David Bindel, Cornell University
Matt Landreman, University of Maryland
Benjamin Faber, University of Wisconsin-Madison
Nathan Duignan, University of Sydney
Daniel Ginsberg, Princeton University
Rogerio Jorge, IST Lisbon, Portugal
Egemen Kolemen, Princeton University
Gabriel Plunk, Max Planck Institute for Plasma Physics
Josefine Proll, Technical University of Eindhoven
Adelle Wright, PPPL
Fusion offers the promise of carbon-free, safe, dispatchable baseload energy production with widely available fuel. Among fusion concepts, the stellarator is unique in its three-dimensional complexity, leading to a heavy reliance on numerical computation for design and a particularly rich mathematics of the underlying equations.
The 2023 annual meeting of the Simons Collaboration on Hidden Symmetries and Fusion Energy brought together an interdisciplinary and international group of experts in the areas of plasma physics, dynamical systems, partial differential equations, numerical methods, and optimization. A set of important questions for stellarator design was discussed that cut across these disciplines: In the absence of symmetry, how can the singularities in magnetohydrodynamic (MHD) equilibria be understood and resolved, and how should these equilibria be represented? How can transport across magnetic fields be understood in the absence of magnetic surfaces? How can the various types of transport across magnetic fields, including turbulence, be optimized? What are effective algorithms for optimizing the shapes of MHD equilibria and of the magnets that confine them? With progress on these questions, there is opportunity to greatly advance the stellarator fusion concept.
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 2023 annual meeting of the Hidden Symmetries and Fusion Energy collaboration was held in New York City, March 23–24, 2023. The attendees, numbering over 100, included faculty, staff, postdocs and students from a diverse range of institutions around the world. This meeting was the first of the renewal period of the group, and the talks represented exciting new directions building on work done in the first four years of the collaboration.
In magnetic confinement fusion concepts, magnetic fields are used to confine a hot, dense plasma. This system is frequently modeled using the magneto-hydro-dynamic (MHD) equations, and MHD computations play a key role in stellarator design. Talks by Prof. Egemen Kolemen and Dr. Adelle Wright described two state-of-the-art MHD codes: the DESC code for finding ideal MHD equilibria, and the M3D-C1 code for high-fidelity time-domain simulations of the extended MHD equations. The DESC code can compute stellarator equilibrium states fast enough for use in an optimization iteration. It supports a variety of optimization tasks, enabled through modern automatic differentiation tools. In contrast, the M3D-C1 code runs on leadership-class parallel computers, allowing it to include far more detailed physics models in order to provide high-fidelity validation of optimized stellarator equilibria.
Though detailed computational models are important to finding and understanding stellarator equilibria, the collaboration has fostered the cooperation of mathematicians and physicists working on carefully constructed analytical models. This was highlighted in the talk by Dr. Daniel Ginsberg that rigorously analyzed an anisotropic diffusion model for the transport of heat within a stellarator plasma. Dr. Nathan Duignan presented work on understanding the confinement of charged particles from the perspective of Hamiltonian mechanics, and Dr. Gabriel Plunk described the use of near-axis expansions in finding optimized stellarators of the quasi-isodynamic (QI) type.
In earlier years, the collaboration focused largely on finding stellarator equilibria with “hidden symmetries” in order to minimize loss of heat from the plasma due to neoclassical transport. With the gains made in minimizing neoclassical transport during the first part of the collaboration, the group has increasingly begun to focus on loss of energy due to plasma turbulence, as discussed in talks by Prof. Josefine Proll and Dr. Benjamin Faber. Plasma turbulence in stellarators is driven by micro-scale instabilities. Though nonlinear effects cause the growth of these instabilities to eventually saturate, the instabilities still limit how hot and dense the plasma can get. Out of a host of possible micro-instabilities, a few key families like the ion temperature gradient (ITG) and trapped electron mode (TEM) instabilities modes have been shown to be particular problems for stellarator performance. Fully-resolved simulations of plasma turbulence remain a challenge mostly tackled with the use of massive supercomputers. However, the group has done significant work on building reduced models that can capture important aspects of the dynamics, as well as understanding the balance of energy available in the system to drive plasma turbulence. This work suggests a path toward finding inexpensive objectives that capture enough detail of turbulent transport that they can be used to find optimized stellarator designs that minimize turbulent energy transport.
