In this work, we have formulated and implemented a mixed unstructured mesh-based finite element (FE)-Fourier decomposition scheme for gyrokinetic simulations in realistic tokamak geometry. An efficient particle positioning (particle-triangle mapping) scheme for the charge deposition and field scattering using an intermediate grid as the search index for triangles has been implemented and a significant speed-up by a factor of $\sim30$ is observed as compared with the brute force scheme for a medium-size simulation. The TRIMEG (TRIangular MEsh based Gyrokinetic) code has been developed. As an application, the ion temperature gradient (ITG) mode is simulated using the simplified gyrokinetic Vlasov-Poisson model. Our simulation and that using the ORB5 code for the DIII-D Cyclone case show reasonable agreement. As an additional application, ITG simulations using an ASDEX Upgrade equilibrium have been performed with density and temperature gradient profiles similar to the Cyclone case. Capabilities of the TRIMEG code for simulations with realistic experimental equilibria in the plasma core and in the whole plasma volume with open field lines are demonstrated.
We present an energy-conserving discontinuous Galerkin scheme for the full-$f$ electromagnetic gyrokinetic system in the long-wavelength limit. We use the symplectic formulation and solve directly for $\partial A_\parallel/\partial t$, the inductive component of the parallel electric field, using a generalized Ohm's law derived directly from the gyrokinetic equation. Linear benchmarks are performed to verify the implementation and show that the scheme avoids the Ampère cancellation problem. We perform a nonlinear electromagnetic simulation in a helical open-field-line system as a rough model o…
The first linear global electromagnetic gyrokinetic particle simulation on the excitation of toroidicity induced Alfven eigenmode (TAE) by energetic particles is reported. With an increase in the energetic particle pressure, the TAE frequency moves down into the lower continuum.
Boundary plasma physics plays an important role in tokamak confinement, but is difficult to simulate in a gyrokinetic code due to the scale-inseparable nonlocal multi-physics in magnetic separatrix and open magnetic field geometry. Neutral particles are also an important part of the boundary plasma physics. In the present paper, noble electrostatic gyrokinetic techniques to simulate the flux-driven, low-beta electrostatic boundary plasma is reported. Gyrokinetic ions and drift-kinetic electrons are utilized without scale-separation between the neoclassical and turbulence dynamics. It is found …
The development of a quasi-linear gyrokinetic transport model for tokamak plasmas, ultimately designed to provide physically comprehensive predictions of the time evolution of the thermodynamic relevant quantities, is a task that requires tight links among theoretical, experimental and numerical studies. The framework of the model here proposed, which operates a reduction of complexity on the nonlinear self-organizing plasma dynamics, allows in fact multiple validations of the current understanding of the tokamak micro-turbulence. The main outcomes of this work stem from the fundamental steps …
This comment clarifies the relation of the research in a recently published article [Phys. Plasmas 14, 042503 (2007)] to other prior publications addressing the inclusion of electromagnetic and drift-kinetic electron physics in gyrokinetic simulation, raises a concern related to the inclusion of kinetic electrons in a system with magnetic shear, and discusses alternatives in the face of an important limitation on the general applicability of the algorithm described therein.
Recently, the electrostatic gyrokinetic Hamiltonian and change of coordinates have been computed to order $ε^2$ in general magnetic geometry. Here $ε$ is the gyrokinetic expansion parameter, the gyroradius over the macroscopic scale length. Starting from these results, the long-wavelength limit of the gyrokinetic Fokker-Planck and quasineutrality equations is taken for tokamak geometry. Employing the set of equations derived in the present article, it is possible to calculate the long-wavelength components of the distribution functions and of the poloidal electric field to order $ε^2$. These h…
Global electromagnetic gyrokinetic simulations show the existence of near threshold conditions for both a high-$n$ kinetic ballooning mode (KBM) and an intermediate-$n$ kinetic version of peeling-ballooning mode (KPBM) in the edge pedestal of two DIII-D H-mode discharges. When the magnetic shear is reduced in a narrow region of steep pressure gradient, the KPBM is significantly stabilized, while the KBM is weakly destabilized and hence becomes the most-unstable mode. Collisions decrease the KBM's critical $β$ and increase the growth rate.
