Drift-kinetic Electrons Research Articles

Effects of collisional ion orbit loss on tokamak radial electric field and toroidal rotation in an L-mode plasma

Ion orbit loss has been used to model the formation of a strong negative radial electric field E r in the tokamak edge, as well as edge momentum transport and toroidal rotation. To quantitatively measure ion orbit loss, an orbit-flux formulation has been developed and numerically applied to the gyrokinetic particle-in-cell code XGC. We study collisional ion orbit loss in an axisymmetric DIII-D L-mode plasma using gyrokinetic ions and drift-kinetic electrons. Numerical simulations, where the plasma density and temperature profiles are maintained through neutral ionization and heating, show the formation of a quasisteady negative E r in the edge. We have measured a radially outgoing ion gyrocenter flux due to collisional scattering of ions into the loss orbits, which is balanced by the radially incoming ion gyrocenter flux from confined orbits on the collisional time scale. This suggests that collisional ion orbit loss can shift E r in the negative direction compared to that in plasmas without orbit loss. It is also found that collisional ion orbit loss can contribute to a radially outgoing (counter-current) toroidal-angular-momentum flux, which is not balanced by the toroidal-angular-momentum flux carried by ions on the confined orbits. Therefore, the edge toroidal rotation shifts in the co-current direction on the collisional time scale.

Orb5: A global electromagnetic gyrokinetic code using the PIC approach in toroidal geometry

This paper presents the current state of the global gyrokinetic code Orb5 as an update of the previous reference (Jolliet et al., 2007). The Orb5 code solves the electromagnetic Vlasov–Maxwell system of equations using a PIC scheme and also includes collisions and strong flows. The code assumes multiple gyrokinetic ion species at all wavelengths for the polarization density and drift-kinetic electrons. Variants of the physical model can be selected for electrons such as assuming an adiabatic response or a “hybrid” model in which passing electrons are assumed adiabatic and trapped electrons are drift-kinetic. A Fourier filter as well as various control variates and noise reduction techniques enable simulations with good signal-to-noise ratios at a limited numerical cost. They are completed with different momentum and zonal flow-conserving heat sources allowing for temperature-gradient and flux-driven simulations. The code, which runs on both CPUs and GPUs, is well benchmarked against other similar codes and analytical predictions, and shows good scalability up to thousands of nodes.

A conservative scheme of drift kinetic electrons for gyrokinetic simulation of kinetic-MHD processes in toroidal plasmas

A conservative scheme of drift kinetic electrons for gyrokinetic simulations of kinetic-magnetohydrodynamic processes in toroidal plasmas has been formulated and verified. Both vector potential and electron perturbed distribution function are decomposed into adiabatic part with analytic solution and non-adiabatic part solved numerically. The adiabatic parallel electric field is solved directly from the electron adiabatic response, resulting in a high degree of accuracy. The consistency between electrostatic potential and parallel vector potential is enforced by using the electron continuity equation. Since particles are only used to calculate the non-adiabatic response, which is used to calculate the non-adiabatic vector potential through Ohm's law, the conservative scheme minimizes the electron particle noise and mitigates the cancellation problem. Linear dispersion relations of the kinetic Alfvén wave and the collisionless tearing mode in cylindrical geometry have been verified in gyrokinetic toroidal code simulations, which show that the perpendicular grid size can be larger than the electron collisionless skin depth when the mode wavelength is longer than the electron skin depth.

Gyrokinetic neoclassical study of the bootstrap current in the tokamak edge pedestal with fully non-linear Coulomb collisions

