A. Reiman

Calculation of three-dimensional MHD equilibria with islands and stochastic regions

Abstract A three-dimensional MHD equilibrium code is described that does not assume the existence of good flux surfaces. Given an initial guess for the magnetic field, the code proceeds by calculating the pressure driven current and then by updating the field using Amperes law. The numerical algorithm to solve the magnetic differential equation for the pressure driven current is described, and demonstrated for model fields having islands and stochastic regions. The numerical algorithm which solves Amperes law in three dimensions is also described. Finally, the convergence of the code is illustrated for a particular stellarator equilibrium with no large islands.

THE PROSPECTS FOR MAGNETOHYDRODYNAMIC STABILITY IN ADVANCED TOKAMAK REGIMES

Stability analysis of advanced regime tokamaks is presented. Here advanced regimes are defined to include configurations where the ratio of the bootstrap current, IBS, to the total plasma current, Ip, approaches unity, and the normalized stored energy, βN* = 80π〈p2〉1/2a/IpB0, has a value greater than 4.5. Here, p is the plasma pressure, a the minor radius in meters, Ip is in mega‐amps, B0 is the magnetic field in Tesla, and 〈⋅〉 represents a volume average. Specific scenarios are discussed in the context of Toroidal Physics Experiment (TPX) [Proceedings of the 20th European Physical Society Conference on Controlled Fusion and Plasma Physics, Lisbon, 1993, edited by J. A. Costa Cabral, M. E. Manso, F. M. Serra, and F. C. Schuller (European Physical Society, Petit‐Lancy, 1993), p. I‐80]. The best scenario is one with reversed shear, in the q profile, in the central region of the tokamak. The bootstrap current obtained from the plasma profiles provides 90% of the required current, and is well aligned with the...

Island formation and destruction of flux surfaces in three-dimensional MHD equilibria

The physics involved in the appearance of islands caused by resonant pressure driven currents in three‐dimensional magnetohydrodynamic equilibria is described. Estimates of island widths are obtained by an expansion in β, with an auxiliary expansion about the magnetic axis. The theory is applied to Princeton’s heliac reference design.

Comparisons of linear and nonlinear plasma response models for non-axisymmetric perturbationsa)

With the installation of non-axisymmetric coil systems on major tokamaks for the purpose of studying the prospects of ELM-free operation, understanding the plasma response to the applied fields is a crucial issue. Application of different response models, using standard tools, to DIII-D discharges with applied non-axisymmetric fields from internal coils, is shown to yield qualitatively different results. The plasma response can be treated as an initial value problem, following the system dynamically from an initial unperturbed state, or from a nearby perturbed equilibrium approach, and using both linear and nonlinear models [A. D. Turnbull, Nucl. Fusion 52, 054016 (2012)]. Criteria are discussed under which each of the approaches can yield a valid response. In the DIII-D cases studied, these criteria show a breakdown in the linear theory despite the small 10−3 relative magnitude of the applied magnetic field perturbations in this case. For nonlinear dynamical evolution simulations to reach a saturated non...

Physics issues in the design of high-beta, low-aspect-ratio stellarator experiments

High-beta, low-aspect-ratio ~‘‘compact’’ ! stellarators are promising solutions to the problem of developing a magnetic plasma configuration for magnetic fusion power plants that can be sustained in steady state without disrupting. These concepts combine features of stellarators and advanced tokamaks and have aspect ratios similar to those of tokamaks ~2‐4!. They are based on computed plasma configurations that are shaped in three dimensions to provide desired stability and transport properties. Experiments are planned as part of a program to develop this concept. A b54% quasi-axisymmetric plasma configuration has been evaluated for the National Compact Stellarator Experiment ~NCSX!. It has a substantial bootstrap current and is shaped to stabilize ballooning, external kink, vertical, and neoclassical tearing modes without feedback or close-fitting conductors. Quasi-omnigeneous plasma configurations stable to ballooning modes at b54% have been evaluated for the Quasi-Omnigeneous Stellarator ~QOS! experiment. These equilibria have relatively low bootstrap currents and are insensitive to changes in beta. Coil configurations have been calculated that reconstruct these plasma configurations, preserving their important physics properties. Theory- and experiment-based confinement analyses are used to evaluate the technical capabilities needed to reach target plasma conditions. The physics basis for these complementary experiments is described.

