Simons Collaboration on Hidden Symmetries and Fusion Energy Annual Meeting
The most compelling transformational use of magnetically confined, hightemperature plasma is to realize sustained fusion energy. Despite impressive progress, net energy production has not yet been achieved. The tokamak, which is the leading magnetic confinement concept in the world today, has the topology of a torus and continuous symmetry with respect to the toroidal angle, giving it good confinement properties. In the stellarator, which is the leading alternative to the tokamak, the confining magnetic field is mostly produced by external currentcarrying coils. In contrast to the tokamak, stellarators rely on symmetry breaking to realize the magnetic field needed to confine particles.
Over the last few decades, a new concept has emerged in the design of stellarators, giving rise to a renaissance—the remarkable discovery that it is possible to design 3D magnetic confinement devices with hidden symmetries that can have the same virtues as tokamaks while overcoming some of the inherent drawbacks of the latter. An example of a hidden symmetry, known as “quasisymmetry,” is that the magnitude B of the magnetic vector field B has a negligible coordinate dependence (in a special curvilinear coordinate system) even though B does not. The primary purpose of this Simons Collaboration in Mathematical and Physical Sciences is to create and exploit an effective mathematical and computational framework for the design of stellarators with hidden symmetries.
The challenge of finding 3D optimum magnetic fields with hidden symmetries encompasses mathematical and computational problems of great subtlety, straddling optimization theory, plasma physics, dynamical systems, and the analysis of partial differential equations. In this Workshop, we are bringing together a diverse interdisciplinary group of applied mathematicians, computer scientists, and plasma physicists to discuss the fundamental challenges and transformational potential of the stellarator approach to magnetic confinement. Presentations at the Annual Meeting, which is by invitation only, include eight plenary lectures, a poster session, and ample time for discussion.
Talks
Amitava Bhattacharjee
Princeton University
Hidden Symmetries and Fusion Energy
The most compelling transformational use of magnetically confined, hightemperature plasma is to realize sustained fusion energy. The tokamak, which is the leading magnetic confinement concept in the world today, has the topology of a torus and continuous symmetry with respect to the toroidal angle, giving it good confinement properties. In the stellarator, which is the leading alternative to the tokamak, the confining magnetic field is mostly produced by external currentcarrying coils. In contrast to the tokamak, stellarators rely on symmetry breaking to realize the magnetic field needed to confine particles.
Over the last few decades, a new concept has emerged in the design of stellarators, giving rise to a renaissance—the remarkable discovery that it is possible to design 3D magnetic confinement devices with hidden symmetries that can have the same virtues as tokamaks while overcoming some of the inherent drawbacks of the latter. An example of a hidden symmetry, known as “quasisymmetry,” is that the magnitude B of the magnetic vector field B has an ignorable coordinate (in a special curvilinear coordinate system) even though B does not. The primary purpose of this collaboration, supported by the Simons Foundation, is to create and exploit an effective mathematical and computational framework for the design of stellarators with hidden symmetries.
Existing experiments show that stellarator design optimization holds the key to successful performance. Results from the Helically Symmetric Experiment (HSX) at the University of WisconsinMadison and the Wendelstein 7X (W7X) experiment at the Max Planck Institute of Plasma Physics in Greifswald (Germany), which started opation about three years ago, are very encouraging, compelling the design of the next generation stellarator.
The challenge of finding 3D optimum magnetic fields with hidden symmetries encompasses mathematical and computational problems of great subtlety, straddling optimization theory, plasma physics, dynamical systems, and the analysis of partial differential equations. We have, therefore, assembled a multidisciplinary team of world experts in these disciplines, reflecting the breadth and importance of our challenge. Synthesizing the understanding developed through our research, our goals include a modern stellarator optimization code that can exploit the full power of petascale and exascale computers, and a few optimal designs of nextgeneration stellarator experiments.
Thomas Sunn Pedersen and Eve Stenson
Max Planck Institute for Plasma Physics
Experimental stellarator research: Recent results and nearterm plans
Stellarators provide a potentially attractive concept for fusion power production, owing to their intrinsic steadystate capabilities and a lack of violent plasma instabilities. However, high confinement at high ion temperatures has in the past been an elusive goal, primarily owing to poorly confined singleparticle orbits despite the magnetic field lines being confined to a toroidal shape. A new generation of stellarators has started demonstrating that this and other issues of importance for a fusion reactor can be addressed successfully by a careful optimization of the magnetic field.
