Session BO08: MFE: Stellarators

Expected performance of the upgraded HSX stellarator experiment

The Helically Symmetric eXperiment (HSX) at UW Madison, Wisconsin is the world's first neoclassically optimized stellarator. It started operation in 2001 and has since then significantly contributed to the understanding of neoclassical and turbulent transport in 3D magnetic field geometries. To further extend the operational space of HSX, the device is currently undergoing a major upgrade, consisting of a new 70 GHz electron cyclotron resonance heating system and an increase of the magnetic field strength to 1.25T. This upgrade will allow plasma operation at three times higher densities, which will significantly increase the ion temperature thanks to better coupling between electrons and ions and reduced charge exchange losses. Hence, studies of ion-temperature gradient modes, as well as the investigation of neoclassical transport of ions in the long mean free path regime, might become possible. In addition, stronger flows in the helical direction thanks to reduced neutral damping and smaller density gradient lengths are expected. This will likely reduce the growth rates and non-linear heat fluxes of collisionless trapped electron modes, which are believed to be the dominant type of instability in HSX plasmas limiting plasma performance.   

Effect of the Parallel Flow on the Ion Resonance in HSX

Experimental data to date have shown no evidence of the ion resonance in poloidal viscosity with respect to the radial electric field in HSX.  To explain this, we use a coupled set of momentum balance equations and Shaing's [1] model of nonlinear viscosity to examine the influence of the parallel flow on the ion resonance in HSX. Due to quasihelical symmetry (QHS), the damping of the parallel flow from viscosity is very low. The parallel flow can become large compared to the ion thermal velocity. Including the neutral damping, the calculations show that the resonant electric field (E_res) can be a factor of 2.5 higher compared to the solution assuming that Vpar = 0.  By increasing the neutral damping in these calculations by a factor of 5, E_res decreased by a factor of 2. Calculating the E_res in a Mirror configuration, where the parallel viscous damping is increased, resulted in a factor of 2 decrease from the QHS solution with standard neutral profiles.  A charge exchange recombination spectroscopy (CHERS) system has been used in HSX to determine E_r and the mean ion parallel flow from measurements of the inboard/outboard asymmetry of the C+6 parallel flow [2]. An experimental program using the CHERS system and physical probes is detailed that will enable comparisons to the model.   

Regulating Trapped Electron Mode Turbulence by Modifying Coil Currents in the HSX Stellarator

In the Helically Symmetric eXperiment (HSX), heat transport outside the mid-radius is believed to be dominated by Trapped Electron Mode (TEM) turbulence. The growth rates of TEMs depend strongly on the magnetic geometry, thus making them susceptible to adjustments in the main and auxiliary coil currents.  Using the VMEC equilibrium code, we have generated a database of >10⁶ coil-current configurations. Using a hierarchy of models, we aim to predict which configurations will exhibit reduced heat transport. Configurations are first selected for their helical symmetry, low epsilon effective profiles, satisfaction of the Mercier criterion, and avoidance of magnetic resonances. Then, configurations are selected using three flux surface averaged field geometry metrics, including magnetic shear, F_κ and G_ss, where F_κ describes the overlap of magnetic wells with regions bad curvature, and G_ss describes the overlap of the flux surface expansion with regions of bad curvature.  Finally, we use gyrokinetic simulations to calculate TEM growth rates and nonlinear heat fluxes. These results indicate significant flexibility in HSX for modifying both TEM growth rates and the nonlinear heat flux. Work is ongoing to identify 2-3 configurations whose heat flux will be measured experimentally.     

Impurity Transport Studies at the HSX Stellarator Using Active and Passive CVI Spectroscopy

The transport of intrinsic carbon impurities has been studied in the helically symmetric stellarator experiment (HSX) using active and passive charge exchange recombination spectroscopy (CHERS). For the analysis of the CHERS signals, the STRAHL impurity transport code has been re-written in the python programming language and optimized for the application in stellarators. This optimization is done via additional considerations of 3D effects on impurity fueling and diagnostic line of sight emission. In addition, neutral densities both along the NBI line of sight as well as for the background plasma have been calculated using the FIDASIM code. By using the basinhopping algorithm to minimize the difference between experimental and predicted active and passive signals, significant levels of anomalous impurity diffusion are observed. Comparisons with necoclassical calculations from DKES/PENTA show that the inferred levels exceed the neoclassical transport by about a factor of four in the core and more than 100 times towards the plasma edge. This conclusion is compatible with heat transport calculations and measurements as well as recent experiments measuring the impurity decay times of laser ablation injected Aluminum.   

Synthetic Phase Contrast Imaging at the Wendelstein 7-X Stellarator

As the largest neoclassically optimized stellarator, Wendelstein 7-X (W7-X) plays a crucial role in stellarator turbulence studies. The main core turbulence diagnostic at W7-X is the phase contrast imaging (PCI) system, which measures the line-integrated fluctuating electron density along a central line-of-sight. The line-integrated nature of the PCI measurements complicates the interpretation of experimental data. It is thus advantageous to utilize synthetic PCI diagnostics, capable of computing the expected PCI signal based on model electron density fluctuations. This contribution describes results obtained with such a synthetic diagnostic at W7-X, including input from the first nonlinear, global gyrokinetic simulations of stellarators with kinetic electrons.   

