A UNIFIED DESCRIPTION OF FREE-ELECTRON MASERS——GYROKINETICS

The instabilities of gyrotron, gyro-peniotron and cyclotron auto-resonance masers(CARM) and their relationship are analysed in detail. By introducing gyrokinetic variablesin the Vlasov equation, a unified description, i.e. the gyrokinetics, of these free-electronmasersisgiven.

Solving Poisson equation with slowing-down equilibrium distribution for global gyrokinetic simulation

Fusion-born alpha particles in burning plasmas are usually regarded as have a slowing-down distribution,which differs significantly from the Maxwellian distribution of thermal particles in velocity space.A generalized multi-point average method has been developed for gyrokinetic Poisson equation with slowing-down equilibrium distribution using optimization in Fourier space.Its accuracy is verified in both long and short wavelength limits.The influence of changing equilibrium distribution from Maxwellian to slowing-down on gyrokinetic Poisson equation is analyzed to illustrate the significance of the new method.The effect of critical speed in the slowingdown distribution on the field solver is also presented.This method forms an important basis for global gyrokinetic simulation of low-frequency drift Alfvénic turbulence in burning plasmas.

Simulation on the transition of electrostatic instabilities in EAST steady-state scenario

The parameter dependence of transition between electrostatic instabilities is studied using gyrokinetic simulation based on a real discharge of steady-state scenario in the Experimental Advanced Superconducting Tokamak.The scan of radial locations shows that trapped electron mode(TEM)dominates around the core while the ion temperature gradient mode(ITG)simultaneously dominates outside.The maximum growth rate of TEM appears aroundρ=0.24,where the maximum electron temperature gradient R/LTelocates,ρis the normalized poloidal flux.Effects of the parameters on the transition between TEM and ITG instability are studied atρ=0.24.It is found that TEM dominates in the scanning with individually changing R/LTe from 2.50 to 25.02 or the density gradient R/L_(n)from 1.38 to 13.76.Meanwhile,the electron-ion temperature ratio T_(e)/T_(i)is found to destabilize TEM,the effect of Teis more sensitive than that of T_(i).The dominant instability diagrams in the(R/L_(Te),R/L_(Ti))plane at different T_(e)/T_(i)and R/Lnare numerically obtained,which clearly show the parameter range of the dominant TEM or dominant ITG instability region.It is found that the dominant TEM region becomes narrower in the plane by decreasing R/L_(n)when T_(e)/T_(i)>0.5.

Gyrokinetic simulation of low-n Alfvénic modes in tokamak HL-2A plasmas

The turbulence characteristics of plasmas with internal transport barriers in the HL-2A tokamak are analyzed by means of linear gyrokinetic simulations. It is found that turbulence is dominated by the ion temperature gradient(ITG)mode together with large-scale modes characterized by high-frequency electromagnetic fluctuation, which are destabilized by the steep ion temperature gradient in the weak magnetic shear regime. Comparison with solutions of analytical dispersion relations shows that their linear features match well with the beta-induced Alfvén eigenmode branch of the shear Alfvénic spectrum. It is further clarified that the large population of fast ions in these plasmas plays a stabilization role through the dilution mechanism in high-n ITG mode regimes.

Self-consistent kinetic theory with nonlinear wave-particle resonances

We have developed, based on the oscillating-center transformation, a general theoretical approach for self-consistent plasma dynamics including, explicitly, effects of nonlinear(higherorder) wave-particle resonances. A specific example is then given for low-frequency responses of trapped particles in axisymmetric tokamaks. Possible applications to transport as well as nonlinear wave growth/damping are also briefly discussed.

The Implementation of Magnetic Islands in Gyrokinetic Toroidal Code

The implementation of magnetic islands in gyrokinetic simulation has been verified in the gyrokinetic toroidal code（GTC）.The ion and electron density profiles become partially flattened inside the islands.The density profile at the low field side is less flattened than that at the high field side due to toroidally trapped particles in the low field side,which do not move along the perturbed magnetic field lines.When the fraction of trapped particles decreases,the density profile at the low field becomes more flattened.

Gyrokinetic simulations of nonlinear interactions between magnetic islands and microturbulence

A conservative scheme of kinetic electrons for gyrokinetic simulations in the presence of magnetic islands has been implemented and verified in the gyrokinetic toroidal code, where zonal and nonzonal components of all perturbed quantities are solved together. Using this new conservative scheme, linear simulation of kinetic ballooning mode has been successfully benchmarked with the electromagnetic hybrid model. Simulations of nonlinear interactions between magnetic islands and the ion temperature gradient(ITG) mode in a tokamak show that the islands rotate at the electron diamagnetic drift velocity. The linear ITG structure shifts from the island O-point toward the X-point due to the pressure flattening effect inside the islands, and the nonlinear ITG structure peaks along the magnetic island separatrix because of the increased pressure gradient there.

