The Stellarator Seminar Series is a set of talks with the goals of discussing stellarator research at or in collaboration with PPPL and facilitating better communication between the experimental, engineering, and theory groups working on stellarators. Please contact Ken Hammond (khammond [at] pppl [dot] gov) if you would like to present your work or to join our mailing list to receive announcements for upcoming seminars. The talks need not contain polished material -- ongoing research is welcome as well. Feel free to give a talk on the challenges you have encountered in your current project, or on the motivation for work that has just started. Students are strongly encouraged to participate.
Particles continuously repeating the same orbit facilitate the excitation of instabilities by resonating with the frequency of an associated magnetic perturbation. If these resonances match the periodicity of the equilibrium, two problems exist even without the occurrence of unstable modes. First, resonance islands appear in the particle orbits, with a periodicity matching that of the equilibrium. They increase in size with particle energy and compromise confinement if they overlap. Second, these resonances also produce local magnetic wells giving unacceptable levels of alpha particle prompt loss through ripple trapping Low period resonances are prime locations for the destabilization of high frequency Alfv ́en modes of the form δB~ = ∇×αB~ by high energy particles. In a stellarator without symmetry there is a random part of the step in flux ψ and in parallel velocity ρk given by ψ ̇ =mδ(B)vkα(ψ), ̇ρk = v2k δ(∂ψB)ξ(ψ) with δX equal to X minus the mean value of X over ζ. ξ is the ideal displacement associated with α.
Arbitrarily small modes produce prompt loss and strong diffusion of particles of all energies and pitch, not local resonance islands restricted in particle energy as in a tokamak. This problem can be alleviated by designing the stellarator with a symmetry, either toroidal or with some helicity. However, symmetry is required in ∂B/∂ψ as well as in B to avoid velocity diffusion.
European efforts and advances in Stellarator power plant studies
Stellarator research in Europe concentrates on the exploitation and extension of the advanced Stellarator Wendelstein 7-X in Greifswald, Germany, which provides unprecedented insights into complex 3D plasma physics. With the success of W7-X, there is also an increasing interest in Stellarators as potential power plants, a research field which received very little attention over the last decades. In 2021, within the new EUROfusion framework program, a small Task was established reinvigorating the field of stellarator power plant studies (SPPS). I will discuss activities that have been started in the frame of this Task and some initial progress focusing on systems studies and stellarator-specific engineering.
Assessment of information flow measures for transport investigations in W7-X
Juan Fernando Guerrero Arnaiz, Max Planck Institute for Plasma Physics, Greifswald, abstract, slides
20 Jan 2023]
Quantifying spatio-temporal behavior of fluctuating quantities may reveal transport characteristics in high temperature plasmas. In fusion, specific interest on techniques for assessing the spatio-temporal coupling result from research into signatures of particle and energy transport affecting the plasma confinement, and thus the performance of a fusion device. Therefore, additional insights from respective analyses may contribute to the identification of operational space of a fusion device. Information-theoretic data analysis methods have been applied in a wide range of scientific fields and, in recent years, these tools have been gaining traction in the field of high temperature plasma physics to study e.g. instabilities and heat transport. Nevertheless, given the complexity of the dynamics within the plasma and the purely mathematical-statistical definition of these methods, the interpretation of these techniques can be convoluted. Here, different techniques are implemented and assessed in terms of physical significance to gain insight on heat transport processes in Wendelstein 7-X, exploring the added value and limitations of the methods investigated.
Wendelstein 7-X high-performance experiments: electromagnetic modification of "stability valley" and kinetic ballooning modes
Ksenia Aleynikova, Max-Planck-Institut für Plasmaphysik, Greifswald, abstract
09 Dec 2022]
Wendelstein 7-X stellarator (W7-X) aims to demonstrate steady state operation at high beta (ratio of kinetic to magnetic pressure) values. In recent W7-X experiments (reported in ), hydrogen pellet core fuelling was realized in order to study effects of plasma pressure. The central beta of above 3.5% was demonstrated. Such plasmas feature peaked density profiles and may potentially be prone to pressure-driven instabilities. During high-beta phases of these discharges MHD-like events were observed, which may indicate a stability limit. In addition, linear GENE simulations suggest that the density and temperature gradients in those phases were large enough to destabilise kinetic ballooning modes (KBMs). Although these plasmas are stable to ideal-MHD instabilities, including ballooning modes, gyrokinetic effects on the latter render them unstable. The possibility of KBMs limiting the performance motivates an extensive study of different W7-X configurations with regard to, first, electromagnetic modifications of microinstabilities and the so-called “stability valley”, and, second, the connection between global MHD configuration properties and local gyrokinetic stability. The threshold of KBM is investigated for a number of W7-X magnetic configurations. We consider the effects of the vacuum rotational transform, iota, and the mirror ratio. The analysis is instrumental in the design of high-beta operation scenarios for the upcoming experimental campaign. In particular, it is demonstrated that some Wendelstein 7-X magnetic configurations have a relatively low kinetic ballooning mode threshold.
