R. Kleiber

Energy conservation in a nonlinear gyrokinetic particle-in-cell code for ion-temperature-gradient-driven modes in θ-pinch geometry

A global nonlinear simulation code for the time evolution of ion-temperature-gradient-driven modes in θ-pinch geometry as a first approximation to the stellarator Wendelstein 7-X (W7-X) [Grieger et al., Proceedings of the 13th International Conference on Plasma Physics and Controlled Nuclear Fusion Research, Washington, DC, 1990 (International Atomic Energy Agency, Vienna, 1991), Vol. 3, p. 525] has been developed. A δf particle-in-cell (PIC) method is used to solve the coupled system of gyrokinetic equations for the ions, in the electrostatic approximation, and the quasineutrality equation, assuming adiabatically responding electrons. The focus has been on adherence to conservation laws, i.e., particle number and energy conservation. Besides other improvements it has been shown that a well-chosen initial distribution of the markers in reduced phase space makes optimal use of the δf PIC method to reduce the statistical noise for a given number of markers. In a model including all (1351) physically relevan...

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 global three-dimensional simulations of linear ion-temperature-gradient modes in Wendelstein 7-X

Using a global approach for solving an ion gyrokinetic model in three-dimensional geometry the linear stability and structure of ion-temperature-gradient (ITG) modes in the configuration of the stellarator Wendelstein 7-X (W7-X) [G. Grieger , in Plasma Physics and Controlled Nuclear Fusion Research 1990 (International Atomic Energy Agency, Vienna, 1991), Vol. 3, p. 525.] is studied. The time evolution of electrostatic perturbations is solved as an initial value problem with a particle-in-cell deltaf method. The vacuum magnetohydrodynamic equilibrium is calculated by the code VMEC [S. P. Hirshman and D. K. Lee, Comput. Phys. Commun. 39, 161 (1986)]. In this work the most unstable ITG mode in W7-X is presented. This mode has a pronounced ballooning-type structure; however, it is not tokamak-like. A driving mechanism analysis using the energy transfer shows that the contribution of curvature effects is non-negligible. The growth rate and the mixing-length estimate for transport are compared with those for ITG modes found in axisymmetric geometries

An improved control-variate scheme for particle-in-cell simulations with collisions

Particle-in-cell methods combined with af approach constitute an established and powerful method for simulating collisionless kinetic equations in e.g. plasma physics. Including collisions in such simulations requires a modified approach leading to a two-weight scheme, which has the drawback of giving a statistical error that increases with time. As in the collisionless case, this scheme can be interpreted as an application of an ordinary control variate. Using an enhanced control variate approach, an improved scheme is constructed. This approach has been applied to a model problem with the result of a much better behaviour of the error, which, instead of growing indefinitely, becomes bounded by the error of a full-f scheme. In addition, the application of the enhanced control variate is illustrated for a collisionless simulation of ITG turbulence. Here it can be used both as a diagnostic tool and as a means to eliminate the spurious violation of particle number conservation inherent tof simulations due to statistical noise.

Gyrokinetic global electrostatic ion-temperature-gradient modes in finite β equilibria of Wendelstein 7-X

The standard scenario for the Wendelstein 7-X (W7-X) stellarator supposes equilibria with average β up to 4.3%. Here, the gyrokinetic description for full-radius ion-temperature-gradient (ITG) driven instabilities in three-dimensional configurations is used to investigate the linear stability and global structure of electrostatic ITG modes for the finite-β W7-X configurations. A massively-parallel initial-value particle-in-cell code has been modified to include the influence of a finite β on ITG modes due to equilibrium effects. We considered a sequence of W7-X equilibria with different plasma β, and found drastic changes in the structure and drive of the ITG instability but a minor influence on the value of the growth rate. The results presented are physically explained; differences between β-effects on ITG modes in a conventional tokamak and in W7-X are discussed, as well as the different structures and physical mechanisms of ITG modes for β = 0 and finite-β W7-X configurations.

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.

Pullback transformation in gyrokinetic electromagnetic simulations

It is shown that a considerable mitigation of the cancellation problem can be achieved by a slight modification of the simulation scheme. The new scheme is verified, simulating a Toroidal Alfven Eigenmode in tokamak geometry at low perpendicular mode numbers, the so-called “MHD limit.” Also, an electromagnetic drift mode has been successfully simulated in a stellarator.

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.

Electrostatic potential variations along flux surfaces in stellarators (Letter)

Direct observations of electrostatic potential variations along the flux surfaces of the TJ-II stellarator are presented. Measurements taken with two distant Langmuir probe arrays show differences in the edge floating potentials profiles of several tens of volts in electron-root wave-heated plasmas. The differences are reduced for higher densities and lower electron temperatures after the ion-root electric field forms at the plasma edge. Neoclassical Monte Carlo simulations estimate the correct order of magnitude for the overall variation in potential and predict the trend observed with the radial electric field. However, for the specific location of the probes, the simulations give differences smaller than those observed experimentally.

New variables for gyrokinetic electromagnetic simulations

A new approach to electromagnetic gyrokinetic simulations based on modified gyrokinetic theory is described. The method is validated using a particle-in-cell code. The Toroidal Alfven Eigenmode at low perpendicular mode numbers, the so-called “magnetohydrodynamical limit,” has been successfully simulated using this method.