Calculation of continuum damping of Alfv\'en eigenmodes in 2D and 3D cases

Abstract

In ideal MHD, shear Alfv\'{e}n eigenmodes may experience dissipationless damping due to resonant interaction with the shear Alfv\'{e}n continuum. This continuum damping can make a significant contribution to the overall growth/decay rate of shear Alfv\'{e}n eigenmodes, with consequent implications for fast ion transport. One method for calculating continuum damping is to solve the MHD eigenvalue problem over a suitable contour in the complex plane, thereby satisfying the causality condition. Such an approach can be implemented in three-dimensional ideal MHD codes which use the Galerkin method. Analytic functions can be fitted to numerical data for equilibrium quantities in order to determine the value of these quantities along the complex contour. This approach requires less resolution than the established technique of calculating damping as resistivity vanishes and is thus more computationally efficient. The complex contour method has been applied to the three-dimensional finite element ideal MHD code CKA . In this paper we discuss the application of the complex contour technique to calculate the continuum damping of global modes in tokamak as well as torsatron, W7X and H1-NF stellarator cases. To the authors' knowledge these stellarator calculations represent the first calculation of continuum damping for eigenmodes in fully three-dimensional equilibria. The continuum damping of global modes in W7X and H1-NF stellarator configurations investigated is found to depend sensitively on coupling to numerous poloidal and toroidal harmonics.