A poloidal tilting symmetry of the local nonlinear $δf$ gyrokinetic model is demonstrated analytically and verified numerically. This symmetry shows that poloidally rotating all the flux surface shaping effects with large poloidal mode number by a single tilt angle has an exponentially small effect on the transport properties of a tokamak. This is shown using a generalization of the Miller local equilibrium model to specify an arbitrary flux surface geometry. With this geometry specification we find that, when performing an expansion in large flux surface shaping mode number, the governing equ…
Transport barrier formation and its relation to sheared flows in fluids and plasmas are of fundamental interest in various natural and laboratory observations and of critical importance in achieving an economical energy production in a magnetic fusion device. Here we report the first observation of an edge transport barrier formation event in a gyrokinetic simulation carried out in a realistic tokamak edge geometry. The results show that turbulent Reynolds stress driven sheared ExB flows act in concert with neoclassical orbit loss to quench turbulent transport and form a transport barrier just…
Geodesic acoustic modes (GAMs) are studied by means of the gyrokinetic global particle-in-cell code ORB5. Linear electromagnetic simulations in the low electron beta limit have been performed, in order to separate acoustic and Alfvénic time scales and obtain more accurate measurements. The dependence of the frequency and damping rate on several parameters such as the safety factor, the GAM radial wavenumber and the plasma elongation is studied. All simulations have been performed with kinetic electrons with realistic electron/ion mass ratio. Interpolating formulae for the GAM frequency and dam…
A 1-D linear gyrokinetic code called AWECS is developed to study the kinetic excitation of Alfvenic instabilities in a high-beta tokamak plasma, with beta being the ratio of thermal to magnetic pressure. It is designed to describe physics associated with a broad range of frequencies and wavelengths. For example, AWECS is capable of simulating kinetic ballooning modes, Alfvenic ion-temperature-gradient-driven modes, as well as Alfven instabilities due to energetic particles. In addition, AWECS may be used to study drift-Alfven instabilities in the low-beta regime. Here, the layout of the code a…
The linear dynamics of Alfven modes in tokamaks is investigated here by means of the global gyrokinetic particle-in-cell code NEMORB. The model equations are shown and the local shear Alfven wave dispersion relation is derived, recovering the continuous spectrum in the incompressible ideal MHD limit. A verification and benchmark analysis is performed for continuum modes in a cylinder and for toroidicity-induced Alfven Eigenmodes. Modes in a reversed-shear equilibrium are also investigated, and the dependence of the spatial structure in the poloidal plane on the equilibrium parameters is descri…
TRIMEG-GKX: an electromagnetic gyrokinetic particle code with a Piecewise Field-Aligned Finite Element Method for Micro- and Macro-Instability Studies in Tokamak Core Plasmas
In this work, we modify the standard flux tube simulation domain to include arbitrary ion gyroradius-scale variation in the radial profile of the safety factor. To determine how to appropriately include such a modification, we add a strong ion gyroradius-scale source (inspired by electron cyclotron current drive) to the Fokker-Planck equation, then perform a multi-scale analysis that distinguishes the fast electrons driven by the source from the slow bulk thermal electrons. This allows us to systematically derive the needed changes to the gyrokinetic model. We find new terms that adjust the io…
Gyrokinetic simulation of the effect of transient fueling on plasma turbulence in ADITYA-U tokamak
Indian Institute of Science Bangalore（印度科学研究所班加罗尔分院）；Institute for Plasma Research, Bhat（巴特等离子体研究所）；Homi Bhabha National Institute（霍米·巴巴国立研究所）；Centre for Development of Advanced Computing（高级计算发展中心）；Department of Physics and Astronomy, University of California Irvine（加州大学尔湾分校物理与天文学系）
A gyrokinetic model is presented that can properly describe strong flows, large and small amplitude electromagnetic fluctuations occurring on scale lengths ranging from the electron Larmor radius to the equilibrium perpendicular pressure gradient scale length, and large deviations from thermal equilibrium. The formulation of the gyrokinetic model is based on a second order description of the single charged particle dynamics, derived from Lie perturbation theory, where the fast particle gyromotion is decoupled from the slow drifts, assuming that the ratio of the ion sound Larmor radius to the p…
