Neoclassical Polarization Research Articles

Active generation and control of radial electric field by local neutral beamlets injection in tokamaks

The radial electric field plays an important role in plasma confinement in tokamaks and can be generated through neutral beam injection. In this study, we propose a model for calculating the radial electric field resulting from tangential local neutral beamlet injection, aiming to externally control and improve plasma confinement. The Neutral beamlet ion and Energetic particles Orbit mover and Electric field solver code has been developed to analyze this issue, and its simulation results have been validated against results from other codes as well as measurements from correlation reflectometers. The charge separation is primarily caused by the redistribution and loss of beam ions due to magnetic gradient and curvature drift as well as collision effects, and it is maintained through continuous beamlet injection. The electric field is calculated using Poisson’s equation, taking into account both classical and neoclassical polarization effects. The results demonstrate that despite the high losses and low heating efficiency associated with localized beamlets, they are capable of generating a significant radial electric field characterized by a steep gradient. This presents opportunities for external control of the electric field, potentially enhancing plasma confinement.

The role of fast ions in spontaneous neoclassical tearing mode instabilities in NSTX

Spontaneous neoclassical tearing modes (spontaneous NTMs) have been observed in the National Spherical Torus Experiment (NSTX). Recently, a TRANSP based model for the modified Rutherford equation analysis was developed to accurately predict the effect of fast ions on the growth of magnetic islands. The goal of this paper is to utilize the TRANSP NTM model to understand the role of fast ions in NSTX spontaneous NTMs. Our analysis shows that when the passing fast ion driven kinetic neoclassical polarization current term is larger than 1% of the bootstrap current term in the modified Rutherford equation, fast ions play a decisive role in the early growth phase of spontaneous NTM. This result is applicable to any tokamak plasmas with a fast ion drift orbit width, or the fast ion poloidal Larmor radius, comparable to the magnetic island width.

Zonal flow screening in negative triangularity tokamaks

This paper presents a comparative study of zonal flow screening in positive and negative triangularity tokamaks. Neoclassical screening sets the strength of zonal flow shear for fixed drive and damping. Orbit calculations show that the radial excursions of trapped particles are reduced in negative triangularity tokamaks, as compared to positive triangularity. Yet surprisingly, the neoclassical dielectric susceptibility actually increases with decreasing triangularity, such that the susceptibility is higher for negative triangularity than for positive triangularity. This is because the reduction in neoclassical polarization by shrinking the banana width is offset by the increase in neoclassical polarization by the enhancement of trapped fraction for negative triangularity. As a result, the zonal flow screening length is actually enhanced for negative triangularity, as compared to positive triangularity. Hence, the zonal flow residual is smaller for negative triangularity than for positive triangularity. Results from gyrokinetic simulations support the analytic calculations.

Generation of E × B flow shear by finite orbit width effects from heat sources in tokamaks

We present a possible mechanism for the generation of strong E × B flow shear relevant to internal transport barrier formation in tokamak plasmas. From gyrokinetic calculations, we show that strong E × B flow shear can be generated by finite orbit width (FOW) effects associated with a non-uniform heat source and is sufficient to lead to transport barrier formation in the core region with a moderate power level. Two FOW effects inducing neoclassical polarization are shown to be responsible for this: (1) the radial drift of particle orbit center due to the variation of the heat source within orbit width and (2) the non-uniformly evolved orbit width by the non-uniform heating.

Simulations of the radial electric field induced by neutral beam injection in a tokamak

A model based on Poisson’s equation has been proposed to study the radial electric field induced by neutral beam injection (NBI). The radial electric field is produced by charge separation of beam induced electrons and beam ions. The charge separation is produced by redistribution and loss of NBI ions, which is due to the magnetic drift and collision effects. The birth and steady-state distributions of NBI fast ions related to charge separation were numerically computed by the orbit code GYCAVA and the NBI code TGCO. According to our model, the neoclassical polarization density of bulk ions has a great shielding effect on the radial electric field due to charge separation. The simulation results show that NBI ions can produce a significant radial electric field (∼10 kV m−1) in the core region. The counter-current NBI (injection anti-parallel to the plasma current) can produce a negative radial electric field, which is due to the magnetic drift and collision effects of NBI ions. While the co-current injection (injection parallel to the plasma current) produces a positive radial electric field in the core region, which is mainly related to the magnetic drift of NBI ions. The dependence of the radial electric field on the injection direction and the beam energy of NBI has been investigated. As the beam energy increases, the magnitudes of the electric field for counter-injections increase and the magnitude of the electric field for co-current perpendicular injection decrease.

