C. D. Beidler

Physics and engineering design for Wendelstein VII-X

AbstractThe future experiment Wendelstein VII-X (W VII-X) is being developed at the Max-Planck-Institut fur Plasmaphysik. A Helical Advanced Stellarator (Helias) configuration has been chosen because of its confinement and stability properties. The goals of W VII-X are to continue the development of the modular stellarator, to demonstrate the reactor capability of this stellarator line, and to achieve quasi-steady-state operation in a temperature regime >5 keV. This temperature regime can be reached in W VII-X if neoclassical transport plus the anomalous transport found in W VII-A prevail. A heating power of 20 MW will be applied to reach the reactor-relevant parameter regime.The magnetic field in W VII-X has five field periods. Other basic data are as follows: major radius R0 = 6.5 m, magnetic induction B0 = 3 T, stored magnetic energy W ≈ 0.88 GJ, and average plasma radius a = 0.65 m. Superconducting coils are favored because of their steady-state field, but pulsed water-cooled copper coils are also bei...

Characterization of energy confinement in net-current free plasmas using the extended International Stellarator Database

International collaboration on development of a stellarator confinement database has progressed. More than 3000 data points from nine major stellarator experiments have been compiled. Robust dependences of the energy confinement time on the density and the heating power have been confirmed. Dependences on other operational parameters, i.e. the major and minor radii, magnetic field and the rotational transform , have been evaluated using inter-machine analyses. In order to express the energy confinement in a unified scaling law, systematic differences in each subgroup are quantified. An a posteriori approach using a confinement enhancement factor on ISS95 as a renormalizing configuration-dependent parameter yields a new scaling expression ISS04; . Gyro–Bohm characteristic similar to ISS95 has been confirmed for the extended database with a wider range of plasma parameters and magnetic configurations than in the study of ISS95. It has also been discovered that there is a systematic offset of energy confinement between magnetic configurations, and its measure correlates with the effective helical ripple of the external stellarator field. Full documentation of the International Stellarator Confinement Database is available at http://iscdb.nifs.ac.jp/ and http://www.ipp.mpg.de/ISS.

Major results from the stellarator Wendelstein 7-AS (Review Article)

Wendelstein 7-AS was the first modular stellarator device to test some basic elements of stellarator optimization: a reduced Shafranov shift and improved stability properties resulted in β-values up to 3.4% (at 0.9 T). This operational limit was determined by power balance and impurity radiation without noticeable degradation of stability or a violent collapse. The partial reduction of neoclassical transport could be verified in agreement with calculations indicating the feasibility of the concept of drift optimization. A full neoclassical optimization, in particular a minimization of the bootstrap current was beyond the scope of this project. A variety of non-ohmic heating and current drive scenarios by ICRH, NBI and in particular, ECRH were tested and compared successfully with their theoretical predictions. Besides, new heating schemes of overdense plasmas were developed such as RF mode conversion heating—Ordinary mode, Extraordinary mode, Bernstein-wave (OXB) heating—or 2nd harmonic O-mode (O2) heating. The energy confinement was about a factor of 2 above ISS95 without degradation near operational boundaries. A number of improved confinement regimes such as core electron-root confinement with central Te ≤ 7 keV and regimes with strongly sheared radial electric field at the plasma edge resulting in Ti ≤ 1.7 keV were obtained. As the first non-tokamak device, W7-AS achieved the H-mode and moreover developed a high density H-mode regime (HDH) with strongly reduced impurity confinement that allowed quasi-steady-state operation (τ ≈ 65 · τE) at densities (at 2.5 T). The first island divertor was tested successfully and operated with stable partial detachment in agreement with numerical simulations. With these results W7-AS laid the physics background for operation of an optimized low-shear steady-state stellarator.

Benchmarking of the mono-energetic transport coefficients—results from the International Collaboration on Neoclassical Transport in Stellarators (ICNTS)

Numerical results for the three mono-energetic transport coefficients required for a complete neoclassical description of stellarator plasmas have been benchmarked within an international collaboration. These transport coefficients are flux-surface-averaged moments of solutions to the linearized drift kinetic equation which have been determined using field-line-integration techniques, Monte Carlo simulations, a variational method employing Fourier–Legendre test functions and a finite-difference scheme. The benchmarking has been successfully carried out for past, present and future devices which represent different optimization strategies within the extensive configuration space available to stellarators. A qualitative comparison of the results with theoretical expectations for simple model fields is provided. The behaviour of the results for the mono-energetic radial and parallel transport coefficients can be largely understood from such theoretical considerations but the mono-energetic bootstrap current coefficient exhibits characteristics which have not been predicted.

