The guiding center and gyrokinetic theory of magnetized particle motion is extended to the regime of large electric field gradients perpendicular to the magnetic field. A gradient in the electric field directly modifies the oscillation frequency and causes the Larmor orbits to deform from circular to elliptical trajectories. In order to retain a good adiabatic invariant, there can only be strong dependence on a single coordinate at lowest order, so that resonances do not generate chaotic motion that destroys the invariant. When the gradient across magnetic flux surfaces is dominant, the guiding center drift velocity becomes anisotropic in response to external forces and additional curvature drifts must be included. The electric polarization density remains gyrotropic, but both the polarization and magnetization are modified by the change in gyrofrequency. The theory can be applied to shear flows that are even stronger than those observed in the edge transport barrier of a high-performance tokamak (H-mode) pedestal, even if the toroidal field is as small as or even smaller than the poloidal field. Yet, the theory retains a mathematical form that is similar to the standard case and can readily be implemented within existing simulation tools.
Higher-order guiding-center polarization and magnetization effects are introduced in gyrokinetic theory by keeping first-order terms in background magnetic-field nonuniformity. These results confirm the consistency of the two-step perturbation analysis used in modern gyrokinetic theory.
The concept of polarization in gyrokinetic theory is clarified and generalized to include contributions from the guiding-center (zeroth-order) polarization as well as the nonlinear (second-order) gyrocenter polarization. The guiding-center polarization, which appears as the antecedent (zeroth-order) of the standard linear (first-order) gyrocenter polarization, is obtained from a modified guiding-center transformation. The nonlinear gyrocenter polarization is derived either variationally from the third-order gyrocenter Hamiltonian or directly by gyrocenter push-forward method.
The effect of varying magnetic shear on scrape-off layer turbulence and profiles is studied via electromagnetic gyrokinetic simulations of a helical scrape-off layer model. We develop a model helical geometry with magnetic shear and a corresponding field-aligned coordinate system, which is used for simulations with the Gkeyll code. We find that perpendicular transport is reduced in cases with stronger shear, resulting in higher peak particle and heat fluxes to the endplates. Electromagnetic effects slightly increase transport in strong shear cases.
The cold ion limit of the local gyrokinetic model is rigorously taken to produce a nonlinear system of fluid equations that includes background flow shear. No fluid closure is required. By considering a simple slab geometry with magnetic drifts, but no magnetic shear, these fluid equations reduce to the Charney-Hasegawa-Mima model in the presence of flow shear. Analytic solutions to this model are found to study the impact of ExB flow shear on the stability of a single Parallel Velocity Gradient (PVG) driven mode. Additionally, the model is used to investigate the effect of background ExB flow…
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.
Adopting the theoretical framework for the generalized fishbonelike dispersion relation, an extended hybrid magnetohydrodynamics gyrokinetic simulation model has been derived analytically by taking into account both thermal ion compressibility and diamagnetic effects in addition to energetic particle kinetic behaviors. The extended model has been used for implementing an eXtended version of Hybrid Magnetohydrodynamics Gyrokinetic Code (XHMGC) to study thermal ion kinetic effects on Alfvénic modes driven by energetic particles, such as kinetic beta induced Alfvén eigenmodes in tokamak fusion pl…
Gyrokinetic Theory of Linear Gravitational Flute Interchanges with Flow Shear Stabilization
University of Maryland, College Park（马里兰大学帕克分校）；Institute for Research in Electronics & Applied Physics, University of Maryland, College Park（马里兰大学电子与应用物理研究所）；Gridfire Inc（Gridfire公司）
Previous limiter-biasing experiments on the Texas Helimak, a simple magnetized torus, have been inconclusive on the effect of flow shear on turbulence levels. To investigate this, the first gyrokinetic simulations of limiter biasing in the Helimak using the plasma physics code Gkeyll have been carried out, and results are presented here. For the scenarios considered, turbulence is mostly driven by the interchange instability, which depends on gradients of equilibrium density profiles. An analysis of both experimental and simulation data demonstrates that shear rates are mostly less than than l…
The exact analytical description of non relativistic charge motion in general magnetic fields is, apparently, a simple problem but, it has not been solved up to now apart for rare cases. The key feature of the present derivation is to adopt a non perturbative magnetic field description to find new solutions of motion. Among all solutions, two are particularly important: guiding particle and gyro-particle solutions. The guiding particle has been characterized to be minimally coupled to the magnetic field, the gyro-particle has been defined to be maximally coupled to the magnetic field and, also…