Real-time wall conditioning experiments utilizing the impurity powder dropper on LHD

Controlled injections of B and BN from the PPPL designed Impurity Powder Dropper into LHD have demonstrated positive effects on the wall conditions in both an intra and inter-shot basis.  Reactions to particulate injection depend upon discharge length which governs the effective time for plasma and vessel response to the added material.  In 4s long plasmas, a reduction in both wall recycling and native impurity content is noted over the course of several discharges.  For plasmas greater than 10s in length, improvements are observed in real time as a consequence of extended injection times. For multi-minute plasmas, repeated sets of 3 x 10s boron injection pulses every 3 minutes resulted in prompt reduction of plasma density presumably through ion co-deposition with the main ion species at the first wall surfaces and gettering by the newly applied boron layer.   This ability to demonstrate active control in steady state scenarios is an important initial step in the development of this technique as a real time boronization alternative.  Powder assimilation efficiency is also studied with modelling from coupled EMC3-EIRENE and DUSTT simulations supporting observed results and further indicating scenarios which could lead to promising long pulse real time boronization applications.   

MUSE: A Simple Optimized Stellarator Using Permanent Magnets

A simple, inexpensive quasi-axisymmetric optimized stellarator ‘MUSE’ is being constructed using rare-earth permanent magnets to generate the 3D magnetic field structure combined with simple planar coils to generate the toroidal magnetic flux. Advanced manufacturing is being used to precisely fabricate the magnet support structures, simplifying metrology and the assembly process. High precision magnets are readily available commercially.  The stellarator equilibrium is highly optimized for good neoclassical confinement with $\epsilon_{eff}^{3/2}$, the coefficient characterizing 3D neoclassical transport calculated to be significantly lower than in any previous stellarator. The experiment is designed to be table-top scale with $R = 0.30$ m and $B = 0.15$ T, suitable for basic physics experiments. The design approach, construction status, and plans will be discussed.   

Development Of Advanced Stellarator With Standardized Permanent Magnet Blocks

Recent study indicates the complicated 3D coils of stellarators can be dramatically simplified by introducing permanent magnets. However, the existing designs use permanent magnets with various shapes, sizes and even arbitrary magnetization orientations, thus their fabrication and assembly may be even more difficult and costly than the 3D coils. For designing standardized permanent magnets, we have performed a series of research. The Fourier decomposition method is introduced to design perpendicular magnets [1]. The “two-step” magnet design strategy [2, 3] is proposed based on the idea of “divide and conquer strategy”, which decomposes the design of large number of magnet blocks into independent designs of each magnet, thus the standardized magnet blocks can be easily customized. This strategy can give almost the same design as the Fourier decomposition method, and most importantly, it is successfully applied to design ESTELL stellarator with identical cubic magnet blocks and a minitype stellarator with identical rectangular magnet pieces. These magnet designs will substantially reduce the difficulty and cost of magnet fabrication and assembly and potentially lower the engineering barrier for stellarator construction.   

Three-Dimensional MHD Equilibria

The numerical calculation of three-dimensional MHD equilibria is an essential step in designing and analyzing toroidal magnetic confinement devices. The type of equilibrium is determined by the assumptions used in designing the code that calculates the equilibrium. The VMEC code is an Ideal MHD code that assumes nested toroidal flux surfaces [S. P. Hirshman and H. K. Meier, Phys. Fluids 28, 1387 (1985)] and it has been widely used in stellarator and tokamak physics. The SIESTA code relaxes the assumption of nested toroidal flux surfaces and switches between Ideal and Resistive MHD in the process of searching for an equilibrium [S. P. Hirshman, R. Sanchez and C.R. Cook, Phys. Plasmas 18, 062504 (2011)]. The SPEC code is designed with the concept of multi-region, relaxed MHD [S. R. Hudson, et al., Phys. Plasmas 19, 112502 (2012)]. In this work, these three codes are used to calculate tokamak and stellarator equilibria, with a comparison of the differences and similarities of the results. The challenges and limitations of the codes will be discussed.   

Stellarator Optimization with DESC

Stellarators must be optimized for both physics and engineering objectives, including good particle confinement and simple construction.  The design space for these non-axisymmetric external magnetic fields is very large, which makes this high-dimensional optimization problem computationally challenging.  Existing stellarator optimization codes wrap around other equilibrium solvers and rely on either gradient-free algorithms, finite differences, or adjoint methods to search the phase space.  Each of these traditional methods has its disadvantages: slow convergence, expensive and inaccurate computations, or labor-intensive coding, respectively.  The DESC equilibrium solver utilizes automatic differentiation, and it has been extended into a stellarator optimization code that takes advantage of this capability to quickly and exactly differentiate arbitrary objectives to any order.  This presentation showcases the ability to efficiently explore the phase space of stellarator equilibria through the example of fixed-boundary optimization for quasi-symmetry.  An overview of the optimization approach is presented along with example results and benchmarks against other codes.  It is also demonstrated how the code can easily be extended to accommodate other optimization objectives.     

