Date / Time (ET)  Speaker  Title of Talk (Link) 

July 20(21), 2023 10:0010:05am (EDT) 10:0511:00am (EDT) 
D. Bindel M. Ruth 
Hidden Symmetries Study Updates Dynamics and Invariant Surfaces Abstract: TBA (Video recording of talk) 
July 6(7), 2023 10:0010:05am (EDT) 10:0511:00am (EDT) 
D. Bindel J. Meiss 
Hidden Symmetries Study Updates Weighted Birkhoff Averages Abstract: TBA (Video recording of talk) 
June 22(23), 2023 10:0010:05am (EDT) 10:0511:00am (EDT) 
D. Bindel F. ParraDiaz 
Hidden Symmetries Study Updates Linear Equations for Stellarator Local MHD Equilibria Abstract: TBA (Video recording of talk) 
June 8(9), 2023 10:0010:05am (EDT) 10:0511:00am (EDT) 
D. Bindel N. Mandell 
Hidden Symmetries Study Updates Trinity+GX for Stellarator Profile Prediction Abstract: TBA (Video recording of talk) 
May 25(26), 2023 10:0010:05am (EDT) 10:0511:00am (EDT) 
D. Bindel C. Nührenberg 
Hidden Symmetries Study Updates Linearized Ideal Magnetohydrodynamic Stability in Stellarators In the past, aspects of linearized ideal MHD stability, e.g. the vacuum field magnetic well, reduced parallel current density, or Mercier's criterion, were targeted for in the optimization of stellarator experiments such as W7X and HSX. In this tutorial, the stability of plasma equilibria is discussed based on the energy principle of ideal MHD in the form of global mode analyses and local stability criteria. The fieldline ballooning equation and Mercier's criterion are derived. Example applications typical for various classes of equilibria (tokamak, stellarator, weak or medium magnetic shear) are used to point out relationships between the violation of the local stability criteria and the spatial structure of global modes aiming at a categorization of ideal MHD stability limits. In the light of experimental results of the W7AS stellarator and of the LHD torsatron, experiments are planned in W7X to clarify the question of how important ideal MHD is in stellarator optimization. (Video recording of talk) 
May 11(12), 2023 7:008:00pm (EDT) 
M. Landreman  Hidden Symmetries Study Updates Efficient Calculation of Internal Magnetic Field & SelfForce for Electromagnetic Coils For designing magnetic fusion facilities, it is important to compute the magnetic field inside the electromagnetic coils due to superconducting quench limits. It is also necessary to compute the Lorentz force on coils to ensure a support structure is feasible. For both the internal field and force calculations, the coils cannot be approximated in the usual way as infinitesimally thin filaments due to divergences when the source and evaluation points coincide, so more computationally demanding calculations are usually required, resolving the finite crosssection of the conductors. Here, we present a new alternative method that enables both the internal magnetic field vector and selfforce to be computed rapidly and accurately within a 1D filament model. The method is obtained by rigorous analysis of the singularity, such that the filament model matches the true highdimensional integrals for the field and force at high coil aspect ratio. The new filament model exactly recovers analytic results for a circular coil, and is shown to accurately reproduce direct finitecrosssection calculations for a nonplanar coil of the HSX stellarator. Due to the efficiency of the model here, it is well suited for use inside optimization, such as in the optimization of stellarator coil shapes. (Video recording of talk) 
Apr 13(14), 2023 7:00 8:00pm (EDT) 7:00 8:00am (AWST) 9:0010:00am (AEST) 
D. Bindel D. Perrella 
Hidden Symmetries Study Updates Existence of Global Symmetries of DivergenceFree Fields with First Integrals The relationship between symmetry ﬁelds and ﬁrst integrals of divergencefree vector ﬁelds is explored in three dimensions in light of its relevance to plasma physics and magnetic conﬁnement fusion. A Noethertype Theorem is known: for each such symmetry, there corresponds a conserved quantity. The extent to which the converse is true is investigated. In doing so, a reformulation of this Noethertype Theorem is found for which the converse holds on what is called the toroidal region. Some consequences of the methods presented are quick proofs of the existence of ﬂux coordinates for magnetic ﬁelds in high generality; without needing to assume a symmetry such as in the cases of magnetohydrostatics (MHS) or quasisymmetry. https://arxiv.org/pdf/2303.03191 (Video recording of talk) 
Mar 30(31), 2023 10:0010:05am (EDT) 10:0511:00am (EDT) 
D. Bindel M. Padidar 
Hidden Symmetries Study Updates Direct Optimization of FastIon Confinement in Stellarators Confining energetic ions such as alpha particles is a prime concern in the design of stellarators. However, directly measuring alpha confinement through numerical simulation of guidingcenter trajectories has been considered to be too computationally expensive and noisy to include in the design loop, and instead has been most often used only as a tool to assess stellarator designs post hoc. In its place, proxy metrics, simplified measures of confinement, have often been used to design configurations because they are computationally more tractable and have been shown to be effective. Despite the success of proxies, it is unclear what is being sacrificed by using them to design the device rather than relying on direct trajectory calculations. In this study, we optimize stellarator designs for improved alpha particle confinement without the use of proxy metrics. In particular, we numerically optimize an objective function that measures alpha particle losses by simulating alpha particle trajectories. While this method is computationally expensive, we find that it can be used successfully to generate configurations with low losses. (Video recording of talk) 
