Gyrokinetic Particle Simulation Research Articles

Size Scaling of Turbulent Transport in Magnetically Confined Plasmas

Transport scaling with respect to device size in magnetically confined plasmas is critically examined for electrostatic ion-temperature-gradient turbulence using global gyrokinetic particle simulations. It is found, by varying device size normalized by ion gyroradius while keeping other dimensionless plasma parameters fixed, that fluctuation scale length is microscopic in the presence of zonal flows. The local transport coefficient exhibits a gradual transition from a Bohm-like scaling for device sizes corresponding to present-day experiments to a gyro-Bohm scaling for future larger devices.

Shear-Alfvén waves in gyrokinetic plasmas

It is found that the thermal fluctuation level of the shear-Alfvén waves in a gyrokinetic plasma is dependent on plasma β(≡cs2/vA2), where cs is the ion acoustic speed and vA is the Alfvén velocity. This unique thermodynamic property based on the fluctuation–dissipation theorem is verified in this paper using a new gyrokinetic particle simulation scheme, which splits the particle distribution function into the equilibrium part as well as the adiabatic and nonadiabatic parts. The numerical implication of this property is discussed.

Generation of radial electric field in the process of full reconnection by kinetic kink mode

Effects of density gradient on the kinetic m = 1 (m/n = 1/1) internal kink mode in a cylindrical tokamak plasma are studied by the gyro-kinetic particle simulations. When the density gradient is not large enough to change the full reconnection process, the phenomena after the full reconnection, such as the plasma flow and the secondary reconnection, are changed considerably due to the self-generated radial electric field, i.e. the m/n = 0/0 mode. The growing mechanism is explained by the difference of E × B drift motion between ions and electrons, which is caused by the fast parallel motion of electron.

Nonlinear dynamics of small-scale magnetic islands in high temperature plasmas

The growth and nonlinear evolution of small-scale magnetic islands having a characteristic width of the order of the skin depth and smaller than an ion gyroradius is studied using nonlinear gyrokinetic particle simulations. In the sheared slab geometry it is demonstrated that in the absence of plasma pressure gradients, in a collisionless plasma, skin-size magnetic islands are spontaneously generated and form a nonlinear Bernstein–Greene–Kruskal-type asymptotic state. In the presence of electron thermal gradients skin-size drift-magnetic islands also form. When shorter wavelength electron-temperature-gradient-driven modes are included in the simulation nonlinear coupling to the drift-magnetic island is observed, which has a significant impact on its temporal evolution.

The split-weight particle simulation scheme for plasmas

An efficient numerical method for treating electrons in magnetized plasmas has been developed. The scheme, which is based on the perturbative (δf) gyrokinetic particle simulation, splits the particle electron responses into adiabatic and nonadiabatic parts. The former is incorporated into the gyrokinetic Poisson’s equation, while the latter is calculated dynamically with the aid of the charge conservation equation. The new scheme affords us the possibility of suppressing unwanted high-frequency oscillations and, in the meantime, relaxing the Courant condition for the thermal particles moving in the parallel direction. It is most useful for studying low-frequency phenomena in plasmas. As an example, one-dimensional drift wave simulation has been carried out using the scheme and the results are presented in this paper. This methodology can easily be generalized to problems in three-dimensional toroidal geometry, as well as those in unmagnetized plasmas.

Effect of externally imposed and self-generated flows on turbulence and magnetohydrodynamic activity in tokamak plasmas

The effects of externally imposed and self-generated poloidal flows on turbulence and magnetohydrodynamic (MHD) activity are examined in the context of the possible Electric Tokamak (ET) [Phys. Plasmas 6, 4722 (1999)] plasmas and (circularized) DIII-D-like [Fusion Technol. 8, 441 (1985)] discharges. Global gyrokinetic particle simulations and reduced MHD calculations respectively show that ion temperature gradient driven turbulence (ITGDT) and resistive internal kink MHD activity can be reduced and/or suppressed with experimentally achievable externally imposed flows for possible ET start-up plasmas. Global gyrokinetic particle simulations of ITGDT also serve to demonstrate that self-generated flows are necessary to yield experimentally relevant radial correlation lengths in the case of DIII-D-like discharges.