The program concluded with a talk by Prof. Rogerio Jorge on the frontier of optimized stellarator design. This talk brought together many of the technical tools that were in part developed by the collaboration and were described earlier talks, including near axis expansions of stellarator magnetic fields, direct measures of neoclassical transport and proxies for turbulent transport within the plasma, and combined optimization of the plasma equilibrium and the coils providing the confining magnetic field.
Preceding the annual meeting, a team meeting was held in Princeton on March 20–22, with approximately 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 updates from each Hidden Symmetries institution, additional research talks, and group discussions, including a panel with the five private stellarator companies.
Thursday, March 23rd
9:30 AM Nathan Duignan | Minimizing separatrix crossings in guiding center motion through isoprominence 11:00 AM Gabriel Plunk | The Quasi-isodynamic Stellarator 1:00 PM Josephine Proll | Beyond Ion-temperature-gradient Turbulence – Other Instabilities Worth Optimizing For 2:30 PM Adelle Wright | Innovations in High-fidelity Magnetohydrodynamic Modeling for Advanced Stellarators 4:00 PM Benjamin Faber | Optimizing stellarators for reduced turbulence
Friday, March 24th
9:30 AM Daniel Ginsberg | On the Distribution of Heat in Integrable and Non-Integrable Magnetic Fields 11:00 AM Egemen Kolemen | DESC: Integrated Equilibrium Solver and Stellarator Optimizer with Automatic Differentiation 1:00 PM Rogerio Jorge | Direct Optimization for Enhanced Stellarator Design in Magnetic Confinement Fusion
University of Sydney
Minimizing Separatrix Crossings in Guiding Center Motion through Isoprominence
The quality of a stellarator relies upon its confinement of charged particles via a magnetic field. Depending on the choice of theoretical model for the particle or plasma dynamics, one can derive a variety of properties desired for magnetic fields to ensure good confinement. In this talk, we will discuss the desired properties of a magnetic field derived from the guiding center model for charged particle motion. This includes (but is not limited to) omnigeneity, isodrasticity, quasisymmetry, quasi-isodynamic, and isodynamic. Such properties will be understood through the Hamiltonian perspective of guiding center motion due to Littlejohn. In particular, we will investigate a simple property of magnetic fields that minimizes bouncing to passing type transitions. This property, called isoprominence, will be explored through the framework of a near-axis expansion. Ultimately, it is shown that isoprominent magnetic fields exist to all orders in a formal expansion about a magnetic axis. Some key geometric features of these fields will be described through examples.
Max Planck Institute for Plasma Physics
The Quasi-isodynamic Stellarator
Quasi-isodynamicity (QI) is a property that implies collisionless particle confinement in a topologically toroidal magnetic field, where contours of the field strength close poloidally, i.e. the short way around the torus.
QI is typically accompanied by other beneficial properties such as intrinsic stability of trapped particle modes (max-J-ness), and low plasma currents, making the magnetic field particularly robust, i.e. relatively insensitive to the presence of plasma. These properties, underlying the success of the W7-X stellarator, make the QI stellarator arguably the most mature concept among stellarator optimization lines.
To serve as a basis for a fusion reactor, quasi-isodynamic stellarator designs must be found that overcome shortcomings of previous designs. The degree to which QI itself is satisfied must be sufficient for confinement of fusion-born fast particles. Coils producing the field must be buildable, and plasma stability and engineering constraints must also be accounted for. Finally, micro-turbulence, which ultimately sets the limit for confinement, must be addressed. Balancing such an array of desirable properties depends crucially on the ability to understand and confidently navigate the full space of QI configurations.
