Ralf Kleiber

Transitions to core-electron-root-confinement (CERC) regime are examined in the presence of low-order rational values of the rotational transform (ι) in Wendelstein 7-X. Experiments at densities and heating powers near the bifurcation from pure ion-root to CERC conditions revealed transitions at stationary plasma conditions but evolving plasma currents. Analysis of the effects due to the ramping plasma current indicates a change of the ι profile temporarily crossing low-order rational values. The transition occurs within a narrow range of plasma currents and, supported by simulations of the shielding current evolution, is consistent with the observed rise in central electron temperature. Prior to these transitions, low-frequency fluctuations were detected, ceasing once higher electron temperatures were reached, and the central radial electric field became positive. Experimental and numerical analyses suggest that these fluctuations are not instabilities, but rather sidebands of zonal flow oscillations, Doppler-shifted by the ambient electric field. Although the oscillations are not the trigger, their systematic characterization offers deeper insight into the underlying physics governing the transition and emphasizes the role of the evolving ι.

We introduce a smooth B-spline discretization in polar coordinates on the unit disc that corrects the loss of regularity present at the origin caused by the coordinate singularity in standard tensor-product B-spline formulations. The method constructs “smooth polar splines” via a Galerkin projection of harmonic polar functions S-m (r, θ) ∶= rlsin(mθ) and Sm (r, θ) ∶= rl cos(mθ), derived from the polar representation of Cartesian monomials, onto the central tensor-product B-spline basis in the innermost radial region. The radial component reproduces rl exactly for 0 ≤ l ≤ p, where p is the B-spline degree, satisfying the near-origin regularity condition. However, exact compatibility with C∞ -regularity at the origin is recovered only in the limit Δθ → 0, when the angular component resolves all angular harmonics accurately. The smooth polar splines are linear combinations of standard tensor-product B-splines and lie in the same function space, enabling mapping between the C∞-regular subspace and the original discretization space via an exact prolongation operator and a corresponding restriction operator acting on the discrete variables. They match standard tensor-product B-splines away from the origin, preserve orthogonality among the newly constructed origin-centered basis functions, and maintain local support and sparse matrices. This smoothness and locality improve the conditioning of mass and stiffness matrices, conserve charge, and reduce statistical errors in particle-in-cell simulations near the origin, while eliminating spurious eigenvalues in eigenvalue problems. The approach provides a robust, high-order, and efficient adaptation of tensor-product B-splines for polar coordinates in physics simulations.

A novel model for Alfvén-eigenmode-induced transport of fast ions in stellarators is introduced. A variety of reduced transport models exists for tokamaks, but nothing comparable currently exists for stellarators. The model is built on a mixing-length approximation for relating the linear growth rate of Alfvén eigenmodes to a diffusion coefficient which enters into a nonlinear steady-state radial diffusion equation for the fast-ion density profile. Analytical theory in the local and cylindrical limits is applied for computing growth rates and frequencies of Alfvénic modes at the intersection points of continuum branches. After verifying the new reduced model against the physically more complete hybrid-gyrokinetic CKA-EUTERPE model and finding good agreement, the new transport model is applied to a Wendelstein 7-X case, where modest profile flatting can be observed. Finally, as an outlook, the tool is also applied to a device resembling a future stellarator reactor. In the reactor, due to the high alpha-particle energy, the fast-ion transport caused by the interaction with Alfvénic modes is potentially very strong. However, due to the simplifications made by the model (Landau damping is the only damping channel, no finite-Larmor-radius or finite-orbit-width effects) the transport in the reactor is likely overestimated.