 Bozhenkov, S., et al. Nuclear Fusion 60.6 (2020): 066011.
Energetic particle optimization of quasi-axisymmetric stellarator equilibria
Alex LeViness, Princeton Plasma Physics Laboratory, abstract
11 Nov 2022]
In this work, a fixed-boundary stellarator equilibrium was re-optimized for energetic particle confinement via a two-step process: first, by minimizing deviations from quasi-axisymmetry (QA) on a single flux surface near the mid-radius, and secondly by maintaining this improved quasi-axisymmetry while minimizing the analytical quantity ΓC, which represents the angle between magnetic flux surfaces and contours of J||, the second adiabatic invariant.
This was performed multiple times, resulting in a group of equilibria with significantly reduced energetic particle losses, as evaluated by Monte Carlo simulations of alpha particles in scaled-up versions of the equilibria. This is the first time that energetic particle losses in a QA stellarator have successfully been reduced by optimizing ΓC. The relationship between energetic particle losses and metrics such as QA error (Eqa) and ΓC in this set of equilibria were examined via statistical methods and a nearly linear relationship between volume-averaged ΓC and prompt particle losses was found.
Modeling nonlinear MHD dynamics in stellarators with M3D-C1
We present results from two recent studies of nonlinear MHD properties in stellarators, using the M3D-C1 code’s new capability to perform extended-MHD simulations in stellarator geometry.
The first is a parametric exploration of β-limits in a 10-field period heliotron, where we examine the effect of heating power and transport on MHD dynamics and nonlinear stability, observing low-n core mode activity that is broadly consistent with experimental observations on the Large Helical Device (LHD). This paves the way for quantitative validation with LHD experiments.
In the second, we assess the nonlinear MHD stability properties of a recently published 2-field period configuration [Landreman et al., Phys. Plasmas 29.8 (2022)], which was optimised for quasiaxisymmetry at 2.5% plasma beta, highlighting the utility and potential of M3D-C1 as a high-fidelity design validation tool.
Next-step HTS stellarators with liquid metal walls
Renaissance Fusion strives to make stellarators smaller via High-Temperature Superconducting (HTS) coils. It makes them less radioactive and more resilient to alpha particle losses and other issues, by flowing mesoscale liquid metal walls. And it makes them simpler to build - by simpler coil winding surfaces and HTS manufacturing. Initial results will be presented in the areas of coil force minimization, simplification of the coil winding surface, neutronic optimization of the liquid wall materials, design point of a compact, profitable stellarator reactor and retrofitting of a fission power-plant. Paradigm-shifting ways of manufacturing HTS stellarator coils and extracting Tritium will be presented. 50 job openings will be posted, as well as areas where we solicit and welcome input from the greater stellarator community.
High energy particle resonances can modify particle distributions and even cause significant particle loss. Resonances can be present in any toroidal confinement device, and can easily be found numerically. Many stellarators have weak magnetic shear, so that large islands and large chaotic regions can be produced by resonant perturbations with small amplitudes. While choice of the field line helicity profile in the plasma can limit the presence of resonances at low particle energy, resonance location is energy dependent, and they can move into the plasma at higher energy. If resonances match the toroidal variation of the equilibrium they can produce wide islands in the phase space of orbits even in the absence of perturbations due to instabilities. These islands increase in size with particle energy and can seriously affect the confinement of high energy ions.
Stellarator coil design using cubic splines for improved access on the outboard side
In recent years many efforts have been undertaken to simplify coil designs for stellarators due to the difficulties in fabricating non-planar coils. The FOCUS code removes the need for a winding surface and represents the coils as arbitrary curves in 3D. In the following work, the implementation of a spline representation for the coils in FOCUS is described, along with the implementation of a new engineering constraint to design coils with a straighter outer section. The new capabilities of the code are shown as an example on HSX, NCSX, and a prototype quasi-axisymmetric reactor-sized stellarator. The flexibility granted by splines along with the new constraint will allow for stellarator coil designs with improved accessibility and simplified maintenance.
