Core MHD instabilities and their spectra in advanced tokamak regimes: ideal and resistive physics

Advanced tokamak regimes, featuring extended regions of low magnetic shear, are promising candidates for future fusion reactors (record breaking DT pulses at JET in hybrid mode) but are also more prone to specific MHD instabilities. The proximity to a rational surface in a very low shear region weakens field line bending stabilisation and amplifies the effects of toroidal coupling between modes, leading to the emergence of long-wavelength ideal or resistive infernal modes. These modes can grow collectively as a discrete spectrum, leading to a cascade of different perturbations for single mode numbers (m, n). Subdominant modes with lower growth rates have distinct radial structures, oscillating more (they can be characterized by Bessel functions). These spectra of fast-growing modes are significant for developing stable scenarios for future reactors, and for the understanding of global reconnection events like sawteeth, motivating a deeper investigation into their fundamental physics.

Considering resistive diffusion, compressibility, toroidal effects, and shaping, we explore in this thesis the critical regions and parameters which influence the linear stability of advanced tokamak regimes. To do so, we start from an analytic model, which provides a unified and global description of pressure and current driven internal instabilities, in an inverse aspect ratio expansion. The first part of the thesis details the main steps of this derivation and presents an extension to include some flux surface shaping effects. We then develop a new resistive modular MHD eigensolver based on this analytical description. This tool is introduced and verified across various ideal and resistive configurations in the early part of the thesis. The main results are then divided into ideal and resistive chapters, with the ideal part featuring mainly analytical derivations, which are checked against the solver, in order to verify newly established theoretical stability criteria. The impact of using reduced MHD models is assessed for various types of instabilities.

For non-resonant monotonic $q$ profiles, we derive an analytic dispersion relation with higher-order corrections in $q-q_s$, providing a more accurate description of a non-resonant kink mode spectrum with $q_s=1$. In the presence of resonance, for $q_s