Mar 23  24, 2023  Various  Hidden Symmetries Annual Meeting @ Simons Foundation (NYC) Rogerio Manuel Cabete de Jesus Jorge, IST Lisbon, Portugal Nathan Duignan, University of Sydney Benjamin Faber, University of WisconsinMadison Daniel Ginsberg, Princeton University Egemen Kolemen, Princeton University Gabriel Plunk, Max Planck Institute for Plasma Physics Josefine Proll, Technical University of Eindhoven Adelle Wright, PPPL 
Mar 20  22, 2023  Various  Hidden Symmetries Team Meeting @ Princeton University (Carnegie Center) Agenda: https://hiddensymmetries.princeton.edu/meetings/2023teammeetingmar 
Mar 16(17), 2023 7:007:05pm (EST) 7:058:00pm (EST) 
D. Bindel D. Anderson 
Hidden Symmetries Study Updates History of the HSX, Optimization and Design (Video recording of talk) 
Mar 2(3), 2023 10:0010:05am (EST) 10:0511:00am (EST) 
D. Bindel S. Hudson 
Hidden Symmetries Study Updates A Tutorial/Review of AlmostInvariant Surfaces (QuadraticFluxMinimizing and Ghost Surfaces) and Heat Transport in NonIntegrable Fields An introduction/tutorial on ghost surfaces (GS) and (weighted) quadraticfluxminimizing (QFM) surfaces, perhaps better described as actiongradientsquared surfaces, and why QFM surfaces (without the "weight") are like snowflake fractals on a hot summer's day, and under what conditions ghostsurfaces and QFM surfaces coincide, and how these surfaces are isotherms of the anisotropic diffusion equation in the limit that \kappa_\perp approaches zero, and why don't we use perturbation theory to solve for the anisotropic diffusion equation by expanding about \kappa_\perp = 0. (Video recording of talk) 
Feb 16(17), 2023 7:007:05pm (EST) 7:058:00pm (EST) 
D. Bindel M. Nemec 
Hidden Symmetries Study Updates GradientBased Shape Optimization using HighFidelity Simulations with GoalOriented Error Control Aerodynamic shape optimization is emerging as an indispensable tool in the design of aerospace vehicles. This presentation focuses on several innovations that have made numerical optimization practical in the context of computational fluid dynamics. We begin with the adjoint method. This method provides optimization gradients and is also used to estimate the level of discretization error in the outputs of interest. The benefits are threefold. First, the computational cost of gradient evaluations is essentially independent of the number of design variables. Second, it offers direct control over discretization error through use of adaptive mesh refinement to improve confidence in the optimized designs and to eliminate the requirement of handcrafting a sufficiently general grid that is appropriate for all candidate designs. Third, we obtain additional cost savings by using progressive optimization, where the depth of the adaptive mesh refinement is systematically increased as the design improves. In addition, we highlight a componentbased geometry approach for flexibility in choosing both the shape parameterization and geometric modelers, along with symbolic definition of objectives and constraints for general problem specification. We present design examples involving realworld aircraft configurations in shockdominated flows, including sonic boom shaping for NASA’s new X59 aircraft. (Video recording of talk) 
Feb 2(3), 2023 10:0010:05am (EST) 10:0511:00am (EST) 
D. Bindel I. Farcas / B. Peherstorfer 
Hidden Symmetries Study Updates Leveraging Machine Learning for LargeScale MultiFidelity Uncertainty Quantification Recent advances in computational science and highperformance computing enable the simulation of largescale realworld problems such as turbulent transport in magnetic confinement devices with everincreasing realism and accuracy. However, these simulations remain computationally expensive even on large supercomputers, which prevents straightforward approaches to important manyquery applications such as uncertainty quantification. In contrast, datadriven machinelearning methods such as those based on deep neural networks provide computationally cheaper lowfidelity models, but typically require large training sets of highfidelity model evaluations to be predictive, which hampers a straightforward application in largescale, computationally expensive problems. In this presentation, we demonstrate that datadriven lowfidelity models learned from few data samples can nevertheless be effectively used for largescale uncertainty quantification: The key is to combine these lowfidelity models with the highfidelity model in a multifidelity fashion. The first part of this presentation focuses on a multifidelity Monte Carlo sampling approach in which a hierarchy of datadriven lowfidelity models is constructed using both the full set of uncertain inputs and subsets comprising only selected, important parameters. We illustrate the proposed method in a plasma microturbulence simulation scenario concerning turbulence suppression via energetic particles with $14$ stochastic parameters, demonstrating that it is about two orders of magnitude more efficient than standard