Effect of temperature anisotropy on residual zonal flow level

Residual level (RZF) of long wavelength zonal flows in high aspect ratio collisionless toroidal plasma with a bi-Maxwellian distribution is calculated using the modern bounce-kinetic theory of neoclassical polarization density [L. Wang and T. S. Hahm, Phys. Plasmas 16, 062309 (2009)]. The result which generalizes the well-known formula for the Maxwellian distribution function [M. N. Rosenbluth and F. L. Hinton, Phys. Rev. Lett. 80, 724 (1998)], RZF=[1+1.63q2ϵ−1/2(T⊥/T∥)3/2]−1, shows a strong dependence on the temperature ratio. Here T⊥ and T∥ are the temperatures in the perpendicular and parallel directions with respect to magnetic field, ϵ is the local inverse aspect ratio, and q is the safety factor. This scaling can be understood from the scalings of barely trapped/passing particles' population fraction and their radial orbit width as well as those of the second adiabatic invariant and the bounce/transit frequency.

Physical picture of collisionless neoclassical polarization current

Past work has introduced that the neoclassical polarization drift of the passing ion is in the same direction as that of the trapped ion. Here, based on the model of single particle motion, our analytical results indicate that the neoclassical polarization drifts of the trapped and passing ions are in the opposite directions without collision considered. This work is helpful to easily understand the previous expression of the collisionless neoclassical polarization current based on the kinetic theories, which is only a small quantity of order O(ε3/2) compared to that in the banana collision regime.

A mechanism of neoclassical tearing modes onset by drift wave turbulence

The evolution of neoclassical tearing modes (NTMs) in the presence of electrostatic drift wave turbulence is investigated. In contrast with anomalous transport effect induced by turbulence on NTMs, a new mechanism that turbulence-driven current can affect the onset threshold of NTMs significantly is suggested. Turbulence acts as a source or sink to exchange energy with NTMs. The turbulence-driven current can change the parallel current in magnetic islands and affect the evolution of NTMs, depending on the direction of turbulence intensity gradient. When the turbulence intensity gradient is negative, the turbulence-driven current enhances the onset threshold of NTMs. When the turbulence intensity gradient is positive, it can reduce or even overcome the stabilizing effect of neoclassical polarization current, leading to a small onset threshold of NTMs. This implies that NTMs can appear without noticeable magnetohydrodynamics (MHD) events.

Geodesic mode instability driven during ion cyclotron heating in tokamaks

The effect of a minor concentration of energetic particles produced by ion cyclotron resonance (ICR) heating on the stability of Geodesic Acoustic Modes (GAM) is studied using fully drift kinetic equation. Assuming the ICR layer position on the high field side of the tokamak magnetic axis, a minority ion distribution is modeled by the energetic ion tail and “triangle”-like pitch angle distribution with the maximum at the trapped-passing ion boundary. The resulting GAM instability is found when the hot ion thermal velocity is larger than the effective parallel GAM phase velocity Rqω and the geodesic frequency has the order of the energetic ion bounce frequency. The standard GAM frequency may be strongly reduced by some small concentration of the energetic ions allowing the mode to be unstable. Respective neoclassical polarization corrections (residual flow) driven by energetic ions are found to be negative and larger than that calculated by Rosenbluth and Hinton [Phys. Rev. Lett. 80, 724 (1998)]. Qualitative comparison of the theory with ICR heating experiments in tokamak is discussed.

