Gyro-kinetic simulations of tokamaks and stellarators including collisions

This thesis, titled 'Gyro-kinetic simulations of tokamaks and stellarators including collisions', focusses on the modelling of resonant destabilization of Alfvén eigenmodes by fast ions in fusion plasmas. It especially addresses non-linear simulations of stellarator plasmas in which particle collisions are retained. This is in contrast to investigations done in the past, in which collisions were neglected. Here, it is shown that collisions are required for a realistic description of Alfvéen waves in plasmas relevant to nuclear fusion. In this sense, this thesis helps to enhance the level of realism in numerical simulations. The work presented here was done in a period in which the stellarator Wendelstein 7-X (W7-X) had its first operation phases. The 2018 experimental campaign, for the first time, included neutral beam injection (NBI), which could trigger Alfvénic activity. Alfvén eigenmodes (AEs) present in the plasma can resonantly interact with the NBI ions (or alpha particles) and could lead to enhanced heat loads on the vessel wall. Such behaviour has been observed in the past in experiments with significant alpha-particle heating (e.g. in TFTR), but also in smaller devices such as ASDEX Upgrade. The TFTR example indicates the potentially destructive behaviour of fast-ion-driven AEs in fusion devices. Avoiding such a scenario is, of course, desirable in any machine. For W7-X, there exist numerical predictions of the classical fast-ion losses to the first wall which have been obtained using the ASCOT code. They are currently being validated using data gathered in the 2018 experimental campaign of W7-X. The question is in what way AE activity in the plasma, possibly in uenced by collisions, changes the loss pattern. To answer this question, a coupling of ASCOT and EUTERPE is planned in the future. Before that, the non-linear dynamics and saturation of AEs in W7-X needs to be better understood. Since collisions play an important role in the formation of the fast-ion distribution function, it is only natural to also include them in EUTERPE simulations, especially since the saturation levels of the AEs are a function of collisionality. At W7-X, experimental data (regarding the destabilization of AEs by fast ions) are produced now which means that the modelling tools need to be at a stage where simulations can readily be compared with experimental measurements. In part, this thesis contributed to code development and benchmarking activities in order to prepare the EUTERPE code for realistic cases. Articles A.3 and A.4 included in this thesis, in which realistic profiles and distribution functions were used to describe the plasma, can be viewed as a step in this direction. For simplified cases, analytical theory regarding the interaction of Alfvén eigenmodes with fast ions while in uenced by collisions is available. The well-established Berk-Breizman paradigm is a prime example. However, the reduction of the complexity of the problem comes at a cost: The Berk-Breizman model is formulated in coordinates (e.g. a one-dimensional velocity space) that are not trivially related to tokamaks, let alone a stellarator. Furthermore, numerical simulations, building upon analytical theory by making use of invariants of the particle motion, are often performed in tokamaks. An appropriate description of stellarators is more difficult. Here, we seek to understand if the effect of collisions on the mode non-linear dynamics in stellarators is comparable to tokamaks. Again, this is in particular important in order to assess Alfvén-eigenmode-induced transport of fast ions { an important issue for present-day devices and future reactors. A future goal is that numerical simulations can guide the way towards a reduced transport model for fast ions in stellarators. Such models exist in the form of the so-called 'Kick model' or the 'resonance broadened quasi-linear (RBQ) model' for tokamaks, but are still lacking for stellarators. The primary goals of this dissertation can be summarized as follows: It deals with the implementation of particle collisions into the EUTERPE code, the thorough benchmark of the implementation, and the application to realistic W7-X plasmas, where we confirm the importance of collisions in non-linear simulations. The thesis is structured as follows: In Ch. 1 we give an introduction to the general topics covered. Ch. 2 summarizes the key findings of each paper and provides links to a number of appendices that contain additional unpublished material, also intended to bridge the gaps in between the individual papers. The thesis concludes with a summary and an outlook in Ch. 3.