The neoclassical prescription to use an equation of motion to determine the electrostatic field within a tokamak plasma is fraught with difficulties. Herein we examine two popular expressions for the equilibrium electrostatic field so determined and show that one fails to withstand a formal scrutiny thereof while the other fails to respect the vector nature of the diamagnetic current. Reconsideration of the justification for the presence of the equilibrium electrostatic field indicates that no field is needed for a neutral plasma when considering the net bound current defined as the curl of the magnetization. With any shift in the toroidal magnetic flux distribution, a dynamic electric field is generated with both radial and poloidal components, providing an alternate explanation for any measurements thereof.
Conventional radially-local neoclassical calculations become inadequate if the radial gradient scale lengths of the H-mode pedestal become as small as the poloidal ion gyroradius. Here, we describe a radially global $δf$ continuum code that generalizes neoclassical calculations to allow stronger gradients. As with conventional neoclassical calculations, the formulation is time-independent and requires only the solution of a single sparse linear system. We demonstrate precise agreement with an asymptotic analytic solution of the radially global kinetic equation in the appropriate limits of aspe…
Stronger impurity density in-out poloidal asymmetries than predicted by the most comprehensive neoclassical models have been measured in several tokamaks around the world during the last decade, calling into question the reduction of turbulence by sheared radial electric fields in H-mode tokamak pedestals. However, these pioneering theories neglect the impurity diamagnetic drift, or fail to retain it self-consistently; while recent measurements indicate that it can be of the same order as the ExB drift. We have developed the first self-consistent theoretical model retaining the impurity diamag…
An investigation is made into the effect of the reduction in anomalous perpendicular electron heat transport inside the separatrix of a magnetic island chain associated with a neoclassical tearing mode in a tokamak plasma, due to the flattening of the electron temperature profile in this region, on the overall stability of the mode. The onset of the neoclassical tearing mode is governed by the ratio of the divergences of the parallel and perpendicular electron heat fluxes in the vicinity of the island chain. By increasing the degree of transport reduction, the onset of the mode, as the diverge…
Poloidal asymmetries in the impurity density can be generated by radio frequency heating in the core and by neoclassical effects in the edge of tokamak plasmas. In a pedestal case study, using global neoclassical simulations we find that finite orbit width effects can generate significant poloidal variation in the electrostatic potential, which varies on a small radial scale. Gyrokinetic modeling shows that these poloidal asymmetries can be strong enough to significantly modify turbulent impurity peaking. In the pedestal the ExB drift in the radial electric field can give a larger contribution…
A model system of equations has been derived to describe a toroidally rotating tokamak plasma, unstable to Resistive Wall Modes (RWMs) and metastable to Neoclassical Tearing Modes (NTMs), using a linear RWM model and a nonlinear NTM model. If no wall is present, the NTM growth shows the typical threshold/saturation island widths, whereas a linearly unstable kink mode grows exponentially in this model plasma system. When a resistive wall is present, the growth of the linearly unstable RWM is accelerated by an unstable island: a form of coupled RWM-NTM mode. Crucially, this coupled system has no…
In a tokamak pedestal, radial scale lengths can become comparable to the ion orbit width, invalidating conventional neoclassical calculations of flow and bootstrap current. In this work we illustrate a non-local approach that allows strong radial density variation while maintaining small departures from a Maxwellian distribution. Non-local effects alter the magnitude and poloidal variation of the flow and current. The approach is implemented in a new global delta-f continuum code using the full linearized Fokker-Planck collision operator. Arbitrary collisionality and aspect ratio are allowed a…
