2025 Simons Collaboration on Hidden Symmetries and Fusion Energy Annual Meeting

Fusion has the potential to provide clean baseload energy, and among fusion energy concepts, the stellarator has a uniquely strong reliance on computational design optimization of the plasma and magnet shapes. The Simons Collaboration on Hidden Symmetries and Fusion Energy has brought together an interdisciplinary team of experts on plasma physics, optimization, partial differential equations, and dynamical systems to address these design challenges. The 2025 annual meeting of the Hidden Symmetries and Fusion Energy collaboration was held in New York City, March 20–21, 2023. The attendees, numbering over 130, included faculty, staff, postdocs, and students from a diverse range of institutions around the world.

The meeting opened with a few words from Prof. Amitava Bhattacharjee in memory of Prof. Emeritus Robert (Bob) Leith Dewar, who passed away early last April, soon after attending the 2024 Hidden Symmetries meeting. Bob Dewar was the founding PI from Australia National University and was a giant in the field, with major contributions to the study of magneto-hydro-dynamics (MHD) and dynamical systems. He was a mentor to many in the collaboration and is deeply missed.

In magnetic confinement fusion concepts, magnetic fields are used to confine a hot, dense plasma. There are several different mathematical models of confined plasmas with different levels of physical fidelity, and the first four talks of the meeting touched on several of these models. The opening talk, by Josh Burby of UT Austin, discussed models for the motion of highly energetic alpha particles (helium nuclei) that are produced during the fusion of deuterium and tritium. Such particles usually gyrate rapidly around magnetic field lines, and the so-called guiding center approximation involves an asymptotic expansion in which this rapid gyration is averaged out. Burby pointed out that this model is not always accurate and described a new, non-asymptotic model that he has been working on which shows much better accuracy on several examples.

The second two talks, by Eduardo Rodriguez of IPP Greifswald and Max Ruth of UT Austin, both focused on approximate ideal MHD equilibria computed using another asymptotic expansion which is valid in the vicinity of the magnetic axis, the “core” of the plasma in a stellarator. Rodrigues described using this expansion method as a way to better understand what it means for a stellarator to be quasi-isodynamic (QI), a property that implies good confinement of particle orbits. He then used the expansion numerically to map the space of QI stellarators and assess their properties. Ruth described the convergence behavior of near-axis expansions from a mathematical perspective and showed how the approach can be tweaked by adding a small regularization term that guarantees convergence.

While guiding center dynamics are useful to understand individual particles in a plasma and ideal MHD computations help in understanding the equilibrium states of confined plasmas, neither of these models suffices to describe the important phenomenon of plasma turbulence. For turbulence modeling, the standard approach is a more expensive gyrokinetic simulation, as discussed in the talk of Georgia Acton on the numerical methods used in the stella code. In stellarators, the magnetic field is largely organized through nested “flux surfaces” to which the magnetic field is everywhere tangent. Particles do not easily move across flux surfaces, and so the methods Acton presented are able to simulate turbulence over one flux surface at a time. Focusing on a surface captures more physics than focusing on a quasi-one-dimensional “flux tube,” as has been done with other versions of the stella code.

Two other speakers looked at further aspects of modeling and optimization of stellarators outside the central plasma. Alan Kaptanoglu of NYU spoke on recent work in the collaboration on the design of the electromagnetic coils outside of stellarators that are used to provide the external magnetic fields that confine the plasmas. Beyond describing how to devise appropriate coil shapes, Kaptanoglu discussed other elements that can be used to shape the magnetic field, including permanent magnets and dipole arrays, and he described how to optimize mechanical engineering aspects of the coil design, including adding models of strains, forces, and torques into the optimization. Chris Smiet of EPFL discussed the design of the magnetic field in the “divertor region” between the central core of the plasma and the wall of the vacuum vessel. Smiet’s approach was strongly influenced by the “turnstile” concept described by Jim Meiss in the 2024 Hidden Symmetries meeting. The talk beautifully combined concepts from dynamical systems, topological analysis of magnetic fields, and numerical experiments with configurations from the QUASR database of stellarators — another product of the collaboration that was discussed in last year’s meeting in a presentation by Andrew Giuliani.

The final talks of both days of the meeting showed off the practical side of the collaboration. On the first day, Antoine Baillod described the Columbia Stellarator eXperiment (CSX), which is currently being developed at Columbia University. With only two interlinked superconducting coils to shape the magnetic field in addition to two circular coils, the design of this experiment is strongly constrained by engineering considerations that Baillod discussed. By coupled optimization of the coils and the plasma shape, the Columbia group has devised a set of promising configurations that are currently under investigation. On the second day, Eve Stenson described the design of the Electron-Positron Stellarator experiment (EPOS). Here, too, a single-stage optimization approach has been important for simultaneously designing an appropriate set of coils and plasma shape. Like Baillod, Stenson emphasized the importance of optimizing with respect to engineering constraints as well as physics constraints, and described how the projects used methods for optimization under uncertainty to find designs with good expected performance even in the case of manufacturing imperfections.

