Simons Collaboration on Hidden Symmetries and Fusion Energy Annual Meeting 2021

2020 Collaboration Report
The most compelling transformational use of magnetically confined, hightemperature plasma is to realize sustained fusion energy. In the stellarator, which is the leading alternative to the tokamak, the confining magnetic field is mostly produced by external currentcarrying coils. 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 good confinement properties of tokamaks while overcoming some of the inherent drawbacks of the latter (such as disruptive instabilities). A form of hidden symmetry is quasisymmetry (QS), in which the magnitude of the magnetic field B, when expressed in a particular coordinate system, depends only on two coordinates while B depends on all three. The members of the Collaboration have made significant progress on QS during 2020 on the following fronts:
A mathematical formulation of QS. We have shown for the first time how to formulate precise mathematical conditions for QS based entirely on the properties of the particle Lagrangian in a magnetic field, independent of the properties of the underlying magnetostatic equilibrium. This has enabled us to establish a deep connection between QS and magnetohydrodynamic (MHD) equilibria with anisotropic pressure and formulate a constrained variational principle that yields anisotropic equilibria as relaxed states subject to ideal MHD invariants. (These results are consistent with the ConstantinDrivasGinzberg theorem, proved under some assumptions, for the existence of forced MHD equilibria.)
Asymptotic equilibria. Our team has made many advances in understanding stellarator equilibria using an expansion in large aspect ratio, i.e., the major radius of the torus compared to the minor radius. This expansion enables a reduction of the equations for MHD equilibrium, reducing the solution time by several orders of magnitude. In the last year, we have shown that certain MHD instabilities can be assessed directly from the reduced equations. Furthermore, one can analytically compute many other figures of merit for stellarators, such as the maximum plasma volume and the errors in QS, and all the geometric quantities entering the equations for plasma turbulence.
Experiments on the LHD and W7X stellarators have shown that most of the energy losses are due to plasma turbulence. It would be extremely desirable to reduce these losses by optimizing the magnetic geometry, if at all possible. Most of the turbulence is due to the iontemperature gradient (ITG) instability. Our team has revisited the basic theory needed to understand how to shift the onset of ITG turbulence in stellarators using gyrokinetic theory from first principles.
Direct gradientbased stellarator design for QS. In our new approach to stellarator coil design, we directly optimize coil shapes to produce a QS vacuum magnetic field with a target rotational transform. It differs from the traditional twostage approach in which first a magnetic surface with desirable physics properties is found and then coils that aim at realizing this surface are designed. We compare riskneutral (i.e., good designs on average) and riskaverse (i.e., avoiding poor designs) approaches. In addition to improved robustness, we observe that the coils found from stochastic optimization are significantly less complex and hence easier to manufacture.
Simons Optimization (SIMSOPT) Code. We have made substantial progress on stellarator optimization software. The MANGO library of parallelized optimization algorithms has been incorporated into the community stellarator design codes STELLOPT and ROSE, giving them access to many new algorithms. SIMSOPT, our new and more modular framework for stellarator optimization, is now able to optimize magnetic configurations for QS and to eliminate magnetic islands, i.e., to have good flux surfaces, using the MHD equilibrium codes VMEC and SPEC. As part of this project, we have begun to write flexible C++/python replacements for legacy fortran77 code used in the stellarator community. A set of benchmark stellarator optimization problems has been defined and collected, allowing us to confirm that STELLOPT and SIMSOPT obtain the same answers when they should.
Pressure effects on the magnetic topology in stellarators. The SteppedPressure Equilibrium Code (SPEC) was recently extended to freeboundary and also made capable of calculating equilibria at fixed toroidal current profile. The algorithms in SPEC were also recently optimized, thereby speeding up the code by a factor of ~50 or more. Leveraging these new capabilities, we are exploiting SPEC to assess the robustness of the magnetic topology in stellarators as the plasma pressure is increased – namely, addressing the question: what is the maximum achievable plasma pressure above which magnetic surfaces are degraded into islands and chaos? We have carried out the first extensive and comprehensive study of the equilibrium pressure limit in a classical stellarator geometry, and have identified different regimes depending on the strength of the bootstrap current. Finally, we have started exploiting the flagship code of this collaboration, SIMSOPT, to eliminate the pressureinduced islands predicted in SPEC equilibria by optimizing the shape of the plasma boundary.
Nonexistence of invariant tori: converse KolmogorovArnoldMoser (KAM) theory. Invariant manifolds are of fundamental importance to the qualitative understanding of dynamical systems. In this work, we explore and extend Robert MacKay’s converse KAM condition to obtain a sufficient condition for the nonexistence of invariant surfaces. KAM theory provides conditions under which a nearly integrable Hamiltonian system or symplectic map can be guaranteed to have invariant tori on which the dynamics are conjugate to rigid rotations. These results are applied using a PoincaréLiouville form, including Beltrami flows, which is appropriate to describe the flow of magnetic field lines in a stellarator.
Isodrastic magnetic fields. We are developing conditions for guidingcentermotion of particles in a magnetic field to have no transitions between different classes of bounce motion nor between bouncing and circulating motion and for the motion in each class to be bounded. This promises to be a significant generalization of omnigeneity. We call it isodrasticity, meaning constant action. Neither MHD equilibria nor a flux function is assumed, though for the circulating particles, some invariant tori are required for confinement.
2020 PPPL/Simons Summer School. We held our second successful international summer school, this year virtually, on the theme “Stellarator Optimization.” The summer school attendees used the 140page primer “Introduction to Stellarators: From magnetic fields to symmetries and optimization” written by members of this Collaboration (LiseMarie ImbertGerard, Elizabeth Paul, and Adelle Wright).