Analysis of sawtooth oscillations in simulations of a current-carrying stellarator
A power transfer analysis and fixed point analysis of previous NIMROD simulations [Roberds et al., Phys. Plasmas 23, (2016)] improve the understanding of the impact of 3D (non-axisymmetric equilibrium) magnetic fields on sawtooth oscillations in the Compact Toroidal Hybrid (CTH) experiment. Computing the locations of fixed points, their Greene's residues, and local values for the rotational transform leads to a description of CTH sawteeth consistent with Kadomtsev's model. A power transfer analysis quantifies the energy distribution among toroidal Fourier modes and their interactions. The Lorentz power transfer drives sawtooth growth, and it is interpreted as the flow of energy from toroidal mode n' to mode n, catalyzed by Bn-n'. It has been previously reported that the CTH sawtooth frequency increases with the 3D field strength. This is attributed to an increased growth rate of the resistive internal kink that drives sawtooth oscillations. Here, 3D fields remove energy from the kink, eliminating the possibility that these fields are an additional energy source that drives growth. Instead, 3D fields catalyze the rearrangement of energy within the kink from large-to-small scales, where magnetic reconnection is stronger. It is proposed that this energy rearrangement enhances the effective resistivity, consistent with the increased growth rate observed at higher 3D field strengths.
Winding Non-Planar Coils with Uninsulated HTS Tape: Optimization and Prototype
This work will describe additional design considerations imposed by HTS technology on the winding of uninsulated HTS non-planar magnets and present a novel optimization framework to mitigate these constraints . For an input coil geometry, the described framework ultimately allows a determination of the minimum size, minimum conductor length, or maximum field of an uninsulated HTS non-planar coil using existing tape technology. This framework will be described and applied to existing stellarator designs as well as provide guidance to next-step device optimizations. A small non-planar HTS prototype coil was realized using this framework to validate the methodology. The design, fabrication, assembly, and testing of this coil will be surveyed.
The Helically Symmetric eXperiment (HSX) at UW Madison, Wisconsin is the world’s first quasisymmetric and 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 from 1T to 1.25T. This upgrade will allow plasma operation at three times higher densities (up to 3.e19/m^3), which will significantly increase the ion temperature due to better coupling between electrons and ions and reduced charge exchange losses because of a reduced core neutral content. First modeling results based on the ISS04 scaling predict an increase of the ion temperature from 50 eV to about 150 eV. Moreover, we expect an off-axis shift of the fueling profile during higher density operation, likely resulting in reduced density gradient lengths which will, in turn, reduce Trapped Electron Mode (TEM) growth rates. To this end, the possibility of additionally modifying the magnetic field geometry of HSX for reduced TEM turbulence has been explored. By changing individual coil currents in free boundary VMEC simulations in the range of 5-10%, magnetic field configurations have been identified for which non-linear GENE simulations predict a significant reduction of TEM heat fluxes, while maintaining excellent neoclassical properties. Implementing these new coil-current configurations, in combination with modified density gradient lengths, could result in a strong reduction of TEMs in HSX upgrade plasmas and might provide access to plasmas with significantly improved confinement properties.
Observation of a reduced-turbulence regime with boron powder injection in a stellarator
In recent experiments in the Large Helical Device, we observed a confinement regime that is characterized by increased confinement and reduced turbulent fluctuations. The transition to this regime is driven by the injection of submillimetric boron powder grains into the plasma. With the line-averaged electron density and input power being kept constant, the stored energy, electron and ion temperatures increase substantially. At the same time, the amplitude of the plasma turbulent fluctuations is halved. While lower frequency fluctuations are damped, higher frequency modes in the range between 100 and 200 kHz are excited. We have observed this regime for different heating schemes, namely with both electron and ion cyclotron resonant radio frequencies and neutral beams, for both directions of the magnetic field and both hydrogen and deuterium plasmas. Preliminary results from a new set of experiments will be discussed, including general trends of the performance improvement with input power and amount of boron injected.
3D equilibrium codes are vital for stellarator design and operation, as they provide the base state for experimental plasma operation. High-accuracy equilibria are also necessary for stability studies. Comparisons of two 3D equilibrium codes will be presented, VMEC, which uses a steepest-descent algorithm to reach a minimum-energy plasma state, and DESC, which minimizes the MHD force error in real space directly. Accuracy as measured by final plasma energy and satisfaction of MHD force balance, as well as other metrics, will be presented for each code. Differences between the results and the code methods will be discussed.
EPOS: Seeking a quasi-symmetric configuration for a quintessentially symmetric plasma
Eve Stenson, Max Planck Institute for Plasma Physics, abstract
17 Dec 2021]
Electron-positron plasmas are the prototypical "pair plasmas", comprising positively and negatively charged particles of equal mass. Theoretical and computational treatments of such systems go back more than 40 years and include a number of intriguing predictions. The EPOS (Electrons and Positrons in an Optimized Stellarator) project --- the latest branch of the APEX (A Positron Electron eXperiment) Collaboration --- aims to create and study electron-positron plasmas in the laboratory, magnetically confined on toroidal flux surfaces specifically designed for that purpose. This talk will review some of the relevant physics at this unique combination of plasma parameters (low temperatures and densities, which are common to non-neutral plasmas but unusual for quasineutral plasmas), geometry (toroidal flux surfaces, which are common for quasineutral plasmas but unusual for nonneutral plasmas), and mass ratio (i.e., unity). It will also discuss some key questions and considerations being addressed on the way to designing --- and ultimately building --- a tabletop-sized, superconducting stellarator for pair plasma experiments.
