P. Xanthopoulos

Stellarator and tokamak plasmas: a comparison

An overview is given of physics differences between stellarators and tokamaks, including magnetohydrodynamic equilibrium, stability, fast-ion physics, plasma rotation, neoclassical and turbulent transport and edge physics. Regarding microinstabilities, it is shown that the ordinary, collisionless trapped-electron mode is stable in large parts of parameter space in stellarators that have been designed so that the parallel adiabatic invariant decreases with radius. Also, the first global, electromagnetic, gyrokinetic stability calculations performed for Wendelstein 7-X suggest that kinetic ballooning modes are more stable than in a typical tokamak.

Gyrokinetic microinstabilities in ASDEX Upgrade edge plasmas

Results of linear gyrokinetic simulations of ASDEX Upgrade [O. Gruber et al., Nucl. Fusion 39, 1321 (1999)] edge plasmas, with experimentally determined geometry and input parameters, are presented. It is found that in the near-edge region, microtearing modes can exist under conditions found in conventional tokamaks. As one enters the steep-gradient region, the growth rate spectrum is dominated—down to very low wavenumbers—by electron temperature gradient modes. The latter tend to peak near the X-point(s) and possess properties which may explain the ratios of the density and temperature gradient scale lengths that have been observed in various experiments over the last decade.

A geometry interface for gyrokinetic microturbulence investigations in toroidal configurations

The GENE/GIST code package is developed for the investigation of plasma microturbulence, suitable for both stellarator and tokamak configurations. The geometry module is able to process typical equilibrium files and create the interface for the gyrokinetic solver. The analytical description of the method for constructing the geometric elements is documented, together with several numerical evaluation tests. As a concrete application of this product, a cross-machine comparison of the anomalous ion heat diffusivity is presented.

Clebsch-type coordinates for nonlinear gyrokinetics in generic toroidal configurations

The nonlinear gyrokinetic equations are frequently used as a basis for simulations of small-scale turbulence in magnetized toroidal plasmas. In this context, field-aligned coordinates are usually employed in order to minimize the number of necessary grid points. The present work proposes a system of Clebsch-type coordinates which does not depend on the existence of flux surfaces. The construction and use of these coordinates is explained, and the corresponding formulation of the nonlinear gyrokinetic equations is accomplished. This setup paves the way toward the investigation of nonaxisymmetric toroidal geometries, also in the region of magnetic islands as well as inside the ergodic layer where flux surfaces cease to exist. For testing purposes, in the axisymmetric, large aspect ratio case, the well-known s-α expressions are recovered for closed flux surfaces. Moreover, geometric data for a specific stellarator configuration are computed and discussed.

Gyrokinetic turbulence under near-separatrix or nonaxisymmetric conditions

Linear and nonlinear gyrokinetic simulations with the GENE code [F. Jenko et al., Phys. Plasmas 7, 1904 (2000)] for tokamak edge plasmas as well as for stellarator core plasmas are presented, shedding light on the behavior of plasma microturbulence under near-separatrix or nonaxisymmetric conditions. To this aim, the required geometric coefficients are inferred directly from the magnetohydrodynamic equilibria of three different devices via the newly developed GIST code. It is found that the residual electron heat transport level in the H-mode edge can be explained in terms of high-wave-number fluctuations driven by electron temperature gradient modes. Moreover, the study of adiabatic ion temperature gradient turbulence in optimized stellarators points to the possibility of a systematic geometric optimization with respect to anomalous transport in nonaxisymmetric devices.

Simulating gyrokinetic microinstabilities in stellarator geometry with GS2

The nonlinear gyrokinetic code GS2 has been extended to treat non-axisymmetric stellarator geometry. Electromagnetic perturbations and multiple trapped particle regions are allowed. Here, linear, collisionless, electrostatic simulations of the quasi-axisymmetric, three-field period national compact stellarator experiment (NCSX) design QAS3-C82 have been successfully benchmarked against the eigenvalue code FULL. Quantitatively, the linear stability calculations of GS2 and FULL agree to within ∼10%.

Gyrokinetic analysis of linear microinstabilities for the stellarator Wendelstein 7-X

A linear collisionless gyrokinetic investigation of ion temperature gradient (ITG) modes—considering both adiabatic and full electron dynamics—and trapped electron modes (TEMs) is presented for the stellarator Wendelstein 7-X (W7-X) [G. Grieger et al., Plasma Physics and Controlled Nuclear Fusion Research 1990 (International Atomic Energy Agency, Vienna, 1991), Vol. 3, p. 525]. The study of ITG modes reveals that in W7-X, microinstabilities of distinct character coexist. The effect of changes in the density gradient and temperature ratio is discussed. Substantial differences with respect to the axisymmetric geometry appear in W7-X, concerning the relative separation of regions with a large fraction of helically trapped particles and those of pronounced bad curvature. For both ITG modes and TEMs, the dependence of their linear growth rates on the background gradients is studied along with their parallel mode structure.

Collisionless microinstabilities in stellarators. III. The ion-temperature-gradient mode

We investigate the linear theory of the ion-temperature-gradient (ITG) mode, with the goal of developing a general understanding that may be applied to stellarators. We highlight the Wendelstein 7X (W7-X) device. Simple fluid and kinetic models that follow closely from existing literature are reviewed and two new first-principle models are presented and compared with results from direct numerical simulation. One model investigates the effect of regions of strong localized shear, which are generic to stellarator equilibria. These “shear spikes” are found to have a potentially significant stabilizing affect on the mode; however, the effect is strongest at short wavelengths perpendicular to the magnetic field, and it is found to be significant only for the fastest growing modes in W7-X. A second model investigates the long-wavelength limit for the case of negligible global magnetic shear. The analytic calculation reveals that the effect of the curvature drive enters at second order in the drift frequency, co...

Oscillations of zonal flows in stellarators

The linear response of a collisionless stellarator plasma to an applied radial electric field is calculated, both analytically and numerically. Unlike in a tokamak, the electric field and associated zonal flow develop oscillations before settling down to a stationary state, the so-called Rosenbluth–Hinton flow residual. These oscillations are caused by locally trapped particles with radially drifting bounce orbits. The particles also cause a kind of Landau damping of the oscillations that depends on the magnetic configuration. The relative importance of geodesic acoustic modes and zonal-flow oscillations therefore varies among different stellarators.

Advances in stellarator gyrokinetics

Recent progress in the gyrokinetic theory of stellarator microinstabilities and turbulence simulations is summarized. The simulations have been carried out using two different gyrokinetic codes, the global particle-in-cell code EUTERPE and the continuum code GENE, which operates in the geometry of a flux tube or a flux surface but is local in the radial direction. Ion-temperature-gradient (ITG) and trapped-electron modes are studied and compared with their counterparts in axisymmetric tokamak geometry. Several interesting differences emerge. Because of the more complicated structure of the magnetic field, the fluctuations are much less evenly distributed over each flux surface in stellarators than in tokamaks. Instead of covering the entire outboard side of the torus, ITG turbulence is localized to narrow bands along the magnetic field in regions of unfavourable curvature, and the resulting transport depends on the normalized gyroradius ρ* even in radially local simulations. Trapped-electron modes can be significantly more stable than in typical tokamaks, because of the spatial separation of regions with trapped particles from those with bad magnetic curvature. Preliminary non-linear simulations in flux-tube geometry suggest differences in the turbulence levels in Wendelstein 7-X and a typical tokamak.