X.Q. Xu

A first-principles predictive model of the pedestal height and width: development, testing and ITER optimization with the EPED model

We develop and test a model, EPED1.6, for the H-mode pedestal height and width based upon two fundamental and calculable constraints: (1) onset of non-local peeling–ballooning modes at low to intermediate mode number, (2) onset of nearly local kinetic ballooning modes at high mode number. Calculation of these two constraints allows a unique, predictive determination of both pedestal height and width. The present version of the model is first principles, in that no parameters are fit to observations, and includes important non-ideal effects. Extensive successful comparisons with existing experiments on multiple tokamaks, including experiments where predictions were made prior to the experiment, are presented, and predictions for ITER are discussed.

BOUT++: A framework for parallel plasma fluid simulations

Abstract A new modular code called BOUT++ is presented, which simulates 3D fluid equations in curvilinear coordinates. Although aimed at simulating Edge Localised Modes (ELMs) in tokamak x -point geometry, the code is able to simulate a wide range of fluid models (magnetised and unmagnetised) involving an arbitrary number of scalar and vector fields, in a wide range of geometries. Time evolution is fully implicit, and 3rd-order WENO schemes are implemented. Benchmarks are presented for linear and non-linear problems (the Orszag–Tang vortex) showing good agreement. Performance of the code is tested by scaling with problem size and processor number, showing efficient scaling to thousands of processors. Linear initial-value simulations of ELMs using reduced ideal MHD are presented, and the results compared to the ELITE linear MHD eigenvalue code. The resulting mode-structures and growth-rate are found to be in good agreement ( γ BOUT++ = 0.245 ω A , γ ELITE = 0.239 ω A , with Alfvenic timescale 1 / ω A = R / V A ). To our knowledge, this is the first time dissipationless, initial-value simulations of ELMs have been successfully demonstrated.

Low-to-high confinement transition simulations in divertor geometry

Recent results are presented for turbulence in tokamak boundary plasmas and its relationship to the low-to-high confinement (L–H) transition in a realistic divertor geometry. These results are obtained from a three-dimensional (3D) nonlocal electromagnetic turbulence code, which models the boundary plasma using fluid equations for plasma vorticity, density, electron and ion temperatures and parallel momenta. With sources added in the core-edge region and sinks in the scrape-off layer (SOL), the code follows the self-consistent profile evolution together with turbulence. Under DIII-D [Luxon et al., International Conference on Plasma Physics and Controlled Nuclear Fusion (International Atomic Energy Agency, Vienna, 1986), p. 159] tokamak L-mode conditions, the dominant source of turbulence is pressure-gradient-driven resistive X-point modes. These modes are electromagnetic and curvature-driven at the outside mid-plane region but become electrostatic near X-points due to magnetic shear and collisionality. Cl...

STABILITY AND DYNAMICS OF THE EDGE PEDESTAL IN THE LOW COLLISIONALITY REGIME: PHYSICS MECHANISMS FOR STEADY-STATE ELM-FREE OPERATION

Understanding the physics of the edge pedestal and edge localized modes (ELMs) is of great importance for ITER and the optimization of the tokamak concept. The peeling–ballooning model has quantitatively explained many observations, including ELM onset and pedestal constraints, in the standard H-mode regime. The ELITE code has been developed to efficiently evaluate peeling–ballooning stability for comparison with observation and predictions for future devices. We briefly review recent progress in the peeling–ballooning model, including experimental validation of ELM onset and pedestal height predictions, and nonlinear 3D simulations of ELM dynamics, which together lead to an emerging understanding of the physics of the onset and dynamics of ELMs in the standard intermediate to high collisionality regime. We also discuss new studies of the apparent power dependence of the pedestal, and studies of the impact of sheared toroidal flow. Recently, highly promising low collisionality regimes without ELMs have been discovered, including the quiescent H-mode (QH) and resonant magnetic perturbation (RMP) regimes. We present recent observations from the DIII-D tokamak of the density, shape and rotation dependence of QH discharges, and studies of the peeling–ballooning stability in this regime. We propose a model of the QH-mode in which the observed edge harmonic oscillation (EHO) is a saturated kink/peeling mode which is destabilized by current and rotation, and drives significant transport, allowing a near steady-state edge plasma. The model quantitatively predicts the observed density dependence and qualitatively predicts observed mode structure, rotation dependence and outer gap dependence. Low density RMP discharges are found to operate in a similar regime, but with the EHO replaced by an applied magnetic perturbation.