This talk presents recent theoretical progress underlying QI optimization. Expansion of the equilibrium equations in the distance from the magnetic axis provides the foundation for understanding the basic geometric properties of QI fields. This theory has led to the discovery of new varieties of QI, with unexpected geometric properties, such as unconventional axis helicities, symmetry breaking, and low plasma elongation. Such directly constructed configurations can then be re-optimized using numerical techniques to target the other properties desired in the next generation of QI stellarators.
Technical University of Eindhoven
Beyond Ion-Temperature-Gradient Turbulence – Other Instabilities Worth Optimizing For
When it comes to optimizing stellarators for turbulent transport, the ion-temperature-gradient mode (ITG) has so far taken front and center – with advances in reducing the diffusivity in the strongly-driven regime or more recently in optimizing for high ITG critical gradient. However, while ITGs do lead to large levels of turbulent transport, other instabilities must not be forgotten. Here I want to give a glimpse into what we have already found outside the typical ITG realm.
I will show that for trapped-electron modes, two routes for lower turbulence have emerged: On the one hand it has been found that so-called maximum-J geometry with the majority of electrons (Wendelstein 7-X is an approximation of that) experiencing bounce-averaged good curvature leads to reduced TEM growth rates and thus low TEM turbulence levels. Even in the absence of the maximum-J property, similarly low turbulence levels can be found – in HSX, which boasts very low global shear, a plethora of subdominant and stable eigenmodes are most likely to thank for enhanced saturation efficiency and consequently low turbulence levels. In the absence of the classical trapped electron mode another mode has recently been found to emerge: the so-called universal instability. This mode, with a much broader mode structure than the classical TEM, has been observed to lead to lower heat fluxes than comparable TEM.
Beyond these classically electrostatic modes also effects of increasing plasma pressure need to be taken into account. In W7-X geometry, it has recently been found that an increase in plasma pressure can lead to an early onset of a kinetic ballooning mode far below the typical threshold. This mode, while subdominant to the ITG, can lead to a strong increase in ITG heat flux.
Finally, I want to highlight a tool that might make classifying the instabilities unnecessary – the so-called available energy. The available energy of trapped electrons has previously been shown to work well as an estimate for TEM heat fluxes in different geometries and can now be expanded to also include passing electrons as well as ions, enabling the use of a turbulence proxy for general turbulence.
Innovations in High-fidelity Magnetohydrodynamic Modeling for Advanced Stellarators
The advent of high-fidelity magnetohydrodynamic (MHD) simulations of advanced stellarator configurations has opened a new frontier in understanding the nonlinear, macroscopic characteristics of stellarator plasmas. All stellarator optimization and design activities rely on certain assumptions, such as the existence of magnetic surfaces, MHD stability, pressure profiles, and the dynamical accessibility of equilibria. The M3D-C1 code uses a high-fidelity, macroscopic physics model and provides a unique way of assessing the veracity of these assumptions. This enables high-fidelity validation of optimized equilibria, which has not previously been possible. Specifically, the magnetohydrodynamic evolution of the magnetic field and pressure profiles are simulated subject to anisotropic thermal transport and realistic heating sources, without imposing any constraints on the geometry of the plasma or the existence of magnetic surfaces. This includes the calculation of transport due to MHD instabilities which may saturate at finite amplitude without imposing stiff constraints on the plasma profiles.
M3D-C1 is a flagship extended-MHD code developed primarily at the Princeton Plasma Physical Laboratory that runs on leadership class computers. The code uses a split-implicit time-stepping algorithm with C1-contiuous finite elements to solve a system of nonlinear partial differential equations over disparate timescales to simulate the macroscopic properties of toroidally confined fusion plasmas.