Monte Carlo methods measured in singlecore performance. This translates into a runtime reduction from around eight days to one hour on 240 cores on parallel machines. The second part of this presentation introduces a contextaware multifidelity Monte Carlo method that optimally balances the costs of training lowfidelity models with the costs of Monte Carlo sampling. Our theory shows that lowfidelity models can be overtrained, which is in stark contrast to traditional surrogate modeling and model reduction techniques that construct lowfidelity models with the primary goal of approximating well the highfidelity model outputs. Numerical experiments in a plasma microturbulence simulation scenario with 12 uncertain inputs show speedups of up to two orders of magnitude compared to standard methods, which corresponds to a runtime reduction from 72 days to about four hours on 32 cores on parallel machines. (Video recording of talk) 
Jan 19(20), 2023 7:007:05pm (EST) 7:057:30pm (EST) 7:358:00pm (EST) 
D. Bindel 
Hidden Symmetries Study Updates Talk 1  Meta Variance Reduction Schemes for Estimation of Alpha Particle Confinement Numerical estimation of energetic particle confinement using Monte Carlo methods remains a computationally demanding task in stellarator optimization loops, due to the large number of simulated trajectories required at each iteration. We introduce meta estimators to accelerate the estimation of confinement statistics, by simultaneously leveraging multiple constituent variance reduction techniques including multifidelity Monte Carlo, importance sampling, and information reuse. Each constituent technique takes advantage of a different facet of the estimation problem, leading to quasimultiplicative speedup when combined. We test our meta estimators, using datadriven surrogates and biasing densities, on a coil optimization seeking to reproduce a quasiaxisymmetric configuration (Landreman and Paul, 2022). We observe that each meta estimator outperforms its constituent variance reduction techniques, with the highest performing meta estimator yielding two orders of magnitude speedup compared to standard Monte Carlo estimation at an equivalent computational cost. (Video recording of Law talk) Talk 2  Geometry Induced Activation of Zonal Flows in Stellarator Plasmas One of the prominent physical processes in confined plasmas is spontaneous formation of mesoscopic coherent structures, e.g., zonal flows, geodesic acoustic oscillations, and radially elongated streamers, which are nonlinearly excited in microscopic turbulence. Then, how can we “activate” or “enhance" such nonlinear structures? The diversity of 3D magnetic geometries in stellarators motivates us to explore the capabilities of enhancing the zonalflow formation. This talk presents a recent activity on the geometryinduced activation of the zonal flows in a context of stellarator optimizations with a nonlinear proxy model. (Video recording of NAKATA talk) 
Jan 5(6), 2023 10:0010:05am (EST) 10:0511:00am (EST) 
D. Bindel D. Ginsberg 
Hidden Symmetries Study Updates On the Distribution of Heat in Integrable and NonIntegrable 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 codimension 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 nonintegrability is defined in terms of the temperature defect distribution and related to the nonintegrable structure of the magnetic field, confirming a physical conjecture of PaulHudsonHelander. 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. (Video recording of Ginsberg talk) 
Dec 22(23), 2022 7:007:05pm (EST) 7:058:00pm (EST) 
D. Bindel H. Yamaguchi 
Hidden Symmetries Study Updates Development of Coil Shaping Based Optimization Code at NIFS (Video recording of talk) 
Dec 8(9), 2022 10:0010:05am (EST) 10:0511:00am (EST) 
D. Bindel 
Hidden Symmetries Study Updates Strong Nonlinearity of MHD Stability in Stellarators In stellarators, one of characteristics on MHD is strong nonlinearity. Because, sophisticated mode couplings of the 3D equilibrium and instability must be considered, and those are saturated nonlinearly. For example, in the LHD experiment, although the Mercier criterion predicts the linearly unstable plasma, the nondisruptive discharge could be realized. The dissipation and damping mechanisms are still mysteries. In this talk, some of results from 3D nonlinear simulations will be discussed to understand the strong nonlinearity of MHD in stellarators. (Video recording of Suzuki talk) 
Nov 10(11), 2022 10:0010:05am (EST) 10:0511:00am (EST) 
M. Landreman D. Spong 
Hidden Symmetries Study Updates History of the HSX and QPS Stellarator Designs QPS (Quasi Poloidal Stellarator) design history from 19902008. (Video recording of Spong talk) 
Oct 27(28), 2022 7:007:05pm (EDT) 7:058:00pm (EDT) 
D. Bindel

Hidden Symmetries Study Updates Introduction to Turnstiles & Transports Turnstiles were introduced by MacKay, Meiss and Percival in 1984 as a mechanism to quantify the flux through broken invariant surfaces, such as the homoclinic tangle formed by an unstable periodic orbit or a cantorus that is the remnant of an invariant torus. A turnstile is constructed from the stable and unstable manifolds of a pair of orbits, and allows one to visualize the transport as the flux through a partial barrier, and show that it can be localized to a single “gap” in the orbit pair. MacKay’s renormalization theory gives a scaling law for the growth of the flux as a function of perturbation strength as a torus is broken. A related power law appears to hold for the exittime distribution from a chaotic region containing regular islands, though a complete theoretical justification is still lacking. (Video recording of Meiss talk) 
