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 amplitude envelope, ${A\left(p\right)}$ , which describes how the local results are combined to form the global mode. We employ the so-called CYCLONE base case for circular Miller equilibrium model. Assuming radially varying profiles of ${a/L_{T}}$ and ${a/L_{n}}$, peaked at ${x=0}$, and with all other equilibrium profiles held constant, ${Ω_{0}(x,p)}$ is found to have a stationary point. The reconstructed global mode sits at the outboard mid-plane of the tokamak, with global growth rate, ${γ\sim}$Max${\left[γ_{0}\right]}$. Including the radial variation of other equilibrium profiles like safety factor and magnetic shear, leads to a mode that peaks away from the outboard mid-plane, with a reduced global growth rate. Finally, the influence of toroidal flow shear has also been investigated through the introduction of a Doppler shift, ${ω_{0} \rightarrow ω_{0} - nΩ_ϕ^{\prime} x}$, where ${Ω_ϕ}$ is the equilibrium toroidal flow, and a prime denotes the radial derivative. The equilibrium profile variations introduce an asymmetry into the global growth rate spectrum with respect to the sign of ${Ω_ϕ^{\prime}}$, such that the maximum growth rate is achieved with non-zero shearing, consistent with recent global gyrokinetic calculations.
The gyrokinetic turbulence code GS2 was used to investigate the effects of plasma beta on linear, collisionless ion temperature gradient (ITG) modes and trapped electron modes (TEM) in National Compact Stellarator Experiment (NCSX) geometry. Plasma beta affects stability in two ways: through the equilibrium and through magnetic fluctuations. The first was studied here by comparing ITG and TEM stability in two NCSX equilibria of differing beta values, revealing that the high beta equilibrium was marginally more stable than the low beta equilibrium in the adiabatic-electron ITG mode case. Howeve…
We present a modified gyrokinetic theory to predict the critical gradient that determines the linear onset of the ion temperature gradient (ITG) mode in stellarator plasmas. A coarse-graining technique is applied to the drift curvature, entering the standard gyrokinetic equations, around local minima. Thanks to its simplicity, this novel formalism yields an estimate for the critical gradient with a computational cost low enough for application to stellarator optimization. On comparing against a gyrokinetic solver, our results show good agreement for an assortment of stellarator designs. Insigh…
Effect of static magnetic island on ITG of ADITYA-U tokamak
Indian Institute of Science（印度科学学院）；Institute for Plasma Research（等离子体研究所）；Homi Bhabha National Institute（霍米·巴巴国立研究所）；IPS Academy（IPS学院）；Centre for Development of Advanced Computing（高级计算发展中心）
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.
Abstract Simulation domain in field-aligned coordinates of the electrostatic gyrokinetic nonlinear turbulence global code, NLT, is extended to include the magnetic axis. The artificial boundary near the magnetic axis is replaced by the natural boundary. The singularity at the magnetic axis in Vlasov solver is treated by considering the spatial relation of fixed grid points in field-aligned coordinates. A new Poisson's equation solver is developed, the coefficient matrix of algebraic equations is derived by using Gauss's theorem. Nonlinear relaxation test of the ITG turbulence with adiabatic el…
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…
The global and electromagnetic gyrokinetic particle-in-cell code ORB5 is employed to investigate the self-consistent interactions between Alfvén modes (AM) and ion temperature gradient (ITG) turbulence in a magnetically confined plasma. Here, an axisymmetric magnetic equilibrium with reversed shear and large aspect ratio is considered. An AM with toroidal mode number n=5 is driven unstable by introducing a population of suprathermal ions. Once the AM saturates in the presence of the fully developed turbulence, the ion heat flux is dominated by the AM and its main harmonics. ITG-induced transpo…
We present a study of the linear properties of ion temperature gradient (ITG) modes with collisions modelled by the linearized gyrokinetic (GK) Coulomb collision operator (Frei et al. 2021) in the local limit. The study is based on a Hermite-Laguerre polynomial expansion of the perturbed ion distribution function applied to the linearized GK Boltzmann equation, yielding a hierarchy of coupled equations for the expansion coefficients, referred to as gyro-moments. We explore analytically the collisionless and high-collisional limits of the gyro-moment hierarchy. Parameter scans revealing the dep…
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 …
Energetic particles, Shafranov shift and finite $β$ effects on TAE, KBM and ITG instabilities in global electromagnetic gyrokinetic simulations
Ecole Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）；Max-Planck-Institut für Plasmaphysik（马克斯·普朗克等离子体物理研究所）；Université de Lorraine（洛林大学）；De Vinci Higher Education（德文奇高等教育）；University of Warwick（华威大学）
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…
A first-principles method to calculate the critical temperature gradient for the onset of the ion-temperature-gradient mode (ITG) in linear gyrokinetics is presented. We find that conventional notions of the connection length previously invoked in tokamak research should be revised and replaced by a generalized correlation length to explain this onset in stellarators. Simple numerical experiments and gyrokinetic theory show that localized "spikes" in shear, a hallmark of stellarator geometry, are generally insufficient to constrain the parallel correlation length of the mode. ITG modes that lo…
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…
Naturally stable to ELMs, and with widths larger than EPED predictions, the I-modes are an excellent laboratory for investigating the role of drift micro-instabilities in pedestals since I-mode pedestal are not "limited" by MHD instabilities. We present here a study based on gyrokinetic simulations (using GENE) to model fluctuations and heat transport in the I-mode pedestals in C-Mod. We find the Weakly Coherent Mode observed on C-Mod I-mode to be an electrostatic Ion Temperature Gradient/Impurity density gradient (ITG/Impurity) driven mode. The ITG/Impurity mode match frequency and the impuri…