Spherical tokamaks (STs) have been shown to possess properties desirable for a fusion power plant such as achieving high plasma ? and having increased vertical stability. To understand the confinement properties that might be expected in the conceptual design for a high $β$ ST fusion reactor, a 1GW ST plasma equilibrium was analysed using local linear gyrokinetics to determine the type of micro-instabilities that arise. Kinetic ballooning modes (KBMs) and micro-tearing modes (MTMs) are found to be the dominant instabilities. The parametric dependence of these linear modes was determined and fr…
Energetic particles produced by neutral beams are observed to excite energetic-particle-driven geodesic acoustic modes (EGAMs) in tokamaks. We study the effects of anisotropy of distribution function of the energetic particles on the excitation of such instabilities with ORB5, a gyrokinetic particle-in-cell code. Numerical results are shown for linear electrostatic simulations with ORB5. The growth rate is found to be sensitively dependent on the phase-space shape of the distribution function. The behavior of the instability is qualitatively compared to the theoretical analysis of dispersion r…
A High-order piecewise field-aligned triangular finite element method for electromagnetic gyrokinetic particle simulations of tokamak plasmas with open field lines
Max Planck Institute for Plasma Physics（马克斯·普朗克等离子体物理研究所）；Max Planck Computing and Data Facility（马克斯·普朗克计算与数据设施）
A gyrokinetic simulation of the influence of electron cyclotron current drive and ion kinetic effect on the m/n=2/1 tearing mode (TM) instabilities is presented in HL-2A and DIII-D tokamak configurations. The TM evolution is calculated with a finite mass electron model and the rf current source is obtained by ray-tracing and the Fokker-Planck method. The TMs are found to be perfectly stabilized by a continuous 1MW 68GHz X2-mode in HL-2A tokamak, while instabilities in the DIII-D discharge (with lower value of CR=I_{rf}/I_{0}, where Irf is the wave driven current and I0 is the equilibrium plasm…
In this paper, we extend the multiscale approach developed in [Abel et. al., Rep. Prog. Phys., submitted] by exploiting the scale separation between ions and the electrons. The gyrokinetic equation is expanded in powers of the electron to ion mass ratio, which provides a rigorous method for deriving the reduced electron model. We prove that ion-scale electromagnetic turbulence cannot change the magnetic topology, and argue that to lowest order the magnetic field lies on fluctuating flux surfaces. These flux surfaces are used to construct magnetic coordinates, and in these coordinates a closed …
Understanding turbulent transport physics in the tokamak edge and scrape-off layer (SOL) is critical to developing a successful fusion reactor. The dynamics in these regions plays a key role in achieving high fusion performance by determining the edge pedestal that suppresses turbulence in H-mode. Additionally, the survivability of a reactor is set by the heat load to the vessel walls, making it important to understand turbulent spreading of heat as it flows along open magnetic field lines in the SOL. Large-amplitude fluctuations, magnetic X-point geometry, and plasma interactions with materia…
We are developing a new continuum gyrokinetic code, Gkeyll, for use in edge plasma simulations, and here present initial simulations of turbulence on open field lines with model sheath boundary conditions. The code implements an energy conserving discontinuous Galerkin scheme, applicable to a general class of Hamiltonian equations. Several applications to test problems have been done, including a calculation of the parallel heat-flux on divertor plates resulting from an ELM crash in JET, for a 1x/1v SOL scenario explored previously, where the ELM is modeled as a time-dependent intense upstream…
Verification and validation of the internal kink instability in tokamak have been performed for both gyrokinetic (GTC) and kinetic-MHD codes (GAM-solver, M3D-C1-K, NOVA, XTOR-K). Using realistic magnetic geometry and plasma profiles from the same equilibrium reconstruction of the DIII-D shot #141216, these codes exhibit excellent agreement for the growth rate and mode structure of the internal kink mode when all kinetic effects are suppresed. The simulated radial mode structures agree quantitatively with the electron cyclotron emission measurement after adjusting, within the experimental uncer…
Gyrokinetic field theory is addressed in the context of a general Hamiltonian. The background magnetic geometry is static and axisymmetric, and all dependence of the Lagrangian upon dynamical variables is in the Hamiltonian or in free field terms. Equations for the fields are given by functional derivatives. The symmetry through the Hamiltonian with time and toroidal angle invariance of the geometry lead to energy and toroidal momentum conservation. In various levels of ordering against fluctuation amplitude, energetic consistency is exact. The role of this in underpinning of conservation laws…