Development Of Advanced Stellarator With Standardized Permanent Magnet Blocks
GuoSheng Xu, Institute of Plasma Physics, Hefei Institute of Physical Science, Chinese Academy of Sciences, abstract, slides
10 Dec 2021]
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. The “two-step” magnet design strategy 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.
Stellarator Simplification using Permanent Magnets
We report the status of the permanent magnet stellarator project. Non-planar coils are the most complicated and expensive part of a stellarator. Permanent magnets provide a novel method to produce optimized stellarator configurations using very simple coils. The new concept for generating 3D fields using permanent magnets has led to the world’s first project examining the use of permanent magnets for stellarators, which has been funded by ARPA-E and FES and will be located at Princeton Plasma Physics Laboratory. The project will design and construct a half-period of the magnet structure for a possible stellarator concept that would use components from NCSX, including the toroidal field coils and vacuum vessel, together with an array of neodymium magnets. Two new codes, MAGPIE and FAMUS, have been developed to design the magnets. MAGPIE provides the geometry information and FAMUS optimizes the magnet arrangements. By using the two codes, the target quasi-axisymmetric equilibrium with improved energetic particle confinement can be realized by uniform cuboidal magnets in a limited number of discrete polarizations together with planar toroidal field coils. A post-office-box structure will be used to mount the magnets. An automated system has been developed within the Virtual Engineering framework of the ANSYS suite of codes to integrate the MAGPIE-FAMUS code set with engineering analysis codes. The magnet positions will be adjusted iteratively and an array of correction magnets will be installed to minimize error fields with tolerance. The methods used and the results from the design effort will be described in detail and the status of the construction activity will be summarized. A table-top PM stellarator project (MUSE) has also been designed for basic experiments, and construction is underway.
Demonstration of Reduced Neoclassical Energy Transport in Wendelstein 7-X
Craig Beidler, Max Planck Institute for Plasma Physics, abstract, slides
19 Nov 2021]
Research on magnetic confinement of high-temperature plasmas has the ultimate goal of harnessing nuclear fusion for the production of electricity. Although the tokamak is the leading toroidal magnetic-confinement concept, it is not without shortcomings and the fusion community has therefore also pursued alternative concepts such as the stellarator. Unlike axisymmetric tokamaks, stellarators possess a three-dimensional (3D) magnetic field geometry and the availability of an additional dimension opens up an extensive configuration space for computational optimization of both the field geometry itself and the current-carrying coils which produce it. Such an optimization was undertaken in designing Wendelstein 7-X (W7-X), a large Helical-Axis Advanced Stellarator (HELIAS) which began operation in 2015, at Greifswald, Germany. A significant drawback of 3D magnetic field geometry, however, is the strong temperature dependence which it introduces into the stellarator's non-turbulent "neoclassical" energy transport. Indeed, such energy losses will become prohibitive in high-temperature reactor plasmas unless a strong reduction of the geometrical factor associated with this transport can be achieved, and such a reduction was therefore made a principal goal of the W7-X design. In spite of the rather modest heating power currently available, W7-X has already been able to achieve high-temperature plasma conditions during its 2017 and 2018 experimental campaigns, producing record values of the fusion triple product for such stellarator plasmas. The triple product of plasma density, ion temperature and energy confinement time is used in fusion research as a figure of merit as it must attain a certain threshold value before net-energy-producing operation of a reactor becomes possible. In this seminar, it will be demonstrated that such record values provide evidence for reduced neoclassical energy transport in W7-X, as the plasma profiles which produced these results could not have been obtained in stellarators lacking a comparably high level of neoclassical optimization.
Radial coordinate maps, radial vectors, and binormal vectors for 5/6, 5/5 and 5/4 edge island domains in W7-X
The edge island domain in Wendelstein 7-X consists of divertor islands that coincide with the location of rational values of the rotational transform ι≈(5/6,5/5,5/4)and surround the main confinement volume (the ‘main plasma’). The 5/5 edge consists of 5 individual islands that are unconnected. In contrast, a single island connects onto itself after 6 or 4 toroidal transits in the 5/6 or 5/4 edge, respectively. Many interesting phenomena are related to these islands and diagnostic analyses require a mapping from ‘laboratory’ or real space coordinates to the island coordinate system. A procedure is described here to calculate several scalar and vector quantities for closed island structures which can be utilized in fast interpolation schemes for inverse maps. For the 5/5 edge, a fixed-boundary vacuum (zero beta) magneto-hydrodynamic solution of the 5/5 island is found with VMEC. The solution is compatible with already existing routines which determine the radial and binormal vectors of VMEC solutions at arbitrary laboratory coordinates. VMEC does not support solutions for the 5/4 and 5/6 islands, but the radial and binormal vectors are available in a local 2-D Fourier island coordinate system.