Oct 13(14), 2022 10:0010:10am (EDT) 10:1011:00am (EDT) 
D. Bindel R. MacKay / N. Shibabrat / E. Paul 
Hidden Symmetries Study Updates Update on Isodrasticity (MacKay Slides) (Shibabrat Slides) (Paul Slides) Group Presentation on Isodrasticity (Video recording of MacKay/Shibabrat/Paul talk) 
Sep 29(30), 2022 7:007:10pm (EDT) 7:108:00pm (EDT) 
D. Bindel / A. Bhattacharjee D. Bindel 
Hidden Symmetries Study Updates Optimization Under Stability Constraints In this talk, we survey some prior work on the treatment of optimization problems with stability constraints in fields other than plasma physics, and we describe the types of technical issues that frequently occur when incorporating eigenvaluebased functions in the objectives or constraints of an optimization problem. We then give a very preliminary discussion of how we are thinking of incorporating stability constraints in stellarator optimization problems. (Video recording of Bindel talk) 
Sep 15, 2022 10:0010:10am (EDT) 10:1011:00am (EDT) 
D. Bindel 
Hidden Symmetries Study Updates On The Construction Of 3D MHD Equilibria In General Bounded Domains This talk will discuss a recent paper (https://arxiv.org/abs/2208.11109) concerning the construction of magnetohydrostatic (MHS) equilibria in a general 3D domain via a longtime limit of a suitably regularized MHD system. It is a rigorous justification of “magnetic relaxation” in the context of a regularized MHD system. (Video recording of Pasqualotto talk) 
Sep 1, 2022 8:008:10 (EDT) 8:109:00 (EDT) 
D. Bindel

Hidden Symmetries Study Updates CoarseGrained Gyrokinetics For The Critical Ion Temperature Gradient In Stellarators This talk will present a modified gyrokinetic theory to predict the critical gradient that determines the linear onset of the ion temperature gradient (ITG) mode in stellarator plasmas. A coarsegraining technique is applied to the drift curvature, entering the standard gyrokinetic equations, around local minima. Thanks to its simplicity, this novel formalism yields an estimate for the critical gradient with a computational cost low enough for application to stellarator optimization. When comparing against a gyrokinetic solver, our results show good agreement for an assortment of stellarator designs. Insight gained here into the physics of the onset of the ITGdriven instability enables us to devise a compact configuration, similar to the Wendelstein 7X device, but with almost twice the ITG linear critical gradient, an improved nonlinear critical gradient, and reduced ITG mode transport above the nonlinear critical gradient. Preliminary optimization results applying the coarsegraining technique to quasisymmetric stellarator design using SIMSOPT will also be discussed. (Video recording of RobergClark talk) 
Aug 18, 2022 8:008:10 (EDT) 8:109:00 (EDT) 
A. Bhattacharjee

Hidden Symmetries Study Updates Omnigeneous Toroidal Plasma Equilibria (Slides) Omnigeneous Toroidal Plasma Equilibria (Paper) A simple condition is derived for omnigeneous toroidal plasma equilibria, which means that in a collisionless plasma the turning points of a trapped particle remain on the same magnetic surface. Omnigeneity is important for it assures that collisionless particle trajectories are consistent with achieving ignition in toroidal fusion systems. When the magnetic field strength depends on only one angular coordinate in Boozer coordinates, the magnetic field is quasisymmetric, and drift trajectories are confined by a conserved canonical momentum. It is shown that a magnetic field is omnigeneous when it obeys the singleangle constraint at extrema of the field strength. Elsewhere it can be far from quasisymmetric, but must obey a symmetry in a function R about field strength minima. When the field strength depends only on the poloidal angle near extrema, it is called quasipoloidally symmetric. For this case, it is shown that bootstrap current need not be zero and the sign of the electric potential is more obscure than generally assumed. https://arxiv.org/pdf/2208.02391 (Video recording of Boozer talk) 
Aug 4, 2022 8:008:10 (EDT) 8:109:00 (EDT) 
A. Bhattacharjee

Hidden Symmetries Study Updates Minimizing Island Width Sensitivity to Maximize Stellarator Coil Tolerances Stellarators have advantageous physical properties but require complicated engineering. Stellarator equilibrium magnetic fields are nonintegrable in general, while the integrability of stellarator magnetic fields is targeted in optimization and typically determines coil tolerances. This talk discusses the use of FOCUS to optimize coils using gradient based algorithms in order to minimize magnetic field errors. (Video recording of Kruger talk) 
July 21, 2022 8:008:10 (EDT) 8:109:00 (EDT) 
A. Bhattacharjee A. Zocco 