Implementation of field-aligned coordinates in a semi-Lagrangian gyrokinetic code for tokamak turbulence simulation

Field-aligned coordinates have been implemented in the gyrokinetic semi-Lagrangian code NLT, Ye et al （2016 J. Comput. Phys. 316 180）, to improve the computational efficiency for the numerical simulations of tokamak turbulence and transport. 4D B-spline interpolation in field- aligned coordinates is applied to solve the gyrokinetic Vlasov equation. A fast iterative algorithm is proposed for efficiently solving the quasi-neutrality equation. A pseudo transform method is used for the numerical integration of the gyro-average operator for perturbations with a high toroidal mode number. The new method is shown to result in an improved code performance for reaching a given accuracy. Some numerical tests are presented to illustrate the new methods.

Global and Kinetic MHD Simulation by the Gpic-MHD Code

In order to implement large-scale and high-beta tokamak simulation, a new algorithm of the electromagnetic gyrokinetic PIC （particle-in-cell） code was proposed and installed on the Gpic-MHD code [Gyrokinetic PIC code for magnetohydrodynamic （MHD） simulation]. In the new algorithm, the vorticity equation and the generalized Ohm＇s law along the magnetic field are derived from the basic equations of the gyrokinetic Vlasov, Poisson, and Ampere system and are used to describe the spatio-temporal evolution of the field quantities of the electrostatic potential φ and the longitudinal component of the vector potential Az. The basic algorithm is equivalent to solving the reduced-MHD-type equations with kinetic corrections, in which MHD physics related to Alfven modes are well described. The estimation of perturbed electron pressure from particle dynamics is dominant, while the effects of other moments are negligible. Another advantage of the algorithm is that the longitudinal induced electric field, ETz = -δAz/δt, is explicitly estimated by the generalized Ohm＇s law and used in the equations of motion. Furthermore, the particle velocities along the magnetic field are used （vz-formulation） instead of generalized momentums （pz-formulation）, hence there is no problem of ＇cancellation＇, which would otherwise appear when Az is estimated from the Ampere＇s law in the pz-formulation. The successful simulation of the collisionless internal kink mode by the new Gpic-MHD with realistic values of the large-scale and high-beta tokamaks revealed the usefulness of the new algorithm.

Unexpanded nonlinear electromagnetic gyrokinetic equations for magnetized plasmas

Nonlinear electromagnetic gyrokinetic equations have been constructed without expanding the field variables into background and finite but small-amplitude fluctuating components. At the long-wavelength limit, these unexpanded nonlinear gyrokinetic equations recover the wellknown drift-kinetic equations. At the expanded limit, they recover the usual nonlinear gyrokinetic equations. These equations can therefore be applied to long-term simulations covering from microscopic to macroscopic spatial scales.

Nonlinear excitation of a geodesic acoustic mode by reversed shear Alfvén eignemodes

The parametric decay process of a reversed shear Alfvén eigenmeode(RSAE)into a geodesic acoustic mode and a kinetic RSAE is investigated using nonlinear gyrokinetic theory.The excitation conditions mainly require the pump RSAE amplitude to exceed a certain threshold,which could be readily satisfied in burning plasmas operated in steady-state advanced scenario.This decay process can contribute to thermal plasma heating and confinement improvement.

Linear gyrokinetic simulations of reversed shear Alfvén eigenmodes and ion temperature gradient modes in DIII-D tokamak

Global linear gyrokinetic simulations using realistic DIII-D tokamak geometry and plasma profiles find co-existence of unstable reversed shear Alfvén eigenmodes(RSAE)with low toroidal mode number n and electromagnetic ion temperature gradient(ITG)instabilities with higher toroidal mode number n.For intermediate n?=?[10,12],RSAE and ITG co-exist and overlap weakly in the radial domain with similar growth rates but different real frequencies.Both RSAE and ITG growth rates decrease less than 5%when compressible magnetic perturbations are neglected in the simulations.The ITG growth rates increase less than 7%when fast ions are not included in the simulations.Finally,the effects of trapped electrons on the RSAE are negligible.

Discovering exact,gauge-invariant,local energy–momentum conservation laws for the electromagnetic gyrokinetic system by high-order field theory on heterogeneous manifolds

Gyrokinetic theory is arguably the most important tool for numerical studies of transport physics in magnetized plasmas.However,exact local energy–momentum conservation laws for the electromagnetic gyrokinetic system have not been found despite continuous effort.Without such local conservation laws,energy and momentum can be instantaneously transported across spacetime,which is unphysical and casts doubt on the validity of numerical simulations based on the gyrokinetic theory.The standard Noether procedure for deriving conservation laws from corresponding symmetries does not apply to gyrokinetic systems because the gyrocenters and electromagnetic field reside on different manifolds.To overcome this difficulty,we develop a high-order field theory on heterogeneous manifolds for classical particle-field systems and apply it to derive exact,local conservation laws,in particular the energy–momentum conservation laws,for the electromagnetic gyrokinetic system.A weak Euler–Lagrange(EL)equation is established to replace the standard EL equation for the particles.It is discovered that an induced weak EL current enters the local conservation laws,and it is the new physics captured by the high-order field theory on heterogeneous manifolds.A recently developed gauge-symmetrization method for high-order electromagnetic field theories using the electromagnetic displacement-potential tensor is applied to render the derived energy–momentum conservation laws electromagnetic gauge-invariant.

