Matthias Borchardt

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.

A synthetic Mirnov diagnostic has been developed to investigate the capabilities and limitations of an arrangement of Mirnov coils in terms of a mode analysis. Eight test cases have been developed, with different coil arrangements and magnetic field configurations. Three of those cases are experimental configurations of the stellarator Wendelstein 7-X. It is observed that, for a high triangularity of the flux surfaces, the arrangement of the coils plays a significant role in the exact determination of the poloidal mode number. For the mode analysis, torus and magnetic coordinates have been used. In most cases, the reconstruction of the poloidal mode number of a prescribed mode was found to be more accurate in magnetic coordinates. As an application, the signal of an Alfvén eigenmode, which has been calculated with a three-dimensional magnetohydrodynamics code, is compared to experimental observations at Wendelstein 7-X. For the chosen example, the calculated and measured mode spectra agree very well and additional information on the toroidal mode number and localization of the mode has been inferred.

Research on magnetic confinement of high-temperature plasmas has the ultimate goal of harnessing nuclear fusion for the production of electricity. Although the tokamak¹ is the leading toroidal magnetic-confinement concept, it is not without shortcomings and the fusion community has therefore also pursued alternative concepts such as the stellarator. Unlike axisymmetric tokamaks, stellarators possess a three-dimensional (3D) magnetic field geometry. The availability of this additional dimension opens up an extensive configuration space for computational optimization of both the field geometry itself and the current-carrying coils that produce it. Such an optimization was undertaken in designing Wendelstein 7-X (W7-X)², a large helical-axis advanced stellarator (HELIAS), which began operation in 2015 at Greifswald, Germany. A major drawback of 3D magnetic field geometry, however, is that it introduces a strong temperature dependence into the stellarator’s non-turbulent ‘neoclassical’ energy transport. Indeed, such energy losses will become prohibitive in high-temperature reactor plasmas unless a strong reduction of the geometrical factor associated with this transport can be achieved; such a reduction was therefore a principal goal of the design of W7-X. In spite of the modest heating power currently available, W7-X has already been able to achieve high-temperature plasma conditions during its 2017 and 2018 experimental campaigns, producing record values of the fusion triple product for such stellarator plasmas^3,4. The triple product of plasma density, ion temperature and energy confinement time is used in fusion research as a figure of merit, as it must attain a certain threshold value before net-energy-producing operation of a reactor becomes possible^1,5. Here we demonstrate that such record values provide evidence for reduced neoclassical energy transport in W7-X, as the plasma profiles that produced these results could not have been obtained in stellarators lacking a comparably high level of neoclassical optimization.