## Doctoral Thesis

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In the present thesis, a systematic study of beam driven Alfvén eigenmodes in high-density and low-temperature plasmas of the W7-AS stellarator is performed. The device went out of operation in 2002 and the study is based on stored experimental data. Alfvén instabilities can roughly be divided into ideal MHD Alfvén eigenmodes and those existing due to kinetic effects. The spectrum of ideal MHD Alfvén waves in toroidal fusion devices consists of a continuum of stable waves that are strongly localized. Weakly damped, discrete eigenmodes can exist in gaps of the continuous spectrum which are formed by plasma inhomogeneities and the coupling of Alfvén continua. This allows an identification of ideal MHD Alfvén eigenmodes in terms of their frequency and mode numbers. Kinetic effects can modify this spectrum and cause additional types of eigenmodes, the kinetic Alfvén eigenmodes (KAE) and energetic particle modes (EPM). The goal of this thesis is twofold: (I) identification and description of fast particle driven Alfvén instabilities in W7-AS, and (II) study of energetic particle losses induced by Alfvén instabilities. The reconstruction of the ideal MHD plasma equilibrium for each discharge with sufficient accuracy is the very foundation of all subsequent steps. This is achieved, based on measured plasma parameter profiles that are further refined by validating them to the measurements of other, independent plasma diagnostics. The applied scheme is inspired by an approach of Integrated Data Analysis (IDA) to combine different diagnostic data and provide combined uncertainties. After mode number analysis and eigenmode identification, the theoretically expected, linear growth rate of the instability is calculated where possible, and the various contributions of the fast particle drive to the instability of the mode are identified. Alfvénic activity recorded by the Mirnov diagnostic is analyzed, which consists of a set of spatially distributed coils that measure magnetic fluctuations. On W7-AS, the probes are arranged in three poloidal arrays at different toroidal positions. The spacing between the probes is non-equidistant. In addition, the signals of one probe array are digitized with a different sample rate. These characteristics prohibit the straight-forward use of standard tools available for harmonic analysis. Instead, a new tool has been developed and thoroughly tested. It is a multi-dimensional extension of the Lomb periodogram, able to provide reliable time-resolved frequency and mode number spectra in the case of uneven datapoint spacing. Numerical studies of this periodogram show a good performance with respect to mode number resolution given the low number of available probes, and robustness against perturbations of the signal. Only two of the probe arrays can be used for the analysis of eigenmodes with frequencies >70 kHz, such that for high-frequency phenomena insufficient information about the mode numbers is available. A total of 133 different Alfvén eigenmodes is studied in discharges from different experimental campaigns. A restriction to discharges from various high-beta campaigns with neutral beam heating is required to allow for a realistic reconstruction of plasma equilibrium and velocity distribution functions of energetic particles. The discharges are characterized by high density, ne = 5 x 1019 m-3 to 2.5 x 1020 m-3 at relatively low temperatures of Te = Ti = 150 ... 600 eV. Alfvén eigenmodes often appear transiently in the startup phase of these discharges, where density and heating power are being ramped up. Occasionally, Alfvén eigenmodes are seen in the stationary, high-beta phase in the presence of considerable neutral beam heating. Most of the Alfvén eigenmodes are successfully classified as ideal MHD eigenmodes. 19 global, 47 toroidicity-induced and 8 ellipticity-induced Alfvén eigenmodes (GAEs, TAEs, and EAEs, respectively) are unambiguously identified by their mode numbers and frequencies. Excellent agreement between experimentally observed mode number spectra and theoretically calculated eigenmode structure is shown for a TAE example. Additional 13 events are found to have frequencies inside the EAE gap and could possibly be EAEs. Evidence for high-frequency Alfvén eigenmodes (mirror- and helicity-induced Alfvén eigenmodes) is seen, but can not be proven rigorously due to uncertain mode numbers and the complexity of the Alfvén continuum. The remaining 41 Alfvén eigenmodes can not be classified to be one of the above cases. Reasons are either high frequencies, mode numbers obscured by far-field effects, or mode numbers that could not be related to ideal MHD Alfvén eigenmodes. A selection of these shows indications of strong non-linear wave-particle interactions and are assumed to be EPMs. Kinetic Alfvén eigenmodes are not expected to exist in the experimental conditions that were studied. The radially resolved velocity distribution function is used to describe the parameter regimes in which the modes are observed in terms of the dimensionless parameters vb/vA (beam velocity normalized to the Alfvén velocity) and ßfast/ßth, where beta is the ratio of plasma pressure to magnetic pressure. The first parameter describes through which of the possible resonance velocities particles can interact with the eigenmode. A peculiarity of the fast particle dynamics in fusion devices is that they can resonantly interact with Alfvén eigenmodes through sideband resonances even if v < vA. The second parameter describes the energy content of the destabilizing fast particle population