A combined modeling/experimental approach to interpret H-alpha measurements in Wendelstein 7-X
Victoria Winters, Max Planck Institute for Plasma Physics, abstract, slides
08 Oct 2021]
In the stellarator Wendelstein 7-X (W7-X), the main locations of main particle sources are expected to be the graphite divertors, baffles, and heat shield first wall. Although it is expected that the inner wall heat shield does not receive large incident ion fluxes, its large surface area (60m2) gives it the potential to be a non-negligible contributor to the total recycling source, even if these fluxes are small. To investigate this, a combined modeling/experimental was taken to interpret the H-alpha measurements to this component. On the modeling side, the heat shield surface was implemented in the 3D plasma boundary code EMC3-Eirene. It was found that the heat shield contributes negligibly to the total recycling source (<1% for typical values of cross-field particle diffusion D=1m2s-1) in the standard magnetic field configuration. The recycling fluxes which originate on this component are localized to several small areas which reproduce well plasma footprint patterns seen in post-experimental campaign in-vessel inspection photos. On the experimental side, however, H-alpha line emission was still measured by the diagnostics on locations where EMC3-Eirene does not predict recycling fluxes. Therefore, the synthetic diagnostic module of EMC3-Eirene was used to reconstruct the line integrated H-alpha photon flux which would be measured by the diagnostics. It was found that the line integrated emission levels measured by lines of sight terminating at the inner wall heat shield can be accounted for purely from the ambient neutral Hydrogen density levels in the main chamber. This is true regardless of whether the synthetic diagnostic views a region of expected recycling on this component or not, since the local recycling flux is so small in this location. The ambient neutral hydrogen density levels in the main chamber originate from the divertor/baffle region, and thus it is the neutral leakage level in the divertor which determines line integrated H-alpha signal levels at the first wall. The results show that great care must be taken when dealing with line-integrated H-alpha signals viewing components with low recycling fluxes. Although these signals may be used as part of a total recycling source calculation, it cannot be used to localize the distribution of these recycling sources on individual components.
SIMSOPT: A new flexible framework for stellarator optimization
SIMSOPT is a new open-source software framework being developed for stellarator optimization, prioritizing flexibility, modularity, and testing. High-level code is in python, with C++ extensions where performance is important, and interfaces to fortran codes including VMEC and SPEC. So far the framework also includes several other components: tools for defining objective functions, surface and curve objects, a variety of magnetic field types, and tools for parallelized finite differences. Among the examples, we demonstrate an optimization in which outputs from both VMEC and SPEC are simultaneously included in the objective function, so both quasisymmetry and good flux surface quality are achieved.
GTC simulation of microturbulence in W7-X and LHD with full flux-surface and kinetic electrons
Zhihong Lin and Javier Nicolau, University of California at Irvine, abstract, slides
04 Jun 2021]
This talk reports a new driftwave eigenmode in the W7-X from the first ever nonlinear gyrokinetic simulation of microturbulence in the stellarator incorporating both full flux-surface and kinetic electrons. Global simulations are necessary to study key physics of the non-axisymmetric stellarator such as linear toroidal coupling of multiple-n toroidal harmonics (e.g., localization of eigenmodes to discrete magnetic field lines, linear coupling between zonal flows and low-n harmonics etc), turbulence spreading, and secular radial drift of helically-trapped particles. In this work, global gyrokinetic simulations using the GTC code find a new electrostatic helically trapped electron mode (HTEM) driven by a realistic density gradient in the W7-X. The HTEM is excited by helically trapped electrons at the toroidal section with a weak magnetic field. The linear eigenmode is localized to discrete field lines on the inner side of the torus. Nonlinear simulations find that the HTEM saturates by inverse cascade of the toroidal harmonics from a linear range of n=[100,300] to a nonlinear range of n=[0,200] and by turbulence spreading to the damped region across the whole flux-surface and in the radial direction. Zonal flows play a secondary role in the HTEM saturation. The saturated HTEM turbulence drives a large particle diffusivity comparable to the heat conductivity driven by the ion temperature gradient (ITG) turbulence with similar gradient scale lengths. The HTEM can only be captured by full flux-surface simulations since helically trapped electrons residing in different flux-tubes can either drive or damp the HTEM. The full flux-surface simulations with kinetic electrons only become feasible thanks to the GTC global field-aligned mesh in real space, which reduces the number of parallel grid points by a factor of 150 in these W7-X simulations.