PIES free boundary stellarator equilibria with improved initial conditions

The MFBE procedure developed by Strumberger (1997 Nucl. Fusion 37 19) is used to provide an improved starting point for free boundary equilibrium computations in the case of W7-X (Nuhrenberg and Zille 1986 Phys. Lett. A 114 129) using the Princeton iterative equilibrium solver (PIES) code (Reiman and Greenside 1986 Comput. Phys. Commun. 43 157). Transferring the consistent field found by the variational moments equilibrium code (VMEC) (Hirshmann and Whitson 1983 Phys. Fluids 26 3553) to an extended coordinate system using the VMORPH code, a safe margin between plasma boundary and PIES domain is established. The new EXTENDER_P code implements a generalization of the virtual casing principle, which allows field extension both for VMEC and PIES equilibria. This facilitates analysis of the 5/5 islands of the W7-X standard case without including them in the original PIES computation.

Recent advances in the design of quasiaxisymmetric stellarator plasma configurations

Strategies for the improvement of quasiaxisymmetric stellarator configurations are explored. Calculations of equilibrium flux surfaces for candidate configurations are also presented. One optimization strategy is found to generate configurations with improved neoclassical confinement, simpler coils with lower current density, and improved flux surface quality relative to previous designs. The flux surface calculations find significant differences in the extent of islands and stochastic regions between candidate configurations. (These calculations do not incorporate the predicted beneficial effects of perturbed bootstrap currents.) A method is demonstrated for removing low order islands from candidate configurations by relatively small modifications of the configuration. One configuration is identified as having particularly desirable properties for a proposed experiment.

Experiments close to the beta-limit in W7-AS

A major objective of the experimental program in the last phase of the W7-AS stellarator was to explore and demonstrate the high-β performance of advanced stellarators. MHD-quiescent discharges at low impurity radiation levels with volume averaged β-values of up to β = 3.4% have been achieved. A very important prerequisite was the attainment of the high density H-Mode (HDH) regime. This was made possible by the installation of extensive graphite plasma facing components designed for island divertor operation. The co-directed neutral beam injection provided increased absorbed heating power of up to 3.2 MW in high-β plasmas with B ≤ 1.25 T. The anticipated improved features concerning equilibrium and stability at high plasma β could be verified experimentally by the comparison of x-ray data with free boundary equilibrium calculations. The maximum β found in configurations with a rotational transform around is determined by the available heating power. No evidence of a stability limit has been found in the accessible configuration space, and the discharges are remarkably quiescent at maximum β, most likely due the increase of the magnetic well depth. An increase in low m/n MHD activity is typically observed during the transition towards high β. The beneficial stability properties of net-current-free configurations could be demonstrated by comparison with configurations where a significant inductive current drive was involved. Current driven instabilities such as tearing modes and soft disruptions can prevent access to β-values as high as in the currentless case. The experimental results indicate that optimized stellarators such as W7-X can be considered as a viable option for an attractive stellarator fusion reactor.

Effect of nonaxisymmetric perturbations on the structure of a tokamak poloidal divertor

The effects of n=1 nonaxisymmetric perturbations on a tokamak poloidal divertor are described. Despite the existence of a region of chaotic field line trajectories outside the last closed flux surface, the footprint of the trajectories on the divertor plates is found to be largely coherent, forming a spiral structure. At a fixed toroidal angle, the footprint exhibits a bifurcation similar to that seen experimentally on DIII‐D [Nucl. Fusion 28, 902 (1988)]. For field errors of the magnitude that exist in present‐day tokamaks, the width of the nonaxisymmetric structure in the divertor footprint is comparable to the width of the scrape‐off layer.

Heliac parameter study

Helical axis stellarators (heliacs) with zero net current are found to possess very good stability properties. Helically symmetric or ‘‘straight’’ heliacs with bean‐shaped cross sections have a first region of stability that reaches to 〈 β〉 of 30% or more. Those with circular cross sections have a second region of stability to Mercier modes. In addition, the stability properties of these plasma configurations as functions of pressure profile, helical aspect ratio, and helical period length are also reported.