GTC Simulation of Ideal Ballooning Mode in Tokamak Plasmas

In the present paper,we first derive the eigenmode equation of the ideal ballooning mode in tokamak plasmas using a gyrokinetic equation.It is shown that the gyrokinetic eigenmode equation can be reduced to the magnetohydrodynamic（MHD） form in the long wavelength limit when kinetic effects are ignored.Then,the global gyrokinetic toroidal code（GTC） is applied for simulations of the edge-localized ideal ballooning modes.The obtained mode structures are compared with the results of ideal MHD simulations.The observed scaling of the linear growth rate with the toroidal mode number is consistent with the ideal MHD theory.The simulation results verify the GTC capability of simulating MHD processes in toroidal plasmas.

Kinetic micro-instabilities in the presence of impurities in toroidal magnetized plasmas

The theoretical and numerical studies on kinetic micro-instabilities,including ion temperature gradient（ITG） driven modes,trapped electron modes（TEMs） in the presence of impurity ions as well as impurity modes（IMs）,induced by impurity density gradient alone,in toroidal magnetized plasmas,such as tokamak and reversed-field pinch（RFP） are reviewed briefly.The basic theory for IMs,the electrostatic instabilities in tokamak and RFP plasmas are discussed.The observations of hybrid and coexistence of the instabilities are categorized systematically.The effects of impurity ions on electromagnetic instabilities such as ITG modes,the kinetic ballooning modes（KBMs） and kinetic shear Alfvén modes induced by impurity ions in tokamak plasmas of finite β（=plasma pressure/magnetic pressure） are analyzed.The interesting topics for future investigation are suggested.

Cyclokinetic models and simulations for high-frequency turbulence in fusion plasmas

Cyrokinetics is widely applied in plasma physics. However, this framework is limited to weak turbulence levels and low drift-wave frequencies because high-frequency gyro-motion is reduced by the gyro-phase averaging. In order to test where gyrokinetics breaks down, Waltz and Zhao developed a new theory, called cyclokinetics [R. E. Waltz and Zhao Deng, Phys. Plasmas 20, 012507 （2013）]. Cyclokinetics dynamically follows the high-frequency ion gyro-motion which is nonlineaxly coupled to the low-frequency drift-waves interrupting and suppressing gyro-averaging. Cyclokinetics is valid in the high-frequency （ion cyclotron frequency） regime or for high turbulence levels. The ratio of the cyclokinetic perturbed distribution function over equilibrium distribution function δf/F can approach 1. This work presents, for the first time, a numerical simulation of nonlinear cyclokinetic theory for ions, and describes the first attempt to completely solve the ion gyro-phase motion in a nonlinear turbulence system. Simulations are performed [Zhao Deng and R. E. Waltz, Phys. Plasmas 22（5）, 056101 （2015）] in a local flux-tube geometry with the parallel motion and variation suppressed by using a newly developed code named rCYCLO, which is executed in parallel by using an implicit time-advanced Eulerian （or continuum） scheme [Zhao Deng and R. E. Waltz, Comp. Phys. Comm. 195, 23 （2015）]. A novel numerical treatment of the magnetic moment velocity space derivative operator guarantee saccurate conservation of incremental entropy. By comparing the more fundamental cyclokinetic simulations with the corresponding gyrokinetic simulations, the gyrokinetics breakdown condition is quantitatively tested. Gyrokinetic transport and turbulence level recover those of cyclokinetics at high relative ion cyclotron frequencies and low turbulence levels, as required. Cyclokinetic transport and turbulence level are found to be lower than those of gyrokinetics at high turbulence levels and low-Ω＊ values with stable ion cyclotron modes. The gyrokinetic approximation is found to break down when the density perturbation exceeds 20%, or when the ratio of nonlinear E x B frequency over ion cyclotron frequency exceeds 20%. This result indicates that the density perturbation of the Tokamak L-mode near-edge is not sufficiently large for breaking the gyro-phase averaging. For cyclokinetic simulations with sufficiently unstable ion cyclotron （IC） modes and sufficiently low Ω^＊ ～10, the high-frequency component of the cyclokinetic transport can exceed that of the gyrokinetic transport. However, the low-frequency component of the cyclokinetic transport does not exceed that of the gyrokinetic transport. For higher and more physically relevant Ω^＊ ≥50 values and physically realistic IC driving rates, the low-frequency component of the cyclokinetic transport remains smaller than that of the gyrokinetic transport. In conclusion, the ＂L-mode near-edge short-fall＂ phenomenon, observed in some low-frequency gyrokinetic turbulence transport simulations, does not arise owing to the nonlinear coupling of high-frequency ion cyclotron motion to low-frequency drift motion.