compared to the potentially stabilizing thermal plasma component. These parameters contain relevant information about the instability of an eigenmode and such diagrams are given for all observed modes. In addition to that, the expected linear growth rate of gap modes is calculated based on a theoretical model that extends the ideal MHD by a perturbative, drift-kinetic description of the energy exchange between waves and circulating particles, neglecting the effects of trapped particles. For the discharges under consideration the thermal electron speed is comparable to vA and the electrons provide a significant Landau damping contribution. Due to strong density gradients near the plasma boundary in most of the discharges, the thermal ions can provide a small drive via the spatial inhomogeneity which does not overcome the electron damping, however. The drive by spatial inhomogeneity of thermal ions requires a certain propagation direction of the mode and is equally stabilizing for opposite mode numbers. The fast particles also contribute to the growth rate via spatial inhomogeneity, velocity gradients and velocity anisotropy terms are negligible in W7-AS. Most of the observed GAE or EAE modes have negative mode numbers, which correspond to a propagation direction for which the spatial inhomogeneity of thermal and beam ions is predicted to be stabilizing. A fast particle drive of these modes is not confirmed, whereas the TAEs are found to be strongly destabilized by neutral beam injection. The distribution of plasma parameters for discharges showing TAEs in terms of the dimensionless stability parameters suggests an instability threshold that is qualitatively confirmed by an exploration of the parameter space with the theoretical model. Wave-induced, resonant losses of energetic ions scale linearly with the wave amplitude. To identify them, correlations between ion loss probe signals and wave amplitudes are searched, where correlation times in the order of the slowing-down time of energetic particles are expected. Significant correlations can be established only exceptionally for 3 of the identified ideal MHD Alfvén eigenmodes. Those Alfvén eigenmodes, however, which are assumed to be EPMs frequently show severe losses of energetic ions that are visible in the time traces of the plasma energy as well.

The present experimental work investigates plasma turbulence in the edge region of magnetized high-temperature plasmas. A main topic is the turbulent dynamics parallel to the magnetic field, where hitherto only a small data basis existed, especially for very long scale lengths in the order of ten of meters. A second point of special interest is the coupling of the dynamics parallel and perpendicular to the magnetic field. This anisotropic turbulent dynamics is investigated by two different approaches. Firstly, spatially and temporally high-resolution measurements of fluctuating plasma parameters are investigated by means of two-point correlation analysis. Secondly, the propagation of signals externally imposed into the turbulent plasma background is studied. For both approaches, Langmuir probe arrays were utilized for diagnostic purposes. The main findings can be summarized as follows: Greatly elongated fluctuation structures exist in plasma edge turbulence. The structures are aligned along the confining magnetic field (k|| = 0). The correlation degree of fluctuations for a short connection length of 0.75m is greater than 80%. For much longer connection lengths of 23m and 66m, the correlation degree is reduced to approximately 40%. A conceptual interpretation of these observations is the coexistence of two different fluctuation components. One component has a correlation length parallel to the magnetic field below 20m and the other component a correlation length greater than 70m. Sine signals in the frequency range 1-100 kHz were injected into the turbulent plasma background. The propagation parallel and perpendicular to the magnetic field of the signals was studied. In poloidal direction, an asymmetry is observed, that can be explained by a copropagation of the signal with the background E × B-rotation of the plasma. The signal propagation parallel to the magnetic field shows no such asymmetry. As an advanced approach, spatio-temporal wave patters were injected into the edge plasma. The waves launched that way can be seen as test waves' in a turbulent background. The coupling strength of the imposed wave patterns to the background turbulence relies on the match of the imposed waves to the dynamics of turbulent structures. If the propagation direction of the imposed waves is parallel to the propagation direction of the background plasma, improved coupling is observed. This finding underlines the importance of the background plasma rotation for future attempts of controlling the plasma edge turbulence. Further optimization of frequency and wave vector of the imposed waves is probably a promising approach for achieving a significant and systematic influence of turbulence. Taking into account the present experimental state-of-the-art, for a deeper insight into the mechanism of the plasma edge turbulence of magnetized high-temperature plasmas a joint effort of numerical modeling and experimental results is a valuable approach. Such a cooperation should cover the explanation of the correlation observations as well as the experiments on signal injection into background turbulence. A quantitative comparison between the results presented in this work and a dedicated numerical drift wave simulation would be a significant step forward to a better understanding of plasma edge turbulence.