On the gyrokinetic equilibrium

Recent developments in gyrokinetic-magnetohydrodynamics (MHD) theory and in electromagnetic gyrokinetic particle simulations raise the question of consistency between the gyrokinetic model and the fluid model. Due to the special characteristics of the guiding center coordinates, it is a nontrivial exercise to show this consistency. In this paper it is shown, in a very general setting, that the gyrokinetic theory and the fluid equations do give an equivalent description of plasma equilibrium (∂/∂t=0). The fluid continuity equation and momentum equation for equilibrium plasmas are recovered entirely from the gyrokinetic theory. However, it was Spitzer who first realized the importance of consistency between guiding-center motion and fluid equations. In particular, he studied the “apparent paradoxical result” regarding the difference between perpendicular particle flow and guiding-center flow, which will be referred to as the Spitzer paradox in this paper. By recovering the fluid equations from the gyrokinetic theory, we automatically resolve the Spitzer paradox, whose essence is how the perpendicular current and flow are microscopically generated from particles’ guiding-center motion. The mathematical construction in the gyrokinetic theory which relates observable quantities in the laboratory frame to the distribution function in the guiding-center coordinates is consistent with Spitzer’s original physical picture, while today’s gyrokinetic-MHD theory covers a much wider range of problems in a much more general and quantitative way.

Effects of Collisional Zonal Flow Damping on Turbulent Transport

Results from 3D global gyrokinetic particle simulations of ion-temperature-gradient driven microturbulence in a toroidal plasma show that the ion thermal transport level in the interior region exhibits significant dependence on the ion-ion collision frequency even in regimes where the instabilities are collisionless. This is identified as arising from the Coulomb collisional damping of turbulence-generated zonal flows.

Gyro-Kinetic Particle Simulation of m=1 Internal Kink Mode in the Presence of Density Gradient.

Effects of density gradient on the kinetic m=1(m/n=1/1) internal kink mode in a cylindrical tokamak plasma are studied by the gyro-kinetic particle simulations. When the density gradient is not large enough to change the full reconnection process, the phenomena after the full reconnection, such as the secondary reconnection and the evolution of the safety factor profile, are changed considerably due to the self-generated radial electric field, i.e. the m/n=0/0 mode. The radial electric field grows up to the same level as the m/n=1/1 mode, and drives an E×B plasma rotation in the ion diamagnetic direction, which breaks the symmetry of the current reconcentration. The density and current distributions, and therefore minimum safety factor qmin after the full reconnection, are found to be affected by the rotation.

Gyrokinetic Particle Simulation Using the Orbit Averaged Electron Drift-Kinetic Equation.

A new gyrokinetic particle simulation method using the orbit averaged electron drift-kinetic equation has been developed in a three dimensional slab geometry. Using the large discrepancy between the time scale of the low frequency fluctuations and the Courant-Friedlichs-Lewy condition determined by the electron ballistic mode, the transit time ordering, ω/ωtrθ(e), is introduced. The motion of the high energy transit electron is averaged over the periodic unperturbed orbit by means of the action-variational Lie perturbation method. In a new gyrokinetic Vlasov-Maxwell system, the drift-kinetic equation for the high energy part of the electron distribution function reduces to a two dimensional problem, which involves only the E×B nonlinearity, and the adiabatic response to the low frequency fluctuation is renormalized in the field equation. The ballistic mode arising from the high energy transit electron is removed and a longer simulation time step is enabled.