Particle transport due to Ion Temperature Gradient/Trapped Electron (ITG/TE) mode turbulence is investigated using the gyrokinetic code GENE. Both a reduced quasilinear (QL) treatment and nonlinear (NL) simulations are performed for typical tokamak parameters corresponding to ITG dominated turbulence. A selfconsistent treatment is used, where the stationary local profiles are calculated corresponding to zero particle flux simultaneously for electrons and trace impurities. The scaling of the stationary profiles with magnetic shear, safety factor, electron-to-ion temperature ratio, collisionalit…
This paper combines results from a local gyrokinetic code with analytical theory to reconstruct the global eigenmode structure of the linearly unstable ion-temperature-gradient (ITG) mode with adiabatic electrons. The simulations presented here employ the s-$α$ tokamak equilibrium model. Local gyrokinetic calculations, using GS2 have been performed over a range of radial surfaces, x, and for ballooning phase angle, p, in the range -$π {\leq} p {\leqπ}$, to map out the complex local mode frequency, ${Ω_0(x, p) = ω_0(x, p) + iγ_0(x, p)}$. Assuming a quadratic radial profile for the drive, namely…
In magnetically confined plasmas used in Tokamak, turbulence is responsible for specific transport that limits the performance of this kind of reactors. Gyrokinetic simulations are able to capture ion and electron turbulence that give rise to heat losses, but require also state-of-the-art HPC techniques to handle computation costs. Such simulations are a major tool to establish good operating regime in Tokamak such as ITER, which is currently being built. Some of the key issues to address more realistic gyrokinetic simulations are: efficient and robust numerical schemes, accurate geometric des…
The generation of intrinsic rotation by turbulence and neoclassical effects in tokamaks is considered. To obtain the complex dependences observed in experiments, it is necessary to have a model of the radial flux of momentum that redistributes the momentum within the tokamak in the absence of a preexisting velocity. When the lowest order gyrokinetic formulation is used, a symmetry of the model precludes this possibility, making small effects in the gyroradius over scale length expansion necessary. These effects that are usually small become important for momentum transport because the symmetry…
Compact formulas for bounce and transit orbit averaging of the fluctuation-amplitude eikonal factor in axisymmetric tokamak geometry, which is frequently encountered in bounce-gyrokinetic description of microturbulence, are given in terms of the Jacobi elliptic functions and elliptic integrals. These formulas are readily applicable to the calculation of the neoclassical susceptibility in the framework of modern bounce-gyrokinetic theory. In the long-wavelength limit for axisymmetric electrostatic perturbations, we recover the expression for the Rosenbluth-Hinton residual zonal flow [Rosenbluth…
In this paper the global mode structures of linear ion-temperature-gradient (ITG) modes in tokamak plasmas are obtained by combining results from the local gyrokinetic code GS2 with analytical theory. Local gyrokinetic calculations, using GS2, are performed for a range of radial flux surfaces, ${x}$, and ballooning phase angles, ${p}$, to map out the local complex mode frequency, ${Ω_{0}(x,p)=ω_{0}(x,p)+iγ_{0}(x,p)}$ for a single toroidal mode number, ${n}$. Taylor expanding ${Ω_{0}}$ about ${x=0}$, and employing the Fourier-ballooning representation leads to a second order ODE for the amplitu…
This paper presents a complete theoretical framework for plasma turbulence and transport in tokamak plasmas. The fundamental scale separations present in plasma turbulence are codified as an asymptotic expansion in the ratio of the gyroradius to the equilibrium scale length. Proceeding order-by-order in this expansion, a framework for plasma turbulence is developed. It comprises an instantaneous equilibrium, the fluctuations driven by gradients in the equilibrium quantities, and the transport-timescale evolution of mean profiles of these quantities driven by the fluctuations. The equilibrium d…
XGC (X-point Gyrokinetic Code) is a whole-volume, total-f gyrokinetic particle-in-cell code developed for modelling tokamaks. In recent work, XGC has been extended to model more general 3D toroidal magnetic configurations, such as stellarators. These improvements have resulted in the XGC-S version. In this paper, XGC-S is benchmarked for linear electrostatic ion temperature gradient-driven microinstabilities, which can underlie turbulent transport in stellarators. An initial benchmark of XGC-S in tokamak geometry shows good agreement with the XGC1, ORB5, and global GENE codes. A benchmark betw…
Internal kink instabilities have been studied in straight tokamak geometry employing an electromagnetic gyrokinetic particle-in-cell (PIC) code. The ideal-MHD internal kink mode and the collisionless $m = 1$ tearing mode have been successfully simulated with the PIC code. Diamagnetic effects on the internal kink modes have also been investigated.