Preceding the annual meeting, a team meeting was held in Princeton on March 17–19, with over 80 attendees. Taking advantage of the fact that many stellarator researchers were traveling to the New York area for the annual meeting later that week, the team meeting provided additional time for the attendees to share results and collaborate. The team meeting included “elevator pitches” for research ideas, additional research talks, and a discussion of future opportunities for the collaboration to continue in to work together after the funding period ends in August 2025.

Josh Burby
University of Texas at Austin

The non-perturbative adiabatic invariant is all you need
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Developing reduced models for highly-oscillatory dynamical systems traditionally proceeds by applying asymptotic averaging methods. However, the quality of asymptotic averaging degrades as timescale separation decreases. In studying a classical application of asymptotic averaging methods, charged particles moving in a strong inhomogeneous magnetic field, we identified a regime of marginal timescale separation where asymptotic averaging fails quantitatively in spite of strong indications that a good averaged model ought to exist. We developed a non-perturbative, data-driven averaging method for the marginal regime and found the resulting non-perturbative averaged model significantly outperforms asymptotic averaging, even when accounting for corrections from higher-order averaging. I will explain the method in general and in the charged particle context.
 

Eduardo Rodriguez
IPP Max Planck Institute – Greifswald

Understanding quasi-isodynamicity
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Quasi-isodynamicity (QI) is a property of magnetic field-plasma systems that enables the confinement of particle orbits necessary to undergo thermonuclear fusion. The class of QI stellarators, as an alternative to quasisymmetric ones, is garnering ever increasing interest due to its distinct advantages. However, this class is more complex, requiring additional effort to understand the broad implications of QI on the design of stellarator devices.

This presentation aims to explore the nature of QI fields from a fundamental perspective, focusing on how the defining characteristics of QI influence magnetic field properties. By doing so, we provide a broader understanding of how various aspects of the system are interconnected. In particular, we will identify which properties are easily achievable and which ones present challenges. While much of the discussion will be framed in general terms based on robust physical principles, many statements become rigorously true in the context of the near-axis description of the field, which we shall carry along through the discussion.

This talk will also go beyond purely theoretical considerations, applying the developed concepts — particularly those from the near-axis model — to a more practical context. We present a systematic survey of the space of QI stellarators grounded in recent advancements in their near-axis theory. These developments enable us to assess such things as neoclassical transport within the near-axis framework, and offer a rapid method for designing MHD-stable fields with minimal plasma boundary shaping.
 

Max Ruth
University of Texas at Austin

Regularization and Convergence of the Near-Axis Expansion
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Through the course of the Simons collaboration, the near-axis expansion has become a ubiquitous technology for efficiently investigating stellarator configurations. The benefits of the near-axis expansion include its speed, ease of use, and simple expressions for quasisymmetry. In this talk, we analyze the near-axis expansion from a numerical point of view. We show that the vacuum near-axis expansion can be proven to converge when properly regularized. On the other hand, we find that no reasonable conditions can be given on the input to guarantee convergence in the unregularized case. We confirm this by demonstrating the convergence with real coil fields. These tests further suggest a link between the radius of convergence and the axis-coil distance, akin to the L-grad-B metric.
 

Georgia Acton
University of Oxford

Full Flux Surface of Gyrokinetic Code, stella
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In the current landscape of gyrokinetic modelling, there exists a need for codes that can efficiently capture turbulent phenomena in complex magnetic geometries. As the fusion community progresses toward more sophisticated devices, such as stellarators and advanced tokamaks, the demand for reliable simulation tools grows. Our code aims to provide a robust algorithm for investigating turbulence under varied magnetic geometries, ultimately contributing to a more comprehensive understanding of plasma behaviour and improved design of future fusion reactors.

Our code leverages a pseudo-spectral method that preserves spectral accuracy in the perpendicular direction while employing an implicit algorithm to effectively model dynamics along the magnetic field. This approach not only retains spectral accuracy in perpendicular derivatives and gyro-averages but also captures fast parallel dynamics.

Here we present the full flux surface version of stella, which incorporates an iterative-implicit treatment that offers fully-implicit and mixed implicit-explicit options. This approach allows for larger time steps in the simulation of kinetic electron behaviour, significantly enhancing computational efficiency compared with fully explicit codes, and ultimately aims to reduce the computational cost of electrostatic turbulent simulations with kinetic electron effects.

To demonstrate the effectiveness of the new approach, we will present a comparative analysis of flux-tube simulations against the full flux version of stella, using both adiabatic and kinetic electrons. The hope is this tool will contribute to ongoing discussions regarding the effects of zonal modes in three-dimensional geometries and contribute to future advancements in turbulence modelling within the fusion research community.

