R. Hatzky

A global collisionless PIC code in magnetic coordinates

A global plasma turbulence simulation code, ORB5, is presented. It solves the gyrokinetic electrostatic equations including zonal flows in axisymmetric magnetic geometry. The present version of the code assumes a Boltzmann electron response on magnetic surfaces. It uses a Particle-In-Cell (PIC), delta f scheme, 3D cubic B-splines finite elements for the field solver and several numerical noise reduction techniques. A particular feature is the use of straight-field-1 line magnetic coordinates and a field-aligned Fourier filtering technique that dramatically improves the performance of the code in terms of both the numerical noise reduction and the maximum time step allowed. Another feature is the capability to treat arbitrary axisymmetric ideal MHD equilibrium configurations. The code is heavily parallelized, with scalability demonstrated up to 4096 processors and 109 marker particles. Various numerical convergence tests are performed. The code is validated against an analytical theory of zonal flow residual, geodesic acoustic oscillations and damping, and against other codes for a selection of linear and nonlinear tests. (c) 2007 Elsevier B.V. All rights reserved.

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...

Full radius linear and nonlinear gyrokinetic simulations for tokamaks and stellarators: zonal flows, applied E x B flows, trapped electrons and finite beta

The aim of this paper is to report on recent advances made in global gyrokinetic simulations of ion temperature gradient (ITG) modes and other microinstabilities. The nonlinear development and saturation of ITG modes and the role of E × B zonal flows are studied with a global nonlinear δf formulation that retains parallel nonlinearity and thus allows for a check of the energy conservation property as a means of verifying the quality of the numerical simulation. Due to an optimized loading technique, the conservation property is satisfied with an unprecedented quality well into the nonlinear stage. The zonal component of the perturbation evolves to a quasi-steady state with regions of ITG suppression, strongly reduced radial energy flux and steepened effective temperature profiles alternating with regions of higher ITG mode amplitudes, larger radial energy flux and flattened effective temperature profiles. A semi-Lagrangian approach free of statistical noise is proposed as an alternative to the nonlinear δf formulation. An ASDEX-Upgrade experiment with an internal transport barrier is analysed with a global gyrokinetic code that includes trapped electron dynamics. The weakly destabilizing effect of trapped electron dynamics on ITG modes in an axisymmetric bumpy configuration modelling W7-X is shown in global linear simulations that retain the full electron dynamics. Finite β effects on microinstabilities are investigated with a linear global spectral electromagnetic gyrokinetic formulation. The radial global structure of electromagnetic modes shows a resonant behaviour with rational q values.

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.

On the definition of a kinetic equilibrium in global gyrokinetic simulations

Nonlinear electrostatic global gyrokinetic simulations of collisionless ion temperature gradient (ITG) turbulence and ExB zonal flows in axisymmetric toroidal plasmas are examined for different choices of the initial distribution function. Using a local Maxwellian leads to the generation of axisymmetric ExB flows that can be so strong as to prevent ITG mode growth. A method using a canonical Maxwellian is shown to avoid this spurious generation of ExB flows. In addition, a revised delta f scheme is introduced and compared to the standard delta f method. (c) 2006 American Institute of Physics.

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.

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

Nonlinear low noise particle-in-cell simulations of electron temperature gradient driven turbulence

In this Letter, it is shown that global, nonlinear, particle-in-cell (PIC) simulations of electron temperature driven turbulence recover the same level of transport as flux-tube codes when the level of statistical noise, associated with the PIC discretization, is sufficiently small. An efficient measure of the signal-to-noise ratio, applicable to every PIC code, is introduced. This diagnostic provides a direct measure of the quality of PIC simulations and allows for the validation of analytical estimates of the numerical noise. Global simulations for values of rho(*)(e)< 1/450 (normalized electron gyroradius) show no evidence of a gyro-Bohm scaling. (c) 2007 American Institute of Physics.

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.

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.