Stabilization of Ion-Temperature-Gradient–Driven Tokamak Modes by Magnetic-Field Gradient Reversal

Global gyrokinetic particle simulations have been used to search tokamak configurations which are stable against the ion-temperature-gradient--driven (ITG) modes commonly held responsible for the core anomalous ion heat transport. The stable configurations are characterized by strongly reduced or reversed $▽B$ drifts on the low-field side. Since excellent transport properties have been observed in high ${\ensuremath{\beta}}_{p}$ (poloidal beta) tokamak experiments, we conjecture, as a result of our simulations, that this may be due to the stabilization of ITG modes by the reduction of the $▽B$ drifts.

Method for solving the gyrokinetic Poisson equation in general geometry.

A generalized gyrokinetic Poisson solver has been developed and implemented in gyrokinetic particle simulation of low frequency microinstabilities in magnetic fusion plasmas. This technique employs local operations in the configuration space to compute the polarization density response and automatically takes into account the background profile effects contained in the gyrokinetic Poisson equation. It is useful for the global gyrokinetic simulation of magnetized plasmas in general equilibria, where the traditional spectral method is not applicable. The numerical scheme is also most amenable to massively parallel algorithms.

Toroidal gyrokinetic particle simulations of core fluctuations and transport

The generation of low frequency electrostatic fluctuations and the formation of anomalous particle and heat diffusion in the core region of tokamak plasmas is investigated using gyrokinetic particle-in-cell simulations in toroidal geometry. We consider the case of H-mode tokamak plasmas characterized by nearly flat core density profiles. The principle source of fluctuations include ion temperature gradient (ITG)-driven instabilities and ITG-dissipative trapped electron modes. We compare the evolution of both branches of instability and find that the electron diffusivity can be larger than the ion thermal diffusivity in the latter case. An externally imposed radial electric field, which gives rise to sheared poloidal flows, is demonstrated to reduce the radial correlation length of the turbulent transport and hence reduce the anomalous thermal flux.

Gyrokinetic particle simulation of neoclassical transport

A time varying weighting (δf ) scheme for gyrokinetic particle simulation is applied to a steady-state, multispecies simulation of neoclassical transport. Accurate collision operators conserving momentum and energy are developed and implemented. Simulation results using these operators are found to agree very well with neoclassical theory. For example, it is dynamically demonstrated that like-particle collisions produce no particle flux and that the neoclassical fluxes are ambipolar for an ion–electron plasma. An important physics feature of the present scheme is the introduction of toroidal flow to the simulations. Simulation results are in agreement with the existing analytical neoclassical theory. The poloidal electric field associated with toroidal mass flow is found to enhance density gradient-driven electron particle flux and the bootstrap current while reducing temperature gradient-driven flux and current. Finally, neoclassical theory in steep gradient profile relevant to the edge regime is examined by taking into account finite banana width effects. In general, in the present work a valuable new capability for studying important aspects of neoclassical transport inaccessible by conventional analytical calculation processes is demonstrated.

Visualization of plasma turbulence

We have developed a three-dimensional toroidal gyrokinetic particle simulation to study tokamak turbulence. The gyrokinetic equations are a reduced set derived from the Vlasov-Maxwell equations by phase averaging over the ion gyromotion and keeping only the time and space scales relevant for describing tokamak plasmas. These large-scale simulations in complex geometry can produce gigabytes of data consisting of large 3D arrays evolving in time. Visualization plays a critical role in going from the raw nonlinear solution of these complex equations to a simplified theoretical model explaining the essential underlying physics.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Nonlinear electromagnetic gyrokinetic equations for rotating axisymmetric plasmas

The influence of sheared equilibrium flows on the confinement properties of tokamak plasmas is a topic of much current interest. A proper theoretical foundation for the systematic kinetic analysis of this important problem has been provided here by presenting the derivation of a set of nonlinear electromagnetic gyrokinetic equations applicable to low-frequency microinstabilities in a rotating axisymmetric plasma. The subsonic rotation velocity considered is in the direction of symmetry, with the angular rotation frequency being a function of the equilibrium magnetic flux surface. In accordance with experimental observations, the rotation profile is chosen to scale with the ion temperature. The results obtained represent the shear flow generalization of the earlier analysis by Frieman and Chen [Phys. Fluids 25, 502 (1982)], where such flows were not taken into account. In order to make it readily applicable to gyrokinetic particle simulations, this set of equations is cast in a phase-space-conserving continuity equation form.

