@phdthesis{Feher2014,
  author    = {Tam{\´a}s B{\´e}la Feh{\´e}r},
  title     = {Simulation of the interaction between Alfv{\´e}n waves and fast particles},
  journal    = {Simulation der Wechselwirkung zwischen Alfv{\´e}n-Wellen und schnelle Teilchen},
  url       = {https://nbn-resolving.org/urn:nbn:de:gbv:9-001742-4},
  year      = {2014},
  abstract  = {There is a wide variety of Alfv{\´e}n waves in tokamak and stellarator plasmas. While most of them are damped, some of the global eigenmodes can be driven unstable when they interact with energetic particles. By coupling the MHD code CKA with the gyrokinetic code EUTERPE, a hybrid kinetic-MHD model is created to describe this wave–particle interaction in stellarator geometry. In this thesis, the CKA-EUTERPE code package is presented. This numerical tool can be used for linear perturbative stability analysis of Alfv{\´e}n waves in the presence of energetic particles. The equations for the hybrid model are based on the gyrokinetic equations. The fast particles are described with linearized gyrokinetic equations. The reduced MHD equations are derived by taking velocity moments of the gyrokinetic equations. An equation for describing the Alfv{\´e}n waves is derived by combining the reduced MHD equations. The Alfv{\´e}n wave equation can retain kinetic corrections. Considering the energy transfer between the particles and the waves, the stability of the waves can be calculated. Numerically, the Alfv{\´e}n waves are calculated using the CKA code. The equations are solved as an eigenvalue problem to determine the frequency spectrum and the mode structure of the waves. The results of the MHD model are in good agreement with other sophisticated MHD codes. CKA results are shown for a JET and a W7-AS example. The linear version of the EUTERPE code is used to study the motion of energetic particles in the wavefield with fixed spatial structure, and harmonic oscillations in time. In EUTERPE, the gyrokinetic equations are discretized with a PIC scheme using the delta-f method, and both full orbit width and finite Larmor radius effects are included. The code is modified to be able to use the wavefield calculated externally by CKA. Different slowing-down distribution functions are also implemented. The work done by the electric field on the particles is measured to calculate the energy transfer between the particles and the wave and from that the growth rate is determined. The advantage of this approach is that the full magnetic geometry is retained without any limiting assumptions on guiding center orbits. Extensive benchmarks have been performed to test the new CKA-EUTERPE code. Three tokamak benchmarks are presented, where the stability of TAE modes are studied as a function of fast particle energy, or in one case as a function of the fast particle charge. The benchmarks show good agreement with other codes. Stellarator calculations were performed for Wendelstein 7-AS and the results demonstrate that the finite orbit width effects tend to be strongly stabilizing.},
  language  = {en}
}