Hidden Symmetries Study Updates Kinetic Infernal Modes and Magnetic Reconnection in Magnetically Confined Fusion Plasmas (Abstract) Magnetically confined fusion plasmas must meet some basic magnetohydrodynamic (MHD) stability requirements in order to be of any use in a reactor. In particular, they must not experience long wavelength instabilities. These can involve the whole plasma column, and have to be either operationally avoided or altogether inhibited by design. The free energy needed for such instabilities to grow is provided by current density and plasma pressure gradients. In this talk I will review the basic concepts of the theory of infernal modes, paying attention to their relation to the reconnecting internal kink mode (especially in W7X) and discuss their extension to fusion relevant regimes in which both electron and ion kinetics are important. (Video recording of Zocco talk) 
June 9, 2022 8:008:15 (EDT) 8:159:00 (EDT) 
P. Helander/ A. Bhattacharjee E. Paul 
Greifswald Retreat, June 27July 8 (15 minutes) Energetic Particle Loss Mechanisms In ReactorScale Equilibria Close To Quasisymmetry The transport of fusionborn alpha particles in 3D equilibria is largely determined by collisionless physics. Several transport mechanisms have been implicated in stellarator configurations, including stochastic diffusion due to transitions, ripple trapping, and superbanana orbits. On longer time scales, many lost trajectories undergo transitions between trapping classes, either with periodic or irregular behavior. Possible optimization strategies for each of the relevant transport mechanisms and perform a comparison between classified guiding center losses and metrics for superbanana transport will be discussed. (Video recording of Helander/Bhattacharjee/Paul talk) 
May 12, 2022 8:008:10 (EDT) 8:109:00 (EDT) 
A. Bhattacharjee W. Sengupta 
Hidden Symmetries Study Updates Preferred Magnetic Axes For Optimal QuasiAxisymmetry This talk will discuss the preferred choice of the magnetic axes for optimal quasisymmetry, which is evident from the numerical optimization of asymptotic expansions near the magnetic axis. The talk will show that the magnetic axis is well described for small rotational transforms by the same equations that govern EulerKirchhoff elastic rod centerlines, and present analytical and numerical evidence applicable for a broad range of quasiaxisymmetric stellarators. (Video recording of Sengupta talk) 
Apr 28, 2022 8:008:10 (EDT) 8:109:00 (EDT) 
A. Bhattacharjee F. Volpe & C. Smiet 
Hidden Symmetries Study Updates Renaissance Fusion  Technology and Plans Renaissance Fusion strives to make stellarators smaller via HighTemperature Superconducting (HTS) coils. It makes them less radioactive and more resilient to alpha particle losses by flowing mesoscale liquid metal walls and simpler to build  by using simpler coil winding surfaces and HTS manufacturing. Initial results will be presented in the areas of coil force minimization, simplification of the coil winding surface, neutronic optimization of the liquid wall materials, design point of a compact, profitable stellarator reactor and retrofitting of a fission powerplant. Paradigmshifting ways of manufacturing HTS stellarator coils and extracting Tritium will be presented. (Video recording of Volpe & Smiet talk) 
Apr 14, 2022 8:008:10 (EDT) 8:109:00 (EDT) 
A. Bhattacharjee R. Mackenbach 
Hidden Symmetries Study Updates Available Energy and its Relation to Turbulent Transport Any collisionless plasma possesses some "available energy" (AE), which is that part of the thermal energy that may be converted into instabilities and turbulence. This talk investigates the AE carried by electrons, which are trapped in a magnetic mirror, and the ability to quickly and cheaply assess the turbulence levels, driven by these trapped electrons. (Video recording of Mackenbach talk) 
Mar 17, 2022 8:008:10 (EDT) 8:109:00 (EDT) 
A. Bhattacharjee R. Dewar 
Hidden Symmetries Study Updates QuasiRelaxed Magnetohydrodynamics (QRxMHD) incorporating Ideal Ohm's Law Constraint (IOL) (part 2) The gap between a recently developed dynamical version of relaxed magnetohydrodynamics (RxMHD) and ideal MHD (IMHD) is bridged by approximating the zeroresistivity "Ideal" Ohm's Law (IOL) constraint using an augmented Lagrangian method borrowed from optimization theory. The augmentation combines a pointwise vector Lagrange multiplier method and global penalty function method and can be used either for iterative enforcement of the IOL to arbitrary accuracy, or for constructing a continuous sequence of magnetofluid dynamics models running between RxMHD (no IOL) and weak IMHD (IOL almost everywhere). This is illustrated by deriving dispersion relations for linear waves on an MHD equilibrium. (Video recording of Dewar talk part 2) 
Mar 3, 2022 8:008:10 (EST) 8:109:00 (EST) 
A. Bhattacharjee R. Dewar 
Hidden Symmetries Study Updates QuasiRelaxed Magnetohydrodynamics (QRxMHD) incorporating Ideal Ohm's Law Constraint (IOL) (part 1) The gap between a recently developed dynamical version of relaxed magnetohydrodynamics (RxMHD) and ideal MHD (IMHD) is bridged by approximating the zeroresistivity "Ideal" Ohm's Law (IOL) constraint using an augmented Lagrangian method borrowed from optimization theory. The augmentation combines a pointwise vector Lagrange multiplier method and global penalty function method and can be used either for iterative enforcement of the IOL to arbitrary accuracy, or for constructing a continuous sequence of magnetofluid dynamics models running between RxMHD (no IOL) and weak IMHD (IOL almost everywhere). This is illustrated by deriving dispersion relations for linear waves on an MHD equilibrium. (Video recording of Dewar talk part 1) 
Feb 17, 2022 8:008:10 (EST) 8:109:00 (EST) 
