Beyond Axisymmetry in Tokamak Plasmas

H-mode tokamak plasmas are characterised by quasi-periodic instabilities, called edge localised modes (ELMs), driven by unstable peeling-ballooning modes inside the pedestal region. For large scale tokamaks, like ITER, the resulting particle and heat fluxes are predicted to be unacceptable and ELM control methods are required. One promising method relies on the application of 3D resonant magnetic perturbations (MPs), and ELM mitigation or even complete suppression is observed. A computational framework is presented that aims to understand the effect of MPs on both plasma equilibria and stability. The ELITE stability code is used to find the linearised plasma response, i.e. the 3D part of the equilibrium, and compute the axisymmetric peeling-ballooning eigenmodes. This information is used to calculate the 3D stability under a perturbative and a variational formulation of the MHD energy principle. In practice, the axisymmetric peeling-ballooning modes are used as trial functions for the minimisation of the 3D energy functional. The symmetry breaking of the toroidal geometry leads to the coupling of toroidal modes which has a direct impact on the linear growth rates of unstable peeling-ballooning modes. This mechanism results in the modification of the plasma stability above a critical value of the applied MP field and field-line localisation of the peeling-ballooning eigenmode. It is observed that intermediate to high n ballooning modes are in general destabilised by the applied MP field, while external peeling-ballooning modes reorganise to an internal ballooning structure. In addition, extrema in the growth rate spectrum, due to low n kink modes, are observed to be strongly destabilised as predicted by perturbation theory. This work provides proof of principle examination of the 3D peeling-ballooning instability as well as a framework for the optimisation of MP coil configuration.