The deceiving Δ ′: on the equilibrium dependent dynamics of nonlinear magnetic islands

Abstract

The linear stability parameter Δ ′ is commonly used as a figure of merit for the nonlinear dynamics of the tearing mode. It is shown, through numerical simulations, that factors other than Δ ′ can play a very important role in determining the evolution of nonlinear magnetic islands, even relatively close to marginal stability. In particular, two different equilibria are analysed and it is shown that, once perturbed, they have a qualitatively and quantitatively different response despite the fact that they are characterised by the same Δ ′. However, the different behaviour can still be associated with linear properties of the equilibrium. It is also studied how the nonlinear and saturation phase are affected by an increasing Δ ′ in the two equilibria. As the instability drive is increased, the systems move from a dynamics characterised by a "universal" generalised Rutherford equation to a Y-point configuration and then to a plasmoid unstable Y-point. Remarkably, in certain configurations the Rutherford phase is absent and the system forms a current ribbon without an X-point collapse.