A. Bhattacharjee A. Punjabi 
Hidden Symmetries Study Updates Magnetic Turnstiles in Nonresonant Stellarator Divertor Nonresonant stellarator divertors have a pair of magnetic flux tubes. One has field lines that go from just outside the outermost confining surface to the surrounding chamber wall, and the other has lines that come inward from the wall. This outwardinward action led to the name magnetic turnstile. Plasma is diverted along both tubes of the pair. The pair of flux tubes cross the annulus between the outermost confining surface and the walls through holes in magnetic cantori, which are the fractal remnants of magnetic surfaces. The exiting and entering flux tubes can be adjacent as in the literature on turnstiles. But, tubes were also found that have the unexpected feature of the entering and the exiting the region near the outermost confining surface at separated locations. Not only can there be two types of turnstiles but pseudoturnstiles can also exist. A pseudoturnstile is formed when an outer surface has a sufficiently large, although limited, radial excursion to strike the wall. The existence of nonadjacent and adjacent turnstiles and pseudoturnstiles resolves issues that arose in earlier simulations of nonresonant stellarator divertors. (Video recording of Punjabi talk) 
Feb 03, 2022 8:008:10 (EST) 8:109:00 (EST) 
A. Bhattacharjee N. Nikulsin 
Hidden Symmetries Study Updates Models and Methods for Nonlinear Magnetohydrodynamic Simulations of Stellarators The JOREK code has recently been extended to allow nonlinear fully 3D stellarator simulations. This is made possible by generalizing the JOREK reduced MHD model to support stellarator geometries, and by allowing the grid to be nonaxisymmetric, so that it can be aligned to the flux surfaces in a stellarator. The stellarator reduced model differs mainly in that the magnetic field can be represented as any curlfree field plus a perturbation in the stellarator model, whereas in the tokamak model it is a toroidal field plus a perturbation. It is shown that this model conserves energy, but introduces an error into momentum conservation. An alternate model, which does not guarantee energy conservation, but has a smaller momentum conservation error is also derived. The energy and momentum conservation properties of the main and alternate models are then studied numerically in the tokamak limit. The main model was then tested on a set of l=2 stellarator equilibria based on Wendelstein 7A. The simulations demonstrate that stable full MHD equilibria are preserved in the reduced model: the flux surfaces do not move throughout the simulation, and closely match the full MHD flux surfaces. Further, both tearing and ballooning modes were simulated, and their growth rates benchmarked against the linear full MHD code CASTOR3D, showing good agreement. (Video recording of Nikulsin talk) 
Jan 20, 2022 8:008:10 (EST) 8:109:00 (EST) 
A. Bhattacharjee E. Rodriguez 
Hidden Symmetries Study Updates Understanding the Space of Quasisymmetric Configurations: Phases and Phase Transitions Optimisation plays a central role in the pursuit of viable stellarator designs. Customarily, such designs result from a search in a large parameter space, guided by desirable physics requirements. This approach has proven useful but the complexity of the space makes the approach, to a large extent, a 'black box'. The talk presents our attempt to shed light on the space of quasisymmetric configurations. We identify designs with a reduced set of functions and parameters that describe the configurations approximately by expansions about their magnet axes. This allows us to structure the space in an effective and powerful way. The presentation will focus on the first step in this approach, which identifies configurations at the most basic level through fundamental geometric properties of their magnetic axes. We show that this reduced level of description is sufficient to naturally divide the space of configurations into quasisymmetric phases and phase transitions. The basic structure of the space can be leveraged to contextualise typical quasisymmetric designs and describe several important properties. We will also present some results on extensions of the basic reduced model. (Video recording of Rodriguez talk) 
Dec 9, 2021 8:008:10 (EST) 8:109:00 (EST) 
A. Bhattacharjee Z. Qu 
Hidden Symmetries Study Updates On The NonExistence Of SteppedPressure Equilibria Far From Symmetry The Stepped Pressure Equilibrium Code (SPEC) has been successful in the construction of equilibria in 3D configurations that contain a mixture of flux surfaces, islands and chaotic magnetic field lines. In this model, the plasma is sliced into subvolumes separated by ideal interfaces, and in each volume the magnetic field is a Beltrami field. In the cases where the system is far from possessing a continuous symmetry, such as in stellarators, the existence of solutions to a steppedpressure equilibrium with given constraints, such as a multiregion relaxed MHD minimum energy state, is not guaranteed but is often taken for granted. Using SPEC, we have studied two different scenarios in which a solution fails to exist in a slab with analytic boundary perturbations. We found that with a large boundary perturbation, a certain interface becomes fractal, corresponding to the breakup of a Kolmogorov–Arnold–Moser (KAM) surface. Moreover, an interface can only support a maximum pressure jump while a solution of the magnetic field consistent with the force balance condition can be found. An interface closer to breakup can support a smaller pressure jump. We discovered that the pressure jump can push the interface closer to being nonsmooth through force balance, thus significantly decreasing the maximum pressure it can support. Our work shows that a convergence study must be performed on a SPEC equilibrium with interfaces close to breakup. These results may also provide insights into the choice of interfaces and have applications in finding out the maximum pressure a machine can support. (Video recording of Qu talk) 