The results of the analysis of the deviation of the force equilibrium for ions from the neoclassical theory prediction, calculated using the direct measurements of the radial electric field, in the view of its possible local and nonlocal correlation with the profiles of electron, Te, and ion, Ti, temperatures in the T-10 tokamak are presented. Local correlations are analyzed by means of the Pearson's correlation. Nonlocal correlations are treated with an inverse problem under the assumption of an integral equation relationship between the deviation and Te and Ti profiles. The discharges with z…
The characteristic profile and magnitude are predicted in theory for the neoclassical toroidal viscosity (NTV) torque induced by plasma response to the resonant magnetic perturbation (RMP) in a tokamak with an edge pedestal. For a low-$β$ equilibrium, the NTV torque is dominated by the toroidal component with the same dominant toroidal mode number of RMP. The NTV torque profile is found to be localized, whose peak location is determined by profiles of both the equilibrium temperature (pressure) and the plasma response. In general, the peak of NTV torque profile is found to track the pedestal p…
Ion orbit loss is considered important for generating the radially inward electric field $E_r$ in a tokamak edge plasma. In particular, this effect is emphasized in diverted tokamaks with a magnetic X point. In neoclassical equilibria, Coulomb collisions can scatter ions onto loss orbits and generate a radially outward current, which in steady state is balanced by the radially inward current from viscosity. To quantitatively measure this loss-orbit current in an edge pedestal, an ion-orbit-flux diagnostic has been implemented in the axisymmetric version of the gyrokinetic particle-in-cell code…
The effect of the perturbed ion polarization current on the stability of neoclassical tearing modes is calculated using an improved, neoclassical, four-field, drift-MHD model. The calculation involves the self-consistent determination of the pressure and scalar electric potential profiles in the vicinity of the associated magnetic island chain, which allows the chain's propagation velocity to be fixed. Two regimes are considered. First, a regime in which neoclassical ion poloidal flow damping is not strong enough to enhance the magnitude of the polarization current (relative to that found in s…
A four-field system of equations which includes the neoclassical flow damping effects and the lowest-order finite-Larmor-radius (FLR) corrections is deduced from a system of gyrofluid equations. The FLR corrections to the poloidal flow damping are calculated by solving a simplified version of the gyrokinetic equation. This system of equations is applied to the study of a chain of freely rotating magnetic islands in a tokamak, resulting from the nonlinear evolution of a resistive tearing mode, to determine the islands rotation velocity consistently with the fields radial profiles close to the r…
The neoclassical polarization current, generated when a magnetic island propagates through a tokamak plasma, is believed to influence the initial stage of the neoclassical tearing mode evolution. Understanding the strength of its contribution in the relevant plasma collision frequency regimes for future tokamaks such as ITER is crucial for the successful control and/or avoidance of the neoclassical tearing mode. A nonlinear drift kinetic theory is employed to determine the full collision frequency dependence of the neoclassical polarization current in the small island limit, comparable to the …
The guiding-center kinetic neoclassical transport code, XGC0, [C.S. Chang et. al, Phys. Plasmas 11, 2649 (2004)] is used to compute the heat fluxes and the heat-load width in the outer divertor plates of Alcator C-Mod and DIII-D tokamaks. The dependence of the width of heat-load fluxes on neoclassical effects, neutral collisions and anomalous transport is investigated using the XGC0 code. The XGC0 code includes realistic X-point geometry, a neutral source model, the effects of collisions, and a diffusion model for anomalous transport. It is observed that width of the XGC0 neoclassical heat-loa…
It is discovered that the tokamak field geometry generates a pitch-angle scattering effect for runaway electrons. This neoclassical pitch-angle scattering is much stronger than the collisional scattering and invalidates the gyro-center model for runaway electrons. As a result, the energy limit of runaway electrons is found to be larger than the prediction of the gyro-center model and to depend heavily on the background magnetic field.
