Global simulations of electromagnetic turbulence in circular-flux-surface tokamak and ASDEX-Upgrade geometry, tearing instabilities and their combination with the electromagnetic turbulence, nonlinear Alfvénic modes in the presence of fast particles and their combination with the electromagnetic turbulence, and global electromagnetic turbulence in Wendelstein 7-X stellarator geometry are carried out using the gyrokinetic particle-in-cell codes ORB5 (E. Lanti et al, Comp. Phys. Comm., 251, 107072 (2020)) and EUTERPE (V. Kornilov et al, Phys. Plasmas, 11, 3196 (2004)). Computational feasibility of simulating such complex coupled systems is demonstrated. For simplicity, reduced mass ratio is used throughout the paper.

The 2018 operation phase (OP 1.2b) of the stellarator Wendelstein 7-X (W7-X) included, for the first time, neutral beam injection (NBI) to heat the plasma. Since the injection geometry at W7-X is not parallel, this generates both passing and trapped fast particles. During longer phases of NBI injection, with the primary purpose to study the heating efficiency of this system, Alfvén eigenmodes (AEs) were observed by a number of diagnostics, including the phase contrast imaging (PCI) system, the magnetic pick-up coils (Mirnov coils), and the soft X-ray multi-camera tomography system (XMCTS).Alfvén eigenmodes are of great interest for future fusion reactors as it has been shown that the resonant interaction of fast ions with self-excited AEs can lead to enhanced transport of fast ions and potentially to energy losses. This is especially true for so-called gap-modes, Alfvén eigenmodes with frequencies in gaps of the continuous spectrum, since they lack continuum damping. These modes are commonly known to be excited by fast ions, but other destabilizing mechanisms, e.g. the electron-pressure gradient are also possible.In this article we present a first analysis of the experimentally observed frequencies from the theoretical side. The calculation of shear Alfvén wave continua for selected cases and the assignment of observed frequencies to the gaps of the continuous spectra are presented. Using the ideal-MHD code CKA [1], we find gap modes that match the experimental measurements in terms of the observed frequencies. We emphasize the crucial roles played by the coupling of sound and Alfvén waves as well as of the Doppler shift arising as a consequence of the radial electric field in W7-X.We employ the perturbative gyrokinetic code CKA-EUTERPE [2], using a slowing-down distribution function for the fast ions as calculated by the Monte-Carlo particle following code ASCOT [3] to assess the fast-ion drive. We find that the fast-ion drive is insufficient to overcome the background-plasma damping. The fact that unstable modes were observed experimentally may point to problems with the modelling or indicate the existence of other destabilizing mechanisms, e.g. associated with the electron-pressure gradient [4] that sensitively depend on the profiles of the background plasma.

The effects of a radial electric field, which is ubiquitous in stellarators, are considered when a system of reduced-MHD equations is derived from a gyro-kinetic Vlasov–Maxwell system. The resulting equations for the MHD continuum have been implemented into the continuum code CONTI. For a tokamak case, the MHD continuum is calculated and compared with a gyro-kinetic continuum calculated using the EUTERPE code, where good agreement of the obtained continua could be shown. Finally, MHD continua including the radial electric field are calculated for Wendelstein 7-X. Interestingly, for these examples, most of the gaps in the continuous spectrum of Alfvén waves below 200 kHz disappear.

A numerical approach is devised to calculate the shear Alfvén continuum inside and outside magnetic islands in cylindrical and stellarator plasmas. Equations for an appropriate set of coordinates and the arising equations for the continuum are derived and implemented in the CONTI code. An experiment-oriented representation of the results is chosen to allow a radial localization of the modes and a comparison of different magnetic configurations. Comparison is made with results of earlier analytic work for validation. Agreement is good but more details of the spectrum, such as the generation of island induced gaps inside and outside the separatrix, are found. While the code is easily usable and can be applied to any magnetic equilibrium accessible with VMEC, the calculations are plagued with convergence issues close to the separatrix. A calculation for a realistic W7-X equilibrium with islands is done where the island width is estimated with the HINT code.