Gyrokinetic Theory Research Articles

A very general electromagnetic gyrokinetic formalism

We derive a gyrokinetic formalism which is very generally valid: the ordering allows both large inhomogeneities in plasma flow and magnetic field at long wavelength, such as typical drift-kinetic theories, as well as fluctuations at the gyro-scale. The underlying approach is to order the vorticity to be small, and to assert that the timescales in the local plasma frame are long compared to the gyrofrequency. Unlike most other derivations, we do not treat the long and short wavelength components of the fluctuating fields separately; the single-field description defines the particle motion and their interaction with the electromagnetic field at small-scale, the system-scale, and intermediate length scales in a unified fashion. As in earlier literature, the work consists of identifying a coordinate system where the gyroangle-dependent terms are small, and using a near-unity transform to systematically find a set of coordinates where the gyroangle dependence vanishes. We derive a gyrokinetic Lagrangian which is valid where the vorticity |∇×(E×B/B)| is small compared to the gyrofrequency Ω, and the magnetic field scale length is long compared to the gyroradius; we also require that time variation be slow in an appropriately chosen reference frame. This appears to be a minimum set of constraints on a gyrokinetic theory and is substantially more general than earlier approaches. It is the general-geometry electromagnetic extension of Dimits, Phys. Plasmas 17, 055901 (2010) (which is an electrostatic formalism with a homogeneous background magnetic field). This approach also does not require a separate treatment of fluctuating and background components of the magnetic field, unlike much of the previous literature. As a consequence, the “cross terms” due to a combination of long- and short-wavelength variation, which were ignored in the earlier work (but derived in a more restrictive ordering in Parra and Calvo, Plasma Phys. Controlled Fusion 53, 045001 (2011)), also appear naturally.

Effects of energetic particles on zonal flow generation by toroidal Alfvén eigenmode

Generation of zonal flow (ZF) by energetic particle (EP) driven toroidal Alfvén eigenmode (TAE) is investigated using nonlinear gyrokinetic theory. It is found that nonlinear resonant EP contribution dominates over the usual Reynolds and Maxwell stresses due to thermal plasma nonlinear response. ZF can be forced driven in the linear growth stage of TAE, with the growth rate being twice the TAE growth rate. The ZF generation mechanism is shown to be related to polarization induced by resonant EP nonlinearity. The generated ZF has both the usual meso-scale and micro-scale radial structures. Possible consequences of this forced driven ZF on the nonlinear dynamics of TAE are also discussed.

Fine structure zonal flow excitation by beta-induced Alfvén eigenmode

Nonlinear excitation of low frequency zonal structure (LFZS) by beta-induced Alfvén eigenmode (BAE) is investigated using nonlinear gyrokinetic theory. It is found that electrostatic zonal flow (ZF), rather than zonal current, is preferentially excited by finite amplitude BAE. In addition to the well-known meso-scale radial envelope structure, ZF is also found to exhibit fine radial structure due to the localization of BAE with respect to mode rational surfaces. Specifically, the zonal electric field has an even mode structure at the rational surface where radial envelope peaks.

Second order gyrokinetic theory for particle-in-cell codes

The main idea of the gyrokinetic dynamical reduction consists in a systematical removal of the fast scale motion (the gyromotion) from the dynamics of the plasma, resulting in a considerable simplification and a significant gain of computational time. The gyrokinetic Maxwell–Vlasov equations are nowadays implemented in for modeling (both laboratory and astrophysical) strongly magnetized plasmas. Different versions of the reduced set of equations exist, depending on the construction of the gyrokinetic reduction procedure and the approximations performed in the derivation. The purpose of this article is to explicitly show the connection between the general second order gyrokinetic Maxwell–Vlasov system issued from the modern gyrokinetic theory and the model currently implemented in the global electromagnetic Particle-in-Cell code ORB5. Necessary information about the modern gyrokinetic formalism is given together with the consistent derivation of the gyrokinetic Maxwell–Vlasov equations from first principles. The variational formulation of the dynamics is used to obtain the corresponding energy conservation law, which in turn is used for the verification of energy conservation diagnostics currently implemented in ORB5. This work fits within the context of the code verification project VeriGyro currently run at IPP Max-Planck Institut in collaboration with others European institutions.