Rotation of tokamak-plasmas, not at the mechanical equilibrium, is investigated utilizing a theorem of thermodynamics, established by Prigogine. This theorem, suitably applied to toroidally confined plasmas, suggests that the global barycentric rotations of the plasma, in the toroidal and poloidal directions, are pure reversible processes. In case of negligible viscosity and by supposing the validity of the balance equation for the internal forces, we show that the plasma, even not in the mechanical equilibrium, may freely rotate as a rigid body in the toroidal direction with an angular freque…
An application of the thermodynamic field theory (TFT) to transport processes in L-mode tokamak plasmas is presented. The nonlinear corrections to the linear (Onsager) transport coefficients in the collisional regimes are derived. A quite encouraging result is the appearance of an asymmetry between the Pfirsch-Schlueter (P-S) ion and electron transport coefficients: the latter presents a nonlinear correction, which is absent for the ions, and makes the radial electron coefficients much larger than the former. Explicit calculations and comparisons between the neoclassical results and the TFT pr…
The component of the neoclassical electrostatic potential that is non-constant on the magnetic surface, that we denote by $\tildeφ$, can affect radial transport of highly charged impurities, and this has motivated its inclusion in some modern neoclassical codes. The number of neoclassical simulations in which $\tildeφ$ is calculated is still scarce, partly because they are usually demanding in terms of computational resources, especially at low collisionality. In this paper the size, the scaling with collisionality and with aspect ratio, and the structure of $\tildeφ$ on the magnetic surface a…
A change of the particle density (by gas puff, pellets or impurity seeding) during the plasma discharge in tokamak produces a radial current and implicitly a torque and rotation that can modify the state of confinement. After ionization the newly born ions will evolve toward the periodic neoclassical orbits (trapped or circulating) but the first part of their excursion, which precedes the periodicity, is an effective radial current. It is short, spatially finite and unique for each new ion, but multiplied by the rate of ionization and it can produce a substantial total radial current. The asso…
In this paper, the secular full-orbit simulations of runaway electrons with synchrotron radiation in tokamak fields are carried out using a relativistic volume-preserving algorithm. Detailed phase-space behaviors of runaway electrons are investigated in different dynamical timescales spanning 11 orders. When looking into the small timescale, i.e., the characteristic timescale imposed by Lorentz force, the severely deformed helical trajectory of energetic runaway electron is witnessed. A qualitative analysis of the neoclassical scattering, a kind of collisionless pitch-angle scattering phenomen…
In evaluating neoclassical transport by radially-local simulations, the magnetic drift tangential to a flux surface is usually ignored in order to keep the phase-space volume conservation. In this paper, effect of the tangential magnetic drift on the local neoclassical transport are investigated. To retain the effect of the tangential magnetic drift in the local treatment of neoclassical transport, a new local formulation for the drift kinetic simulation is developed. The compressibility of the phase-space volume caused by the tangential magnetic drift is regarded as a source term for the drif…
From a common expression for the poloidal electrostatic field of a tokamak, in the limit of large aspect ratio and concentric circular flux surfaces, one may determine the associated potential. This potential satisfies Poisson's equation, which reduces to Laplace's equation when the medium has vanishing charge density, in axial geometry but not toroidal geometry. A simple transformation takes the potential over to the correct harmonic form for tokamak coordinates, and the resulting electrostatic field is calculated. From the radial field one may estimate the supporting charge density on the bo…
The neoclassical transport theory is applied to calculate electron cyclotron current drive (ECCD) efficiency in an axisymmetric tokamak in the low-collisionality regime. The tokamak ordering is used to obtain a system of equations that describe the dynamics of the plasma where the nonlinear ponderomotive (PM) force due to high-power RF waves is included. The PM force is produced around an electron cyclotron resonant surface at a specific poloidal location. The ECCD efficiency is analyzed in the cases of first and second harmonics (for different impinging angles of the RF waves) and it is valid…
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 …
In tokamak transport barriers, the radial scale of profile variations can be comparable to a typical ion orbit width, which makes the coupling of the distribution function across flux surfaces important in the collisional dynamics. We use the radially global steady-state neoclassical δf code Perfect to calculate poloidal and toroidal flows, and radial fluxes, in the pedestal. In particular, we have studied the changes in these quantities as the plasma composition is changed from a deuterium bulk species with a helium impurity to a helium bulk with a deuterium impurity, under specific profile s…