How can model reduction help reduce turbulent transport?
Turbulent transport is a key mechanism behind energy loss in fusion plasmas. Unfortunately, models of turbulent transport via the electro-magnetic Vlasov-Maxwell equations require fine discretizations leading to high-dimensional models, thus taking long computing times. Therefore, optimization systems for the design of fusion devices usually rely on less expensive proxies for turbulent transport. In order to accurately compute turbulent transport fast enough to use the computation in an optimization loop, we need to build a reduced model. Even more importantly, this reduced model should share the same physical structure/conservations laws; otherwise, the reduced model could be unphysical and yield poor approximation quality. In this talk, we describe preliminary steps towards model reduction methods for transport equations (like Vlasov-Maxwell) that are able to preserve the Hamiltonian structure of the underlying equations, and that are able to assess and control the error in the reduced model (w.r.t. the high-dimensional model) via an error estimator.
Updates and improvements to the DESC code for finding and optimizing stellarator equilibria
DESC is a pseudo-spectral code that computes 3D MHD equilibria by directly solving the force balance equations JxB=grad(p). We present a number of recent improvements to the code for both finding and optimizing stellarator equilibria. Automatic differentiation allows fast and accurate computation of derivatives used for solving the force balance equations and determining sensitivity to inputs. These calculations are further accelerated by leveraging GPUs. Accurate derivatives are used in a perturbation method to explore the phase space of stellarators and efficiently compute high resolution equilibria. We also demonstrate using DESC to optimize equilibria for physics objectives such as quasisymmetry.
Development and simulation of fast ion loss detectors for Wendelstein 7-X
Samuel Lazerson, Max Planck Institute for Plasma Physics, abstract
30 Apr 2021]
The demonstration of adequate fast ion confinement in magnetically confined fusion devices is a necessary milestone on the path to an energy producing reactor. The Wendelstein 7-X (W7-X) stellarator has such a milestone, seeking to demonstrate the theoretically predicted improvement in fast ion confinement which comes with quasi-omnigeneity. In the high mirror configuration, contours of maximum-J (the second adiabatic invariant) close on flux surfaces as beta increases, resulting in confinement of deeply trapped particles. To assess such an effect, W7-X is equipped with both neutral beam injection and ion-cyclotron resonance heating, providing sources for fast ions. Given the three-dimensional nature of the stellarator magnetic fields, an array of diagnostics is planned to assess fast ion confinement. Simulations of fast ions are helping to inform placement diagnostics while validation of those models is being conducted. Key among those diagnostics are the in-tile Faraday cup fast ion loss detectors (FC-FILD). These energy discriminating sensors provide microchip-like packaging allowing almost any wall tile in W7-X to be instrumented for the detection of lost fast ions. A prototype sensor will be installed on W7-X for the next campaign, along with a scintillating detector to come later.
Adjoint calculation of magnetic island width sensitivity
Alessandro Geraldini, École Polytechnique Fédérale de Lausanne, abstract, slides
26 Mar 2021]
Minimising the presence and size of magnetic islands in the core of stellarators is necessary to improve confinement. However, the size of certain islands in a configuration may be extremely sensitive to small perturbations to the magnetic field, which is undesirable and can lead to tight coil tolerances during construction. It is therefore also crucial to reliably and efficiently quantify the sensitivity of island size. To this end, I will present an adjoint approach to calculate the gradient of two quantities related to magnetic island size. The first quantity is the full radial width of an island calculated using a method developed by Cary & Hanson (1991), which relies on the island size being small compared to the system size and only requires finding the island center. The second quantity, known as Greene’s residue, can be calculated for any periodic field line, and for an island centre it is strongly correlated with the island width. Convergence tests for the gradients of island width and residue are presented. The result of a gradient-based optimisation of residues in a magnetic configuration produced by a pair of helical coils are presented. Furthermore, the shape gradient — a measure of sensitivity — of the island width with respect to coils is calculated and numerically verified for a magnetic island in the NCSX configuration. A key advantage of the adjoint method relative to a finite difference calculation is that the island center need not be recomputed for every possible magnetic field perturbation.
Using recently developed adjoint methods for computing the shape derivatives of functions that depend on MHD equilibria (Antonsen et al. 2019; Paul et al. 2020), we present the first example of analytic gradient-based optimization of fixed-boundary stellarator equilibria. We take advantage of gradient information to optimize figures of merit of relevance for stellarator design, including the rotational transform, magnetic well, and quasisymmetry near the axis. With the application of the adjoint method, we reduce the number of equilibrium evaluations by the dimension of the optimization space (∼50−500) in comparison with a finite-difference gradient-based method. We discuss regularization objectives of relevance for fixed-boundary optimization, including a novel method that prevents self-intersection of the plasma boundary. We present several optimized equilibria, including a vacuum field with very low magnetic shear throughout the volume. Finally, extensions of this adjoint method to other equilibrium models will be discussed.
