Non-linear modelling of saturated internal and external MHD instabilities in tokamaks

Macroscopic instabilities and their unfavourable effects on plasma confinement pose a central challenge for the development of reactor relevant tokamak scenarios. Some promising operation scenarios feature extended regions of low magnetic shear. These are in the core region for hybrid scenarios, and in the pedestal-edge region for e.g. the ELM-free quiescent H-mode (QH-mode). This thesis presents a non-linear study of macroscopic magnetohydrodynamic (MHD) instabilities with a focus on plasmas with regions of low magnetic shear. In the first part, we investigate the triggering of fast growing resistive modes in hybrid tokamak plasmas, where infernal modes can couple to neoclassical tearing modes (NTMs). Numerical simulations with the non-linear resistive initial value code XTOR-2F with and without inclusion of bootstrap current effects allow us to determine the evolution of MHD modes from the linear to the late non-linear phase. An analytical model is developed to describe the vanishing of mode coupling in the early non-linear regime. In this context, we extend the tearing stability parameter $\Delta'$ from the linear to the non-linear phase and calculate the individual contributions to the growth of the resistive mode. This allows for an identification of the triggering mechanism in the initial phase and the dominant terms in the non-linear phase. A comparison with the numerical results shows that infernal mode coupling can destabilise otherwise stable NTMs and thus seed $2/1$ magnetic islands. The helically perturbed bootstrap current is found to further destabilise the magnetic islands in the non-linear phase. In the second part of the thesis, 3D free-boundary equilibrium computations are employed to describe saturated external kink-type modes. The approach is first demonstrated to capture the salient features of non-linearly saturated external kink modes in standard baseline tokamak scenarios and is then applied to QH-mode plasmas. A method to conveniently extract the saturated displacement amplitude from edge-corrugated VMEC equilibria in terms of straight field line Fourier modes is presented. For standard current-driven modes the amplitude is compared with a non-linear analytical model, for which a numerical solver is implemented. In QH-mode plasmas, dangerous edge localised modes (ELMs) are replaced by benign edge harmonic oscillations (EHOs). The latter are commonly assumed to be connected to saturated external kink states. In our study of QH-mode plasmas, we consider two different driving mechanisms for external kink type-modes separately. We find that standard current-driven external kinks are linearly unstable, and non-linearly stable in a wide parameter range, especially where $q_{\text{edge}} \lesssim m/n$. But, where standard current-driven kinks are linearly stable we find that coupling of pressure-driven infernal modes can cause instability, and their upper sideband drives edge corrugations that appear to have external kink features. Both types of modes are identified with the VMEC equilibrium code, and the spectra are compared favourably with those of linear numerical approaches and analytic methods. EHOs could be connected to both type of modes.