As a follow-up on the drift-kinetic study of the non-local bootstrap current in the steep edge pedestal of tokamak plasma by Koh et al. [Phys. Plasmas 19, 072505 (2012)], a gyrokinetic neoclassical study is performed with gyrokinetic ions and drift-kinetic electrons. Besides the gyrokinetic improvement of ion physics from the drift-kinetic treatment, a fully non-linear Fokker-Planck collision operator—that conserves mass, momentum, and energy—is used instead of Koh et al.'s linearized collision operator in consideration of the possibility that the ion distribution function is non-Maxwellian in the steep pedestal. An inaccuracy in Koh et al.'s result is found in the steep edge pedestal that originated from a small error in the collisional momentum conservation. The present study concludes that (1) the bootstrap current in the steep edge pedestal is generally smaller than what has been predicted from the small banana-width (local) approximation [e.g., Sauter et al., Phys. Plasmas 6, 2834 (1999) and Belli et al., Plasma Phys. Controlled Fusion 50, 095010 (2008)], (2) the plasma flow evaluated from the local approximation can significantly deviate from the non-local results, and (3) the bootstrap current in the edge pedestal, where the passing particle region is small, can be dominantly carried by the trapped particles in a broad trapped boundary layer. A new analytic formula based on numerous gyrokinetic simulations using various magnetic equilibria and plasma profiles with self-consistent Grad-Shafranov solutions is constructed.

Particle simulation of lower hybrid wave propagation in fusion plasmas

Global particle simulations of the lower hybrid (LH) waves have been carried out using fully kinetic ions and drift kinetic electrons with a realistic electron-to-ion mass ratio. The LH wave frequency, mode structure, and electron Landau damping from the electrostatic simulations agree very well with the analytic theory. Linear simulation of the propagation of a LH wave-packet in the toroidal geometry shows that the wave propagates faster in the high field side than the low field side, in agreement with a ray tracing calculation. This poloidal asymmetry arises from the non-conservation of the poloidal mode number due to the non-uniform magnetic field. In contrast, the poloidal mode number is conserved in the cylindrical geometry with the uniform magnetic field.

Gyrokinetic simulations of collisionless tearing modes

We investigate the effects of temperature and density gradients on collisionless magnetic reconnection by means of PIC-δf simulations with standard gyrokinetic equations using gyrokinetic ions and drift kinetic electrons. In a slab model ion and electron temperature gradients act stabilising on the tearing mode instability in the cases that have been considered. An additional ion and electron density gradient completely stabilises the mode below a critical η (ratio of temperature gradient to density gradient). The critical η becomes larger with decreasing equilibrium current.Nonlinear single-mode simulations show that the island width saturates at the level predicted by [1], if the linear instability growth rate is sufficiently small. Including ion and electron temperature and density gradients in nonlinear calculations the saturated island width vanishes below a critical η.

Particle-in-cell simulation with Vlasov ions and drift kinetic electrons

There are certain limitations in using gyrokinetic ions for simulations of turbulent transport in tokamak plasmas. Applications where Vlasov ions might be more appropriate include the electron temperature gradient driven turbulence, edge turbulence with steep density gradient, and magnetic reconnection in a weak guide field. In such cases the ion gyrokinetic model presently used in simulations needs to be extended, but a satisfactory extension valid for fully electromagnetic turbulence is not presently available. Even if an accurate model is found, its numerical implementation could be very challenging. We propose a kinetic model that combines Vlasov ions with gyrokinetic electrons to avoid the difficulties with gyrokinetic ions. The field equations of this model are the Faraday’s equation and the Ampere’s equation without the displacement current. The perturbed fields B1 and E1 rather than the scalar and vector potentials are used to formulate the field equations. We have devised an implicit scheme for this model, demonstrated in three-dimensional slab for the Alfvén waves, the drift Alfvén instability and the ion acoustic waves.

Particle-in-cell guiding center models of ion velocity space shell instabilities

Particle-in-cell simulations with full dynamics ions and drift kinetic electrons are essential tools to model kinetic instabilities in strongly magnetized plasmas. In these guiding center models, parallel acceleration of the electrons is retained while their perpendicular velocity is given to lowest order by the drift. These simulations are applied to a study of the electrostatic stability of energetic velocity-space ion shells observed in the Earth's plasma sheet boundary layer by the Cluster satellites.