In this Letter, we demonstrate by comparison of computer simulations and experimental results that excitation of turbulence-driven electromagnetic secondary instabilities can serve as a paradigm to explain the observed activity of otherwise stable modes in fusion devices. In particular, we discuss the excitation of Alfvén eigenmodes in electron cyclotron heated plasmas in the Wendelstein 7-X (W7-X) stellarator. In more than 700 experimental programs of W7-X Operational Phase 1.2b (from which 25 were selected and discussed in this paper), poloidal magnetic fluctuation measurements with a clear Alfvénic nature are seen to correlate with turbulence density fluctuations. In response, we perform nonlinear, global, gyrokinetic, electromagnetic simulations using profiles consistent with the experimental conditions. The results agree remarkably well with the measurements and reveal excitation of zonal flow activity and long wavelength Alfvénic modes by ion temperature gradient turbulence. Integer ratios between the primary and secondary growth rates suggest an excitation mechanism similar to the forced-driven excitation of zonal flow by Alfvén eigenmodes.

The neoclassical ambipolarity condition governing the radial electric field in stellarators can have several solutions, and sudden transitions (in radius) between these can then take place. The radial position and structure of such a transition cannot be determined from local transport theory, and instead a non-rigorous model based on a diffusion equation for the electric field is usually employed for this purpose (Turkin et al 2011 Phys. Plasmas 18 022505). We compare global (full plasma volume) drift-kinetic simulations of neoclassical transport in the Wendelstein 7-X stellarator with this model and find significant discrepancies. The position r ₀ of the transition is not predicted correctly by the diffusion model, but the radial structure of the transition layer is in reasonable agreement if the diffusion coefficient is chosen appropriately. In particular, it should depend on the plasma temperature in the same way as the plateau-regime coefficient of neoclassical transport theory or the gyro-Bohm diffusion coefficient. In the small-gyroradius limit, the prediction of r ₀ by the diffusion model simplifies to the so-called Maxwell construction (Shaing 1984 Phys. Fluids 27 1567–9; Shaing 1984 Phys. Fluids 27 1924–6). However, this property also emerges from a wide range of other mathematical models in the appropriate limit. The basic assumption underlying these models is that the diffusion, or generalisations thereof, is independent of the radial electric field, which is however unlikely to be the case in practice. Presumably this fact explains the discrepancy between the diffusion model and the drift-kinetic simulations. Finally, it is found that global simulations replicate the phenomenon of spontaneous root transitions driven by variations in the electron-to-ion temperature ratio, as predicted by local theory in the small-gyroradius limit.

Alfvén eigenmodes are of great interest in any fusion device as they can be excited by fast ions in the plasma. If the modes grow to large amplitudes, they can cause transport and redistribution of the fast ions, thus limiting fusion performance. To save computational resources, the resonant kinetic interaction between the fast-particle species and the modes is often modeled by MHD-kinetic hybrid codes. Here, we present such a hybrid model which is applicable to three-dimensional magnetic fields, accounts for a finite parallel electric field and multiple MHD modes present at the same time. The model extends the one previously implemented in the CKA-EUTERPE code allowing for a better estimate of the damping due to the parallel electric field and nonlinear mode-mode interaction. The capabilities of our model are illustrated by applying the code to model nonlinear frequency chirping and fast-ion profile flattening.

We calculate Alfvén eigenmodes within a magnetic island (MiAE) which have been conjectured over a decade ago. Starting from a cylindrical plasma equilibrium, we calculate the complete metric of the island interior assuming an iota profile with a constant shear for Wendelstein 7-X parameters. Then, we solve the resulting Magneto-Hydrodynamic equations inside the island optionally considering Finite Larmor Radius corrections. We find various eigenmodes in the lowest gaps for n = 0. The eigenmode with the lowest frequency shows a weakly non-linear dependence on the island width which deviates considerably from an earlier estimate.

The radial electric field in the Wendelstein 7-X stellarator is computed by means of self-consistent, global, neoclassical simulations using the gyrokinetic particle-in-cell code EUTERPE. The simulation results are compared with local predictions obtained from a transport code using locally computed neoclassical transport coefficients. The analysis focuses on ion-electron-root transitions and investigates their dependence on collisionality, normalised ion gyroradius, and the electron-ion temperature ratio. Several of the results cannot be reproduced using conventional, local neoclassical transport theory. An approximate criterion for root transitions is derived, which results in an analytical scaling law that is useful for understanding how the position of the transition layer varies with plasma parameters.