Finite Orbit Effects Research Articles

Drift kinetic theory of alpha transport by tokamak perturbations

Upcoming tokamak experiments fuelled with deuterium and tritium are expected to have large alpha particle populations. Such experiments motivate new attention to the theory of alpha particle confinement and transport. A key topic is the interaction of alpha particles with perturbations to the tokamak fields, including those from ripple and magnetohydrodynamic modes like Alfvén eigenmodes. These perturbations can transport alphas, leading to changed localization of alpha heating, loss of alpha power and damage to device walls. Alpha interaction with these perturbations is often studied with single-particle theory. In contrast, we derive a drift kinetic theory to calculate the alpha heat flux resulting from arbitrary perturbation frequency and periodicity (provided these can be studied drift kinetically). Novel features of the theory include the retention of a large effective collision frequency resulting from the resonant alpha collisional boundary layer, correlated interactions over many poloidal transits and finite orbit effects. Heat fluxes are considered for the example cases of ripple and the toroidal Alfvén eigenmode (TAE). The ripple heat flux is small. The TAE heat flux is significant and scales with the square of the perturbation amplitude, allowing the derivation of constraints on mode amplitude for avoidance of significant alpha depletion. A simple saturation condition suggests that TAEs in one upcoming experiment will not cause significant alpha transport via the mechanisms in this theory. However, saturation above the level suggested by the simple condition, but within numerical and experimental experience, which could be accompanied by the onset of stochasticity, could cause significant transport.

Drift kinetic response of ions to magnetic island perturbation and effects on NTM threshold

Controlling neoclassical tearing modes (NTMs) is vital for future tokamaks, such as ITER. An NTM control system relies on the magnetic island threshold physics. In this paper, new results for the ion response to the island perturbation and its influence on the island evolution are presented. Considering a small island, w ≪ r, where r is the minor radius, but crucially retaining the ordering ρbi ∼ w (relevant for threshold) to capture the finite orbit width effect, we determine the ion response using the drift kinetic equation. Momentum conservation and quasineutrality are taken into account, which are crucial for determining the current perturbations. The results show that the finite particle orbit effects are significant even for a moderately small ratio of ρθi/w (ρθi is the ion poloidal Larmor radius; ρbi ∼ ϵ1/2ρθi). When w ∼ ρθi, the flattening of the pressure gradient across the island is substantially restored, implying that the bootstrap current drive for the island growth is suppressed. Moreover, we find that for a sufficiently small island, w ≪ ρθi, the contribution can be negative, meaning that it can stabilize small seed islands, providing a threshold. This will have significant impact on our understanding of the NTM threshold physics.

A full wave model of high-harmonic EMIC wave propagation in sheared magnetic fields

ABSTRACT A new dispersion relation, with finite Larmor orbit effects, for oblique propagating electromagnetic ion cyclotron (EMIC) waves in a magnetized plasma medium, is derived including the magnetic shear effect. The approximate, yet accurate, dispersion relation is used to implement the ray tracing model. A parabolic magnetic field is considered to model the geomagnetic field in the magnetosphere. Energetic protons are also considered as resonant particles. The propagation characteristics of EMIC waves in the vicinity of the ion cyclotron resonances are investigated in some detail. The results reveal adiabatic oscillating motion for wave and magnetic field fluctuations where high harmonics limit the wave damping and confines the magnetic fluctuations. For inward propagating EMIC waves we find (1) turning points which depend on the wave launch position, and (2) wave trapped areas playing a role in quasi-coherent wave-particle interaction in agreement with the observational and theoretical studies. This wave trapping is an effective process for particle acceleration in the context of space plasmas.

Integrated modelling of steady-state scenarios and heating and current drive mixes for ITER

Recent progress on ITER steady-state (SS) scenario modelling by the ITPA-IOS group is reviewed. Code-to-code benchmarks as the IOS group's common activities for the two SS scenarios (weak shear scenario and internal transport barrier scenario) are discussed in terms of transport, kinetic profiles, and heating and current drive (CD) sources using various transport codes. Weak magnetic shear scenarios integrate the plasma core and edge by combining a theory-based transport model (GLF23) with scaled experimental boundary profiles. The edge profiles (at normalized radius ρ = 0.8–1.0) are adopted from an edge-localized mode-averaged analysis of a DIII-D ITER demonstration discharge. A fully noninductive SS scenario is achieved with fusion gain Q = 4.3, noninductive fraction fNI = 100%, bootstrap current fraction fBS = 63% and normalized beta βN = 2.7 at plasma current Ip = 8 MA and toroidal field BT = 5.3 T using ITER day-1 heating and CD capability. Substantial uncertainties come from outside the radius of setting the boundary conditions (ρ = 0.8). The present simulation assumed that βN (ρ) at the top of the pedestal (ρ = 0.91) is about 25% above the peeling–ballooning threshold. ITER will have a challenge to achieve the boundary, considering different operating conditions (Te/Ti ≈ 1 and density peaking). Overall, the experimentally scaled edge is an optimistic side of the prediction. A number of SS scenarios with different heating and CD mixes in a wide range of conditions were explored by exploiting the weak-shear steady-state solution procedure with the GLF23 transport model and the scaled experimental edge. The results are also presented in the operation space for DT neutron power versus stationary burn pulse duration with assumed poloidal flux availability at the beginning of stationary burn, indicating that the long pulse operation goal (3000 s) at Ip = 9 MA is possible. Source calculations in these simulations have been revised for electron cyclotron current drive including parallel momentum conservation effects and for neutral beam current drive with finite orbit and magnetic pitch effects.