Carbon and nitrogen impurity transport coefficients are determined from gas puff experiments carried out during repeat L-mode discharges on the Mega-Amp Spherical Tokamak (MAST) and compared against a previous analysis of helium impurity transport on MAST. The impurity density profiles are measured on the low-field side of the plasma, therefore this paper focuses on light impurities where the impact of poloidal asymmetries on impurity transport is predicted to be negligible. A weak screening of carbon and nitrogen is found in the plasma core, whereas the helium density profile is peaked over t…
The purpose of this document is to find an analytical solution for the gyrokinetic equation under specific, simplificative hypotheses. The case I am considering is that of a collisional plasma in the presence of a chain of magnetic islands. The presence of the magnetic islands causes the onset of perturbative fields, in particular an electrostatic field, with a gradient length-scale comparable with the island's width. When the island's width w becomes comparable with the ion Larmor radius rho_i , the drift-kinetic equation is inadequate to treat the transport and the calculation of the neoclas…
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…
The Stacey-Sigmar model for tokamak equilibrium as presented in the literature relies heavily on the neoclassical theory of the electrostatic field. Its neglect of Gauss's law is inconsistent with the potential formulation of electrodynamics. Its treatment of the dynamic electric field generated by the central heating and poloidal field coils also is suspect. Its derivation of the viscosity term remains incomplete and does not account for the varying pitch angle of the magnetic field. A derivation of the viscosity term which respects the pitch angle can account for the radial force commonly as…
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…
First-principles numerical simulations are used to describe a transport bifurcation in a differentially rotating tokamak plasma. Such a bifurcation is more probable in a region of zero magnetic shear than one of finite magnetic shear because in the former case the component of the sheared toroidal flow that is perpendicular to the magnetic field has the strongest suppressing effect on the turbulence. In the zero-magnetic-shear regime, there are no growing linear eigenmodes at any finite value of flow shear. However, subcritical turbulence can be sustained, owing to the transient growth of mode…
The ion temperature behavior in the plasma core of the spherical tokamak Globus-M (major radius 0.36 m, minor radius 0.24 m, torus aspect ratio 1.5, toroidal magnetic field near the plasma axis 0.4 T, plasma current up to 0.3 MA) was studied by means of 12-channels neutral particle analyzer (NPA) ACORD-12. The experiments were performed in ohmic regimes as well as in regimes with the ion cyclotron resonance heating (ICRH) in the vicinity of fundamental harmonic for hydrogen minority in deuterium bulk plasma and the neutral beam injection (NBI). The total auxiliary power exceeded the magnitude …
The impurity dynamics in stellarators has become an issue of moderate concern due to the inherent tendency of the impurities to accumulate in the core when the neoclassical ambipolar radial electric field points radially inwards (ion root regime). This accumulation can lead to collapse of the plasma due to radiative losses, and thus limit high performance plasma discharges in non-axisymmetric devices.\\ A quantitative description of the neoclassical impurity transport is complicated by the breakdown of the assumption of small $\mathbf{E}\times \mathbf{B}$ drift and trapping due to the electros…
Despite high demonstrated efficiency, lower-hybrid current drive (LHCD) has not been considered localized enough for neoclassical tearing mode (NTM) stabilization in tokamaks. This assessment must be reconsidered in view of the RF current condensation effect. We show that an island with a central hot spot induces significant localization of LHCD. Furthermore, in steady state tokamaks where a significant amount of current is provided by LHCD, passive stabilization of NTMs may occur automatically, particularly as islands become large, without requiring precise aiming of the wave power.
The radial flux of toroidal angular momentum is needed to determine tokamak intrinsic rotation profiles. Its computation requires knowledge of the gyrokinetic distribution functions and turbulent electrostatic potential to second-order in $ε= ρ/L$, where $ρ$ is the ion Larmor radius and $L$ is the variation length of the magnetic field. In this article, a complete set of equations to calculate the radial transport of toroidal angular momentum in any tokamak is presented. In particular, the $O(ε^2)$ equations for the turbulent components of the distribution functions and electrostatic potential…
Improved Particle Confinement with Resonant Magnetic Perturbations in DIII-D Tokamak H-Mode Plasmas
Lawrence Livermore National Laboratory（劳伦斯利弗莫尔国家实验室）；Princeton Plasma Physics Laboratory（普林斯顿等离子体物理实验室）；General Atomics（通用原子能公司）；University of California, Los Angeles（加州大学洛杉矶分校）；Massachusetts Institute of Technology（麻省理工学院）