Gyrokinetic Theory and Dynamics of the Tokamak Edge

AbstractThe validity of modern gyrokinetic field theory is assessed for the tokamak edge. The basic structure of the Lagrangian and resulting equations and their conservation laws is reviewed. The conventional microturbulence ordering for expansion is small potential/arbitrary wavelength. The equilibrium ordering for expansion is long wavelength/arbitrary amplitude. The long‐wavelength form of the conventional Lagrangian is derived in detail. The two Lagrangians are shown to match at long wavelength if the E × B Mach number is small enough for its corrections to the gyroaveraging to be neglected. Therefore, the conventional derivation and its Lagrangian can be used at all wavelengths if these conditions are satisfied. Additionally, dynamical compressibility of the magnetic field can be neglected if the plasma beta is small. This allows general use of a shear‐Alfvén Lagrangian for edge turbulence and self consistent equilibrium‐scale phenomena for flows, currents, and heat fluxes for conventional tokamaks without further modification by higher‐order terms. Corrections in polarisation and toroidal angular momentum transport due to these higher‐order terms for global edge turbulence computations are shown to be small. (© 2016 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Nonlinear gyrokinetic theory and its application to computation of the gyrocenter motion in ripple field

The nonlinear gyrokinetic equation with full electromagnetic potential perturbations is derived by using the two-step transform procedure. The second-order transformed Hamiltonian can be simplified as 12δA∥2, instead of 12δA2 in the long-wave-length limit. A numerical code based on the I-transform method is improved to compute the gyrocenter orbit in the TFTR tokamak with a ripple field, and the numerical results indicate that the collisionless stochastic diffusion criterion agrees well with the theoretical prediction.

Gyrokinetic stability theory of electron–positron plasmas

The linear gyrokinetic stability properties of magnetically confined electron–positron plasmas are investigated in the parameter regime most likely to be relevant for the first laboratory experiments involving such plasmas, where the density is small enough that collisions can be ignored and the Debye length substantially exceeds the gyroradius. Although the plasma beta is very small, electromagnetic effects are retained, but magnetic compressibility can be neglected. The work of a previous publication (Helander, Phys. Rev. Lett., vol. 113, 2014a, 135003) is thus extended to include electromagnetic instabilities, which are of importance in closed-field-line configurations, where such instabilities can occur at arbitrarily low pressure. It is found that gyrokinetic instabilities are completely absent if the magnetic field is homogeneous: any instability must involve magnetic curvature or shear. Furthermore, in dipole magnetic fields, the stability threshold for interchange modes with wavelengths exceeding the Debye radius coincides with that in ideal magnetohydrodynamics. Above this threshold, the quasilinear particle flux is directed inward if the temperature gradient is sufficiently large, leading to spontaneous peaking of the density profile.

Scaling of up–down asymmetric turbulent momentum flux with poloidal shaping mode number in tokamaks

Breaking the up–down symmetry of tokamaks removes a constraint limiting intrinsic momentum transport, and hence toroidal rotation, to be small. Using gyrokinetic theory, we study the effect of different up–down asymmetric flux surface shapes on the turbulent transport of momentum. This is done by perturbatively expanding the gyrokinetic equation in large flux surface shaping mode number. It is found that the momentum flux generated by shaping that lacks mirror symmetry (which is necessarily up–down asymmetric) has a power law scaling with the shaping mode number. However, the momentum flux generated by mirror symmetric flux surface shaping (even if it is up–down asymmetric) decays exponentially with large shaping mode number. These scalings are consistent with nonlinear local gyrokinetic simulations and indicate that low mode number shaping effects (e.g. elongation, triangularity) are optimal for creating rotation. Additionally it suggests that breaking the mirror symmetry of flux surfaces may generate significantly more toroidal rotation.

On the Validation of Gyrokinetic L-Mode Simulations

Over the last decade, plasma turbulence simulations based on gyrokinetic theory have reached an amazing degree of physical comprehensiveness and realism. In contrast to early gyrokinetic studies, which were restricted to qualitative statements, state-of-the-art investigations may now be compared quantitatively, therefore enabling validation and detailed analysis of their predictive capabilities. Here, particular attention is paid to outer-core L-mode discharges for which some previous gyrokinetic studies have found an underprediction of ion heat transport by almost one order of magnitude, the so-called shortfall. Carrying out radially local and nonlocal GENE simulations using actual plasma profiles and parameters and magnetohydrodynamic equilibria, and employing as much physics as available, only a mild underprediction is found, which can, furthermore, be overcome by varying the ion temperature gradient within the error bars associated with the experimental measurement. The significance and reliability of these simulations is furthermore demonstrated by extensive comparison with experimental measurements. The latter involve sophisticated synthetic beam emission spectroscopy and correlation electron cyclotron emission data analysis. The agreement found between the measurements and the state-of-the-art postprocessed simulation data confirms the high degree of realism.