/spl delta/f Simulation of the collisionless tearing mode instability with a gyrokinetic ion response

The evolution of the collisionless tearing mode instability is studied using an electromagnetic gyrokinetic /spl delta/f particle-in-cell simulation model. Drift-kinetic electrons are used. High resolution, small box (less than 10 ion gyroradii) simulations have been well benchmarked with eigenmode analysis and their nonlinear evolution agrees well with theory. In this regime, the ion response is not important and can be either fixed, adiabatic or fully gyrokinetic. Here, results are presented with larger box sizes (64 ion gyroradii radially) where the ion gyrokinetic response is important and cannot be neglected. In these larger box simulations, the instability exhibits an odd parity, different than the even tearing parity.

Gyrokinetic δf simulation of the collisionless and semicollisional tearing mode instability

The evolution of collisionless and semicollisional tearing mode instabilities is studied using an electromagnetic gyrokinetic δf particle-in-cell simulation model. Drift-kinetic electrons are used. Linear eigenmode analysis is presented for the case of fixed ions and there is excellent agreement with simulation. A double peaked eigenmode structure is seen indicative of a positive Δ′. Nonlinear evolution of a magnetic island is studied and the results compare well with existing theory in terms of saturation level and electron bounce oscillations. Electron-ion collisions are included to study the semicollisional regime. The algebraic growth stage is observed and compares favorably with theory. Nonlinear saturation following the algebraic stage is observed.

Kinetic electron closures for electromagnetic simulation of drift and shear-Alfvén waves. II

An electromagnetic hybrid scheme (fluid electrons and gyrokinetic ions) is elaborated in example calculations and extended to toroidal geometry. The scheme includes a kinetic electron closure valid for βe>me/mi (βe is the ratio of the plasma electron pressure to the magnetic field energy density). The new scheme incorporates partially linearized (δf ) drift-kinetic electrons whose pressure and number density moments are used to close the fluid momentum equation for the electron fluid (Ohm’s law). The test cases used are small-amplitude kinetic shear-Alfvén waves with electron Landau damping, the ion-temperature-gradient instability, and the collisionless drift instability (universal mode) in an unsheared slab as a function of the plasma βe. Attention is given to resolution and convergence issues in simulations of turbulent steady states.

Gyrokinetic turbulence simulations with fully kinetic electrons

It is now possible to include kinetic electron physics in low-beta gyrokinetic simulations of tokamak plasma turbulence. These are three-dimensional (3-D) toroidal simulations using gyrokinetic ions and drift-kinetic electrons. Including electrons increases the linear growth rate and nonlinear energy transport for typical H-mode parameters. Here, we briefly discuss our results and low-cost high-performance stereoscopic visualization techniques for analyzing the large-scale 3-D gyrokinetic turbulence simulation data.

Kinetic electron closures for electromagnetic simulation of drift and shear-Alfvén waves. I.

The electromagnetic hybrid scheme of Chen and Parker (fluid electrons and gyrokinetic ions) [Phys. Plasmas 8, 441 (2001)] is extended to include a kinetic electron closure valid for βe>me/mi (βe is the ratio of the plasma electron pressure to the magnetic field energy density). The new schemes incorporate partially linearized (δf ) drift-kinetic electrons whose pressure and number density moments are used to close the fluid momentum equation for the electron fluid (Ohm’s law) using the departure of the perturbed δf kinetic pressure from the isothermal perturbed pressure response. Comparisons are made between the results of the hybrid schemes with kinetic electron closure and a conventional δf algorithm for drift-kinetic electrons and gyrokinetic ions in a two-dimensional slab model. The test cases used are small-amplitude kinetic shear-Alfvén waves with electron Landau damping, the ion-temperature-gradient instability, and the collisionless drift instability (universal mode) in an unsheared slab as a function of the plasma βe. The hybrid schemes have the desirable properties that they do not require that the mesh size perpendicular to the applied magnetic field be smaller than the collisionless skin depth c/ωpe and naturally accommodate zonal flow physics (radial modes) with nonadiabatic electron effects. The most successful of the new algorithms introduced gives very good results for βe>me/mi.

