Simons Collaboration on Hidden Symmetries and Fusion Energy Annual Meeting 2022

Amitava Bhattacharjee
Princeton University

Hidden symmetries and Fusion Energy: An Overview

The Simons 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 confronted by this collaboration, which brings together a team of applied mathematicians, computer scientists and plasma physicists drawn from multiple institutions. 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 Framework (SIMSOPT), which is a state-of-the-art computational optimization tool. In this talk, we will provide an overview of some of the new developments during approximately the first three years of the collaboration. We will also discuss plans towards a conceptual design of a stellarator fusion reactor.
 

Joaquim Loizu
EPFL, Lausanne

Fast 3D MHD equilibrium and stability calculations in the presence of islands and chaos

Fast calculations of 3D magnetohydrodynamic (MHD) equilibria are crucial for stellarator optimization. The evaluation of macroscopic plasma stability, neoclassical and turbulent transport, and the design of coils and heat exhaust solutions all require the knowledge of the magnetic equilibrium. While stellarator coils can be designed such that the vacuum magnetic field lines approximately lie on nested toroidal foliations, the presence of finite plasma pressure and currents makes the field generally non-integrable, consisting of an intricate combination of magnetic surfaces, islands and field-line chaos. The accurate and fast computation of 3D MHD equilibria is a fascinating challenge, involving the solution of an intrinsically nonlinear problem subject to undesirable pathologies. Moreover, only certain classes of numerically tractable solutions are guaranteed to exist. Our collaboration has made tremendous progress in developing efficient numerical tools that supply the required MHD input to optimization codes. This talk reviews some of the main achievements in this area. These are, for example, the understanding of equilibrium pressure limits including the effect of bootstrap current, the possibility of performing free-boundary optimization to preserve good magnetic surfaces at finite pressure, the unified calculation of linear ideal and resistive MHD stability, the possibility of rapidly finding nonlinear saturation of MHD instabilities, the definition of criteria for the existence of stepped-pressure equilibria far from symmetry, the understanding of how quasisymmetry and magnetic islands may coexist in finite-pressure equilibria, or the formulation of combined plasma-coil optimization algorithms.
 

Per Helander
Max Planck Institute for Plasma Physics

Optimizing plasma confinement

Energy losses from stellarator plasmas occur through radiation, plasma turbulence and so-called neoclassical transport. The latter dominates at high temperatures and is successfully reduced by magnetic-field optimization in several stellarators, including HSX and W7-X. The highest-performance discharges in W7-X would not have been possible without such optimization [C.D. Beidler et al., Nature 596, 221 (2021)], but most losses are due to turbulence.

Confinement would thus be improved further if the turbulent transport could be reduced by additional magnetic field optimization. Turbulence in magnetically confined plasmas is caused by various microinstabilities, the most important of which are believed to be ion-temperature-gradient (ITG) modes and trapped-electron modes (TEM).

One way of suppressing ITG turbulence is to increase the threshold for linear instability. For this method to be useful in stellarator optimization, a method must be found to calculate the critical temperature gradient at which the plasma becomes unstable rapidly enough to be done inside the optimization loop. The critical gradient is determined by the gyrokinetic system of equations, whose solution usually takes too long, but a simpler method has been found to estimate the instability threshold from basic geometry parameters in flux-tube geometry. This method finds the onset of both ITG modes, which extend broadly along a magnetic field and those that are localized within a well in the drift curvature. The calculation is simple, rapid, and can be applied to all stellarator cases considered so far.

Another way of weakening ITG turbulence is to allow linear instability but reduce the nonlinear saturation amplitude. Efforts to understand how 3D shaping impacts turbulence saturation processes are therefore being pursued. The theoretical approach relies on the premise that the dominant saturation mechanism is a nonlinear energy transfer from unstable to damped modes at comparable wavelengths via a three-way interaction. Reduced fluid models and analytical theory are used to derive quantitative metrics for turbulent transport. These metrics are being tested and refined using nonlinear gyrokinetic simulations and are starting to be implemented in stellarator optimization schemes.