Intrinsic rotation drive by collisionless trapped electron mode turbulence

Both the parallel residual stress and parallel turbulent acceleration driven by electrostatic collisionless trapped electron mode (CTEM) turbulence are calculated analytically using gyrokinetic theory. Quasilinear results show that the parallel residual stress contributes an outward flux of co-current rotation for normal magnetic shear and turbulence intensity profile increasing outward. This may induce intrinsic counter-current rotation or flattening of the co-current rotation profile. The parallel turbulent acceleration driven by CTEM turbulence vanishes, due to the absence of a phase shift between density fluctuation and ion pressure fluctuation. This is different from the case of ion temperature gradient turbulence, for which the turbulent acceleration can provide co-current drive for normal magnetic shear and turbulence intensity profile increasing outward. Its order of magnitude is predicted to be the same as that of the divergence of the residual stress [L. Wang and P. H. Diamond, Phys. Rev. Lett. 110, 265006 (2013)]. A possible connection of these theoretical results to experimental observations of electron cyclotron heating effects on toroidal rotation is discussed.

Turbulence spreading in gyro-kinetic theory

In this letter a new operative definition for the turbulence intensity in connection with magnetized plasmas is given. In contrast to previous definitions the new definition satisfies a Fisher–Kolmogorov–Petrovskii–Piskunov type equation. Furthermore, explicit expressions for the turbulence intensity and the turbulence intensity flux, that allow for the first time direct numerical evaluation, are derived. A carefully designed numerical experiment for the case of a tokamak is performed to study the impact of turbulence spreading. The effective turbulence diffusion coefficient is measured to be smaller than the heat conduction coefficient and the turbulence spreading length is found to be of the order of the turbulence correlation length. The results show that turbulence spreading can play a role in the non-local flux gradient relation, or in the scaling of transport coefficients with the normalized Larmor radius, only over lengths scale of the order of the turbulence correlation length. A new turbulence convection mechanism, due to the drift connected with the magnetic field inhomogeneities, is described. The convective flux integrates to zero under the flux surface average unless there is an up–down asymmetry in the tubulence intensity. The latter asymmetry can be generated through a radial inhomogeneity or plasma rotation. It is shown that the turbulence convection can lead to a spreading of the order of the correlation length.

Energetically consistent collisional gyrokinetics

We present a formulation of collisional gyrokinetic theory with exact conservation laws for energy and canonical toroidal momentum. Collisions are accounted for by a nonlinear gyrokinetic Landau operator. Gyroaveraging and linearization do not destroy the operator's conservation properties. Just as in ordinary kinetic theory, the conservation laws for collisional gyrokinetic theory are selected by the limiting collisionless gyrokinetic theory.

Trapped electron mode driven electron heat transport in JET: experimental investigation and gyro-kinetic theory validation

The main purpose of this work is to study the dependence of trapped electron modes (TEM) threshold and of electron stiffness on the most relevant plasma parameters. Dedicated transport experiments based on heat flux scans and Te modulation have been performed in JET in TEM dominated plasmas with pure ICRH electron heating and a numerical study using gyrokinetic simulations has been performed with the code GKW. Using multilinear regressions on the experimental data, the stabilizing effect of magnetic shear predicted by theory for our plasma parameters is confirmed while no significant effect of safety factor was found. Good quantitative agreement is found between the TEM thresholds found in the experiments and calculated with linear GKW simulations. Non-linear simulations have given further confirmation of the threshold values and allowed comparison with the values of stiffness found experimentally. Perturbative studies using RF power modulation indicate the existence of an inward convective term for the electron heat flux. Adding NBI power, ion temperature gradient (ITG) modes become dominant and a reduction of with respect to pure ICRH, TEM dominant discharges has been experimentally observed, in spite of increased total electron power. Possible explanations are discussed.

Effects of collisions on conservation laws in gyrokinetic field theory

Effects of collisions on conservation laws for toroidal plasmas are investigated based on the gyrokinetic field theory. Associating the collisional system with a corresponding collisionless system at a given time such that the two systems have the same distribution functions and electromagnetic fields instantaneously, it is shown how the collisionless conservation laws derived from Noether's theorem are modified by the collision term. Effects of the external source term added into the gyrokinetic equation can be formulated similarly with the collisional effects. Particle, energy, and toroidal momentum balance equations including collisional and turbulent transport fluxes are systematically derived using a novel gyrokinetic collision operator, by which the collisional change rates of energy and canonical toroidal angular momentum per unit volume in the gyrocenter space can be given in the conservative forms. The ensemble-averaged transport equations of particles, energy, and toroidal momentum given in the present work are shown to include classical, neoclassical, and turbulent transport fluxes which agree with those derived from conventional recursive formulations.