After a short introduction into magnetic fusion research and stellarators, we will review key achievements from advanced stellarators in the world, in particular from the largest and most optimized stellarator, Wendelstein 7X (W7X), which recently started operation (2015). With a 30cubicmeter plasma volume, a superconducting coil system operating at 2.5 T, and steadystate plasma heating capability of up to 10 MW, it will be capable of demonstrating the benefits of optimized stellarators at parameters approaching those of a fusion power plant. Some of the first results already demonstrate key features of the optimization, including strongly reduced toroidal currents [1], and long confinement times at relatively high ion temperatures (τ>200 msec at T_{i}>3.5 keV) [2].
The last part of the talk will describe nearfuture plans for optimized stellarator experiments, in particular for a tabletopsized, highly optimized stellarator experiment to confine and study electronpositron plasmas. Due to the perfect symmetry in mass between the two chargedparticle species, electronpositron plasmas have unique properties [3, 4], and are particularly sensitive to the optimization of singleparticle orbits in stellarators.
[1] A. Dinklage et al. Nature Physics (2018)
[2] T. Sunn Pedersen et al., Plasma Phys. Contr. Fusion (2018)
[3] Tsytovic and Wharton Comm. Plasma Phys. Contr. Fusion (1978)
[4] E. V. Stenson et al, Journal of Plasma Physics (2017)
Per Helander
Max Planck Institute for Plasma Physics, Greifswald, Germany
Mathematical description of Stellarators
In this talk, an overview of the theory of stellarator plasmas for mathematically able nonspecialists will be given. Gaps in our present understanding and open mathematical problems will be particularly emphasized.
On the simplest useful level, magnetically confined plasmas plasma can be described by magnetohydrodynamic equations. The question of existence and regularity of 3D timeindependent solutions to these equations is an old and difficult problem. Most solutions are believed to possess chaotic magnetic field lines and/or a countable infinity of tangential discontinuities, which should be taken into account when constructing approximate numerical solutions. Recent progress and outstanding problems in this area will be reviewed.
A more detailed description of the plasma is furnished by the Boltzmann equation, augmented by an electromagnetic force and (usually) simplified to account for the fact that the magnetic field is relatively large, resulting in the socalled driftkinetic or gyrokinetic equations. Without the collision term, the characteristics of these equations describe collisionless particle orbits. Because the collision frequency is relatively small in hot plasmas, the great majority of these orbits should stay within the plasma if the latter is to be well confined by the magnetic field. This condition, that most collisionless particle orbits should not leave the plasma region, is not satisfied by most magnetic fields and thus calls for optimization. It is strictly ensured if the magnetic field has an exact continuous symmetry, and it is approximately (suitably defined) ensured if the field has an approximate symmetry, e.g., if it is “quasisymmetric’’ of “omnigenous”. These concepts will be explained, and existence theorems (or the lack thereof) discussed.
Regardless of whether the magnetic field possesses these symmetries, it is possible to find nearly timeindependent solutions to the full kinetic equation by means of a ChapmanEnskogtype expansion, giving valuable information about the energy losses from the plasma. As expected, these losses are relatively small in quasisymmetric or omnigenous plasmas, and it also turns out that these symmetries affect other experimentally important properties of the plasma, such as the electric current that flows through the plasma and its rotation. Fully timedependent solutions of the kinetic equation usually show that the plasma is turbulent, with effectively random fluctuations in density, temperature and other parameters causing a degradation in confinement. A great unknown is how the magnetic geometry affects this turbulence and whether the latter can be substantially reduced by magnetic field optimization.
Robert MacKay
University of Warwick
The search for hidden symmetries and quantification of effects of imperfect symmetry on confinement
The study of confinement of a charged particle in a strong magnetic field has from the beginning been greatly aided by an excellent approximate symmetry, namely rotation of the particle’s “gyroradius vector” around its “guiding center.” This reduces the problem (to high accuracy) to that of confining the guiding center. If the guidingcenter motion has a continuous symmetry then the guiding center is confined to a specific region associated to its initial condition, generically the projection of an invariant 2torus to positionspace. More formally, the guidingcenter motion is “integrable.”
An obvious example of such symmetry occurs if the magnetic field has rotation symmetry about some axis. This is the principle of tokamaks. They have the defects, however, that they need a strong toroidal current to make the confinement tight, driving such a current for long times is not easy, and it generates instabilities.