Reduction of neoclassical polarization current contribution to NTM evolution

The neoclassical polarization current, which can be generated by a time-dependent electric field resulting from magnetic island rotation, is believed to play an important role in the initial stage of the neoclassical tearing mode (NTM) evolution in tokamak plasmas. In the previous analytical description of the neoclassical polarization current contribution to the evolution of NTMs in the limit of low collision frequency (νii≪εω, νii is ion collision frequency, ε is the inverse aspect ratio, and ω is the island propagation frequency in the plasma rest frame), the width of magnetic islands has been assumed to be much larger than the finite-banana-width (FBW) of the trapped ions in order to solve the drift-kinetic equation of ions by using the perturbation method. In this paper, we introduce a new analytical approach to investigate the neoclassical polarization current contribution to the NTM evolution without the assumption of the large island width by solving the drift-kinetic equation in a so-called ion-banana-center coordinate system. The results show that, when the island width is comparable to the FBW of the thermal ion, the neoclassical polarization current term in the equation of the NTM evolution is much smaller than the previous analytical expression but matches well with the empirical anticipation commonly adopted in experiments.

Method to integrate full particle orbit in toroidal plasmas

It is important to integrate full particle orbit accurately when studying charged particle dynamics in electromagnetic waves with frequency higher than cyclotron frequency. We have derived a form of the Boris scheme using magnetic coordinates, which can be used effectively to integrate the cyclotron orbit in toroidal geometry over a long period of time. The new method has been verified by a full particle orbit simulation in toroidal geometry without high frequency waves. The full particle orbit calculation recovers guiding center banana orbit. This method has better numeric properties than the conventional Runge-Kutta method for conserving particle energy and magnetic moment. The toroidal precession frequency is found to match that from guiding center simulation. Many other important phenomena in the presence of an electric field, such as E×B drift, Ware pinch effect and neoclassical polarization drift are also verified by the full orbit simulation.

Effect of Finite-Ion-Banana-Width on the Polarization Contribution to the Neoclassical Tearing Modes Evolution

In the previous analytical description of the neoclassical polarization current effect on the neoclassical tearing modes (NTMs), it is usually assumed that the magnetic island is much larger than the finite-ion-banana-width (FBW). This assumption is questionable when the experimentally observed seed island width of the NTMs is comparable to the FBW. We introduce a simple and direct theoretical method to investigate the FBW effect on the neoclassical polarization contribution to the NTM evolution in collisional plasmas. The results show that, the FBW effect can strongly modify the neoclassical polarization current profile near the island separatrix, and thus weaken its probably stabilizing effect on the NTMs.

Collision frequency dependence of polarization current in neoclassical tearing modes

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 trapped ion orbit width. Focusing on the region away from the island separatrix (where a layer with a complex mix of physics processes exists), we evaluate for the first time the variation of the neoclassical ion polarization current in the transition regime between the analytically tractable collisionless and collisional limits. In addition, the island propagation frequency-dependence of the neoclassical polarization current and its contribution to the island evolution is revealed. For a range of propagation frequencies, we find that the neoclassical polarization current is maximum in the intermediate collision frequency regime analyzed here—a new and unexpected result.

Neoclassical polarization drift of collisionless single ion in tokamaks with arbitrary time-varying radial electric field

An improved theory for the neoclassical polarization drift of the collisionless single ion with the arbitrary time-varying radial electric field is presented. A guiding-center motion code is used to numerically study the neoclassical polarization drift of a collisionless single ion in a model tokamak geometry. The results from the improved theory are compared with the numerical results and they agree with each other very well. The improved theory can be used to study the neoclassical polarization drift due to the fast time-varying radial electric field.

Influence of collision frequency on neoclassical polarization current

A kinetic theory for the evolution of magnetic islands is considered in a tokamak plasma, in both the low (νi ≪ ϵω) and high (νi ≫ ϵω) collision frequency limits (νi is the ion collision frequency, ϵ is the inverse aspect ratio and ω is the island propagation frequency in the E × B rest frame). The calculation of the bootstrap current perturbation in the presence of a magnetic island is reviewed, and is confirmed to be independent of ω and the collision frequency regime. The neoclassical polarization current perturbation is calculated in the two collision frequency limits (within the banana regime). The result in the collisional limit is in agreement with a fluid theory. The effect of collisions in the ‘dissipation layer’ at the trapped/passing boundary is also considered, for νi ≪ ϵω. It is found that the dissipation layer provides an additional contribution to the neoclassical polarization current perturbation. Consequently, if the polarization current is stabilizing, it provides a critical island width for instability, which is found to scale as , where r is a weak logarithmic function of .