Agenda
12:00 PM Amitava Bhattacharjee
Hidden Symmetries and Fusion Energy—Overview12:30 PM Joaquim Loizu
Pressure effects on the topology of magnetic fields in stellarators12:45 PM Matt Landreman
Advances in stellarator optimization within the Hidden Symmetries collaboration1:00 PM Q&A 
Abstracts & Slides
Amitava Bhattacharjee
Princeton UniversityHidden Symmetries and Fusion Energy—Overview
The Collaboration on “Hidden Symmetries and Fusion Energy” is animated by the beautiful idea that magnetic fields with hidden symmetries provide a unique pathway to realize thermonuclear fusion under controlled laboratory conditions. An example of a hidden symmetry, known as “quasisymmetry,” is that the magnitude B of the vector magnetic field B has an ignorable coordinate in a particular curvilinear coordinate system, even though B may not. The quest for designing quasisymmetric “stellarators” is a grand computational challenge that is confronted by this Collaboration, which brings together a team of applied mathematicians, computer scientists, and plasma physicists drawn from multiple institutions. During the last year, this team has broken new ground on elucidating fundamental mathematical and physical implications of quasisymmetry and producing new methodologies for various components of the Simons Optimization (SIMSOPT) software, which we envision as a stateoftheart computational optimization tool that can potentially exploit the power of supercomputers. In this talk, we will provide an overview of the new developments during approximately the first two years of the Collaboration (with more emphasis on the second year). We will also describe work carried out by some affiliates of the Collaboration in the Princeton Plasma Physics Laboratory on a possible new experiment on stellarators with permanent magnets, which, if successful, could lead to dramatic reductions in the cost for fusion power.
Matt Landreman
University of MarylandAdvances in stellarator optimization within the Hidden Symmetries collaboration
View Slides (PDF)The Hidden Symmetries collaboration is advancing the stellarator fusion concept in part by applying recent ideas from numerical optimization and scientific software design. One area of progress has been the calculation of electromagnetic coil shapes using stochastic optimization. This approach ensures that the magnetic field still provides good confinement when there are small errors in the coil shapes; the resulting widened tolerances will reduce costs. We have also applied Bayesian methods to accelerate the global optimization of coil shapes. Simultaneously, we have been developing the SIMSOPT software framework to pose stellarator optimization problems and connect the required physics codes. In this modular and extensible framework, python scripting allows flexible problem formulation, with compiled physics modules where performance is needed. As a demonstration, we will show the optimization of stellarator magnetic fields for integrability, exploiting Greene’s residue from the theory of dynamical systems.
Joaquim Loizu
Ecole Polytechnique Fédérale de LausannePressure effects on the topology of magnetic fields in stellarators
Threedimensional magnetic fields in a torus generally trace out an intricate combination of magnetic surfaces, magnetic islands, and magnetic fieldline chaos. By exploiting the BiotSavart law, vacuum magnetic fields can be designed such that magnetic field lines approximately lie on nested toroidal foliations. At finite plasma pressure, however, currents generated by the plasma itself perturb the vacuum magnetic field, thus potentially breaking the magnetic surfaces and ultimately impairing confinement. To date, there is no theory nor extensive numerical study characterizing the maximum achievable pressure above which magnetic surfaces are destroyed, nor a theory on the dependency of this critical pressure on the other relevant operational parameters. The main reason is that the accurate and fast computation of 3D magnetohydrodynamic (MHD) equilibria is an outstanding challenge, involving the solution to an intrinsically nonlinear problem that is subject to undesirable pathologies and for which only certain classes of numerically tractable solutions are guaranteed to exist. Exploiting the latter, the SteppedPressure Equilibrium Code (SPEC) has been recently developed and shown to be capable of rapidly and accurately computing 3D MHD equilibria with magnetic surfaces, islands, and chaos. Here, we present the first extensive and comprehensive study of the equilibrium pressure limit in a classical stellarator geometry with a selfgenerated bootstrap current. We identify different regimes depending on the strength of the bootstrap current and provide theoretical explanations for the mechanisms behind these pressure limits.

Public Lecture (Wednesday March 24)
Plasma, Plasma, Burning Bright
Amitava Bhattacharjee, Ph.D.
Head of Theory Department, Princeton Plasma Physics Laboratory, Princeton University4:45 – 5:00 PM ET Webinar waiting room opens
5:00 – 6:15 PM ET Talk + Q&APlasmas permeate most of the visible universe. Utilizing plasmas, scientists hope to make thermonuclear fusion a viable energy source, offering effectively unlimited energy production with no carbon footprint. That quest requires better understanding, control and optimization of plasmas and has sparked the creation of the international ITER megaproject, an axisymmetric toroidal plasma device known as the tokamak that carries an axial current.
In this lecture, Amitava Bhattacharjee will discuss progress in nuclear fusion plasma research and alternatives to tokamaks called stellarators. Stellarators are threedimensional toroidal plasma devices that are nearly currentless and are hence free of disruptive instabilities. Such devices can have a subtle form of hidden symmetry called ‘quasisymmetry.’ Designing quasisymmetric stellarators is an outstanding problem straddling plasma physics, applied mathematics, dynamical systems theory and computational physics. The knowledge gained over the last few decades in fusion plasma physics has directly or indirectly inspired major spinoffs in other areas of science and engineering, ranging from fundamental insights on unsolved problems in the laboratory and the cosmos to applications in materials and medicine.More information is available at the lecture’s page.

Virtual Poster Talks
A collection of 10minute virtual poster talk videos and slide PDFs may be viewed here.