The collisional dynamics of test impurity ions in spherical tokamak plasmas with strongly-sheared radial electric fields is investigated by means of a test particle full orbit simulation code. The strength of the shear is such that the standard drift ordering can no longer be assumed and a full orbit approach is required. The effect of radial electric field shear on neoclassical particle transport is quantified for a range of test particle mass and charge numbers and electric field parameters. It is shown that the effect of a sheared electric field is to enhance the confinement of impurity spe…
A comprehensive description of methods, suitable for solving the kinetic equation for fast ions and impurity species in tokamak plasmas using Monte Carlo approach, is presented. The described methods include Hamiltonian orbit-following in particle and guiding center phase space, test particle or guiding center solution of the kinetic equation applying stochastic differential equations in the presence of Coulomb collisions, neoclassical tearing modes and Alfvén eigenmodes as electromagnetic perturbations relevant to fast ions, together with plasma flow and atomic reactions relevant to impurity …
A two channel oblique electron cyclotron emission (ECE) radiometer was installed on the DIII-D tokamak and interfaced to 4 gyrotrons. Oblique ECE was used to toroidally and radially localize rotating magnetic islands and so assist their electron cyclotron current drive (ECCD) stabilization. In particular, after manipulations operated by the interfacing analogue circuit, the oblique ECE signals directly modulated the current drive in synch with the island rotation and in phase with the island O-point, for a more efficient stabilization. Apart from the different toroidal location, the diagnostic…
Time-dependent helium gas puff experiments have been performed on the Mega Ampere Spherical Tokamak (MAST) during a two point plasma current scan in L-mode and a confinement scan at 900 kA. An evaluation of the He II spectrum line induced by charge exchange suggests anomalous rates of diffusion and inward convection in the outer regions of both L-mode plasmas. Similar rates of diffusion are found in the H-mode plasma, however these rates are consistent with neoclassical predictions. The anomalous inward pinch found in the core of L-mode plasmas is also not apparent in the H-mode core. Linear g…
In general, the orbit-averaged radial magnetic drift of trapped particles in stellarators is non-zero due to the three-dimensional nature of the magnetic field. Stellarators in which the orbit-averaged radial magnetic drift vanishes are called omnigeneous, and they exhibit neoclassical transport levels comparable to those of axisymmetric tokamaks. However, the effect of deviations from omnigeneity cannot be neglected in practice. For sufficiently low collision frequencies (below the values that define the $1/ν$ regime), the components of the drifts tangential to the flux surface become relevan…
An impurity hole observed in the Large Helical Device (LHD) is a hollow density profile of an impurity ion species formed in the core plasma where the negative (inward-pointing) ambipolar radial electric field ($E_r$) exists. Although local neoclassical models have predicted the sign of $E_r$ in impurity hole plasmas is negative for the entire minor radius, an experimental measurement of an impurity hole plasma has shown that the $E_r$ changes the sign from negative to positive along the minor radius. In the present work, we investigate neoclassical impurity transport in an impurity hole plasm…
Impurity temperature screening is a favorable neoclassical phenomenon involving an outward radial flux of impurity ions from the core of fusion devices. Quasisymmetric magnetic fields are intrinsically ambipolar and give rise to temperature screening for low enough $η^{-1}\equiv d\ln n/d\ln T$. In contrast, neoclassical fluxes in generic stellarators will depend on the radial electric field, which is predicted to be inward for ion-root plasmas, potentially leading to impurity accumulation. Here we examine the impurity particle flux in a number of approximately quasisymmetric stellarator config…
In transport barriers, particularly H-mode edge pedestals, radial scale lengths can become comparable to the ion orbit width, causing neoclassical physics to become radially nonlocal. In this work, the resulting changes to neoclassical flow and current are examined both analytically and numerically. Steep density gradients are considered, with scale lengths comparable to the poloidal ion gyroradius, together with strong radial electric fields sufficient to electrostatically confine the ions. Attention is restricted to relatively weak ion temperature gradients (but permitting arbitrary electron…
Tokamak plasmas rotate even without external injection of momentum. A Doppler backscattering system installed at MAST has allowed this intrinsic rotation to be studied in Ohmic L-mode and H-mode plasmas, including the first observation of intrinsic rotation reversals in a spherical tokamak. Experimental results are compared to a novel 1D model, which captures the collisionality dependence of the radial transport of toroidal angular momentum due to the effect of neoclassical flows on turbulent fluctuations. The model is able to accurately reproduce the change in sign of core toroidal rotation, …