A Gyrokinetic simulation model for low frequency electromagnetic fluctuations in magnetized plasmas

We present a new model for simulating the electromagnetic fluctuations with frequencies much lower than the ion cyclotron frequency in plasmas confined in general magnetic configurations.This novel model(termed as GK-E&B)employs nonlinear gyrokinetic equations formulated in terms of electromagnetic fields along with momentum balance equations for solving fields.It,thus,not only includes kinetic effects,such as wave-particle interaction and microscopic(ion Larmor radius scale)physics;but also is computationally more efficient than the conventional formulation described in terms of potentials.As a benchmark,we perform linear as well as nonlinear simulations of the kinetic Alfvén wave;demonstrating physics in agreement with the analytical theories.

Electromagnetic High Frequency Gyrokinetic Particle-in-Cell Simulation

Using the gyrocenter-gauge kinetic theory,an electromagnetic version of the high frequency gyrokinetic numerical algorithm for particle-in-cell simulation has been developed.The new algorithm,being an alternative to a direct Lorentz-force simulation,offers an efficient way to simulate the dynamics of plasma heating and current drive with radio frequency waves.Gyrokinetic formalism enables separation of gyrocenter and gyrophase motions of a particle in a strong magnetic field.From this point of view,a particlemay be viewed as a combination of a slow gyrocenter and a quickly changing Kruskal ring.In this approach,the nonlinear dynamics of high frequency waves is described by the evolution of Kruskal rings based on first principles physics.The efficiency of the algorithm is due to the fact that the simulation particles are advanced along the slow gyrocenter orbits,while the Kruskal rings capture fast gyrophase physics.Moreover,the gyrokinetic formalism allows separation of the cold response from kinetic effects in the current,which allows one to use much smaller number of particles than what is required by a direct Lorentz-force simulation.Also,the new algorithm offers the possibility to have particle refinement together with mesh refinement,when necessary.To illustrate the new algorithm,a simulation of the electromagnetic low-hybrid wave propagating in inhomogeneous magnetic field is presented.

Plasma Edge Kinetic-MHD Modeling in Tokamaks Using Kepler Workflow for Code Coupling,Data Management and Visualization

A new predictive computer simulation tool targeting the development of the H-mode pedestal at the plasma edge in tokamaks and the triggering and dynamics of edge localizedmodes(ELMs)is presented in this report.This tool brings together,in a coordinated and effective manner,several first-principles physics simulation codes,stability analysis packages,and data processing and visualization tools.A Kepler workflow is used in order to carry out an edge plasma simulation that loosely couples the kinetic code,XGC0,with an ideal MHD linear stability analysis code,ELITE,and an extended MHD initial value code such as M3D or NIMROD.XGC0 includes the neoclassical ion-electron-neutral dynamics needed to simulate pedestal growth near the separatrix.The Kepler workflow processes the XGC0 simulation results into simple images that can be selected and displayed via the Dashboard,a monitoring tool implemented in AJAX allowing the scientist to track computational resources,examine running and archived jobs,and view key physics data,all within a standard Web browser.The XGC0 simulation is monitored for the conditions needed to trigger an ELM crash by periodically assessing the edge plasma pressure and current density profiles using the ELITE code.If an ELM crash is triggered,the Kepler workflow launches the M3D code on a moderate-size Opteron cluster to simulate the nonlinear ELM crash and to compute the relaxation of plasma profiles after the crash.This process is monitored through periodic outputs of plasma fluid quantities that are automatically visualized with AVS/Express and may be displayed on the Dashboard.Finally,the Kepler workflow archives all data outputs and processed images using HPSS,as well as provenance information about the software and hardware used to create the simulation.The complete process of preparing,executing and monitoring a coupled-code simulation of the edge pressure pedestal buildup and the ELM cycle using the Kepler scientific workflow system is described in this paper.

Full Torus Electromagnetic Gyrokinetic Particle Simulations with Kinetic Electrons

The full torus electromagnetic gyrokinetic particle simulations using the hybrid model with kinetic electrons in the presence of magnetic shear is presented.The fluid-kinetic electron hybrid model employed in this paper improves numerical properties by removing the tearing mode,meanwhile,preserves both linear and nonlinear wave-particle resonances of electrons with Alfven wave and ion acoustic wave.