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…
Local gyrokinetic simulations solve the gyrokinetic equations with homogeneous background gradients, typically using a doubly periodic domain in the (x,y) plane (i.e. perpendicular to the field line). Spatial Fourier representations are almost universal in local gyrokinetic codes, and the wavevector-remap method was introduced in [Hammett et. al., Bull Am Phys Soc VP1 136, (2006)] as a simple method for expressing the local gyrokinetic equations with a background shear flow in a Fourier representation. Although extensively applied, the wavevector-remap method has not been formally shown to con…
Detecting Shearless Phase-Space Transport Barriers in Global Gyrokinetic Turbulence Simulations with Test Particle Map Models
The University of Texas at Austin（德克萨斯大学奥斯汀分校）；Zhejiang University（浙江大学）；Princeton University（普林斯顿大学）；University of São Paulo（圣保罗大学）；Princeton Plasma Physics Laboratory（普林斯顿等离子体物理实验室）
The gyrokinetic stability of electron-positron plasmas contaminated by ion (proton) admixture is studied in slab geometry. The appropriate dispersion relation is derived and solved. The ion-temperature-gradient driven instability, the electron-temperature-gradient driven instability, the universal mode, and the shear Alfven wave are considered. The contaminated plasma remains stable if the contamination degree is below some threshold, and it is found that the shear Alfven wave can be present in a contaminated plasma in cases where it is absent without ion contamination.
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…
Local gyrokinetic simulations use a field-aligned domain that twists due to the magnetic shear of the background magnetic equilibrium. However, if the magnetic shear is strong and/or the domain is long, the twist can become so extreme that it fails to properly resolve the turbulence. In this work, we derive and implement the "non-twisting flux tube," a local simulation domain that remains rectangular at all parallel locations. Convergence and runtime tests indicate that it can calculate the heat flux more efficiently than the conventional flux tube. For one test case, it was 30 times less comp…
Axisymmetric ($m=0$) gyrokinetic and extended-MHD simulations of sheared-flow Z-pinch plasma are performed with the high-order finite volume code COGENT. The present gyrokinetic model solves the long-wavelength limit of the gyrokinetic equation for both ion and electron species coupled to the electrostatic gyro-Poisson equation for the electrostatic potential. The electromagnetic MHD model includes the effects of the gyro-viscous pressure tensor, diamagnetic electron and ion heat fluxes, and generalized Ohm's law. A prominent feature of this work is that the radial profiles for the plasma dens…
The gyrocenter phase-space transformation used to describe nonlinear gyrokinetic theory is rediscovered by a recursive solution of the Hamiltonian dynamics associated with the perturbed guiding-center Vlasov operator. The present work clarifies the relation between the derivation of the gyrocenter phase-space coordinates by the guiding-center recursive Vlasov method and the method of Lie-transform phase-space transformations.
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…
Differential rotation is known to suppress linear instabilities in fusion plasmas. However, even in the absence of growing eigenmodes, subcritical fluctuations that grow transiently can lead to sustained turbulence. Here transient growth of electrostatic fluctuations driven by the parallel velocity gradient (PVG) and the ion temperature gradient (ITG) in the presence of a perpendicular ExB velocity shear is considered. The maximally simplified case of zero magnetic shear is treated in the framework of a local shearing box. There are no linearly growing eigenmodes, so all excitations are transi…
Upon combining Northrop's picture of charged particle motion with modern liquid crystal theories, this paper provides a new description of guiding center dynamics (to lowest order). This new perspective is based on a rotation gauge field (gyrogauge) that encodes rotations around the magnetic field. In liquid crystal theory, an analogue rotation field is used to encode the rotational state of rod-like molecules. Instead of resorting to sophisticated tools (e.g. Hamiltonian perturbation theory and Lie series expansions) that still remain essential in higher-order gyrokinetics, the present approa…
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…
A combined analytic and computational gyrokinetic approach is developed to address the question of the scaling of pedestal turbulent transport with arbitrary levels of $E \times B$ shear. Due to strong gradients and shaping in the pedestal, the instabilities of interest are not curvature-driven like the core instabilities. By extensive numerical (gyrokinetic) simulations, it is demonstrated that pedestal modes respond to shear suppression very much like the predictions of a basic analytic decorrelation theory. The quantitative agreement between the two provides us with a new dependable, first …
