Axisymmetric Tokamak Geometry Research Articles

Representation and modeling of charged particle distributions in tokamaks

Experimental diagnostics, analysis tools and simulations represent particle distributions in various forms and coordinates. Algorithms to manage these data are needed on platforms like the ITER Integrated Modeling & Analysis Suite (IMAS), performing tasks such as archiving, modeling, conversion and visualization. A method that accomplishes some of the required tasks for distributions of charged particles with arbitrarily large magnetic drifts in axisymmetric tokamak geometry is described here. Given a magnetic configuration, we first construct a database of guiding center orbits, which serves as a basis for representing particle distributions. The orbit database contains the geometric information needed to perform conversions between arbitrary coordinates, modeling tasks, and resonance analyses. Using that database, an imported or newly modeled distribution is mapped to an exact equilibrium, where the dimensionality is reduced to three constants of motion (CoM). The orbit weight is uniquely given when the input is a true distribution: one that measures the number of physical particles per unit of phase space volume. Less ideal inputs, such as distributions estimated without drifts, or models of particle sources, can also be processed. As an application example, we reconstruct the drift-induced features of a distribution of fusion-born alpha particles in a large tokamak, given only a birth profile, which is not a function of the alpha's CoM. Repeated back-and-forth transformations between CoM space and energy-pitch-cylinder coordinates are performed for verification and as a proof of principle for IMAS.

An asymptotic-preserving 2D-2P relativistic drift-kinetic-equation solver for runaway electron simulations in axisymmetric tokamaks

We propose an asymptotic-preserving (AP), uniformly convergent numerical scheme for the relativistic collisional Drift-Kinetic Equation (rDKE) to simulate runaway electrons in axisymmetric toroidal magnetic field geometries typical of tokamak devices. The approach is derived from an exact Green's function solution with numerical approximations of quantifiable impact, and results in a simple, two-step operator-split algorithm, consisting of a collisional Eulerian step, and a Lagrangian orbit-integration step with analytically prescribed kernels. The AP character of the approach is demonstrated by analysis of the dominant numerical errors, as well as by numerical experiments. We demonstrate the ability of the algorithm to provide accurate answers regardless of plasma collisionality on a circular axisymmetric tokamak geometry.

Effect of non-axisymmetric perturbations on the ambipolar Er and neoclassical particle flux inside the ITER pedestal region

The transport dynamics of impurities in the pedestal region of ITER plasmas is of crucial interest since this regulates the penetration of impurities from the edge into the core plasma, where an excessive accumulation of impurities can degrade their fusion performance. In the pedestal region of H-mode tokamak plasmas, anomalous transport is highly reduced and impurity transport is found to be well described by the neoclassical theory. Under these conditions, perturbations to the axisymmetric tokamak geometry can strongly affect both the radial electric field and particle transport. In this work, we describe the results of numerical studies performed to quantify the effects on the pedestal ambipolar electric field and radial particle fluxes of the non-axisymmetric fields, associated with both the intrinsic toroidal field ripple and extrinsic fields applied for ELM control, for ITER Q = 10 plasma conditions with emphasis on high Z impurity transport. It is found that the effect of the ITER toroidal field ripple on high Z impurity transport is negligible. On the contrary, extrinsic three-dimensional fields applied for ELM control cause a strong modification of the pedestal ambipolar electric (to less negative values) and the appearance of multi-valued solutions for the pedestal electric field, analogue to core stellarator transport significantly increasing the outward character of neoclassical pedestal transport for both the main plasma ions (D and T in ITER) and high Z impurities, suggesting a strong modification of the background plasma profiles. Finally, it is found that for the Z impurity, its quantitative evaluation has uncertainties (with important implications for the radial flow direction) associated with the high poloidal Mach number ~ 1, due to the high pedestal electric field and large ion mass, which renders the first order neoclassical theory questionable. Detailed ITER MHD 3D equilibria and a benchmark of SFINCS against NEO and NCLASS codes has also been obtained.

Runaway electron generation in axisymmetric tokamak geometry

The generation of relativistic electrons in tokamak plasmas has been the subject of extensive research due to their intrinsic interest as well as the threat they pose to reactor-scale tokamak devices. Runaway electron generation becomes particularly robust for the very large inductive electric fields that are often present during tokamak disruptions. In this letter we show that when such large inductive electric fields are present, the physics of how tokamak geometry impacts runaway generation processes is qualitatively modified compared to the more commonly studied limit of weak inductive electric-field strengths. This modification leads to a significant increase in the efficiency of all runaway generation processes.