Partially Linearized Algorithms in Gyrokinetic Particle Simulation

In this paper, particle simulation algorithms with time-varying weights for the gyrokinetic Vlasov-Poisson system have been developed. The primary purpose is to use them for the removal of the selected nonlinearities in the simulation of gradient-driven microturbulence so that the relative importance of the various nonlinear effects can be assessed. It is hoped that the use of these procedures will result in a better understanding of the transport mechanisms and scaling in tokamaks. Another application of these algorithms is for the improvement of the numerical properties of the simulation plasma. For instance, implementations of such algorithms (1) enable us to suppress the intrinsic numerical noise in the simulation, and also (2) make it possible to regulate the weights of the fast-moving particles and, in turn, to eliminate the associated high frequency oscillations. Examples of their application to drift-type instabilities in slab geometry are given. We note that the work reported here represents the first successful use of the weighted algorithms in particle codes for the nonlinear simulation of plasmas.

Implementation of a Semi-implicit Orbit-Averaged Gyrokinetic Particle Code

A semi-implicit orbit-averaged time-integration algorithm has been successfully implemented in a gyrokinetic particle simulation code for the study of self-consistent phenomena in a strongly magnetized plasma. The semi-implicit aspect of the integration scheme relaxes the timestep constraints required to ensure numerical stability. The orbit averaging is useful in reducing statistical noise and relaxes the statistical constraints for kinetic simulation. For appropriate applications, the semi-implicit orbit-averaged algorithm should be more efficient than are traditional particle-in-cell plasma simulation algorithms with explicit time-integration schemes. Both a linear numerical dispersion analysis and illustrative simulation examples are presented.

Nonlinear mechanisms for drift wave saturation and induced particle transport

In previous studies, it was shown that, in the nonlinearly saturated phase of the evolution of drift instabilities in gyrokinetic particle simulations, the saturation levels and especially the particle fluxes are significantly less than predicted by quasilinear theory, and have an unexpected dependence on collisionality. In this paper, a theory is developed that explains these phenomena. The key features of the theory are as follows: The saturation level is determined by a balance between the steady-state electron and ion fluxes. The ion flux is small for levels of the potential below an E×B-trapping threshold and increases sharply once this threshold is crossed. Because of the presence of resonant electrons, the electron flux has a much smoother dependence on the potential. In the 2 (1)/(2) - dimensional (‘‘pseudo-3-D’’) geometry, the electrons are accelerated away from the resonance as they diffuse spatially, resulting in an inhibition of their diffusion, and thereby reducing the values of the potential and flux at which the fluxes can balance. Collisions and three-dimensional effects can repopulate the resonance thereby increasing the value of the particle flux.

Numerical simulation of ion-temperature-gradient-driven modes

Ion-temperature-gradient-driven modes in the presence of ion–ion collisions in a toroidal geometry with trapped ions have been studied by using a one-and-one-half-dimensional (11/2-D) linearized gyrokinetic particle simulation code in the electrostatic limit. The purpose of the investigation is to try to understand the physics of flat density discharges, in order to test the marginal stability hypothesis. Results giving threshold conditions of LTi/R0, and linear growth rates and mode frequencies over all wavelengths for the collisionless ion-temperature-gradient-driven modes are obtained. The behavior of ion-temperature-gradient-driven instabilities in the transition from slab to toroidal geometry, with trapped ions, is shown. A Monte-Carlo scheme for the inclusion of ion–ion collisions, in which ions can undergo Coulomb collisional dynamical friction, velocity space diffusion, and random walk of guiding centers, has been constructed. The effects of ion–ion collisions on the long wavelength limit of the ion modes is discussed.