## Doctoral Thesis

Experience in the construction of optimized stellarators shows the coil system is a significant challenge. The precision necessary allow the generation of accurate flux surfaces in recent experiments affected both cost and schedule negatively. Moreover, recent experiments at Wendelstein 7-X have shown that small field corrections were necessary for the operation of specific desired magnetic configurations. Therefore, robust magnetic configurations in terms of coil geometry and assembly tolerances have a high potential to facilitate swifter and less expensive construction of future, optimized stellarators. We present a new coil optimization technique that is designed to seek out coil configurations that are resilient against 3D coil displacements. This stochastic version of stellarator coil optimization uses the sampling average approach to incorporate an iterative perturbation analysis into the optimization routine. The result is a robust magnetic configuration that simultaneously reproduces the target magnetic field more accurately and leads to a better fusion performing coil configuration.

With this thesis, studies which form the bedrock for the long term goal of first wall heat load control and optimization for the advanced stellarator Wendelstein 7-X are developed, described and put into context. It is laid out how reconstruction of features of the edge magnetic field from plasma facing component heat loads is an important first step and can successfully be achieved by artificial neural networks. A detailed study of plasma facing component heat load distribution, potential overloads and overload mitigation possibilities is made in first order approximation of the impact of the main plasma dynamic effects.

This dissertation focusses on the numerical modelling of resonant destabilization of AlfvÃ©n eigenmodes by fast ions in fusion plasmas. It especially addresses non-linear simulations of stellarator plasmas in which particle collisions are retained. It is shown that collisions are required for a realistic description of AlfvÃ©n waves in plasmas relevant to nuclear fusion.
We start by carefully verifying the implementation of the collision operators into the electromagnetic version of the gyro-kinetic delta-f particle-in-cell code EUTERPE. After these initial benchmarks are completed successfully, the code is in a position to be applied to realistic tokamak and stellarator scenarios.
Since every collision operator needs to fulfil conservation laws, a momentum-conserving version of the pitch-angle scattering operator is implemented. This is in particular important for neoclassical transport simulations aimed at computing flux-surface variations of the electrostatic potential in stellarators.
Using the simplified CKA-EUTERPE model (employing a fixed-mode-structure approximation), we perform non-linear simulations in tokamaks and stellarators. We show that the non-linear dynamics of fast-ion-driven AlfvÃ©n eigenmodes is significantly influenced by collisions. They have the potential to enhance the saturation level and to affect the frequency chirping of the modes.
It is thus concluded that collisions play an essential role in determining AlfvÃ©n-eigenmode-induced fast-ion transport - an important issue for future fusion devices. In order to address this issue the CKA-EUTERPE model is extended to evolve multiple modes at the same time. First results of this multi-mode version (which enhances the level of realism of the simulations) are shown in the Appendix of the 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.

The confinement of energy has always been a challenge in magnetic confinement fusion devices. Due to their toroidal shape there exist regions of high and low magnetic field, so that the particles are divided into two classes - trapped ones that are periodically reflected in regions of high magnetic field with a characteristic frequency, and passing particles, whose parallel velocity is high enough that they largely follow a magnetic field line around the torus without being reflected. The radial drift that a particle experiences due to the field inhomogeneity depends strongly on its position, and the net drift therefore depends on the path taken by the particle. While the radial drift is close to zero for passing particles, trapped particles experience a finite radial net drift and are therefore lost in classical stellarators. These losses are described by the so-called neoclassical transport theory. Recent optimised stellarator geometries, however, in which the trapped particles precess around the torus poloidally and do not experience any net drift, promise to reduce the neoclassical transport down to the level of tokamaks. In these optimised stellarators, the neoclassical transport becomes small enough so that turbulent transport may limit the confinement instead. The turbulence is driven by small-scale-instabilities, which tap the free energy of density or temperature gradients in the plasma. Some of these instabilities are driven by the trapped particles and therefore depend strongly on the magnetic geometry, so the question arises how the optimisation affects the stability. In this thesis, collisionless electrostatic microinstabilities are studied both analytically and numerically. Magnetic configurations where the action integral of trapped-particle bounce motion, J, only depends on the radial position in the plasma and where its maximum is in the plasma centre, so-called maximum-J configurations, are of special interest. This condition can be achieved approximately in quasi-isodynamic stellarators, for example Wendelstein 7-X. In such configurations the precessional drift of the trapped particles is in the opposite direction from the direction of propagation of drift waves. Instabilities that are driven by the trapped particles usually rely on a resonance between these two frequencies. Here it is shown analytically by analysing the electrostatic energy transfer between the particles and the instability that, thanks to the absence of the resonance, a particle species draws energy from the mode if the frequency of the mode is well below the charateristic bounce frequency. Due to the low electron mass and the fast bounce motion, electrons are almost always found to be stabilising. Most of the trapped-particle instabilities are therefore predicted to be absent in maximum- J configurations in large parts of parameter space. Analytical theory thus predicts enhanced linear stability of trapped-particle modes in quasi-isodynamic stellarators compared with tokamaks. Moreover, since the electrons are expected to be stabilising, or at least less destabilising, for all instabilities whose frequency lies below the trapped-electron bounce frequency, other modes might benefit from the enhanced stability as well. In reality, however, stellarators are never perfectly quasi-isodynamic, and the question thus arises whether they still benefit from enhanced stability. Here the stability properties of Wendelstein 7-X and a more quasi-isodynamic configuration, QIPC, are investigated numerically and compared with another, non-quasiisodynamic stellarator, the National Compact Stellarator Experiment (NCSX) and a typical tokamak. In gyrokinetic simulations, performed with the gyrokinetic code GENE in the electrostatic and collisionless approximation, several microinstabilities, driven by the density as well as both ion and electron temperature gradients, are studied. Wendelstein 7-X and QIPC exhibit significantly reduced growth rates for all simulations that include kinetic electrons, and the latter are indeed found to be stabilising when the electrostatic energy transfer is analysed. In contrast, if only the ions are treated kinetically but the electrons are taken to be in thermodynamic equilibrium, no such stabilising effect is observed. These results suggest that imperfectly optimised stellarators can retain most of the stabilising properties predicted for perfect maximum-J configurations. Quasi-isodynamic stellarators, in addition to having reduced neoclassical transport, might therefore also show reduced turbulent transport, at least in certain regions of parameter space.