Verification and Validation of XICS Ion-Temperature and Plasma Flow Measurements at W7-X
X-ray raytracing is used to examine the accuracy of the analysis techniques used for the X-ray Imaging Crystal Spectrometer (XICS) used at W7-X, and in particular the accuracy of the 1D tomographic inversion in stellarator geometry. In course of performing these verification exercises an ion-temperature correction has been developed for the W7-X system with general applicability to all XICS diagnostics. The XICS is a powerful diagnostic able to measure ion-temperature, electron-temperature, plasma flow and impurity charge state densities. While these systems are relatively simple in design, accurate characterization of the instrumental response and validation of analysis techniques are difficult to perform experimentally due the requirement of extended x-ray sources. For this reason a new x-ray raytracing code has been developed, XICSRT, that can be used to simulate and raytrace both complex plasma sources and detailed diagnostics layouts. A complete raytracing scenario has been developed for this new raytracing code that allows characterization of the spectrometer and verification of the analysis methods while fully considering the real geometry of the XICS system and W7-X plasma. Through the use of raytracing, several important corrections have been found that must be accounted for in order to accurately reconstruct the ion-temperature profiles. Finally, the accuracy of the tomographic inversion technique in stellarator geometry is investigated, providing for the first time a verification exercise for inversion accuracy with stellarator geometry and a complete XICS analysis tool-chain.
Impurity flow measurements with Coherence Imaging Spectroscopy at Wendelstein 7-X
Valeria Perseo, Max Planck Institute for Plasma Physics, abstract, slides
12 Feb 2021]
The Scrape-Off Layer (SOL) of the Wendelstein 7-X (W7-X) stellarator is characterized by the presence of magnetic islands, exploited for the so-called island divertor conﬁguration. The intersection of the magnetic islands with the modular divertor targets results in a complex 3D SOL configuration, characterized by the presence of counter-streaming flows of particles and energy within each island. Previously predicted, these flow channels have been directly visualized for first time with the use of a Coherence Imaging Spectroscopy (CIS) diagnostic during the last experimental campaign (OP1.2b). CIS is a camera-based interferometric system capable of measuring Doppler particle ﬂows associated with a selected visible emission line from the plasma. In contrast to other flow diagnostics such as dispersive spectroscopy and Mach probes, CIS is distinguished by providing high time and velocity resolution together with high spatial coverage, allowing the measurements of flow velocities for a full module of W7-X simultaneously. A CIS diagnostic has been fully designed for W7-X with the aim of monitoring flows parallel to the SOL open magnetic field lines. Thanks to a newly implemented calibration source, based on a continuous-wave-emission tunable laser, the intrinsic carbon impurity behavior has been investigated for in different plasma scenarios, characterized by impurity gas puffs, development of bootstrap current, or detachment. Moreover, CIS can be used as a powerful tool to test the limits of the current theoretical models, for example in the case of forward and reversed field experiments. CIS observations of velocities varying in the range 0-35 km/s are compared with other diagnostics and EMC3-EIRENE simulations.
Algorithms for combined plasma and coil optimization
Sophia Henneberg, Max Planck Institute for Plasma Physics, abstract, slides
05 Feb 2021]
Combined plasma-coil optimization approaches for designing stellarators are discussed and a new method for calculating free-boundary equilibria is proposed. Four distinct categories of stellarator optimization, two of which are novel approaches, are the fixed-boundary optimization, the generalized fixed-boundary optimization, the quasi free-boundary optimization, and the free-boundary (coil) optimization. These are described using the multi-region relaxed magnetohydrodynmics (MRxMHD) energy functional, the Biot-Savart integral, the coil-penalty functional and the virtual casing integral, and their derivatives. The proposed free-boundary equilibrium calculation differs from existing methods in how the boundary-value problem is posed, and for the new approach it seems that there is not an associated energy minimization principle because a non-symmetric functional arises. We propose to solve the weak formulation of this problem using a spectral-Galerkin method, and this will reduce the free-boundary equilibrium calculation to something comparable to a fixed-boundary calculation. In our discussion of combined plasma-coil optimization algorithms, we emphasize the importance of the stability matrix.