Drift-kinetic simulation of neoclassical transport with impurities in tokamaks

Plasmas in modern tokamak experiments contain a significant fraction of impurity ions in addition to the main deuterium background ions. A new multiple ion-species δf particle simulation capability has been developed to self-consistently study the nonlocal effects of impurities on neoclassical transport in toroidal plasmas. A new algorithm for an unlike-particle collision operator, including test-particle and conserving field-particle parts, is described. Effects of the carbon impurity on the main deuterium species heat flux as well as an ambipolar radial electric field in a National Spherical Torus Experiment (NSTX) [M. Ono, S. M. Kaye, Y.-K. M. Peng et al., Nucl. Fusion 40, 557 (2000)] configuration were studied. A difference between carbon poloidal rotation found from simulation and from conventional theoretical estimates has been investigated and was identified to be a nonlocal finite orbit effect. In the case of large-aspect ratio tokamak configurations with steep toroidal flow profiles, we propose a theoretical model to describe this nonlocal effect. The dominant mechanisms captured by the model are associated with ion parallel velocity modification due to steep toroidal flow and radial electric field profiles. We present simulation results for carbon poloidal velocity in NSTX. Comparisons with neoclassical theory are discussed.

Resonant excitation of shear Alfvén perturbations by trapped energetic ions in a tokamak

A new analytic expression is derived for the resonant drive of high n Alfvénic modes by particles accelerated to high energy by ion cyclotron resonance heating. This derivation includes finite orbit effects, and the formalism is completely nonperturbative. The high-n limit is used to calculate the complex particle response integrals along the orbits explicitly. This new theory is applied to downward sweeping Alfvén cascade quasimodes completing the theory of these modes and making testable predictions. These predictions are found to be consistent with experiments carried out on the Joint European Torus [P. H. Rebut and B. E. Keen, Fusion Technol. 11, 13 (1987)].

Exploration of finite ion orbit effects in the ion cyclotron range of frequencies

The rf-SciDAC collaboration is developing computer simulations to predict the damping of radio frequency (rf) waves in fusion plasmas. Here we extend self-consistent quasi-linear calculations of ion cyclotron resonant heating to include the finite drift of ions from magnetic flux surfaces and rf induced spatial transport. The all-orders spectral wave solver AORSA is iteratively coupled with a particle based update of the plasma distribution function.

Comparison of the Monte Carlo ion cyclotron heating model with the full-wave linear absorption model

To fully account for the wave-particle interaction physics in ion cyclotron resonant frequency (ICRF) heating experiment, finite orbit effects and non-Maxwellian distribution have to be self-consistently coupled with full-wave solutions. For this purpose, the five-dimensional Monte Carlo code ORBIT-RF [M. Choi et al., Phys. Plasmas 12, 1 (2005)] is being coupled with the two-dimensional full-wave code AORSA [E. F. Jaeger et al., Phys. Plasmas 13, 056101 (2006)] to iteratively evolve the ion distribution in four-dimensional spatial velocity space that is used to update the dielectric tensor in AORSA for evaluating the full-wave fields. In this paper, it is demonstrated that using the full-wave fields from a Maxwellian dielectric tensor in AORSA and confining the resonant ions to their initial orbits in ORBIT-RF, ORBIT-RF largely reproduces the AORSA linear wave absorption profiles for fundamental and higher harmonic ICRF heating. An exception is an observed inward shift in the ORBIT-RF absorption peak for high harmonics near the magnetic axis compared with that of AORSA, which can be attributed to a finite orbit width effect. The success of this verification supports the validity of the Monte Carlo wave-particle interaction model and the readiness of the iterative coupling between ORBIT-RF and AORSA for an improved modeling of ICRF heating experiments.