Trapped-electron modes can also cause turbulence and substantial energy losses but are somewhat easier to control. This is because they rely on magnetically trapped electrons in regions of unfavorable magnetic curvature (convex field lines), which can be avoided by increasing the field strength in these places. Importantly, it is possible to combine this goal with the suppression of neoclassical transport by optimizing the field to be quasi-isodynamic. W7-X is, to a first approximation, optimized in this way and therefore enjoys reduced TEM activity and relatively good confinement in plasmas with steep density profiles.
 

Elizabeth Paul
Princeton University

Advancing the stellarator with adjoint-based optimization

Modern stellarator design requires numerical optimization to navigate the high-dimensional spaces used to describe their geometry. Physical insight into the self-adjointness properties of the underlying partial differential equations enables advanced optimization methods through the efficient calculation of sensitivity information. However, these so-called adjoint methods, widely applied in the fluid dynamics community, have only recently begun to impact stellarator design.

We present the first applications of adjoint methods for stellarator design. An adjoint drift-kinetic equation is derived based on the self-adjointness property of the Fokker-Planck collision operator. This adjoint method allows one to understand the sensitivity of neoclassical quantities, such as the radial collisional transport and bootstrap current, to perturbations of the magnetic field strength. The self-adjointness property of the magnetohydrodynamic force operator enables the evaluation of the sensitivity of equilibrium properties to perturbations of coil shapes or the plasma boundary. Adjoint methods enable recent advances in stellarator design, including the efficient calculation of equilibria with excellent hidden symmetry.
 

Peter Constantin
Princeton University

On magnetohydrostatic equilibria

We discuss MHD equilibria and free boundary MHD equilibria. We review notions of quasisymmetry and the construction of smooth, nearly quasisymmetric magnetohydrostatic solutions (MHS) with good flux surfaces sustained by a small force. We consider the issue of constructing free surface MHS with prescribed nested flux surfaces. We prove strong rigidity results in 2D: if the magnetic and velocity fields are not commensurate, the only possible domain is a disk, and the velocity and magnetic fields are circular. We show how it is possible to break this rigidity by imposing external magnetic fields. We discuss also penalized variational and dynamical methods of construction of MHS in 3D.
 

Matt Landreman
University of Maryland

Innovations in stellarator optimization for quasisymmetry

Quasisymmetry is a property of some magnetic fields that can enable the confinement of charged particles and plasma. Quasisymmetry can be realized in toroidal devices called stellarators. Here we present several new stellarator configurations that possess quasisymmetry throughout a volume to significantly higher precision than demonstrated previously. As a result, these new configurations have much-improved confinement of charged particles and plasma. The tool used to generate the new designs is SIMSOPT, a software framework developed through the collaboration. Finally, a new method is presented for including the electric current in the plasma, known as the bootstrap current, self-consistently in the optimizations.
 

Florian Wechsung
New York University

Derivative-based algorithms to find coil designs that are based on manufacturing errors

When designing a new stellarator, the coils that generate the magnetic field used to contain the plasma are crucial. Once designed, building these coils is extremely expensive, and tight manufacturing tolerances only further increase the cost. We present novel derivative-based algorithms implemented in SIMSOPT to incorporate manufacturing errors during the design process, show the improved robustness of the resulting designs, and systematically study the optimization landscape. We also give an overview of SIMSOPT capabilities regarding the handling of geometric objects such as coils, coil perturbations, magnetic axis, and surfaces, and discuss the graph-based representation of objective functions in SIMSOPT and how it is used to compute derivatives automatically.
 

Thomas Sunn Pedersen
Max Planck Institute for Plasma Physics

Towards an optimized stellarator design

Analytic, numerical and experimental results have been pouring out in recent years, verifying that the optimization of stellarators for fusion energy production can successfully deliver an attractive steady-state, energy-producing power plant. This is further boosted by recent technological advances, both internal and external to fusion energy research, such as breakthroughs in high-temperature superconductor technology, advances in stellarator coil design tools, improved 3D/additive manufacturing capabilities in industry, and advances in neutron blanket engineering solutions. We will present plans to extend the existing physics optimization tools of SIMSOPT with code capabilities that capture engineering and manufacturing aspects of stellarator fusion power plants to enable swift and low-capital-cost stellarator reactor development.