Stellarators are a class of magnetic confinement devices in which no toroidal current is required. To achieve integrability without axisymmetry, however, requires a hidden symmetry, called quasisymmetry. Although this can be achieved to a good approximation, it is widely believed that exact quasisymmetry is impossible. Yet it is worth keeping in mind the amazing result of Kovalevskaya that in addition to the axisymmetric integrable tops of Lagrange there is a family of nonaxisymmetric integrable tops. I will report on our search for magnetic fields with exact quasisymmetry.
Even if quasisymmetry is achievable only approximately, all is not lost. Integrability is not necessary for confinement of guiding centers. If the magnetic field is smooth and close enough to having quasisymmetry there is generically a fat Cantor set of invariant 2tori in the 3D space representing guiding centers with given energy and magnetic moment. Guiding centers trapped between two of these 2tori are confined almost as well as those exactly on a 2torus. I will report on the development of methods to determine the complement of the fat Cantor set. This is a set to be minimised by the optimisers in our team, at least in the interior of the plasma. Regions not on invariant tori are important, however, for the design of “divertors”, structures in the magnetic field that scrape off particles from outside the main plasma region to keep a clear gap between the plasma and the vacuum vessel and to control removal of impurities and fusion products.
In reality, there are deviations from the above picture. Firstly, the guiding center approximation is not perfect, in particular for the highenergy alphaparticles that would be produced by fusion of tritium with deuterium, and if the magnetic field is not very smooth (as could result from nearsingular current distributions). Secondly, the magnetic (and electric) field is timedependent, for example due to turbulence, and this leads to diffusion of particles from their ideal motion. Thirdly, the particles in a machine interact, which causes exchange of energy and magnetic moment between guiding centers and relative shifts of the guiding centers, hence additional diffusive effects. Such effects have been and are being addressed by others in our team and elsewhere, but we believe there is value in pinning down the singleparticle guidingcenter motion more precisely, as here.
Stuart Hudson
Australian National University
The numerical construction of stellarator equilibria and coil design
This talk shall describe the numerical construction of a magnetically confined plasma in macroscopic force balance with an externally applied magnetic field that is produced by an arrangement of external currents, commonly referred to as coils. The magnetohydrodynamic equilibrium state of the plasma and the coil geometry are given as the solution.
The plasma equilibrium is obtained as an appropriately constrained minimum of the plasma energy functional. A restriction upon the boundary conditions supplied to the resulting differential equation is required to avoid nonphysical and nontractable solutions.
The coil geometry is obtained by minimizing an error functional that quantifies how well the coils provide the magnetic field required to hold the plasma in equilibrium. An adjustable penalty on the length allows the user to control how geometrically deformed the coils are.
Matt Landreman
University of Maryland
Constructing magnetic fields with hidden symmetry
Historically, magnetic fields with a hidden symmetry implying good particle confinement (quasisymmetry) have been found using optimization. However, a direct geometric construction of quasisymmetric fields, pioneered by Garren and Boozer, is also possible. While the construction is limited to high aspect ratio (shapes resembling a skinny bicycle tire), it has several advantages compared to optimization, such as ordersofmagnitude reduced computational cost and opportunities for analytic insight. We prove an existence and uniqueness theorem for the construction, enabling precise understanding of the size of the solution space. We also demonstrate that the central regions of quasisymmetric fields obtained by optimization match the construction. It is shown that the construction can be modified for a more general confinement condition, omnigenity. The speed of the construction enables wide and highresolution numerical searches to identify and map the interesting regions of parameter space.
Chris Hegna
University of WisconsinMadison
Turbulent Transport Optimization in Stellarators
A frontier research area in stellarator optimization is to use 3D shaping of the magnetic field geometry to reduce turbulent transport. Microinstability induced turbulent transport is the dominant transport channel in presentday optimized stellarators. The ability to define means to control the turbulence properties of magnetic confinement devices through external magnetic fields has the potential to considerably improve the attractiveness of magnetic fusion.
As directly coupling nonlinear turbulence simulation codes into stellarator optimization schemes is impractical, the approach pursued here is to develop an analytically based “metrics” for turbulence transport rates that can be easily incorporated into optimization tools. Recent work has focused on understanding turbulent saturation properties. The theory relies on a paradigm of nonlinear energy transfer from unstable to damped eigenmodes at comparable wavelength as the dominant turbulent saturation process. This mechanism is enabled by a threewave interaction where the identity of the dominant nonlinear energy process is controlled by the properties of the 3D geometry. The nonlinear energy transfer physics is quantified by the product of a turbulent correlation lifetime and a geometric coupling coefficient. The turbulent correlation time is determined by a threewave frequency mismatch, which can be calculated from the sum of the linear eigenfrequencies of the three modes. Larger turbulent correlation times denote larger levels of nonlinear energy transfer and hence smaller turbulent transport.