Single-stage stellarator coil optimisation for quasi-symmetry under uncertainty

When designing a new stellarator, it is crucial to design coils that properly contain the plasma. Once designed, building these coils is extremely expensive and tight manufacturing tolerances only increase the cost further. The classical way of designing coils is a two-stage approach in which first a magnetic configuration with desirable physics properties is found, and then coils that aim at realizing this magnetic configuration are designed. Recently, a single-stage approach for vacuum fields was introduced that directly optimises coils for their physics properties. We combine this approach with a model for coil manufacturing errors to find coil designs that are robust with respect to small perturbations of the coils.

As opposed to existing approaches, the new method

1) is gradient based, enabling the use of faster converging optimisation algorithms;

2) separates the discretisation of the coils and the model for the perturbations; and

3) directly targets quasi symmetry of the magnetic field.

We show that the resulting stochastic optimisation problem is better conditioned than the original deterministic problem, and that the coil configurations obtained from the stochastic problem perform better as measured in terms of quasi symmetry and particle confinement.   

Single-stage gradient-based stellarator coil design: optimization for quasi-symmetry on surfaces

We will present a single-stage approach to optimize current-carrying coils to produce quasisymmetric vacuum magnetic fields on surfaces.  First, we describe a new technique to compute magnetic surfaces in Boozer coordinates.  Then, we optimize for quasisymmetry on these surfaces.  Use gradient-based optimization methods, we can directly optimize the coils to favor quasisymmetry on surfaces.  We will present numerical results that demonstrate the performance of this approach.   

Applications of soliton theory to the study of 3D magnetic fields with nested flux surfaces

The theory of solitons has been successfully applied in the past to understand various nonlinear plasma behavior. For example, the steady-state MHD equilibrium with the isodynamic constraint is described by the nonlinear Schrodinger equation (NLS) [1]. Soliton theory can shed light on how singular currents and island formation may be avoided in MHD. In this work, we show that the appearance of NLS-like soliton equations and their hierarchy of conserved quantities are ubiquitous in ideal MHD. In particular, we study three classes of MHD equilibrium: i) a circular cross-section near an arbitrary magnetic axis, ii) exact quasisymmetry near a planar flux-surface, iii) quasisymmetry close to isodynamic. In case i), we employ near-axis expansion to show that the rotational transform and its derivatives are related to the NLS invariants. In ii), we use near-surface expansion to obtain a class of quasisymmetric vacuum magnetic fields near a planar flux surface described by a reflectionless potential. Finally in iii), we study perturbations of exact isodynamic MHD equilibrium that are approximately quasisymmetric.

1. Schief, W. K. JPP. 69.6 (2003): 465-484   

Approach to nonlinear magnetohydrodynamic simulations in stellarator geometry

The capability to model the nonlinear magnetohydrodynamic (MHD) evolution of stellarator plasmas is developed by extending the M3D-C¹ code to allow non-axisymmetric domain geometry. We introduce a set of logical coordinates, in which the computational domain is axisymmetric, to utilize the existing finite-element framework of M3D-C¹. A C¹ coordinate mapping connects the logical domain to the non-axisymmetric physical domain, where we use the M3D-C¹ extended MHD models essentially without modifications. We present several numerical verifications on the implementation of this approach, including simulations of the heating, destabilization, and equilibration of a stellarator plasma with strongly anisotropic thermal conductivity, and of the relaxation of stellarator equilibria to integrable and non-integrable magnetic field configurations in realistic geometries.   

Quasisymmetric Equilibria in Anisotropic Magnetohydrodynamics

We study the existence of quasisymmetric magnetohydrodynamic equilibria in the presence of pressure anisotropy. When compared with isotropic magnetohydrodynamics, the tensorial nature of pressure enlarges the class of equilibrium magnetic fields. First, we show that any solenoidal vector field satisfying tangential boundary conditions solves anisotropic force balance provided that the pressure tensor is appropriately chosen. Hence, a quasisymmetric equilibrium can be achieved by constructing a quasisymmetric vector field in the domain of interest. Then, we derive a system of two coupled nonlinear first order partial differential equations expressing a family of quasisymmetric anisotropic magnetohydrodynamic equilibria in bounded domains, and exhibit regular quasisymmetric vector fields corresponding to local solutions of anisotropic magnetohydrodynamics such that boundary conditions are satisfied on a portion of the boundary. The problem of locality is also discussed: we find that the topological obstruction encountered in the derivation of global quasisymmetric magnetic fields is reflected by the local nature of the solutions of the governing nonlinear first order partial differential equations.