2024 Simons Collaboration on Hidden Symmetries and Fusion Energy Annual Meeting
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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.
2024 Simons Collaboration on Hidden Symmetries and Fusion Energy Annual Meeting
Summary
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 2024 annual meeting of the Hidden Symmetries and Fusion Energy collaboration was held in New York City, March 21–22, 2024. The attendees, numbering over 140, included faculty, staff, postdocs and students from a diverse range of institutions and companies around the world.
Stellarator design and analysis is a rich source of problems for mathematicians, though the vocabulary barrier between physicists and mathematicians has been challenging. The meeting began with three talks related to the mathematics of stellarators. In the first talk of the program, Lise-Marie Imbert-Gerard described many opportunities for mathematicians interested in working on fusion plasmas. Imbert-Gerard also advertised a book on stellarators for an applied math audience, co-authored with Elizabeth Paul and Adelle Wright and expected to be published in 2025.
One mathematical challenge in the analysis of stellarators is the existence of desired
non-axisymmetric solutions to the ideal magnetohydrodynamics (MHD) equilibrium problem. While the workhorse codes in the field assume solutions involving smooth fields with nested magnetic surfaces, and standard constructions such as Boozer coordinates only exist in such situations, exact solutions of this form have long been conjectured not to exist in the absence of axisymmetry. David Pfefferlé described progress on some of these issues in his talk on the “fine print” of the MHD equilibrium problem. In magnetic fields that are not foliated by surfaces, efficiently distinguishing regions of chaotic fields from regions with regular surfaces can be a challenge. The weighted Birkhoff average, described in the talk by Jim Meiss, provides an efficient computational procedure to distinguish these cases.
Though the assumption of nested flux surfaces may not exactly hold, codes based on this assumption remain highly useful for stellarator optimization. A talk by Rory Conlin highlighted some of the optimization capabilities in DESC, a relatively recent equilibrium and optimization code of this sort that has seen widespread adoption since its introduction a few years ago.
Compared to prior codes of the same sort, DESC offers a number of attractive features, such as spectral discretization in the radial and poloidal direction for better resolution of the near-axis magnetic field, analytic derivatives computed through auto-differentiation, as well as GPU acceleration. Modern stellarator equilibrium and optimization tools developed over the past few years, such as DESC, SIMSOPT and various tools based on near-axis expansions of MHD equilibria, have immensely expanded the range of optimized stellarator configurations available for us to explore. In her talk, Sophia Henneberg described one new family of hybrid
stellarator-tokamak configurations, with the simple toroidal field coils characteristic of a tokamak together with a novel “banana coil” to provide non-axisymmetric shaping. Somewhat remarkably, only one type of auxiliary coil is needed for these configurations. Andrew Giuliani then described the QUAsi-symmmetric Stellarator Repository (QUASR) of over 300,000
vacuum-field stellarators and associated coils, along with visualization tools for exploring the repository.
Once an optimized design has been studied with an ideal MHD equilibrium code like DESC, it is helpful to study that purported equilibrium with time-domain codes that include additional physics, such as resistivity. One such code is NIMSTELL, a stellarator variant of the NIMROD code widely used to study tokamaks, described in a talk by Carl Sovinec. Simulations in NIMSTELL remain quite expensive, enough to make them awkward for use within an optimization loop. However, in the machine learning community there has been immense recent interest in physics-informed learning of Fourier neural operators (FNOs) from complicated physics simulations, including modeling plasma evolution in the MAST spherical tokamak, and the program concluded with a talk by Anima Anandkumar on these types of learning models. Such models must be trained on observational or simulation data (as well as incorporating training signals based on the underlying PDEs), but once they are trained, they can run several orders of magnitude faster than conventional simulations.
Preceding the annual meeting, a team meeting was held in Princeton on March 18–20, with approximately 85 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 three sessions of five-minute “elevator pitch” talks to prompt further research discussions, updates from each Hidden Symmetries institution and several additional research talks. There was also a tour of fusion experiments at the Princeton Plasma Physics Laboratory.
