Electrostatic potential variations on stellarator magnetic surfaces in low collisionality regimes

The component of the neoclassical electrostatic potential that is non-constant on the magnetic surface, that we denote by $\tilde\varphi$, can affect radial transport of highly charged impurities, and this has motivated its inclusion in some modern neoclassical codes. The number of neoclassical simulations in which $\tilde\varphi$ is calculated is still scarce, partly because they are usually demanding in terms of computational resources, especially at low collisionality. In this paper the size, the scaling with collisionality and with aspect ratio, and the structure of $\tilde\varphi$ on the magnetic surface are analytically derived in the $1/\nu$, $\sqrt{\nu}$ and superbanana-plateau regimes of stellarators close to omnigeneity; i. e. stellarators that have been optimized for neoclassical transport. It is found that the largest $\tilde\varphi$ that the neoclassical equations admit scales linearly with the inverse aspect ratio and with the size of the deviation from omnigeneity. Using a model for a perturbed omnigeneous configuration, the analytical results are verified and illustrated with calculations by the code KNOSOS. The techniques, results and numerical tools employed in this paper can be applied to neoclassical transport problems in tokamaks with broken axisymmetry.Comment: 30 pages, 12 figures, 1 table. Published versio

Stellarator impurity flux driven by electric fields tangent to magnetic surfaces

The control of impurity accumulation is one of the main challenges for future stellarator fusion reactors. The standard argument to explain this accumulation relies on the, in principle, large inward pinch in the neoclassical impurity flux caused by the typically negative radial electric field in stellarators. This simplified interpretation was proven to be flawed by Helander et al. [Phys. Rev. Lett. 118, 155002 (2017)], who showed that in a relevant regime (low-collisionality main ions and collisional impurities) the radial electric field does not drive impurity transport. In that reference, the effect of the component of the electric field that is tangent to the magnetic surface was not included. In this Letter, an analytical calculation of the neoclassical radial impurity flux incorporating such effect is given, showing that it can be very strong for highly charged impurities and that, once it is taken into account, the dependence of the impurity flux on the radial electric field reappears. Realistic examples are provided in which the inclusion of the tangential electric field leads to impurity expulsion.Comment: 10 pages, 1 figure. Published versio

Nonlinear gyrokinetic PIC simulations in stellarators with the code EUTERPE

In this work, the first nonlinear particle-in-cell simulations carried out in a stellarator with the global gyrokinetic code EUTERPE using realistic plasma parameters are reported. Several studies are conducted with the aim of enabling reliable nonlinear simulations in stellarators with this code. First, EUTERPE is benchmarked against ORB5 in both linear and nonlinear settings in a tokamak configuration. Next, the use of noise control and stabilization tools, a Krook-type collision operator, markers weight smoothing and heating sources is investigated. It is studied in detail how these tools influence the linear growth rate of instabilities in both tokamak and stellarator geometries and their influence on the linear zonal flow evolution in a stellarator. Then, it is studied how these tools allow improving the quality of the results in a set of nonlinear simulations of electrostatic turbulence in a stellarator configuration. Finally, these tools are applied to a W7-X magnetic configuration using experimental plasma parameters.Comment: 24 pages, 19 figure

Turbulent transport of impurities in 3D devices

A large diffusive turbulent contribution to the radial impurity transport in Wendelstein 7-X (W7-X) plasmas has been experimentally inferred during the first campaigns and numerically confirmed by means of gyrokinetic simulations with the code stella. In general, the absence of strong impurity accumulation during the initial W7-X campaigns is attributed to this diffusive term. In the present work the diffusive contribution is also calculated in other stellarator plasmas. In particular, the diffusion (D) and convection (V) coefficients of carbon and iron impurities produced by ion-temperature-gradient (ITG) turbulence are obtained for W7-X, LHD, TJ-II and NCSX. The results show that, although the size of D and V can differ across the four devices, inward convection is found for all of them. For W7-X, TJ-II and NCSX the two coefficients are comparable and the turbulent peaking factor is surprisingly similar. In LHD, appreciably weaker diffusive and convective impurity transport and significantly larger turbulent peaking factor are predicted. All this suggests that ITG turbulence, although not strongly, would lead to negative impurity density gradients in stellarators. Then, considering mixed ITG/Trapped Electron Mode (TEM) turbulence for the specific case of W7-X, it has been quantitatively assessed to what degree pellet fueled reduced turbulence scenarios feature reduced turbulent transport of impurities as well. The results for trace iron impurities show that, although their turbulent transport is not entirely suppressed, a significant reduction of V and a stronger decrease of D are found. Although the diffusion is still above neoclassical levels, the neoclassical convection would gain under such conditions a greater specific weight on the dynamics of impurities in comparison with standard ECRH scenarios.Comment: 16 pages, 8 figure

On neoclassical impurity transport in stellarator geometry

The impurity dynamics in stellarators has become an issue of moderate concern due to the inherent tendency of the impurities to accumulate in the core when the neoclassical ambipolar radial electric field points radially inwards (ion root regime). This accumulation can lead to collapse of the plasma due to radiative losses, and thus limit high performance plasma discharges in non-axisymmetric devices.\\ A quantitative description of the neoclassical impurity transport is complicated by the breakdown of the assumption of small $\mathbf{E}\times \mathbf{B}$ drift and trapping due to the electrostatic potential variation on a flux surface $\tilde{\Phi}$ compared to those due to the magnetic field gradient. The present work examines the impact of this potential variation on neoclassical impurity transport in the Large Helical Device (LHD) stellarator. It shows that the neoclassical impurity transport can be strongly affected by $\tilde{\Phi}$. The central numerical tool used is the $\delta f$ particle in cell (PIC) Monte Carlo code EUTERPE. The $\tilde{\Phi}$ used in the calculations is provided by the neoclassical code GSRAKE. The possibility of obtaining a more general $\tilde{\Phi}$ self-consistently with EUTERPE is also addressed and a preliminary calculation is presented.Comment: 11 pages, 15 figures, presented at Joint Varenna-Lausanne International Workshop on Theory of Fusion Plasmas, 2012. Accepted for publication to Plasma Phys. and Control. Fusio