The XGC1 edge gyrokinetic code is used for a high fidelity prediction for the width of the heat-flux to divertor plates in attached plasma condition. The simulation results are validated against the empirical scaling $λ_q \propto B_P^{-γ}$ obtained from present tokamak devices, where $λ_q$ is the divertor heat-flux width mapped to the outboard midplane and $γ_q=1.19$ as defined by T. Eich et al. [Nucl. Fusion 53 (2013) 093031], and $B_P$ is the magnitude of the poloidal magnetic field at outboard midplane separatrix surface. This empirical scaling predicts $λ_q \leq 1mm$ when extrapolated to I…
A two-fold analysis of electromagnetic core tokamak instabilities in the framework of the gyrokinetic theory is presented. First principle theoretical foundations of the gyrokinetic theory are used to explain and justify the numerical results obtained with the global electromagnetic particle-in-cell code Orb5 whose model is derived from the Lagrangian formalism. The energy conservation law corresponding to the Orb5 model is derived from the Noether theorem and implemented in the code as a diagnostics for energy balance and conservation verification. An additional Noether theorem based diagnost…
Electromagnetic turbulence is addressed in tokamak and stellarator plasmas with the global gyrokinetic particle-in-cell codes ORB5 [E. Lanti et al, Comp. Phys. Comm, vol. 251, 107072 (2020)] and EUTERPE [V. Kornilov et al, Phys. Plasmas, vol. 11, 3196 (2004)]. The large-aspect-ratio tokamak, down-scaled ITER, and Wendelstein 7-X geometries are considered. The main goal is to increase the plasma beta, the machine size, the ion-to-electron mass ratio, as well as to include realistic-geometry features in such simulations. The associated numerical requirements and the computational cost for the ca…
It is shown that a considerable improvement in the global gyrokinetic electromagnetic simulations can be achieved by a slight modification of the simulation scheme. The new scheme is verified, simulating a Toroidal Alfvén Eigenmode in tokamak geometry at low perpendicular mode numbers, the so-called "MHD limit". Also, an electromagnetic drift mode has been successfully simulated in a stellarator.
First-principles simulations of tokamak turbulence have proven to be of great value in recent decades. We develop a pseudo-spectral velocity formulation of the turbulence equations that smoothly interpolates between the highly efficient but lower resolution 3D gyrofluid representation and the conventional but more expensive 5D gyrokinetic representation. Our formulation is a projection of the nonlinear gyrokinetic equation onto a Laguerre-Hermite velocity-space basis. We discuss issues related to collisions, closures, and entropy. While any collision operator can be used in the formulation, we…
Gyrokinetic simulations have greatly improved our theoretical understanding of turbulent transport in fusion devices. Most gyrokinetic models in use are delta-f simulations in which the slowly varying radial profiles of density and temperature are assumed to be constant for turbulence saturation times, and only the turbulent electromagnetic fluctuations are calculated. New massive simulations are being built to self-consistently determine the radial profiles of density and temperature. However, these new codes have failed to realize that modern gyrokinetic formulations, composed of a gyrokinet…
In the linear collisionless limit, a zonal potential perturbation in a toroidal plasma relaxes, in general, to a non-zero residual value. Expressions for the residual value in tokamak and stellarator geometries, and for arbitrary wavelengths, are derived. These expressions involve averages over the lowest order particle trajectories, that typically cannot be evaluated analytically. In this work, an efficient numerical method for the evaluation of such expressions is reported. It is shown that this method is faster than direct gyrokinetic simulations performed with the GENE and EUTERPE codes. C…
An analytic equilibrium, the Toroidal Bessel Function Model, is used in conjunction with the gyrokinetic code GYRO to investigate the nature of microinstabilities in a reversed field pinch (RFP) plasma. The effect of the normalized electron plasma pressure (β) on the characteristics of the microinstabilities is studied. A transition between an ion temperature gradient (ITG) driven mode and a microtearing mode as the dominant instability is found to occur at a β value of approximately 4.5%. Suppression of the ITG mode occurs as in the tokamak, through coupling to shear Alfven waves, with a crit…
Gyrokinetic simulations of ion temperature gradient mode and trapped electron mode driven impurity transport in a realistic tokamak geometry are presented and compared with results using simplified geometries. The gyrokinetic results, obtained with the GENE code in both linear and non-linear modes are compared with data and analysis for a dedicated impurity injection discharge at JET. The impact of several factors on heat and particle transport is discussed, lending special focus to tokamak geometry and rotational shear. To this end, results using s-alpha and concentric circular equilibria are…