Anima Anandkumar
California Institute of Technology
AI Accelerating Science: Neural Operators for Learning on Function Spaces View Slides (PDF)
Anima Anandkumar will present exciting developments in the use of AI for scientific applications such as solving PDEs. Neural operators yield 4–5 orders of magnitude speedups over traditional simulations. They learn mappings between function spaces that makes them ideal for capturing multi-scale processes. Anima Anandkumar will demonstrate the utility of using the Fourier neural operator (FNO) to model the plasma evolution in simulations and experiments. Joint work shows that the FNO can predict magnetohydrodynamic models governing the plasma dynamics, 6 orders of magnitude faster than the traditional numerical solver, while maintaining considerable accuracy. Anandkumar and collaborators also use the FNO approach to model the plasma evolution observed by the cameras positioned within the MAST spherical tokamak and is capable of forecasting the full length of the plasma shot within half the time of the shot duration.
Rory Conlin will present an overview of the new stellarator design and analysis results made possible by the DESC code suite. This software package couples equilibrium and optimization codes together to efficiently solve the numerical optimization problems required for next-generation stellarator designs. Unlike finite differences or adjoint methods, automatic differentiation provides access to exact derivatives of any objective function and allows the inclusion of more physics constraints, such as metrics for particle confinement and stability. We demonstrate optimization of MHD equilibria for omnigenity, ballooning stability, alpha particle confinement, turbulent transport and coil feasibility metrics. Conlin will also show new results from coil and winding surface optimization along with free boundary equilibria and single stage optimization of coils and equilibrium simultaneously.
Andrew Giuliani
Flatiron Institute
Direct Stellarator Coil Optimization for Quasisymmetry in Vacuum Field View Slides (PDF)
Recent advances in global stellarator coil optimization have culminated in an automatic workflow for the direct design of stellarators and their coils for accurate quasi-axisymmetry and quasi-helical symmetry in vacuum field. Using these algorithms, Andrew Giuliani is able to explore the
landscape of stellarator coil designs, and more completely study the trade-off between a range of design targets such as total coil length, device aspect ratio, rotational transform and quality of quasisymmetry. These approaches have culminated in a database called QUASR (for ‘QUAsi-symmetric Stellarator Repository’) of over 300,000 vacuum-field stellarators and associated electromagnetic coils. Andrew Giuliani will also give an overview of an online navigator utility to explore the data set.
The two leading magnetically confined fusion concepts are tokamaks and stellarators. Tokamaks, the current record holder for fusion performance, boast compactness and relative simplicity in their shape and coil design. But this simplicity comes at the price of large plasma currents, which can lead to detrimental instabilities (disruptions), posing a serious impediment to steady state operation. Stellarators on the other hand don’t rely on such currents to create the required magnetic field. Instead, the magnetic field needed is produced by the electromagnetic coils, which can be optimized in shape to achieve desirable physics properties. This generally results in a complex design, having a multitude of non-planar shaped coils, whose challenge to construct has so far been regarded as a drawback of the stellarator.
A natural and compelling question is whether one can merge the strengths of stellarators and tokamaks into a single “hybrid” concept: ideally, such a device would have a compact design, simple coils, good confinement and easily achievable steady state operation.
This presentation will show the result of efforts to achieve this goal, in the form of a compact quasi-axisymmetric stellarator-tokamak hybrid. Its coil set consists of standard tokamak coils with the addition of four simple stellarator coils, all identical in shape. Such a machine can operate as a tokamak, a QA-stellarator or anything in-between.
It is also shown to exhibit flux surfaces in vacuum, which could be exploited for a novel start-up scenario, potentially eliminating the need for the expensive central solenoid of a tokamak. To show flexibility in the concept, several additional examples of low-aspect-ratio, high-field-period quasi-axisymmetric equilibria are presented, which were numerically found by optimizing with a new target. The levels of quasi-axisymmetry are sufficient to ensure good fast-particle confinement.
This presentation will first compare mathematical models for various notions of symmetry in toroidal geometries, exploited to improve confinement properties or simplify the geometric description. It will then discuss different models for stellarator optimization, including both physics and engineering objectives, to introduce examples of optimized stellarator experiments.