Recent experimental observations at JET show evidence of reduced ion temperature profile stiffness, hypothesised to be due to concomitant low magnetic shear (s) and significant toroidal rotational flow shear. Non-linear gyrokinetic simulations are performed, aiming to investigate the physical mechanism behind the observations. A comprehensive set of simulations are carried out, comparing the impact on the ion heat flux of various parameters that differ within the data-set. These parameters include q, s, rotation, effect of rotation on the magnetohydrodynamic (MHD) equilibrium, R/L_n, beta_e, Z…
In the present work, a consistent Lagrangian model that encapsulates fully kinetic ions and gyrokinetic electrons for solar wind electromagnetic turbulence is formulated. Using a consistent method, where both electrons and protons are treated with the same mathematical formalism, we derive and implement a model in which high frequency waves and kinetic electrons effects are described in a computationally cost-efficient way. To that aim, higher order Lie-transform perturbation methods applied to Hamiltonian formulation of guiding center motion are used in order to describe the dynamics of parti…
A gyrokinetic reduction is based on a specific ordering of the different small parameters characterizing the background magnetic field and the fluctuating electromagnetic fields. In this tutorial, we consider the following ordering of the small parameters: $ε\_B=ε\_δ^2$ where $ε\_B$ is the small parameter associated with spatial inhomogeneities of the background magnetic field and $ε\_δ$ characterizes the small amplitude of the fluctuating fields. In particular, we do not make any assumption on the amplitude of the background magnetic field. Given this choice of ordering, we describe a self-co…
The induced electric field in a tokamak drives a parallel electron current flow. In an inhomogeneous, finite beta plasma, when this electron flow is comparable to the ion thermal speed, the Alfven mode wave solutions of the electromagnetic gyrokinetic equation can become nearly purely growing kink modes. Using the new "low-flow" version of the gyrokinetic code GS2 developed for momentum transport studies [Barnes et al 2013 Phys. Rev. Lett. 111, 055005], we are able to model the effect of the induced parallel electric field on the electron distribution to study the destabilizing influence of cu…
We propose a novel derivation of the gyrokinetic field-particle Lagrangian for non-collisional ion-electron plasmas in a magnetic background with strong variations (maximal ordering). Our approach follows the two-step reduction process, where the guiding-center coordinate transformation is followed by the gyrocenter coordinate transformation in the single-particle phase space. For the first time both steps are addressed within a unique methodology, based on near-identity coordinate transformations constructed as polynomial transforms. These are well-defined transformations composed of a finite…
The linear gyrokinetic stability properties of magnetically confined electron-positron plasmas are investigated in the parameter regime most likely to be relevant for the first laboratory experiments involving such plasmas, where the density is small enough that collisions can be ignored and the Debye length substantially exceeds the gyroradius. Although the plasma beta is very small, electromagnetic effects are retained, but magnetic compressibility can be neglected. The work of a previous publication (Helander, 2014) is thus extended to include electromagnetic instabilities, which are of imp…
This work proposes a parametric equilibrium distribution function F_eq to be applied to the gyrokinetic studies of the Finite Orbit Width behavior of guiding centers representing several species encountered in axisymmetric tokamak plasma, as fusion products, thermal bulk and energetic particles from Ion Cyclotron Radiation Heating and Negative Neutral Beam Injections. After the analysis of the basic results of orbit theory obtained with a particularly convenient orbit coordinates set, it is shown how the proposed F_eq satisfies the two conditions that make it an equilibrium distribution functi…
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 purpose of this paper is to develop a simplified model as the modeling of the magnetized plasmas. The starting point is an assumption that the distribution of the ensemble of charged particles in the same species is homogeneous over gyrophase. The particles in this ensemble are located at the same guiding-cneter position $(\mathbf{X},μ,U)$. Then, a fundamental Lagrangian differential 1-form is developed. It contains all particles in the magnetized plasma system as well as the Coulomb pair force between particles instead of field-particle interaction used in conventional gyrokinetic models.…
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…
The first nonlinear gyrokinetic simulations of electron internal transport barriers (e-ITBs) in the National Spherical Torus Experiment show that reversed magnetic shear can suppress thermal transport by increasing the nonlinear critical gradient for electron-temperature-gradient-driven turbulence to three times its linear critical value. An interesting feature of this turbulence is nonlinearly driven off-midplane radial streamers. This work reinforces the experimental observation that magnetic shear is likely an effective way of triggering and sustaining e-ITBs in magnetic fusion devices.
