A. Könies

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...

Coupled principal component analysis

A framework for a class of coupled principal component learning rules is presented. In coupled rules, eigenvectors and eigenvalues of a covariance matrix are simultaneously estimated in coupled equations. Coupled rules can mitigate the stability-speed problem affecting noncoupled learning rules, since the convergence speed in all eigendirections of the Jacobian becomes widely independent of the eigenvalues of the covariance matrix. A number of coupled learning rule systems for principal component analysis, two of them new, is derived by applying Newtons method to an information criterion. The relations to other systems of this class, the adaptive learning algorithm (ALA), the robust recursive least squares algorithm (RRLSA), and a rule with explicit renormalization of the weight vector length, are established.

LIGKA: A linear gyrokinetic code for the description of background kinetic and fast particle effects on the MHD stability in tokamaks

Abstract In a plasma with a population of super-thermal particles generated by heating or fusion processes, kinetic effects can lead to the additional destabilisation of MHD modes or even to additional energetic particle modes. In order to describe these modes, a new linear gyrokinetic MHD code has been developed and tested, LIGKA (linear gyrokinetic shear Alfven physics) [Ph. Lauber, Linear gyrokinetic description of fast particle effects on the MHD stability in tokamaks, Ph.D. Thesis, TU Munchen, 2003; Ph. Lauber, S. Gunter, S.D. Pinches, Phys. Plasmas 12 (2005) 122501], based on a gyrokinetic model [H. Qin, Gyrokinetic theory and computational methods for electromagnetic perturbations in tokamaks, Ph.D. Thesis, Princeton University, 1998]. A finite Larmor radius expansion together with the construction of some fluid moments and specification to the shear Alfven regime results in a self-consistent, electromagnetic, non-perturbative model, that allows not only for growing or damped eigenvalues but also for a change in mode-structure of the magnetic perturbation due to the energetic particles and background kinetic effects. Compared to previous implementations [H. Qin, mentioned above], this model is coded in a more general and comprehensive way. LIGKA uses a Fourier decomposition in the poloidal coordinate and a finite element discretisation in the radial direction. Both analytical and numerical equilibria can be treated. Integration over the unperturbed particle orbits is performed with the drift-kinetic HAGIS code [S.D. Pinches, Ph.D. Thesis, The University of Nottingham, 1996; S.D. Pinches et al., CPC 111 (1998) 131] which accurately describes the particles’ trajectories. This allows finite-banana-width effects to be implemented in a rigorous way since the linear formulation of the model allows the exchange of the unperturbed orbit integration and the discretisation of the perturbed potentials in the radial direction. Successful benchmarks for toroidal Alfven eigenmodes (TAEs) and kinetic Alfven waves (KAWs) with analytical results, ideal MHD codes, drift-kinetic codes and other codes based on kinetic models are reported.

Global particle-in-cell simulations of fast-particle effects on shear Alfvén waves

This paper reports self-consistent global linear gyrokinetic particle-in-cell simulations of shear Alfven waves destabilized by fast particles in tokamak geometry. Resonant excitation of toroidal Alfven eigenmodes by fast particles and their transition to energetic particle modes (when the fast-particle drive is large enough) has been observed in the simulations.

Experiments close to the beta-limit in W7-AS

A major objective of the experimental program in the last phase of the W7-AS stellarator was to explore and demonstrate the high-β performance of advanced stellarators. MHD-quiescent discharges at low impurity radiation levels with volume averaged β-values of up to β = 3.4% have been achieved. A very important prerequisite was the attainment of the high density H-Mode (HDH) regime. This was made possible by the installation of extensive graphite plasma facing components designed for island divertor operation. The co-directed neutral beam injection provided increased absorbed heating power of up to 3.2 MW in high-β plasmas with B ≤ 1.25 T. The anticipated improved features concerning equilibrium and stability at high plasma β could be verified experimentally by the comparison of x-ray data with free boundary equilibrium calculations. The maximum β found in configurations with a rotational transform around is determined by the available heating power. No evidence of a stability limit has been found in the accessible configuration space, and the discharges are remarkably quiescent at maximum β, most likely due the increase of the magnetic well depth. An increase in low m/n MHD activity is typically observed during the transition towards high β. The beneficial stability properties of net-current-free configurations could be demonstrated by comparison with configurations where a significant inductive current drive was involved. Current driven instabilities such as tearing modes and soft disruptions can prevent access to β-values as high as in the currentless case. The experimental results indicate that optimized stellarators such as W7-X can be considered as a viable option for an attractive stellarator fusion reactor.