Nov 24, 2021 8:008:10 (EST) 8:109:00 (EST) 
A. Bhattacharjee R. Nies 
Hidden Symmetries Study Updates Adjoint Methods for Quasisymmetry of Vacuum Fields on a Surface Stellarator optimisation can be significantly sped up by using adjoint methods instead of finitedifferences to obtain derivative information. In this work, we apply adjoint methods to stellarator vacuum fields, considering objective functions targeting quasisymmetry and rotational transform on the boundary. To define quasisymmetry on the surface, a novel way of constructing approximate flux coordinate on an isolated flux surface is proposed. The obtained adjoint equations are of a simple form such that they can be solved with existing numerical tools, and yield highly accurate derivatives. Using adjoint methods, we obtain configurations with highly accurate quasisymmetry on the boundary, and we are able to systematically investigate the interplay between quasisymmetry and other optimisation targets, such as the aspect ratio and rotational transform. (Video recording of Nies talk) 
Oct 28, 2021 8:008:10 (EDT) 8:109:00 (EDT) 
A. Bhattacharjee D. Spong 
Hidden Symmetries Study Updates Suppression of Energetic ParticleDriven Instabilities in Stellarators Good energetic particle (EP) confinement is essential for stellarators to achieve efficient heating and protection of plasma facing components. EP confinement is influenced by several mechanisms: classical orbit confinement, slowingdown timescales, and EPdriven instabilities. Stellarator optimization has recently made significant progress towards improved quasisymmetry (which improves classical EP confinement). However, experimental evidence has shown that EP instabilities can enhance EP transport to levels of the same order as classical orbit losses. While EP instability driven transport can to some extent be influenced by profile control, threedimensional shaping is also expected to affect these instabilities. Shaping can be used to control either the MHD wave structures that EP populations resonate with, or the particlewave resonance dynamics. The physics of EP instabilities will be reviewed and some of the new possibilities for EP instability optimization targets described. (Video recording of Spong talk) 
Oct 14, 2021 8:008:10 (EDT) 8:109:00 (EDT) 
A. Bhattacharjee P. Helander 
Hidden Symmetries Study Updates Upper Bounds on Gyrokinetic Instabilities Energy confinement in Stellarators is mostly limited by turbulence, in particular when the plasma temperature is low or the neoclassical transport has been reduced by optimization of the magnetic field. The turbulence and the underlying microinstabilities are thought to be well described by the gyrokinetic set of equations, which have been the subject of thousands of papers and millions of lines of computer code. Yet, little is known about the general properties of the solutions to these equations. Here, a family of rigorous upper bounds on the growth rate of local gyrokinetic instabilities is derived. These bounds hold for both electrostatic and electromagnetic instabilities, regardless of the number of particle species, their collision frequency, and the geometry of the magnetic field. A large number of results that have earlier been derived in special cases and observed in numerical simulations are thus brought into a unifying framework. These bounds apply not only to linear instabilities but also imply an upper limit to the nonlinear growth of the free energy. (Video recording of Helander talk) 
Sep 30, 2021 8:008:10 (EDT) 8:109:00 (EDT) 
A. Bhattacharjee YM. Huang 
Hidden Symmetries Study Updates Structure of PressureDriven Current Singularities The objective of this study is to work out a prototype problem with p' <> 0 on the resonant surface as a concrete example. For that purpose, we use The HahmKulsrudTaylor (HKT) problem, which 1  is amenable to analytic solutions; 2  has been studied with various codes including a GradShafranov solver, a fully Lagrangian code, and SPEC for the case with p = 0. (Video recording of Huang talk) 
Sep 16, 2021 8:008:30 (EDT) 
A. Giuliani  Optimization For QuasiSymmetry on Surfaces in SingleStage Coil Design In this talk, we'll give an update on our singlestage approach to optimizing for quasisymmetry on surfaces. Previously, we've presented our new technique for computing surfaces in Boozer coordinates, currently available in SIMSOPT. In this talk, we'll give addition details and describe how these surfaces can be used in a gradientbased approach to optimize for quasisymmetry. (Video recording of Giuliani talk) 
Sep 16, 2021 8:309:00 (EDT) 