Unified linear response function for zonal flows with full finite orbit effects

A new formulation of the linear response function of electrostatic potential to nonlinear drive (due to turbulence) is presented in this paper; zonal flows play important roles in the self-regulation of turbulence and their basic physics are contained in response functions. Two branches of zonal flows [stationary zonal flow and geodesic acoustic mode (GAM)] are known to exist in the low and high frequency ranges. However, they have been analyzed separately using different approximations due the difference in their frequency ranges. This paper visits this problem and gives a unified expression of the response function by taking full account of finite orbit effects. The drift kinetic equation is integrated along particle orbits by expanding them in Fourier series. Thus, a separate handling of passing and trapped particles is facilitated revealing some important aspects of zonal flows: (1) neoclassical poloidal mode coupling due to finite orbit effects, (2) enhancement of the nonuniform potential field due to reduced parallel transport, and (3) the presence of two propagation bands of GAM as the quadratic dispersion relation is solved.

Simulation of fast Alfvén wave interaction with beam ions over a range of cyclotron harmonics in DIII-D tokamak

To study fast Alfvén wave damping on fast ions, a Monte-Carlo code, ORBIT-RF, has been coupled with a 2D full wave code, TORIC4. The ORBIT-RF/TORIC4 combination has been applied to DIII-D experimental conditions to investigate fast wave (FW) heating of injected beam ions over a range of ion cyclotron harmonics. ORBIT-RF using a single dominant toroidal and poloidal Fourier wave number qualitatively reproduces the strong FW–beam interaction at 60 MHz (4ΩD and 5ΩD) and the much weaker interaction at 116 MHz (8ΩD) in DIII-D L-mode plasmas, consistent with experimental observations. The result at 8ΩD differs from linear theory prediction using a Maxwellian to model the fast ion distribution function, suggesting the importance of finite orbit effect, Coulomb collisions for transport across flux surfaces and details of the non-Maxwellian fast ion distribution.

Sawtooth control in fusion plasmas

Clear observations of early triggering of neo-classical tearing modes by sawteeth with long quiescent periods have motivated recent efforts to control, and in particular destabilize, sawteeth. One successful approach explored in TCV utilizes electron cyclotron heating in order to locally increase the current penetration time in the core. The latter is also achieved in various machines by depositing electron cyclotron current drive or ion cyclotron current drive close to the q = 1 rational surface. Crucially, localized current drive also succeeds in destabilizing sawteeth which are otherwise stabilized by a co-existing population of energetic trapped ions in the core. In addition, a recent reversed toroidal field campaign at JET demonstrates that counter-neutral beam injection (NBI) results in shorter sawtooth periods than in the Ohmic regime. The clear dependence of the sawtooth period on the NBI heating power and the direction of injection also manifests itself in terms of the toroidal plasma rotation, which consequently requires consideration in the theoretical interpretation of the experiments. Another feature of NBI, expected to be especially evident in the negative ion based neutral beam injection (NNBI) heating planned for ITER, is the parallel velocity asymmetry of the fast ion population. It is predicted that a finite orbit effect of asymmetrically distributed circulating ions could strongly modify sawtooth stability. Furthermore, NNBI driven current with non-monotonic profile could significantly slow down the evolution of the safety factor in the core, thereby delaying sawteeth.

Internal kink mode stabilization and the properties of auxiliary heated ions

The stability of the internal kink mode is analyzed in terms of the essential properties of auxiliary heated ions. A fundamental difference between fusion born alpha particles and auxiliary heated ions is that the latter are typically anisotropically distributed. The effect of anisotropy, which is usually ignored in the fluid contribution of hybrid kinetic-magnetohydrodynamic codes, provides a stabilizing effect for populations that have a large passing fraction. The effect on internal kink mode stability is further modified by crucial finite orbit effects if the distribution is asymmetric in v‖, such as for unbalanced beam ion populations. Additionally, such heating scenarios transfer momentum to the background plasma thereby especially modifying the kinetic response of thermal ions. In contrast, anisotropic distributions with large trapped fractions incur improved kinetic stability, but at the cost of increased fluid instability, which is partially due to the effect of anisotropy on the equilibrium.