University of Wisconsin-Madison
Optimizing Stellarators for Reduced Turbulence
A key component to achieving economical fusion power with stellarators is improving plasma confinement characteristics by reducing turbulence-driven transport losses. This was clearly demonstrated in recent results from the W7-X stellarator, which showed significant ion thermal transport attributed to turbulence that effectively limits ion temperature profiles. Turbulence is inherently nonlinear and complex in nature, making it impossible to include high-fidelity calculations directly in stellarator optimization routines. Moreover, there is not a single reduced model capable of capturing the myriad of plasma instabilities that can drive turbulence. This makes optimizing for turbulence transport an open problem in stellarator design. This presentation will give an overview of recent advances in turbulence optimization, including the development of reduced models intended to capture key elements of turbulence dynamics; increasing the threshold for linear instability, decreasing the energy available to fluctuations, improving the zonal flow response, and enhancing the coupling to damped modes. The results suggest robust optimization for reduced turbulence is now possible across both quasi-isodynamic and quasisymmetric stellarator configurations.
On the Distribution of Heat in Integrable and Non-Integrable Magnetic Fields
We study the equilibrium temperature distribution in a model for strongly magnetized plasmas in dimension two and higher. Provided the magnetic field is sufficiently structured (integrable in the sense that it is fibered by co-dimension one invariant tori, on most of which the field lines ergodically wander) and the effective thermal diffusivity transverse to the tori is small, it is proved that the temperature distribution is well approximated by a function that only varies across the invariant surfaces. The same result holds for “nearly integrable” magnetic fields up to a “critical” size. In this case, a volume of non-integrability is defined in terms of the temperature defect distribution and related to the non-integrable structure of the magnetic field, confirming a physical conjecture of Paul-Hudson-Helander. Our proof crucially uses a certain quantitative ergodicity condition for the magnetic field lines on full measure set of invariant tori, which is automatic in two dimensions for magnetic fields without null points and, in higher dimensions, is guaranteed by a Diophantine condition on the rotational transform of the magnetic field. This is based on joint work with Theodore D. Drivas and Hezekiah Grayer II.
DESC: Integrated Equilibrium Solver and Stellarator Optimizer with Automatic Differentiation
I will explain DESC, a stellarator code suite, that enables very fast full end-end optimization that incorporates a new fast equilibrium solver. The suite is developed with the following principles 1) Easy to use, open source, well documented, python code with extensive test coverage. 2) Pseudo spectral Zernike basis which allows fast calculations with properly handled core. 3) Automatic differentiation with exact fast end-to-end optimization. 4) Hardware agnostic code which runs on CPUs, GPUs, and TPUs. 5) Modular and flexible architecture with easy extensions. Our comparisons show ~2 orders of magnitude faster optimization compared to STELLOPT and ~2 orders of magnitude less force error for the equilibria compared to VMEC. The code is designed to take any variable as a constraint or an optimization parameter. This allows us to specify the Poincare section in phi=constant or the near axis expansion coefficients exactly at the core (instead of the usual rho=1 surface Fourier coefficients), achieving lower QS and force errors. I will show some examples of QS, QI, and coil optimization and discuss recent results on anisotropic pressure equilibrium and end-to-end turbulence optimization.
IST Lisbon, Portugal
Direct Optimization for Enhanced Stellarator Design in Magnetic Confinement Fusion
The pursuit of magnetic confinement fusion calls for magnetic fields of exceptional quality to sustain high-heat plasmas and regulate plasma density, fast particle management, and turbulence. Traditional design methods for stellarator machines optimize magnetic fields and coils independently, leading to narrow engineering tolerances, and often neglect turbulent transport in the optimization process. However, recent advancements in the optimization of stellarator devices, such as direct near-axis designs, integrated plasma-coil algorithms, precise quasisymmetric and quasi-isodynamic fields, and direct turbulence optimization, are transforming the field by allowing for a more comprehensive and efficient optimization process. This presentation will delve into these breakthroughs and their implications for the future of fusion devices in plasma physics.