Jim Meiss
University of Colorado Boulder
The Weighted Birkhoff Average as an Efficient Test for Chaos View Slides (PDF)
Traditional methods to detect chaotic trajectories include visualization (simply looking at the phase portrait or Poincaré section) and the computation of Lyapunov exponents. Both of these require long time computations of the trajectory and — for Lyapunov — its linearization. A number of Lyapunov indicators, including the “Fast Lyapunov Indicator” (FLI) and “Mean Exponential Growth factor of Nearby Orbits” (MEGNO) have been developed to attempt to get around the slow convergence. An alternative technique, “Frequency Analysis” was developed to indicate trajectories that lie on tori with dynamics conjugate to a rotation, thus giving an indicator for lack of chaos. A contrasting idea is the “0-1 Test” which shows chaos if the trajectory can be mapped onto a random walk.
Jim Miess will discuss work with Evelyn Sander and Nathan Duignan on using weighted Birkhoff averages (WBAs) as a test for chaos. Birkhoff’s ergodic theorem implies that when an orbit is ergodic on an invariant set, spatial averages of a phase-space function can be computed as time averages. However, the convergence of a time average can be very slow. In 2016, Das et al. introduced a \(C^\infty\) weighting technique that they later showed gives super-polynomial convergence when the dynamics on the invariant set is conjugate to a rigid rotation with frequency that is sufficiently incommensurate (Diophantine). Using this WBA gives a sharp distinction between chaotic and regular dynamics and allows accurate computation of rotation vectors for regular orbits. This allows one to find critical parameter values at which invariant tori are destroyed in Hamiltonian systems and symplectic maps. We apply these methods to circle, torus, area-preserving, three-dimensional angle-action maps and magnetic field line flows. Comparisons with other techniques show that the WBA is more efficient. It also has the advantage of accurately computing physically relevant quantities like rotation vectors. We also show that the WBA can detect “strange non-chaotic attractors,” invariant sets that are geometrically strange but have zero Lyapunov exponents.
David Pfefferlé
The University of Western Australia
The Fine Print of The Magneto-Hydrostatics Problem View Slides (PDF)
The conjecture by Grad, namely that equilibrium solutions to the magneto-hydrodynamics equations featuring nested toroidal pressure surfaces (and non-recurrent field-lines) can only exist in the presence of symmetry by isometry, is a splinter in the design and optimization of magnetic confinement fusion devices. That the well-posedness of magneto-hydrostatics (MHS) remains unchecked leaves the possibility for unattainable or untameable solutions. The numerical results from various tools are nevertheless remarkably convincing in accuracy and fidelity, and stellarators displaying outstanding confinement properties have been built, and hopefully continue to be improved. To agree on the “fine print” of the MHS problem, namely the explicit and implicit assumptions made to guarantee the desired properties of solutions, David Pfefferlé will review the relationship between magneto-hydrostatics (MHS) states, the existence of symmetries and the availability of straight field-line coordinates. Pfefferlé will delve into peculiar/pathological examples that may occur when detail in assumptions is omitted. In the process, Pfefferlé will highlight a range of robust techniques to analyze magnetic fields.
Carl Sovinec
University of Wisconsin-Madison
Spectral-Element Magnetohydrodynamics for Stellarators View Slides (PDF)
Time-dependent numerical computations can be used to predict the evolution of stellarators, starting from vacuum fields or magnetohydrodynamic (MHD) equilibria, subject to heating or macroscale instabilities. Although non-ideal MHD simulations are widely used for tokamaks and other nominally symmetric configurations, modeling the geometry of stellarators poses greater demands with respect to numerical resolution, hence computational cost. The NIMSTELL code aims to address these challenges through a nodal spectral-element representation, where the degree of polynomials within a 2D plane of elements and the 1D Fourier expansion for a generalized toroidal angle are specified at runtime, allowing either finite-element (h) or spectral (p) refinement. NIMSTELL differs from NIMROD in having a 3D mapping from element to physical coordinates and in using vector potential expanded in the H(curl) space instead of expanding magnetic-field components in the H1 space. Numerical analysis shows that the combination of expansions for the physical fields and the approach for imposing a gauge affect whether the representation admits artificial modes and whether convergence on interchange is from the stable side. However, linear NIMSTELL computations on interchange indicate that representing plasma pressure at greater polynomial degree than flow velocity is needed, which is not expected from the simplified analysis. Three-dimensional computations import equilibria from DESC, and initial applications include a resistive tearing benchmark with the stellarator version of JOREK.
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