ELMs and constraints on the H-mode pedestal: peeling-ballooning stability calculation and comparison with experiment

We review and test the peeling–ballooning model for edge localized modes (ELMs) and pedestal constraints, a model based upon theoretical analysis of magnetohydrodynamic (MHD) instabilities that can limit the pedestal height and drive ELMs. A highly efficient MHD stability code, ELITE, is used to calculate quantitative stability constraints on the pedestal, including constraints on the pedestal height. Because of the impact of collisionality on the bootstrap current, these pedestal constraints are dependent on the density and temperature separately, rather than simply on the pressure. ELITE stability calculations are directly compared with experimental data for a series of plasmas in which the density is varied and ELM characteristics change. In addition, a technique is developed whereby peeling–ballooning pedestal constraints are calculated as a function of key equilibrium parameters via ELITE calculations using series of model equilibria. This technique is used to successfully compare the expected pedestal height as a function of density, triangularity and plasma current with experimental data. Furthermore, the technique can be applied for parameter ranges beyond the purview of present experiments, and we present a brief projection of peeling–ballooning pedestal constraints for burning plasma tokamak designs.

Kinetic effects in tokamak scrape-off layer plasmas

The short mean-free path expansion used in fluid modeling of scrape-off layer plasmas is often violated for typical discharge parameters, especially by the superthermal particles, which carry most of the heat flux. Thus, the tail of the distribution function can strongly depart from Maxwellian due to nonlocal mean-free path effects, which can modify plasma transport, impurity radiation, and plasma–neutral gas interactions. These nonlocal effects become particularly pronounced for detached plasma conditions that are characterized by sharp gradients in the plasma parameters along the magnetic field. These problems are being addressed by developing one spatial dimension and two velocity variables, fully kinetic, collisional, and time-dependent particle-in-cell code, W1 [Contrib. Plasma Phys. 34, 436 (1994)], and its parallel-computer version, PW1 [Contrib. Plasma Phys. 34, 424 (1996)]. Comparisons are made with the Fokker–Planck code ALLA [Phys. Plasmas 3, 1634 (1996)] and with experimental results. Kinetic ...

Nonlinear ELM simulations based on a nonideal peeling–ballooning model using the BOUT++ code

A minimum set of equations based on the peeling–ballooning (P–B) model with nonideal physics effects (diamagnetic drift, E × B drift, resistivity and anomalous electron viscosity) is found to simulate pedestal collapse when using the BOUT++ simulation code, developed in part from the original fluid edge code BOUT. Linear simulations of P–B modes find good agreement in growth rate and mode structure with ELITE calculations. The influence of the E × B drift, diamagnetic drift, resistivity, anomalous electron viscosity, ion viscosity and parallel thermal diffusivity on P–B modes is being studied; we find that (1) the diamagnetic drift and E × B drift stabilize the P–B mode in a manner consistent with theoretical expectations; (2) resistivity destabilizes the P–B mode, leading to resistive P–B mode; (3) anomalous electron and parallel ion viscosities destabilize the P–B mode, leading to a viscous P–B mode; (4) perpendicular ion viscosity and parallel thermal diffusivity stabilize the P–B mode. With addition of the anomalous electron viscosity under the assumption that the anomalous kinematic electron viscosity is comparable to the anomalous electron perpendicular thermal diffusivity, or the Prandtl number is close to unity, it is found from nonlinear simulations using a realistic high Lundquist number that the pedestal collapse is limited to the edge region and the ELM size is about 5–10% of the pedestal stored energy. This is consistent with many observations of large ELMs. The estimated island size is consistent with the size of fast pedestal pressure collapse. In the stable α-zones of ideal P–B modes, nonlinear simulations of viscous ballooning modes or current-diffusive ballooning mode (CDBM) for ITER H-mode scenarios are presented.

