Weakly damped shear Alfvén eigenmodes can occur in toroidal magnetically confined fusion plasmas, which may be destabilised through interaction with fast particles. Unstable modes can increase fast particle transport, degrading plasma facing components and reducing heating of the thermal plasma. The linear growth rate and saturation amplitude of these modes depend on the damping rate. In this presentation the calculation of damping rates of shear Alfvén eigenmodes in complicated two- and three-dimensional cases is discussed.
Continuum damping represents the resonant transfer of energy from the global modes to highly damped and spatially localised continuum modes. Counterintuitively, this damping can be calculated using non-dissipative ideal MHD through appropriate treatment of the continuum resonance poles of the force operator. One analytic technique for calculating continuum damping is by representing the continuum resonance through a perturbation to the Lagrangian describing the linearised dynamics of the wave . A numerical technique is to solve the shear Alfvén eigenvalue problem over a causal complex contour . Kinetic extensions to MHD introduce additional damping due to conversion to kinetic Alfvén waves, particle collisions and Landau damping.
Results of the perturbative technique for calculating continuum damping are compared with those of the complex contour technique. A novel finite element technique is described in which continuum damping is calculated using singular basis functions. The extension of the complex contour technique to two- and three-dimensional magnetic geometries using a finite element method is discussed. This complex contour technique is used to perform the first continuum damping calculations in realistic three-dimensional geometry. Results of continuum damping calculations for shear Alfvén eigenmodes in tokamak, torsatron and H-1NF heliac configurations are presented. Additionally, results of kinetic damping calculations are compared for different models of magnetic geometry and kinetic effects. Damping due to kinetic effects is calculated for an NGAE in the H-1NF heliac.
 Berk et al. Physics of Fluids B 4(7):1806-1835, 1992
 Könies & Kleiber, Physics of Plasmas 19(12):122111, 2012