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…
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…
The first experimental campaigns have proven that, due to the optimization of the magnetic configuration with respect to neoclassical transport, the contribution of turbulence is essential to understand and predict the total particle and energy transport in Wendelstein 7-X (W7-X). This has spurred much work on gyrokinetic modelling for the interpretation of the available experimental results and for the preparation of the next campaigns. At the same time, new stellarator gyrokinetic codes have just been or are being developed. It is therefore desirable to have a sufficiently complete, document…
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 fully implicit particle-in-cell method for handling the $v_\parallel$-formalism of electromagnetic gyrokinetics has been implemented in XGC. By choosing the $v_\parallel$-formalism, we avoid introducing the non-physical skin terms in Ampère's law, which are responsible for the well-known ``cancellation problem" in the $p_\parallel$-formalism. The $v_\parallel$-formalism, however, is known to suffer from a numerical instability when explicit time integration schemes are used due to the appearance of a time derivative in the particle equations of motion from the inductive component of the elec…
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 …
The transport of heat out of tokamak plasmas by turbulence is the dominant mechanism limiting the performance of fusion reactors. Turbulence can be driven by the ion temperature gradient (ITG) and suppressed by toroidal sheared flows. Numerical simulations attempting to understand turbulence are crucial for guiding the design of future reactors. We investigate ion-scale turbulence via gyrokinetic simulations in the outer core of the Mega Ampere Spherical Tokamak (MAST). We perform a parameter scan in the values of the ITG and the flow shear. We show that nonlinear simulations reproduce the exp…
One metric for comparing confinement properties of different magnetic fusion energy configurations is the linear critical gradient of drift wave modes. The critical gradient scale length determines the ratio of the core to pedestal temperature when a plasma is limited to marginal stability in the plasma core. The gyrokinetic turbulence code GS2 was used to calculate critical temperature gradients for the linear, collisionless ion temperature gradient (ITG) mode in the National Compact Stellarator Experiment (NCSX) and a prototypical shaped tokamak, based on the profiles of a JET H-mode shot an…
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…
The impact of electromagnetic stabilization and flow shear stabilization on ITG turbulence is investigated. Analysis of a low-$β$ JET L-mode discharge illustrates the relation between ITG stabilization, and proximity to the electromagnetic instability threshold. This threshold is reduced by suprathermal pressure gradients, highlighting the effectiveness of fast ions in ITG stabilization. Extensive linear and nonlinear gyrokinetic simulations are then carried out for the high-$β$ JET hybrid discharge 75225, at two separate locations at inner and outer radii. It is found that at the inner radius…
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…
The presence of impurities is considered in gyrokinetic calculations of ion temperature gradient (ITG) instabilities and turbulence in the reversed field pinch device RFX-mod. This device usually exhibits hollow Carbon/Oxygen profiles, peaked in the outer core region. We describe the role of the impurities in ITG mode destabilization, and analyze whether ITG turbulence is compatible with their experimental gradients.
To help reveal the complete picture of linear kinetic drift modes, four independent numerical approaches, based on integral equation, Euler initial value simulation, Euler matrix eigenvalue solution and Lagrangian particle simulation, respectively, are used to solve the linear gyrokinetic electrostatic drift modes equation in Z-pinch with slab simplification and in tokamak with ballooning space coordinate. We identify that these approaches can yield the same solution with the difference smaller than 1\%, and the discrepancies mainly come from the numerical convergence, which is the first detai…
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…
A natural fueling mechanism that helps to maintain the main core deuterium and tritium (DT) density profiles in a tokamak fusion reactor is discussed. In H-mode plasmas dominated by ion- temperature gradient (ITG) driven turbulence, cold DT ions near the edge will naturally pinch radially inward towards the core. This mechanism is due to the quasi-neutral heat flux dominated nature of ITG turbulence and still applies when trapped and passing kinetic electron effects are included. Fueling using shallow pellet injection or supersonic gas jets is augmented by an inward pinch of could DT fuel. The…
Nonlinear entropy transfer via zonal flows in gyrokinetic plasma turbulence
The Graduate University for Advanced Studies（高级研究大学院大学）；Faculty of Arts and Sciences, Komazawa University（驹泽大学文理学部）；National Institute for Fusion Science（核聚变科学研究所）
In the present study, turbulent impurity transport in Deuterium tokamak plasmas, driven by Ion Temperature Gradient (ITG) and Trapped Electron (TE) modes, has been investigated using fluid and gyrokinetic models. The impurity diffusivity ($D_Z$) and convective velocity ($V_Z$) are calculated, and from these the zero-flux peaking factor ($PF_0$) is derived. This quantity expresses the impurity density gradient at which the convective and diffusive transport of impurities are exactly balanced. The sign of $PF_0$ is of special interest, as it determines whether the impurities are subject to an in…
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 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…