The WEGA stellarator is used to confine low temperature, overdense (densities exceeding the cut-off density of the heating wave) plasmas by magnetic fields in the range of B=50-500 mT. Microwave heating systems are used to ignite gas discharges using hydrogen, helium, neon or argon as working gases. The produced plasmas have been analyzed using Langmuir and emissive probes, a single-channel interferometer and ultra-high resolution Doppler spectroscopy. For a typical argon discharge in the low field operation, B=56 mT, the maximum electron density is n_e~10^18m^{-3} with temperatures in the range of T=4-12 eV. The plasma parameters are determined by using Langmuir probes and are cross-checked with interferometry. It is demonstrated within this work that the joint use of emissive probes and ultra-high resolution Doppler spectroscopy allows a precise measurement of the radial electric field. Here the floating potential measurements using emissive probes have been compared to measurements of the poloidal rotation of the plasma which is also linked to the radial electric field. In order to alter the plasma parameters a biasing probe setup has been used during this work. The focus of this work is on demonstrating the ability to modify the existing radial electric field in a plasma by using the biasing probe. This technique is in principle not new, as it has been around for decades. Looking at details, it turns out that describing low field operation WEGA argon plasmas in connection with biasing is not covered by the present set of theoretical approaches and experimental cognition. This work will commence with a basic approach and first establishes the diagnostic tools in a well-known discharge. Then the perturbation caused by the biasing probe is assessed. Following the characterization of the unperturbed plasmas, plasma states altered by the operation of the energized biasing probe will be characterized. It is demonstrated that modifying the existing radial electric field can be achieved and reliably diagnosed using spectroscopy and probe measurements. In order to verify the different approaches for determining the radial electric field the diagnostics are cross-checked against another whenever possible. During biasing the plasma two different stable plasma states have been found. Stable here refers to the state existing much longer than the confinement time for WEGA. The presence of a calorimetric limiter placed in the scrape-off layer has an impact on the type of the plasma state. The two observed plasma states differ in plasma parameter profiles, such as density, temperature, electric field and confined energy. The results are compared to two simple models. One model relies on the relevant atomic processes and a second one is based on neoclassical theory. Both models can be used to derive the particle and power flux from the plasma. The losses predicted by the atomic models can be tested using bolometry. It can be shown that both models agree well in the description of the particle balance of the electrons for large regions of the plasma. By comparing the models the neoclassical heat flux turns out to be small compared to the energy fluxes caused by atomic processes. For the reference discharge taking the energy flux due to the atomic processes and balancing it by the input microwave power is satisfying the energy balance, without the need for transport. For the biased discharges it turns out that neoclassical transport can be neglected as well, but the additional biasing power has to be taken into account. A simple model for the biasing power is motivated and tested. An agreement in the energy balance can be reached in this way as far as the models are applicable. The models also allow drawing conclusions on the amount of absorbed microwave power.