Massively parallel Vlasov simulation of electromagnetic drift-wave turbulence

We use a hybrid model of drift-kinetic electrons and fluid ions to treat electromagnetic drift-wave turbulence in an inhomogeneous collisionless plasma confined by a strong external magnetic field. The electrons are described in terms of a distribution function in five-dimensional phase space, and the basic nonlinear equations are solved numerically on a massively parallel computer using explicit finite-difference methods of the upwind and predictor-corrector type. The algorithm as well as its parallelization are explained, the parallel speedup is measured, and various numerical tests are performed.

Implicit, partially linearized, electromagnetic particle simulation of plasma drift-wave turbulence

Implicit, perturbative algorithms [B. I. Cohen, A. M. Dimits, J. J. Stimson, and D. C. Barnes, Phys. Rev. E 53, 2708 (1996)] are extended to address electromagnetic particle simulation of drift-wave instabilities in electron and ion plasmas with a strong applied magnetic field. The algorithms use gyrokinetic ions and drift-kinetic electrons with a realistic mass ratio. Implicit moment and direct implicit solutions of the coupled Maxwell and plasma dynamical equations are demonstrated. Aspects of the numerical dispersion properties are confirmed in test simulations. Applications to ion-temperature-gradient and collisionless drift instabilities in a two-dimensional slab with magnetic shear are presented that illustrate the efficacy of the algorithms and the electromagnetic effects.

Implicit-moment, partially linearized particle simulation of kinetic plasma phenomena.

An algorithm for kinetic simulation of electron and ion plasmas is introduced that combines both implicit and perturbative methods to deal efficiently with disparate time scales and signal-to-noise issues associated with particle discreteness. The algorithm is introduced for an electrostatic model in a strong magnetic field with gyrokinetic ions and drift-kinetic electrons. Some of the numerical dispersion properties are analyzed and confirmed in test simulations. Two-dimensional applications of the algorithm to drift-wave instabilities excited either by an ion temperature gradient (ion-temperature-gradient instability) or by electron inverse Landau damping in the presence of a density gradient (collisionless drift instability) are presented that illustrate the usefulness of the method. Clear gains in resolution and computational efficiency over previous methods are achieved. {copyright} {ital 1996 The American Physical Society.}

Hybrid-kinetic stability properties of high-beta plasmas

A hybrid-kinetic model (Vlasov ions and drift-kinetic electrons) is used to investigate the equilibrium and stability properties of collisionless linear screw-pinch configurations, assuming low-frequency perturbations (|ω|≲ωci) but making no a priori assumptions regarding the size of the ion Larmor radius (arbitrary rLi/L⊥). A general eigenvalue equation is derived for perturbations about a general equilibrium characterized by density profile N(r), isotropic ion equilibrium f0i(H), and locally Maxwellian electrons. This eigenvalue equation clearly exhibits the importance of finite electron temperature and electron kinetic effects parallel to the magnetic field direction, and reduces to the Vlasov-fluid eigenvalue equation of Freidberg in the limit of zero electron temperature.

A hybrid-kinetic model for high-β plasmas

A hybrid-kinetic model (Vlasov ions and drift-kinetic electrons) is developed to describe the equilibrium and stability properties of collisionless high-beta plasmas in arbitrary magnetic field configurations. No ordering of ion properties (such as ω/Ωi and rLi/L⊥) is assumed, so the ions are described by the exact (unexpanded) Vlasov equation. The electrons are assumed to be strongly magnetized with ω/Ωε ∼rLe/L⊥=ε≪1, and the electron Vlasov equation is expanded in ε to give a drift-kinetic equation for fe. Examples of linear screw-pinch equilibria are constructed to demonstrate the ease with which equilibrium solutions are found within this formalism. Also, it is shown that the hybrid-kinetic model correct to O (ε) includes physical effects which are important during the implosion and immediate post-implosion phases of high-density pinch experiments, where stability behavior is characterized by the fast time scale ω−1LH= (ΩeΩi)−1/2. In particular, a local dispersion relation is derived for the modified-two-stream and lower-hybrid-drift instabilities, which have characteristic growth rates of order (ΩeΩi)1/2 when ω2pe/Ω2e≫1.