Design of the Continuous Pellet Fueling System for Wendelstein 7-X
Larry Baylor, Oak Ridge National Laboratory, abstract
04 Dec 2020]
A continuous pellet fueling system (CPFS) is currently being designed as a collaborative effort to provide long pulse fueling of the Wendelstein 7-X (W7-X) stellarator. The purpose of the CPFS is to provide a deep high fueling rate for mitigation of hollow density profiles, sustainment of high-density discharges, and density feedback-controlled operation. The system will provide the capability to inject cylindrical pellets of solid hydrogen or deuterium into the plasma core, with flexibility to dynamically vary the pellet size and injection frequency. Pellets are nominally to be 3 mm in diameter and have a variable length between 1 – 3 mm. The pellets will nominally be injected at 1000 m/s and they can be injected at 10 Hz steady -state and higher rates for shorter periods.
The core of the CPFS is a vertically oriented twin-screw extruder, cooled by three Gifford- McMahon cryocoolers in parallel, capable of forming a continuous flowing filament of solid hydrogen or deuterium. The filament width, which determines the pellet length, can be adjusted by means of a variable nozzle driven by a linear actuator at the base of the extruder nozzle. A prototype extruder has been tested for throughput capability before design of the final extruder is completed and test results will be presented. Coupled to the nozzle is a solenoid operated gas gun and cutter assembly that cuts and accelerates the pellets into W7-X. Three gaps in the injection line provide expansion and pumping locations to pump the helium propellant gas before it reaches the plasma.
The injector will be ideally suited for feedback control on the plasma density or density gradient, depending on the physics optimization desired and feedback signals available. The individual pellets can be chambered and fired with a single trigger signal from a feedback system and subsequent pellets can be fired 100 ms or more later provided the extruder is running fast enough to supply the solid pellet material at that rate.
Challenges for extending this technology to a reactor environment will be described.
Position Tolerances for Permanent Magnets in the MUSE Stellarator
Permanent magnets have been proposed to resolve the challenges and costs imposed by complicated coils and necessarily high-precision requirements in stellarators. The MUSE stellarator is a permanent magnet prototype soon to be constructed at PPPL. We are using the FAMUS code to optimize the permanent magnet configuration for MUSE. To inform the construction of MUSE, tolerance calculations are helpful in identifying the magnets with the largest impact on a particular quantity of interest. We used the shape gradient of the normal field error for dipole positions as a first-order approximation of the permanent magnet position tolerances in MUSE. Further, we implemented the Hessian matrix method as a second-order approximation of tolerance of the normal field error to permanent magnet position. We found that the largest shape gradient values (1.1e-7) were for the dipoles in the thinnest regions on the outboard side of the torus. We found that in the case of small shape gradients, the Hessian matrix method is necessary to strengthen our estimates of position tolerance.
Development of a scenario with combined density and heating control to reduce the impact of the bootstrap current in Wendelstein 7-X
Wendelstein 7-X is a low-shear stellarator with an island divertor, formed by natural magnetic islands at the plasma edge and ten modular divertor units for particle and energy exhaust. For the island divertor concept to work properly, the device is optimized for small internal currents. In particular, the bootstrap current is minimized. Previous studies predicted a thermal overload of the targets at a particular location, due to the slow evolution of the toroidal net current in the initial phase of certain otherwise desirable high-power discharges. The present numerical study explores the neoclassical predictions for the bootstrap current in more detail and demonstrates, as a proof of principle, that a path from low density and low heating power to high density and full heating power exists, on which the bootstrap current remains constant. This offers the possibility to reach the predetermined toroidal net current at low heating power, where no heat overload will occur in the transient phase.
Permanent magnet arrangements for stellarators: a linear least-squares algorithm
A problem arising in several engineering areas is to design magnets outside a volume that produce a desired magnetic field inside it. One instance of this problem is stellarator design, where it has recently been shown that permanent magnets can provide the required shaping of the magnetic field. Here we demonstrate a robust and efficient algorithm REGCOIL_PM to calculate the spatial distribution of these permanent magnets. The procedure involves a small number of fixed-point iterations, with a linear least-squares problem solved at each step. The method exploits the Biot-Savart Law's exact linearity in magnetization density and approximate linearity in magnet size, for magnets far from the target region. No constraint is placed on the direction of magnetization, so Halbach solutions are found naturally, and the magnitude of the magnetization can be made uniformly equal to a target value.
Penalty Functions in FOCUS to Constrain Stellarator Coil Optimization
Finding less complicated coils that have adequately low field errors is a crucial step in stellarator development. One coil complexity metric that is of high importance, is the maximum curvature of the coil center line, or coil single-filament. High coil curvatures can force current densities to become too high in a finite build coil. Additionally, high coil curvatures can cause strains to exceed acceptable levels, especially in superconducting coils. We investigate three ways to optimize a coils curvature and find that applying a constraint on coil curvature solves for the most optimal coils. To apply this constraint, we use penalty functions. Penalty functions are implemented in FOCUS and coil solutions, optimized for an HSX like "plasma boundary" are presented.