The present work is the first work dealing with turbulence in the WEGA stellarator. The main object of this work is to provide a detailed characterisation of electrostatic turbulence in WEGA and to identify the underlying instability mechanism driving turbulence. The spatio-temporal structure of turbulence is studied using multiple Langmuir probes providing a sufficiently high spatial and temporal resolution. Turbulence in WEGA is dominated by drift wave dynamics. Evidence for this finding is given by several individual indicators which are typical features of drift waves. The phase shift between density and potential fluctuations is close to zero, fluctuations are mainly driven by the density gradient, and the phase velocity of turbulent structures points in the direction of the electron diamagnetic drift. The structure of turbulence is studied mainly in the plasma edge region inside the last closed flux surface. WEGA can be operated in two regimes differing in the magnetic field strength by almost one order of magnitude (57mT and 500mT, respectively). The two regimes turned out to show a strong difference in the turbulence dynamics. At 57mT large structures with a poloidal extent comparable to the machine dimensions are observed, whereas at 500mT turbulent structures are much smaller. The poloidal structure size scales nearly linearly with the inverse magnetic field strength. This scaling may be argued to be related to the drift wave dispersion scale. However, the structure size remains unchanged when the ion mass is changed by using different discharge gases. Inside the last closed flux surface the poloidal ExB drift in WEGA is negligible. The observed phase velocity is in good agreement with the electron diamagnetic drift velocity. The energy in the wavenumber-frequency spectrum is distributed in the vicinity of the drift wave dispersion relation. The three-dimensional structure is studied in detail using probes which are toroidally separated but aligned along connecting magnetic field lines. As expected for drift waves a small but finite parallel wavenumber is found. The ratio between the average parallel and perpendicular wavenumber is in the order of 10^-2. The parallel phase velocity of turbulent structures is in-between the ion sound velocity and the AlfvÃ¨nvelocity. In the parallel dynamics a fundamental difference between the two operational regimes at different magnetic field strength is found. At 500mT turbulent structures can be described as an interaction of wave contributions with parallel wavefronts. At 57mT the energy in the parallel wavenumber spectrum is distributed among wavenumber components pointing both parallel and antiparallel to the magnetic field vector. In both cases turbulent structures arise preferable on the low field side of the torus. Some results on a novel field in plasma turbulence are given, i.e. the study of turbulence as a function of resonant magnetic field perturbations leading to the formation of magnetic islands. Magnetic islands in WEGA can be manipulated by external perturbation coils. A significant influence of field perturbations on the turbulence dynamics is found. A distinct local increase of the fluctuation amplitude and the associated turbulent particle flux is found in the region of magnetic islands.

In the present thesis, a systematic study of beam driven AlfvÃ©n 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. AlfvÃ©n instabilities can roughly be divided into ideal MHD AlfvÃ©n eigenmodes and those existing due to kinetic effects. The spectrum of ideal MHD AlfvÃ©n 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 AlfvÃ©n continua. This allows an identification of ideal MHD AlfvÃ©n eigenmodes in terms of their frequency and mode numbers. Kinetic effects can modify this spectrum and cause additional types of eigenmodes, the kinetic AlfvÃ©n eigenmodes (KAE) and energetic particle modes (EPM). The goal of this thesis is twofold: (I) identification and description of fast particle driven AlfvÃ©n instabilities in W7-AS, and (II) study of energetic particle losses induced by AlfvÃ©n 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. AlfvÃ©nic 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 AlfvÃ©n 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. AlfvÃ©n eigenmodes often appear transiently in the startup phase of these discharges, where density and heating power are being ramped up. Occasionally, AlfvÃ©n eigenmodes are seen in the stationary, high-beta phase in the presence of considerable neutral beam heating. Most of the AlfvÃ©n eigenmodes are successfully classified as ideal MHD eigenmodes. 19 global, 47 toroidicity-induced and 8 ellipticity-induced AlfvÃ©n 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 AlfvÃ©n eigenmodes (mirror- and helicity-induced AlfvÃ©n eigenmodes) is seen, but can not be proven rigorously due to uncertain mode numbers and the complexity of the AlfvÃ©n continuum. The remaining 41 AlfvÃ©n 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 AlfvÃ©n eigenmodes. A selection of these shows indications of strong non-linear wave-particle interactions and are assumed to be EPMs. Kinetic AlfvÃ©n 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 AlfvÃ©n velocity) and ÃŸfast/ÃŸth, 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 AlfvÃ©n 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 AlfvÃ©n eigenmodes. Those AlfvÃ©n 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 Ã— 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.