The role of the finite-orbit effect in theory of magnetic islands in tokamaks

The role of the finite-orbit effect in theory of magnetic islands in tokamaks

Stability of long field-reversed configurations

The stability of very elongated field-reversed configurations is solved by an expansion in the small parameter ε (inverse elongation). It is first shown that all possible unstable modes have small growth rates (γ ∼ε). The internal tilt mode is considered in detail. An explicit form for δW in leading order is derived, and leads to a quadratic form including Hall terms. A sufficient condition for stability is obtained by minimizing δW, leading to a field-line ordinary differential equation. Sufficient stability conditions are obtained from this formulation, and indicate stability for S*/E<2 (where S* is the ratio of separatrix radius to collisionless ion skin depth and E the elongation of the separatrix), if the local criterion is used. It is argued that a volume-averaged condition is more appropriate when finite ion orbit effects are included. This leads to S*/E<3.5–4 for stability, independent of separatrix shape or xs (separatrix radius to wall radius at the midplane). This condition for stability compares favorably with experimental observations.

Bifurcation of the radial electric field in the presence of edge polarization in tokamaks

In guiding-center simulations of a tokamak plasma edge, bifurcations of both the radial electric field and the electrode current are observed for a threshold value of the electrode bias voltage, in qualitative agreement with experiments. The simulations are neoclassical and Monte Carlo based, also containing the perpendicular viscosity effect. Above the voltage threshold, an inward moving, solitary radial electric field structure is generated with a concomitant drop in the electrode current. The position, width, and height of the soliton depend on the magnitude and radial variation of the conductivity (and, thus, on local plasma parameters), finite orbit effects, and also on the perpendicular viscosity and neutral density. The obtained Er structures are narrower and higher than the ones measured in the edge polarization experiments in the Jülich tokamak TEXTOR-94 [S. Jachmich et al., Plasma Phys. Controlled Fusion 40, 1105 (1998)].

Loss of fast tritons in JT-6OU reversed magnetic shear discharges

In reversed magnetic sheaf-plasmas, fast ion confinement is anticipated to deteriorate because of a weak poloidal magnetic field in the core. In fact, in experiments at JT-6OU considerable depletion of triton burnup in reversed magnetic shear has been observed, compared with normal (positive) magnetic shear. The experimental triton burnup in the reversed shear was 11 to 39% of that expected, while that in the normal shear was 64 to 87%. Here, the expected burnup is based on one dimensional (1-D) calculations that assume classical slowing down and no diffusion of fast tritons. Orbit following Monte Carlo simulations, which treat finite orbit effects and ripple transport processes properly, match the experimental depletion and indicate that ripple loss is responsible for the enhanced triton loss in reversed shear operations. The results raise concerns about serious megaelectronvolt ion loss in reversed shear operations in steady state tokamak reactors

Magnetic Field Co-Ordinate and Equilibrium Structure of Field-Reversed Configurations

According to the recent study of the magnetohydrodynamic stability for both the global and local modes of field-reversed configurations (FRCs), it is required to develop a theory including the ion`s finite orbit effect which is valid in FRCs. This means that the two dimensional analysis on the poloidal surface is inevitably necessary. As the first step toward the two dimensional stability analysis, the magnetic field co-ordinate is obtained numerically in FRCs for the first time. Using the co-ordinate, the equilibrium properties of FRCs are examined for the preparation of stability analysis. 6 refs., 4 figs.

Modelling of the observed particle and heat fluxes in the X-point region in JET

Particle and heat fluxes, deduced from experimental CCD camera and magnetic measurements in the upper X-point region during Ohmic and auxiliary heated discharges in JET have been analysed. Modelling of the particle and heat transport processes at the edge of the plasma with Monte Carlo orbit following calculations agrees well with the observations. The importance of the finite orbit effects in low collisionality plasmas is assessed. The convection by plasma ions is found to play a very important role in the heat transport from the plasma to the wall

An analytical solution of the linearized Vlasov equation for nearly rigid displacements

Fluid equations are derived for nearly rigid electromagnetic displacements in two-dimensional equilibria described by distribution functions of the form F(H, Pz). Solutions in closed form of the Maxwell-Vlasov equations are given for the m=1 mode in z pinches, as well as for the corresponding free boundary mode in Extrap. If feedback stabilization is excluded, an instability of modes with large axial wavelengths is predicted, regardless of the shape of the equilibrium pressure profile. In z pinches, stability of modes with mod kza mod <0.5 requires that the radius of a conducting cylinder that surrounds the plasma is not more than twice the plasma radius. It is stressed that finite orbit effects at the magnetic O-point in z pinches are rigorously treated, since the formulas are essentially exact in the limit of negligible wall stabilization and small axial wavelength of the perturbation. Even chaotic particle orbits are adequately described in some cases.

Kinetic stability of large-scale MHD modes

Two independent particle methods, which are based on the Vlasov fluid model, have been developed to examine the modification of magnetohydrodynamic instabillities due to finite orbit effects. The relative cost and accuracy of the methods, the dispersion functional analysis and initial value method, are discussed and the results of applying the methods to a field-reversed, magnetic confinement geometry are presented.