Dependence of the L- to H-mode Power Threshold on Toroidal Rotation and the Link to Edge Turbulence Dynamics

The injected power required to induce a transition from L-mode to H-mode plasmas is found to depend strongly on the injected neutral beam torque and consequent plasma toroidal rotation. Edge turbulence and flows, measured near the outboard midplane of the plasma (0.85 < r/a < 1.0) on DIII-D with the high-sensitivity 2D beam emission spectroscopy (BES) system, likewise vary with rotation and suggest a causative connection. The L–H power threshold in plasmas with the ion ∇B drift directed away from the X-point decreases from 4–6 MW with co-current beam injection, to 2–3 MW near zero net injected torque and to <2 MW with counter-injection in the discharges examined. Plasmas with the ion ∇B drift directed towards the X-point exhibit a qualitatively similar though less pronounced power threshold dependence on rotation. 2D edge turbulence measurements with BES show an increasing poloidal flow shear as the L–H transition is approached in all conditions. As toroidal rotation is varied from co-current to balanced in L-mode plasmas, the edge turbulence changes from a uni-modal character to a bi-modal structure, with the appearance of a low-frequency (f = 10–50 kHz) mode propagating in the electron diamagnetic direction, similar to what is observed as the ion ∇B drift is directed towards the X-point in co-rotating plasmas. At low rotation, the poloidal turbulence flow near the edge reverses prior to the L–H transition, generating a significant poloidal flow shear that exceeds the measured turbulence decorrelation rate. This increased poloidal turbulence velocity shear appears to facilitate the L–H transition. No such reversal is observed in high rotation plasmas. The high-frequency poloidal turbulence velocity spectrum exhibits a transition from a geodesic acoustic mode zonal flow to a higher-power, lower frequency zero-mean-frequency zonal flow as rotation varies from co-current to balanced during a torque scan at constant injected neutral beam power, perhaps also facilitating the L–H transition. This reduced power threshold at lower toroidal rotation may benefit inherently low-rotation plasmas such as ITER.

Simulation of edge localized modes using BOUT

The BOUT++ code is used to simulate edge localized modes (ELMs) in a shifted circle equilibrium. Reduced ideal MHD simulations are first benchmarked against the linear ideal MHD code ELITE, showing good agreement. Diamagnetic drift effects are included finding the expected suppression of high toroidal mode-number modes. Nonlinear simulations are performed, making the assumption that the anomalous kinematic electron viscosity is comparable to the anomalous electron thermal diffusivity. This allows simulations with realistically high Lundquist numbers (S = 108), finding ELM sizes of 5–10% of the pedestal stored thermal energy. Scans show a strong dependence of the ELM size on resistivity at low Lundquist numbers, with higher resistivity leading to more violent eruptions. At high Lundquist numbers relevant to high-performance discharges, ELM size is independent of resistivity as hyper-resistivity becomes the dominant dissipative effect.

Edge Gyrokinetic Theory and Continuum Simulations

The following results are presented from the development and application of TEMPEST, a fully nonlinear (full-f) five-dimensional (3d2v) gyrokinetic continuum edge-plasma code. (1) As a test of the interaction of collisions and parallel streaming, TEMPEST is compared with published analytic and numerical results for endloss of particles confined by combined electrostatic and magnetic wells. Good agreement is found over a wide range of collisionality, confining potential and mirror ratio, and the required velocity space resolution is modest. (2) In a large-aspect-ratio circular geometry, excellent agreement is found for a neoclassical equilibrium with parallel ion flow in the banana regime with zero temperature gradient and radial electric field. (3) The four-dimensional (2d2v) version of the code produces the first self-consistent simulation results of collisionless damping of geodesic acoustic modes and zonal flow (Rosenbluth–Hinton residual) with Boltzmann electrons using a full-f code. The electric field is also found to agree with the standard neoclassical expression for steep density and ion temperature gradients in the plateau regime. In divertor geometry, it is found that the endloss of particles and energy induces parallel flow stronger than the core neoclassical predictions in the SOL.