F. Law  Multifidelity Monte Carlo Estimation of Energetic Particle Confinement in Stellarators In the design of Stellarators, energetic particle confinement is a critical point of concern which remains challenging to analyze from a numerical point of view. Due to the absence of fully reliable proxy functions in quantifying the energetic particle confinement properties of magnetic configurations, studies are typically based on a standard Monte Carlo analysis. (Video recording of Law talk) 
Sep 2, 2021 8:009:00 (EDT) 
N. Sato 佐藤直木 
Quasisymmetric Magnetic Fields in Asymmetric Toroidal Domains (Video recording of Sato talk) 
Aug 5, 2021 8:009:00 (EDT) 
M. Landreman  Magnetic Fields With Excellent Quasisymmetry Throughout a Volume (Video recording of Landremantalk) 
July 22, 2021 8:009:00 (EDT) 
S. Henneberg  Representing The Plasma Boundary in Stellarator Optimization (Video recording of Henneberg talk) 
July 8, 2021 8:009:00 (EDT) 
R. White  Particle Resonances in Toroidal Plasmas (Video recording of White talk) 
Jun 24, 2021 8:009:00 (EDT) 
D. PeraltaSalas  MHD Equilibria with NonConstant Pressure in Nondegenerate Toroidal Domains (Video recording of PeraltaSalas talk) 
May 27, 2021 8:009:00 (EDT) 
A. CarltonJones E. Stenson 
Computing The Shape Gradient of Stellarator Coil Complexity With Respect to the Plasma Boundary (Video recording of CarltonJonestalk) Computational Studies of MuBreaking in a Magnetic Dipole and Other Simple Coil Configurations (Video recording of Stenson talk) 
Apr 29, 2021 8:009:00 (EDT) 
M. Landreman  Update on SIMSOPT (Video recording of Landreman talk) 
Apr 15, 2021 8:009:00 (EDT) 
YM. Huang  Numerical Approach to ∂function Current Sheets Arising From Resonant Magnetic Perturbations (Video recording of Huang talk) 
Apr 1, 2021 2:003:00 (EDT) 
A. Geraldini  Adjoint Calculation of Magnetic Island Width Sensitivity (Video recording of Geraldini talk) 
Mar 18, 2021 2:003:00 (EDT) 
R. MacKay / J. Burby 
Isodrastic Magnetic Fields (Video recording of MacKay talk) 
Mar 4, 2021 2:003:00 (EST) 
J. Lion  Generalization of the Systems Code PROCESS to Stellarators 
Feb 18, 2021 2:003:00 (EST) 
F. Wechsung  SingleStage GradientBased Stellarator Coil Design 
Feb 4, 2021 2:003:00 (EST) 
S. Henneberg  Algorithms for Combined Plasma and Coil optimization 
Jan 7, 2021 2:003:00 (EST) 
N. Duignan  A Presymplectic View of Magnetic Fields 
Dec 11, 2020 9:1010:00 (EST) 
W. Sengupta  Obtaining exact quasisymmetry on a single flux surface: a nearsurface expansion approach 
Nov 20, 2020 9:1010:00 (EST) 
A. Guiliani  Singlestage gradientbased stellarator coil design: Optimization for nearaxis quasisymmetry 
Oct 30, 2020 9:2010:00 (EDT) 
E. Rodriguez  Avoiding the problem of overdetermination in quasisymmetric nearaxis 
Oct 2, 2020 9:2010:00Am (EDT) 
J. F. Lobsien  Stochastic Stellarator Coil Optimization 
July 24, 2020 9:3010:00Am (EDT) 
N. McGreivy  FiniteBuild Stellarator Coil Design & Automatic Differentiation 
June 26, 2020 9:2010:00Am (EDT) 
T. Pedersen  Introduction to (Stellarator) Divertors 
June 12, 2020 9:2010:00Am (EDT) 
J. Burby  GradShafranov Equation For NonAxisymmetric MHD Equilibria (slides) https://arxiv.org/pdf/2005.13664.pdf (paper) 
May 29, 2020 9:009:20Am (EDT) 
A. Bhattacharjee  Summer School 2020 Schedule 
May 29, 2020 9:2010:00Am (EDT) 
R. Granetz  Applying High Temperature Superconductor Technology to Stellarators 
May 15, 2020 9:0010:00Am (EDT) 
B. Khesin  Madelung Transform and Binormal Flows in the Euler Hydrodynamics 
May 1, 2020 9:059:30Am (EDT) 
E. Rodriguez  Constructing Quasisymmetry 
May 1, 2020 9:3510:00Am (EDT) 
N. Kallinikos  Approximate Quasisymmetry 
Apr. 17, 2020 9:0010:00Am (EDT) 
S. Glas  Optimization Under Uncertainty 
Apr. 6, 2020 3:00‑4:00Pm (EDT) 
D. Ginsberg  Quasisymmetric Equilibria 
Mar. 23, 2020 3:00‑4:00Pm (EDT) 
P. Helander  Stellarators with Permanent Magnets 
Mar. 9, 2020 3:00‑4:00Pm (EDT) 
A. Bader  Stellarator Optimization in Practice And Where We Can Improve 
Mar. 2, 2020 3:00‑4:00Pm (EST) 
D. Malhotra  Integral Equation Methods for Calculating SteppedPressure Equilibrium in Stellarators 
Feb. 24, 2020 3:00‑4:00Pm (EST) 
A. Giuliani  AdjointBased VacuumField Stellarator Optimization 
Feb. 10, 2020 3:00‑4:00Pm (EST) 
E. Paul  Efficient Stellarator Shape Optimization and Sensitivity Analysis 
Jan. 27, 2020 3:00‑4:00Pm (EST) 
R. Dewar and Z. Qu 
MRxMHD With Flow 
Jan. 10, 2020 9:00‑10:00Am (EST) 
M. Landreman  SIMSOPT phase 1: MANGO 
Dec. 13, 2019 9:00‑10:00Am (EST) 
F. Hindenlang  GVEC: A newly developed 3D ideal MHD Galerkin Variational Equilibrium Code 
Nov. 29, 2019 9:00‑10:00Am (EST) 
J. Loizu  Direct Prediction of Nonlinearly Saturated Tearing Modes with SPEC 
Nov. 15, 2019 9:00‑10:00Am (EST) 
B. Faber  Stellarator Optimization at the University of WisconsinMadison 
Nov. 8, 2019 8:30‑9:30Am (EST) 
Special Discussion on SIMSOPT  
Nov. 1, 2019 9:00‑10:00Am (EDT) 
R. Jorge  NearAxis Expansion Framework at Arbitrary Order in the Distance to the Magnetic Axis 
Sep. 20, 2019 9:00‑10:00Am (EDT) 
Dhariya Malhotra  HigherOrder Integration for Singular Integrals in Magnetostatics 
Sep. 6, 2019 9:00‑10:00Am (EDT) 
Allen Boozer  Curlfree Magnetic Fields for Stellarator Optimization 
Aug. 5, 2019 3:00‑4:00Pm (EDT) 
David Bindel  Surrogate Optimization 
May 13, 2019 3:00‑4:00Pm (EDT) 
Georg Stadler  Optimization 
Apr. 29, 2019 3:00‑4:00Pm (EDT) 
Amitava Bhattacharjee  Summer School Schedule 
Mar. 18, 2019 3:00‑4:00Pm (EDT) 
Amitava Bhattacharjee  Team meeting in PCTS, Annual Meeting at Simons Foundation 