## Doctoral Thesis

There is a wide variety of AlfvÃ©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 waveâ€“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 AlfvÃ©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 AlfvÃ©n waves is derived by combining the reduced MHD equations. The AlfvÃ©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 AlfvÃ©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.

Electromagnetic Drift Waves
(2010)

In the rf-plasma of the linear magnetized VINETA experiment, different types of low-frequency waves are observed. The emphasis in this work is on the interaction mechanism between drift waves on the one and kinetic Alfven waves on the other hand. In the peaked density profile of the plasma column drift waves occur as modulation of the plasma density. As gradient driven instability, they draw their energy from the radial density gradients. Alfven waves as magnetic field fluctuations are stable in the present configuration. They are launched by a magnetic excitation antenna. Parallel conduction currents in the plasma are common to both wave phenoma. A B-dot probe as standard diagnostic tool is used to detect the fluctuating magnetic fields of both wave types. The challenge are the small induced voltages due to the low wave frequency. The probe design with an integrated amplifier close to the probe head takes this into acount. The developed B-dot probe is mounted to different positioning systems to characterize both wave phenomena. For Alfven waves, the dispersion relation is recorded experimentally. It is found to be in good agreement with the prediction of the Hall-MHD theory with included resistive term, accounting for the cold collisional plasma. The fluctuating magnetic field pattern is recorded with azimuthal scans. The current density is obained by Amperes law. It is concentrated in helically twisted current filaments. For the unstable drift waves, similar investigations are done with simultaneously recorded density fluctuations. In the azimuthal plane, the locations of the parallel current filaments and the fluctuating density are found to be in phase, supporting the predicted drive of parallel currents by pressure gradients. A mutual influence of the two wave types is observed in an interaction experiment. Assuming parallel currents as coupling quantity, an interpretation of the experimental findings is given based on the linear theory of drift waves.