Antoine Baillod
Columbia University

Design and Optimization of the Columbia Stellarator eXperiment
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The Columbia Stellarator eXperiment (CSX), currently in the design phase at Columbia University, aims to investigate quasi-axisymmetric plasmas at a small aspect ratio and validate recent advancements in stellarator theory, optimization, and technology. CSX is designed to test key theoretical predictions, including plasma flow damping, magnetohydrodynamic (MHD) stability, and trapped particle confinement. The magnetic field is generated by two circular planar poloidal field (PF) coils and two shaped interlinked (IL) coils, with potential additional coils to enhance shaping and flexibility. The PF coils and vacuum vessel are repurposed from the former Columbia Non-Neutral Torus (CNT) experiment, while the IL coils will be fabricated in-house using non-insulated high-temperature superconducting (HTS) tapes. These coils undergo shape and strain optimization to achieve the desired plasma configuration while meeting engineering constraints. A major challenge in CSX’s design is finding a plasma shape that satisfies physics objectives while being realizable with a limited number of coils. The constrained coil set restricts the range of achievable plasma shapes, making the traditional two-stage stellarator optimization approach impractical. Instead, we employ novel single-stage

optimization techniques, where plasma and coils are optimized concurrently. While this increases problem complexity, it enables the discovery of configurations that satisfy both engineering and physics requirements. We compare two single-stage optimization methodologies and explore their application to CSX’s design, aiming to identify configurations that generate a plasma regime suitable for the experiment’s objectives. A set of promising configurations is then selected and further refined using a multi-filament model that accounts for the finite thickness of the coils. Finally, the robustness and sensitivity to manufacturing and installation errors are assessed for some noteworthy configurations. The shape gradients of key metrics and the performance degradation under random coil perturbations are evaluated. Our results underscore the feasibility of the CSX design and provide critical insights that will inform the coil engineering.
 

Alan Kaptanoglu
New York University

Recent advances in stellarator coil optimization
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We will review recent work in the field of stellarator coil optimization, including adding strains, forces, and torques, as well as new methodologies based on voxels, dipole arrays, passive arrays, and more.
 

Chris Smiet
École Polytechnique Fédérale de Lausanne

Turnstiles and topological index in fusion reactor divertors
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A divertor often exploits the topology of ‘x’-points –– hyperbolic fixed points in the Poincaré map of the magnetic field line trajectories –– to create diversion of field lines and increase the separation between the plasma and wall. Around these x-points the magnetic field can appear chaotic, and trajectories in the Poincaré map do not trace out neat surfaces, but appear randomly distributed. Despite appearing chaotic, field line flow is deterministic, and the amount of field lines that pass through a given boundary can be precisely quantified. In particular, an important quantity for analyzing chaotic transport in Hamiltonian systems is the so-called turnstile, which dictates the exchange of magnetic flux between two well separated regions of space, and whose area quantifies the efficiency with which this exchange happens.

In this talk, we present a numerical algorithm that can evaluate the turnstile area directly from a stellarator coil solution, and which is efficiently calculated using an action principle by Meiss. We also explore how the magnetic topology (i.e. the location and interactions between fixed points) and the magnetic chaos can be controlled in the edge of fusion reactors.

Using this tool, we explore select configurations from the QUASR database which exhibit novel divertor topologies where transport to the wall is mediated through interacting turnstiles. Furthermore, we show that in a low-iota configuration of W7-X, the turnstile mechanism causes intricate features in connection length plots used to analyze divertor heat loads. Finally, we present the first results of optimizations on W7-X and other configurations where fixed-point topology and stochasticity in the divertor region is controlled.
 

Eve Stenson
Max Planck Institute for Plasma Physics

Optimization & engineering for EPOS

Electron-positron plasmas are the quintessential “pair plasmas” (comprising positively and negatively charged particles of equal mass), and a moderately high-field, tabletop-sized stellarator (with ~2-ton axis and ~10-liter confinement volume) is an attractive option for laboratory trapping of these plasmas in the low-temperature (eV or less), strongly magnetized (r_L << λ_D ) regime. This will facilitate experimental comparisons to theoretical and computational predictions of plasma phenomena in these unusually symmetric systems. In turn, positrons and e+e- pair plasma offer a sensitive probe of confinement properties and hence a validation of modern stellarator optimization (e.g., more robust construction tolerances and a high degree of quasisymmetry). The mission of EPOS (Electrons and Positrons in an Optimized Stellarator), part of the APEX (A Positron-Electron eXperiment) Collaboration, is thus to unite and advance these two plasma physics frontiers.

The Simons HSFE collaboration and the SIMSOPT framework have been instrumental to addressing key optimization and engineering questions for EPOS. Despite its exotic target, these questions relate to those for fusion efforts in many ways. Milestones have included, for example: the calculation, fabrication, and successful tests of small (10-cm-scale), non-planar coils made from high-temperature superconducting (HTS) tape; implementation of single-stage, stochastic optimization, with the inclusion of HTS strain and finite-build coils, to produce highly quasisymmetric magnetic field configurations even when subjected to errors/uncertainties; and the integration of more device-specific needs, such as a “weave lane” for e+ injection from an external beam line. This talk will provide an overview of those results, the resulting design for the device, and the plans for building it in the coming year.