The effect of density fluctuations on electron cyclotron beam broadening and implications for ITER

We present state-of-the-art computations of propagation and absorption of electron cyclotron waves, retaining the effects of scattering due to electron density fluctuations. In ITER, injected microwaves are foreseen to suppress neoclassical tearing modes (NTMs) by driving current at the and resonant surfaces. Scattering of the beam can spoil the good localization of the absorption and thus impair NTM control capabilities. A novel tool, the WKBeam code, has been employed here in order to investigate this issue. The code is a Monte Carlo solver for the wave kinetic equation and retains diffraction, full axisymmetric tokamak geometry, determination of the absorption profile and an integral form of the scattering operator which describes the effects of turbulent density fluctuations within the limits of the Born scattering approximation. The approach has been benchmarked against the paraxial WKB code TORBEAM and the full-wave code IPF-FDMC. In particular, the Born approximation is found to be valid for ITER parameters. In this paper, we show that the radiative transport of EC beams due to wave scattering in ITER is diffusive unlike in present experiments, thus causing up to a factor of 2–4 broadening in the absorption profile. However, the broadening depends strongly on the turbulence model assumed for the density fluctuations, which still has large uncertainties.

Optimization of the Angular Orientation for a Fast Ion Loss Detector in a Tokamak

A scintillator-type fast ion loss detector (FILD) measures the gyroradius and pitch angle distribution of superthermal ions escaping from a magnetically confined fusion plasma at a single location. Described here is a technique for optimizing the angular orientation of such a detector in an axisymmetric tokamak geometry in order to intercept losses over useful and interesting ranges of pitch angle. The method consists of evaluating the detector acceptance as a function of the fast ion constants of motion, i.e., energy, canonical toroidal momentum, and magnetic moment. The detector acceptance can then be plotted in a plane of constant energy and compared with the relevant orbit class boundaries and fast ion source distributions. Knowledge of expected or interesting mechanisms of loss can further guide selection of the detector orientation. The example of a FILD for the National Spherical Torus Experiment-Upgrade (NSTX-U) is considered.

The role of zonal flows in the saturation of multi-scale gyrokinetic turbulence

The 2D spectrum of the saturated electric potential from gyrokinetic turbulence simulations that include both ion and electron scales (multi-scale) in axisymmetric tokamak geometry is analyzed. The paradigm that the turbulence is saturated when the zonal (axisymmetic) ExB flow shearing rate competes with linear growth is shown to not apply to the electron scale turbulence. Instead, it is the mixing rate by the zonal ExB velocity spectrum with the turbulent distribution function that competes with linear growth. A model of this mechanism is shown to be able to capture the suppression of electron-scale turbulence by ion-scale turbulence and the threshold for the increase in electron scale turbulence when the ion-scale turbulence is reduced. The model computes the strength of the zonal flow velocity and the saturated potential spectrum from the linear growth rate spectrum. The model for the saturated electric potential spectrum is applied to a quasilinear transport model and shown to accurately reproduce the electron and ion energy fluxes of the non-linear gyrokinetic multi-scale simulations. The zonal flow mixing saturation model is also shown to reproduce the non-linear upshift in the critical temperature gradient caused by zonal flows in ion-scale gyrokinetic simulations.

Advances in stellarator gyrokinetics

Recent progress in the gyrokinetic theory of stellarator microinstabilities and turbulence simulations is summarized. The simulations have been carried out using two different gyrokinetic codes, the global particle-in-cell code EUTERPE and the continuum code GENE, which operates in the geometry of a flux tube or a flux surface but is local in the radial direction. Ion-temperature-gradient (ITG) and trapped-electron modes are studied and compared with their counterparts in axisymmetric tokamak geometry. Several interesting differences emerge. Because of the more complicated structure of the magnetic field, the fluctuations are much less evenly distributed over each flux surface in stellarators than in tokamaks. Instead of covering the entire outboard side of the torus, ITG turbulence is localized to narrow bands along the magnetic field in regions of unfavourable curvature, and the resulting transport depends on the normalized gyroradius ρ* even in radially local simulations. Trapped-electron modes can be significantly more stable than in typical tokamaks, because of the spatial separation of regions with trapped particles from those with bad magnetic curvature. Preliminary non-linear simulations in flux-tube geometry suggest differences in the turbulence levels in Wendelstein 7-X and a typical tokamak.

Compact formulas for bounce/transit averaging in axisymmetric tokamak geometry

Compact formulas for bounce and transit orbit averaging of the fluctuation-amplitude eikonal factor in axisymmetric tokamak geometry, which is frequently encountered in bounce-gyrokinetic description of microturbulence, are given in terms of the Jacobi elliptic functions and elliptic integrals. These formulas are readily applicable to the calculation of the neoclassical susceptibility in the framework of modern bounce-gyrokinetic theory. In the long-wavelength limit for axisymmetric electrostatic perturbations, we recover the expression for the Rosenbluth-Hinton residual zonal flow [M. N. Rosenbluth and F. L. Hinton, Phys. Rev. Lett. 80, 724 (1998)] accurately.