Global gyrokinetic analysis of Wendelstein 7-X discharge: unveiling the importance of trapped-electron-mode and electron-temperature-gradient turbulence
本文首次对类 W7-X 仿星器放电进行含动理学电子、平衡径向电场及电磁碰撞效应的非线性径向全局回旋动理学模拟，揭示芯部湍流电子热通量主要由 TEM 和 ETG 驱动，并强调有限电子温度梯度的关键作用。
Nonlinear gyrokinetic simulations have been conducted to investigate turbulent transport in tokamak plasmas with rotational shear. At sufficiently large flow shears, linear instabilities are suppressed, but transiently growing modes drive subcritical turbulence whose amplitude increases with flow shear. This leads to a local minimum in the heat flux, indicating an optimal E x B shear value for plasma confinement. Local maxima in the momentum fluxes are also observed, allowing for the possibility of bifurcations in the E x B shear. The sensitive dependence of heat flux on temperature gradient i…
Upgraded spectroscopic hardware and an improved impurity concentration calculation allow accurate determination of boron density in the ASDEX Upgrade tokamak. A database of boron measurements is compared to quasilinear and nonlinear gyrokinetic simulations including Coriolis and centrifugal rotational effects over a range of H-mode plasma regimes. The peaking of the measured boron profiles shows a strong anti-correlation with the plasma rotation gradient, via a relationship explained and reproduced by the theory. It is demonstrated that the rotodiffusive impurity flux driven by the rotation gr…
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
The prevailing paradigm for plasma turbulence associates a unique stationary state to given equilibrium parameters. We report the discovery of bistable turbulence in a strongly magnetised plasma. Two distinct states, obtained with identical equilibrium parameters in first-principle gyrokinetic simulations, have turbulent fluxes of particles, momentum and energy that differ by an order of magnitude, with the low-transport state agreeing with experimental observations. Occurrences of the two states are regulated by the competition between an externally imposed mean flow shear and "zonal" flows g…
We present gyrokinetic simulations with the GENE code addressing the near-edge region of an L-mode plasma in the DIII-D tokamak. At radial position $ρ=0.80$, simulations with the ion temperature gradient increased by $40\%$ above the nominal value give electron and ion heat fluxes that are in simultaneous agreement with the experiment. This gradient increase is consistent with the combined statistical and systematic uncertainty $σ$ of the Charge Exchange Recombination Spectroscopy (CER) measurements at the $1.6 σ$ level. Multi-scale simulations are carried out with realistic mass ratio and geo…
The flow shear associated with the differential toroidal rotation of tokamak plasmas breaks an underlying symmetry of the turbulent fluctuations imposed by the up-down symmetry of the magnetic equilibrium. Using experimental Beam-Emission-Spectroscopy (BES) measurements and gyrokinetic simulations, this symmetry breaking in ion-scale turbulence in MAST is shown to manifest itself as a tilt of the spatial correlation function and a finite skew in the distribution of the fluctuating density field. The tilt is a statistical expression of the "shearing" of the turbulent structures by the mean flow…
Non-linear, radially global, turbulence simulations of ASDEX Upgrade (AUG) plasmas are performed and the nonlinear generated intrinsic flow shows agreement with the intrinsic flow gradients measured in the core of Ohmic L-mode plasmas at nominal parameters. Simulations utilising the kinetic electron model show hollow intrinsic flow profiles as seen in a predominant number of experiments performed at similar plasma parameters. In addition, significantly larger flow gradients are seen than in a previous flux-tube analysis (Hornsby et al {\it Nucl. Fusion} (2017)). Adiabatic electron model simula…
The competition between the drive and stabilization of plasma microinstabilities by sheared flow is investigated, focusing on the ion temperature gradient mode. Using a twisting mode representation in sheared slab geometry, the characteristic equations have been formulated for a dissipative fluid model, developed rigorously from the gyrokinetic equation. They clearly show that perpendicular flow shear convects perturbations along the field at a speed we denote by $Mc_s$ (where $c_s$ is the sound speed), whilst parallel flow shear enters as an instability driving term analogous to the usual tem…
