Three-dimensional analysis of tokamaks and stellarators

Abstract

The NSTAB equilibrium and stability code and the TRAN Monte Carlo transport code furnish a simple but effective numerical simulation of essential features of present tokamak and stellarator experiments. When the mesh size is comparable to the island width, an accurate radial difference scheme in conservation form captures magnetic islands successfully despite a nested surface hypothesis imposed by the mathematics. Three-dimensional asymmetries in bifurcated numerical solutions of the axially symmetric tokamak problem are relevant to the observation of unstable neoclassical tearing modes and edge localized modes in experiments. Islands in compact stellarators with quasiaxial symmetry are easier to control, so these configurations will become good candidates for magnetic fusion if difficulties with safety and stability are encountered in the International Thermonuclear Experimental Reactor (ITER) project. magnetic fusion ͉ numerical methods ͉ plasma physics M agnetic fusion reactors confine a plasma of electrons and of deuterium and tritium ions in a strong magnetic field so at high temperatures the ions can fuse to form helium and release neutrons as a practical source of energy. The geometry of such devices is usually toroidal to allow the hot particles to circulate with a mean free path exceeding by far the size of the reactor. The most thoroughly studied configurations are tokamaks, which are axially symmetric and require a destabilizing net current in the plasma to prevent charged particles from drifting out toroidally. There is an alternate concept called the stellarator, which uses fully three-dimensional (3D) coils to create a poloidal field eliminating the toroidal drift. Our interest lies in magnetohydrodynamic (MHD) partial differential equations governing equilibrium and stability of the plasma, and in guiding center orbits of the ions and the electrons that lead to numerical simulations of transport. The method we apply to this problem in computational science is to run the NSTAB equilibrium and stability code and the TRAN Monte Carlo code written by Octavio Betancourt and Mark Taylor (1-4). We calculate physically realistic predictions of limits on the average ratio ␤ ϭ 2p/B 2 of the fluid and magnetic pressures and reliable estimates of the energy confinement time, which is a measure of pressure times volume divided by power in magnetic fusion experiments. Computational Science of Magnetic FusionUsing the Maxwell stress tensor, we write the MHD equations for equilibrium and stability of a plasma in the conservation formwhere B is the magnetic field and p is the fluid pressure. To obtain an accurate numerical approximation of force balance across discontinuities like islands or current sheets, analogous difference equations are represented in the NSTAB code by a similar scheme that telescopes down through addition over grid points to an integral statement of equilibrium related to the divergence theorem. Elementary examples from shock capturing show that ignoring conservation form results in failure ...

Optimized quasisymmetric stellarators are consistent with the Garren–Boozer construction

“…Figure 7. Shapes of the magnetic surfaces at ρ = √ s = 0.1 and 0.2 for QAS2 [13,14] (dark dashed curves) at 8 toroidal locations, and the same surfaces of the corresponding direct-construction quasisymmetric configuration (solid curves). This phenomenon can be understood as follows.…”

Pseudo‐Spectral Methods for the Laplace‐Beltrami Equation and the Hodge Decomposition on Surfaces of Genus One

“…We illustrate the performance of our methods using two (nonaxisymmetric) surfaces. For these, Garabedian coordinates provide a convenient parametrization for stellarator‐like geometries—that is surfaces of genus with a central curve, referred to as the magnetic axis, embedded in the interior. This involves three parameters: the poloidal and toroidal angles, as well as a radius‐like parameter .…”

“…Here, we restrict our attention to smooth, closed surfaces of genus one embedded in , primarily because of their importance in plasma physics and stellarator design . More precisely, we assume that is a smooth toroidal surface: where is a doubly periodic function of .…”

Design of the DEMO Fusion Reactor Following ITER

“…Fully 3D equilibria are calculated by first imposing and later releasing a suitable constraint in runs of the NSTAB code chosen to find bifurcated solutions that cannot be obtained without permitting discontinuous alterations in the topology of the magnetic surfaces [ 6 ]. On relatively crude radial grids the computations capture small islands whose widths are of the same order of magnitude as the mesh size, but are big enough to account for a significant change in topology.…”