Generalized expression for polarization density

A general polarization density which consists of classical and neoclassical parts is systematically derived via modern gyrokinetics and bounce kinetics by employing a phase-space Lagrangian Lie-transform perturbation method. The origins of polarization density are further elucidated. Extending the work on neoclassical polarization for long wavelength compared to ion banana width [M. N. Rosenbluth and F. L. Hinton, Phys. Rev. Lett. 80, 724 (1998)], an analytical formula for the generalized neoclassical polarization including both finite-banana-width and finite-Larmor-radius effects for arbitrary radial wavelength in comparison to banana width and gyroradius is derived. In additional to the contribution from trapped particles, the contribution of passing particles to the neoclassical polarization is also explicitly calculated. The generalized analytic expression agrees very well with the previous numerical results for a wide range of radial wavelength.

Zonal flow in a tokamak pedestal

Neoclassical shielding is the dominant mechanism reducing the collisionless zonal flow in a tokamak. Previously, this phenomenon was analyzed in the case of an essentially homogeneous equilibrium since the wavelength of the zonal flow perturbation was assumed to be much less than the scale length of background plasma parameters. This assumption is not appropriate in a tokamak pedestal. Therefore the pedestal neoclassical polarization and the zonal flow residual differ from the conventional results. This change is due to the strong electric field intrinsic to a subsonic pedestal that modifies neoclassical ion orbits so that their response to a zonal flow perturbation is qualitatively different from that in the core. In addition to orbit squeezing, we find a spatial phase shift between the initial and final zonal flow potentials—an effect absent in previous works. Moreover, we demonstrate that because of orbit modification neoclassical phenomena disappear in the large electric field limit making the residual close to one.

Collisionless residual zonal flow level of toroidally axisymmetric plasmas with arbitrary aspect ratio and elongation

The collisionless residual zonal flow level is calculated for toroidally axisymmetric plasmas with arbitrary aspect ratio and large elongation. For small aspect ratio, this level deviates largely from that predicted by the Rosenbluth–Hinton formula and retains a finite value. Plasma shaping such as elongation is favorable for retaining the residual zonal flow as well. The trapped particles make a major contribution to the neoclassical polarization.

Collisional damping for ion temperature gradient mode driven zonal flow

Zonal flow helps reduce and control the level of ion temperature gradient turbulence in a tokamak. The collisional damping of zonal flow has been estimated by Hinton and Rosenbluth (HR) in the large radial wavelength limit. Their calculation shows that the damping of zonal flow is closely related to the frequency response of neoclassical polarization of the plasma. Based on a variational principle, HR calculated the neoclassical polarization in the low and high collisionality limits. A new approach, based on an eigenfunction expansion of the collision operator, is employed to evaluate the neoclassical polarization and the zonal flow residual for arbitrary collisionality. An analytical expression for the temporal behavior of the zonal flow is also given showing that the damping rate tends to be somewhat slower than previously thought. These results are expected to be useful extensions of the original HR collisional work that can provide an effective benchmark for numerical codes for all regimes of collisionality.

Short wavelength effects on the collisionless neoclassical polarization and residual zonal flow level

Sheared zonal flow helps to reduce the turbulent transport caused by the ion temperature gradient mode. Rosenbluth and Hinton (R-H) calculated the residual zonal flow level for radial wavelengths that are much larger than the ion poloidal gyroradius. Their calculation is extended to treat arbitrary radial wavelengths. For the radial wavelengths that approach the ion poloidal gyroradius, but are much larger than the ion gyroradius, an analytical formula is obtained. For radial wavelengths that are comparable or shorter than the poloidal ion gyroradius and the ion gyroradius a numerical solution is provided. These small radial wavelength results are then extended into the electron temperature gradient regime, where the residual zonal flow level is large but ineffective in regulating the turbulence, indicating that the conventional R-H explanation that zonal flow regulates turbulence is incomplete.