Damping of Alfvén eigenmodes in complicated tokamak and stellarator geometries

A variety of Alfvén wave phenomena are found in toroidal magnetically confined fusion plasmas. Shear Alfvén eigenmodes may exist, which can be driven unstable by interaction with energetic particles. The linear stability of such modes depends on damping through several mechanisms. Continuum resonances cause damping of the modes, which occurs even in non-dissipative ideal magnetohydrodynamic (MHD) theory given appropriate treatment of resulting poles. Additional damping of the modes occurs due to conversion to kinetic Alfvén waves and finite parallel electric fields when kinetic extensions to MHD are considered. In this thesis, methods for calculating the damping of Alfvén eigenmodes are developed, with particular focus on the continuum damping component. Damping of modes in complicated twoand three-dimensional magnetic geometries characteristic of tokamak and stellarator plasmas is considered. In this work, shear Alfvén eigenmodes are analysed based on reduced MHD models. A background is provided, covering relevant theoretical aspects of plasma equilibrium, coordinate systems and linearised MHD waves. A coordinate independent reduced MHD wave equation is derived for Alfvén eigenmodes in low β tokamaks and stellarators. Coupled wave equations in terms of Fourier harmonics of the eigenmode are then derived for large aspect-ratio plasmas. Expressions for continuum damping are derived perturbatively from the coordinate independent and coupled harmonic wave equations. Application of the expressions using Galerkin and shooting methods is described. Damping computed in this manner is compared with values from an accepted method for the benchmark case of a TAE in a large aspect-ratio circular cross-section tokamak. The perturbative technique is shown to produce significant errors, even where continuum damping is small. A novel singular finite element method is developed to compute continuum damping. The Galerkin method adopted employs special basis functions reflecting the asymptotic form of the solution near continuum resonance poles. For particular eigenmodes, the unknown complex eigenvalue and pole location are computed iteratively. The procedure is verified by application to a TAE in a large aspect-ratio circular cross-section tokamak, where well converged and accurate complex eigenvalue and mode structure are obtained. Continuum damping can be computed numerically by solving the ideal MHD eigenvalue problem over a complex contour which circumvents continuum resonance poles according to the causality condition. This calculation is implemented in the ideal MHD eigenvalue code CKA, using analytic continuation of equilibrium quantities. The method is verified through application to a TAE in a tokamak, where the complex eigenvalue computed agrees closely with that found using the accepted resistive method, but converges faster with increasing radial mesh resolution. Continuum damping of shear Alfvén eigenmodes is computed for three-dimensional configurations in torsatron, helias and heliac stellarators. Extensions to the ideal MHD wave equations allow non-ideal kinetic effects to be modelled. The damping of a TAE in a tokamak case through these effects is computed using different models for magnetic geometry and kinetic effects. Choice of the former strongly influences results, while choice of the latter does not. Damping from kinetic effects is also computed for an NGAE in a heliac