The theory provides an analytic prediction for how 3D shaping can be used to lower turbulent transport in stellarators. Initial comparisons of the theoretical predictions with nonlinear turbulent simulations are favorable. In particular, the theory suggests that quasihelically symmetric configurations may have an intrinsic advantage with regard to turbulent saturation relative to other optimized stellarators. An important next step in this work is to implement the turbulent optimization metric into optimization design studies. We will show recent progress on this front.
David Bindel
Cornell University
Global, robust, multiobjective optimization of stellarators
We cannot measure how good a stellarator is with a single number. Among other things, an “optimal” stellarator should approximately satisfy desirable field symmetries, satisfy macroscopic and local stability, minimize different types of transport, and minimize engineering complexity. We also want designs that are robust to small variations and do not require unrealistically tight tolerances for coil manufacturing and placement. These objectives and constraints are complicated, nonconvex functions fo the plasma boundary and coil shapes, and they may be subject to subtle tradeoffs.
In this talk, we describe why stateoftheart stellarator optimization tools do not yet fully explore the range of design tradeoffs to find robustly “optimal” designs. We discuss formulations of robust optimization and show some results achieved by members of our group in the context of stellarator optimization. We then discuss the shortcoming of the “scalarization” formulation of multiobjective optimization used in the current generation of stellarator optimizers, showing why this approach may miss parts of the Pareto frontier of best tradeoffs. Finally, we discuss how to go beyond “black box” approaches with optimization algorithms that use both derivative information and physicsbased approximations to quickly explore and converge to the globally best parts of the design space.

Posters
Aaron Bader University of WisconsinMadison Pitfalls and Paths Forward for Stellarator Optimization Antoine Baillod Ecole Polytechnique Fédérale de Lausanne Constraining the toroidal current profile in SPEC equilibria Allen Boozer Columbia University Turnstiles, Cantori, and Stellarator Divertors Michael Cole Princeton Plasma Physics Lab Global gyrokinetic simulations of ion temperature gradientdriven instabilities in stellarators Wilfred Anthony Cooper Retired from Swiss Plasma Center 3D Free Boundary VMEC Equilibria for SPEC Fixed Boundary Simulations in W7X Robert Dewar Australian National University Variational Formulations of Relaxed Fluid Dynamics and Relaxed MHD with timevarying boundaries Heinke Frerichs University of Wisconsin  Madison Scoping studies for the divertor design in optimized stellarators LiseMarie ImbertGerard University of Maryland Bits and pieces from the PEq'tionary project Arunav Kumar Australian National University JimFelix Lobsien Max Planck Institute for Plasma Physics Stochastic Stellarator Coil Optimization Joaquim Loizu Ecole Polytechnique Fédérale de Lausanne Stability and saturation of current sheets in helicityconserving plasma relaxation Dhairya Malhotra New York University BIEST: A HighOrder Boundary Integral Equation Solver for Computing Taylor States Harry Mynick Princeton Plasma Physics Laboratory Constraining stellarator optimization to the quasisymmetric subspace with QSC Elizabeth Paul University of Maryland Adjoint methods for efficient stellarator optimization and sensitivity analysis Benjamin Peherstorfer Courant Institute of Mathematical Sciences, New York University "Multifidelity methods for uncertainty quantification and optimization under uncertainty" David Pfefferlé University of Western Australia Rigidity of MHD equilibrium states to smooth ideal motion Gabriel Plunk Max Planck Institute for Plasma Physics Hong Qin Princeton Plasma Physics Laboratory Spontaneous paritytime symmetry breaking as an instability mechanism for magnetized plasmas Zhisong Qu Australian National University Donald Spong Oak Ridge National Laboratory Energetic particle physics and optimization issues for stellarators Yasuhiro Suzuki National Institute for Fusion Science Cesare Tronci University of Surrey, UK & Max Planck Institute, Germany A lowfrequency variational model for energetic particle effects in the pressurecoupling scheme Andrew Ware University of Montana Adelle Wright Australian National University Resistive stability of cylindrical MHD equilibria with radially localised pressure gradients Caoxiang Zhu Princeton Plasma Physics Laboratory Quick identification of dangerous coil deviations using a Hessian matrix method