Improved strong scaling of a spectral/finite difference gyrokinetic code for multi-scale plasma turbulence

Abstract Optimization techniques of a plasma turbulence simulation code GKV for improved strong scaling are presented. This work is motivated by multi-scale plasma turbulence extending over multiple spatio-temporal scales of electrons and ions, whose simulations based on the gyrokinetic theory require huge calculations of five-dimensional (5D) computational fluid dynamics by means of spectral and finite difference methods. First, we present the multi-layer domain decomposition of the multi-dimensional and multi-species problem, and segmented MPI-process mapping on 3D torus interconnects, which fully utilizes the bi-section bandwidth for data transpose and reduces the conflicts of simultaneous point-to-point communications. These techniques reduce the inter-node communication cost drastically. Second, pipelined computation-communication overlaps are implemented by using the OpenMP/MPI hybrid parallelization, which effectively mask the communication cost. Careful regulations of the pipeline length and of the thread granularity respectively suppress latencies of MPI, load imbalance and scheduling overheads of OpenMP. Thanks to the above optimizations, GKV achieves excellent strong scaling up to ∼600k cores with high computational performance 782.4 TFlops (8.29% of the theoretical peak) and high effective parallelization rate ∼99.99994% on K, which demonstrates its applicability and efficiency toward a million of cores. The optimized code realizes multi-scale plasma turbulence simulations covering electron and ion scales, and reveals cross-scale interactions of electron- and ion-scale turbulence.

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.

A toroidal momentum transport equation for axisymmetric 1D transport codes

A toroidal momentum transport equation is presented, which is suited for implementation in 1D transport codes in axisymmetric geometry (e.g. tokamak plasmas). Contributions to the total toroidal momentum from the different plasma species are retained. The contributions of E × B, diamagnetic and poloidal flows are also retained. The different pieces appearing in the expression for the turbulence-driven part of the flux are obtained consistently from standard gyro-kinetic theory.

Fourier–Hermite spectral representation for the Vlasov–Poisson system in the weakly collisional limit

We study Landau damping in the 1+1D Vlasov–Poisson system using a Fourier–Hermite spectral representation. We describe the propagation of free energy in Fourier–Hermite phase space using forwards and backwards propagating Hermite modes recently developed for gyrokinetic theory. We derive a free energy equation that relates the change in the electric field to the net Hermite flux out of the zeroth Hermite mode. In linear Landau damping, decay in the electric field corresponds to forward propagating Hermite modes; in nonlinear damping, the initial decay is followed by a growth phase characterized by the generation of backwards propagating Hermite modes by the nonlinear term. The free energy content of the backwards propagating modes increases exponentially until balancing that of the forward propagating modes. Thereafter there is no systematic net Hermite flux, so the electric field cannot decay and the nonlinearity effectively suppresses Landau damping. These simulations are performed using the fully-spectral 5D gyrokinetics code SpectroGK, modified to solve the 1+1D Vlasov–Poisson system. This captures Landau damping via Hou–Li filtering in velocity space. Therefore the code is applicable even in regimes where phase mixing and filamentation are dominant.

Microturbulence study of the isotope effect

The influence of the ion mass on the dynamics of magnetized plasmas is an important challenge in fusion research. The discrepancies between the improvement of the magnetic confinement with the ion mass in tokamak experiments and diffusive turbulent transport predictions have remained unexplained for several decades. We refer to this phenomenon as the isotope effect. In this paper, we study this effect with gyrokinetic theory using the Gene code. We find several sets of plasma parameters that correspond to low wavenumber turbulence for which the isotope effect is present, although the intensity is smaller than the experimental observations. We also relate these results to the zonal flow intensity of the system, which is characterized by the average shear flow rate.

On the drift magnetosonic waves in anisotropic low beta plasmas

A generalized dispersion relation of obliquely propagating drift magnetosonic waves is derived by using the gyrokinetic theory for anisotropic low beta plasmas. The stability analysis applicable to a wide range of plasma parameters is performed to understand the stabilization mechanism of the drift magnetosonic instability and the estimation of the growth rate is also presented. It is noted that the growth rate of the drift instability enhances for small anisotropy (Ae,i = T⊥e,i/T∥e,i < 1) whereas it is suppressed for large anisotropy (Ae,i > 1).