Canonical transformation for trapped/passing guiding-center orbits in axisymmetric tokamak geometry

The generating function for the canonical transformation from the parallel canonical coordinates (s,p||) to the action-angle coordinates (ζ, J) for trapped/passing guiding-center orbits in axisymmetric tokamak geometry is presented. Drawing on the analogy between the phase-space portraits of the librating/rotating pendulum and the trapped/passing guiding-center orbits, the generating function is expressed in terms of the Jacobi zeta function, which can then readily be used to obtain an explicit expression for the bounce-center transformation for trapped/passing-particle guiding-center orbits in axisymmetric tokamak geometry.

Exact momentum conservation laws for the gyrokinetic Vlasov-Poisson equations

The exact momentum conservation laws for the nonlinear gyrokinetic Vlasov-Poisson equations are derived by applying the Noether method on the gyrokinetic variational principle [A. J. Brizard, Phys. Plasmas 7, 4816 (2000)]. From the gyrokinetic Noether canonical-momentum equation derived by the Noether method, the gyrokinetic parallel momentum equation and other gyrokinetic Vlasov-moment equations are obtained. In addition, an exact gyrokinetic toroidal angular-momentum conservation law is derived in axisymmetric tokamak geometry, where the transport of parallel-toroidal momentum is related to the radial gyrocenter polarization, which includes contributions from the guiding-center and gyrocenter transformations.

Full-field drift Hamiltonian particle orbits in axisymmetric tokamak geometry

A Hamiltonian/Lagrangian theory to describe guiding center orbit drift motion that is canonical in Boozer magnetic coordinates is developed to include full electrostatic and electromagnetic perturbed fields in axisymmetric tokamak geometry. Furthermore, the radial component of the equilibrium magnetic field in the covariant representation is retained and the background equilibrium state extends to anisotropic plasma pressure conditions. A gauge transformation on the perturbed vector potential is imposed to guarantee canonical structure in the Boozer frame. Perturbed field nonlinear wave-wave interactions affect only the evolution of the guiding center particle parallel gyroradius. The evolution of the particle coordinate positions retains only linear wave-particle interactions. For particle motion in magnetohydrodynamic (MHD) instability structures, the electrostatic potential is linked mainly to the binormal component of the perturbed displacement vector when finite δA⊥ components are included.

Compact formulas for guiding-center orbits in axisymmetric tokamak geometry

Compact formulas for trapped-particle and passing-particle guiding-center orbits in axisymmetric tokamak geometry are given in terms of the Jacobi elliptic functions and complete elliptic integrals. These formulas can find applications in bounce-center kinetic theory as well as guiding-center Fokker–Planck kinetic theory.

Linear Instability Analysis for Toroidal Plasma Flow Equilibria

The non-self-adjoint Frieman–Rotenberg equation for the linear ideal magnetohydrodynamic modes in flow equilibria is numerically solved in shaped finite-aspect ratio axisymmetric tokamak geometry. A quadratic form is derived from this equation, and, in particular, the self-adjoint force operator with finite toroidal rotation is cast into a manifestly self-adjoint form. The toroidal rotational velocities in the subsonic regime are considered. The quadratic form is discretized by a mixed finite-element procedure in the radial direction and by Fourier modes in the periodic directions. The mode frequency of the unstable mode is located by root searching, and the final root refinement is obtained by a rapid inverse iteration procedure for complex roots. The real part of then= 1 internal kink mode scales linearly with the plasma rotation, and the imaginary part of the unstable mode is at least an order of magnitude higher in the presence of high plasma rotation velocities. The kink mode is also found to be unstable at high rotation velocities, even when the axis safety factor is above unity. The instability characterized by these features is termed here as the “centrifugal” instability. The centrifugal kink instability would have finite real parts, as shown by the plasma rotation observed in plasma devices such as tokamaks. To explain the features of this mode, the plasma rotation should be taken into account. Therein lies the usefulness of the computational analysis presented here.

Nonlinear gyrokinetic Vlasov equation for toroidally rotating axisymmetric tokamaks

The nonlinear gyrokinetic Vlasov equation is derived for an arbitrary magnetized plasma in a local reference frame moving with the nonuniform equilibrium fluid velocity u(r). The derivation of the guiding-center and gyrocenter Hamilton equations, which appear as the characteristics of the gyrokinetic Vlasov equation, is based on the use of Lie-transform perturbation techniques. Although a general form for u is initially used, attention is later focused on an incompressible toroidal equilibrium flow when considering axisymmetric tokamak geometry.