taylor relaxation, where the plasma in each constant-pressure region is hypothesised to preferentially relax to an energy-minimising state while conserving volume-averaged mass, entropy, magnetic helicity and fluxes within each region. To address these questions, the inextricably linked physical concepts of energy-minimising equilibria and relaxation models which describe evolution towards energetically favourable states are interrogated.

A new equilibrium model with radially localised pressure gradients is developed to directly compare physical properties, including linear MHD stability, of smooth and non-smooth equilibria. The results suggest that MRxMHD interfaces can be interpreted as an approximation of highly localised, continuous pressure gradients. Analytic studies and initial-value extended-MHD simulations highlight that volume-localised Taylor relaxation is not guaranteed to be the preferred plasma evolution pathway. A way to explicitly treat non-ideal plasma parameters (e.g. resistivity) as part of the MRxMHD framework is needed to better constrain the dynamical accessibility of MRxMHD equilibria. Nonetheless, these preliminary results are consistent with the occurrence of volume-localised Taylor relaxation, under suitable conditions. This work substantiates conditions under which MRxMHD-like states can form. Combining these findings with the existing literature, a new avalanche scenario is proposed to model partial and global macroscopic relaxation events in magnetically confined fusion plasmas. Starting from an N-volume MRxMHD equilibrium, an external perturbation triggers the break-up of an MRxMHD interface whereupon the intermediate state re-equilibrates via volume-localised Taylor relaxation to an (N-1)-volume state. The procedure is repeated until all remaining interfaces satisfy an established MRxMHD interface existence criterion. A possible scheme for coupling this avalanche process to evolution on a slower transport timescale is outlined. The potential for realising significant gains in computational efficiency (compared to initial-value extended-MHD calculations, for example) opens up opportunities in areas such as optimisation and machine learning.