A General Solution of the Ripple-Averaged Kinetic Equation (GSRAKE)

A general solution of the ripple-averaged kinetic equation, GSRAKE, is presented and used to investigate neoclassical transport in the model magnetic field of a simple stellarator. No assumptions are made as to the relative sizes of the collision frequency, nu , and poloidal precessional frequency, as, so that the solution is valid throughout the entire long-mean-free-path regime. Separate but fully self-consistent treatments of both localized and non-localized particles are provided; the interaction between these two classes of particles is accounted for through a set of appropriate physical boundary conditions. All drift terms present within the framework of the ripple-averaged theory are included; in particular, for localized particles Omega theta = Omega E+ Omega Del B is comprised of both the E*B and Del B precessional frequencies. The solution is thus equally valid in the Omega E>> Omega Del B and the Omega E=0 limits of standard neoclassical theory. A detailed comparison of results with those of the FLOCS code is undertaken; estimates of neoclassical transport coefficients obtained from several codes are also presented. Agreement of results is found in all of these comparisons, GSRAKE requiring but a tiny fraction of the computational time necessary for the other codes.

The Helias reactor HSR4/18

The Helias reactor is an upgraded version of the Wendelstein 7-X experiment. A straightforward extrapolation of Wendelstein 7-X leads to HSR5/22, which has 5 field periods and a major radius of 22 m. HSR4/18 is a more compact Helias reactor with 4 field periods and an 18 m major radius. Stability limit and energy confinement times are nearly the same as in HSR5/22, thus the same fusion power (3000 MW) is expected in both configurations. Neoclassical transport in HSR4/18 is very low, and the effective helical ripple is below 1%. The article describes the power balance of the Helias reactor, and the blanket and maintenance concepts. The coil system of HSR4/18 comprises 40 modular coils with NbTi superconducting cables. The reduction from 5 to 4 field periods and the concomitant reduction in size will also reduce the cost of the Helias reactor.

Stellarator and tokamak plasmas: a comparison

An overview is given of physics differences between stellarators and tokamaks, including magnetohydrodynamic equilibrium, stability, fast-ion physics, plasma rotation, neoclassical and turbulent transport and edge physics. Regarding microinstabilities, it is shown that the ordinary, collisionless trapped-electron mode is stable in large parts of parameter space in stellarators that have been designed so that the parallel adiabatic invariant decreases with radius. Also, the first global, electromagnetic, gyrokinetic stability calculations performed for Wendelstein 7-X suggest that kinetic ballooning modes are more stable than in a typical tokamak.

Neoclassical transport simulations for stellarators

The benchmarking of the thermal neoclassical transport coefficients is described using examples of the Large Helical Device (LHD) and TJ-II stellarators. The thermal coefficients are evaluated by energy convolution of the monoenergetic coefficients obtained by direct interpolation or neural network techniques from the databases precalculated by different codes. The temperature profiles are calculated by a predictive transport code from the energy balance equations with the ambipolar radial electric field estimated from a diffusion equation to guarantee a unique and smooth solution, although several solutions of the ambipolarity condition may exist when root-finding is invoked; the density profiles are fixed. The thermal transport coefficients as well as the ambipolar radial electric field are compared and very reasonable agreement is found for both configurations. Together with an additional W7-X case, these configurations represent very different degrees of neoclassical confinement at low collisionalities. The impact of the neoclassical optimization on the energy confinement time is evaluated and the confinement times for different devices predicted by transport modeling are compared with the standard scaling for stellarators. Finally, all configurations are scaled to the same volume for a direct comparison of the volume-averaged pressure and the neoclassical degree of optimization.

Momentum correction techniques for neoclassical transport in stellarators

In the traditional neoclassical ordering for stellarators, monoenergetic transport coefficients are evaluated using the simplified Lorentz form of the pitch-angle collision operator which violates momentum conservation. In this paper, the parallel momentum balance with radial parallel momentum transport and viscosity terms is analyzed, in particular, with respect to the radial electric field. Next, the impact of momentum conservation in the stellarator long-mean-free-path regime is estimated for the radial transport and the parallel electric conductivity. Two different momentum correction techniques are described based on monoenergetic transport coefficients calculated by the DKES code [W. I. van Rij and S. P. Hirshman, Phys. Fluids B 1, 563 (1989)]. The benchmarking of the parallel electric conductivity and of the bootstrap current is presented for a tokamak as well as for two W7-X stellarator configurations [G. Grieger et al., Phys. Fluids B 4, 2081 (1992)]. Finally, the impact of the momentum correctio...

Ripple transport in helical-axis advanced stellarators: a comparison with classical stellarator/torsatrons

Calculations of the neoclassical transport rates due to particles trapped in the helical ripples of a stellarators magnetic field are carried out, based on solutions of the bounce-averaged kinetic equation. These calculations employ a model for the magnetic field strength, B, which is an accurate approximation to the actual B for a wide variety of stellarator-type devices, among which are Helical-Axis Advanced Stellarators (Helias) as well as conventional stellarators and torsatrons. Comparisons are carried out in which it is shown that the Helias concept leads to significant reductions in neoclassical transport rates throughout the entire long-mean-free-path regime, with the reduction being particularly dramatic in the nu -1 regime. These findings are confirmed by numerical simulations. Further, it is shown that the behaviour of deeply trapped particles in Helias can be fundamentally different from that in classical stellarator/torsatrons; as a consequence, the beneficial effects of a radial electric field on the transport make themselves felt at a lower collision frequency than is usual.