Conventional δf-particle simulations of electromagnetic perturbations with finite elements

The possibility of electromagnetic particle-in-cell simulations with a conventional δf approach is shown in slab geometry using finite elements. Both the ion-temperature-gradient driven mode and the shear Alfven wave are reproduced and benchmarked with the analytical linear dispersion relation. Particularly, the Alfven wave is simulated successfully at the limit k⊥→0.

Global particle-in-cell simulations of Alfvénic modes

Global linear gyrokinetic particle-in-cell (PIC) simulations of electromagnetic modes in pinch and tokamak geometries are reported. The global Alfven eigenmode, the mirror Alfven eigenmode, the toroidal Alfven eigenmode, and the kinetic ballooning modes have been simulated. All plasma species have been treated kinetically (i.e., no hybrid fluid-kinetic or reduced-kinetic model has been applied). The main intention of the paper is to demonstrate that the global Alfven modes can be treated with the gyrokinetic PIC method.

Alfvén eigenmodes measured in the TJ-II stellarator

High frequency modes (150-300 kHz) are found in several magnetic configurations of TJ-II plasmas heated by neutral beam injection (NBI). The clear dependence of mode frequency on plasma density and mass species suggests them to be Alfveigenmodes. The appearance of these modes is linked to the presence of low order rational surfaces close to the rotational transform profile. They can exhibit steady or chirping behaviour depending on the plasma profiles. Frequency chirping is observed in NBI plasmas with broad temperature profiles, but rarely observed with relatively peaked profiles. The Alfvactivity has been characterized in detail with magnetic coils for the standard configuration. Cross analyses with heavy ion beam probe and reflectometer signals have yielded spatial resolution and radial profiles of the perturbation. Correlation between magnetic coil signals and signals from diagnostics sensitive to edge ion losses, namely Langmuir probes and a fast ion loss detector, has been observed in some cases and characterized taking advantage of the chirping nature of the observed Alfvactivity. (Some figures in this article are in colour only in the electronic version)

Particle simulations with a generalized gyrokinetic solver

This paper presents a generalized gyrokinetic solver which can be used for all perpendicular wavelengths of interest and allows to include the nonlinear gyrokinetic polarization density in the simulations. The polarization density, being an integral over the phase space is calculated using “numerical particles” (not to be confused with the marker particles which are used in the charge assignment) and finite elements. Integrals over the gyroangle are calculated using an N-point approximation. The accuracy requirements on the number of the gyropoints and numerical particles are discussed. The linear part of the solver has been implemented numerically and benchmarked with the slab dispersion relation for both the ion temperature gradient driven (ITG) mode and the electron temperature gradient driven (ETG) mode. Additionally, linear ITG and ETG modes are considered in a two-dimensional bumpy pinch configuration.

Diagnostics of dense plasmas from the profile of hydrogen spectral lines in the presence of a magnetic field

A many particle approach to spectral line shapes has been generalized including the influence of a uniform magnetic field. Within this approach the electrons are treated fully quantum mechanically. Ion dynamics have been included via the well-known model microfield method. The developed theory has been applied to calculate the first Balmer lines of hydrogen for plasma parameters as can be found in dense divertor plasmas. It has been shown that the half-width, e.g., of the Hβ line is a useful tool for the determination of the electron densities in such plasmas.