# Stellarator Group Receives $2M Simons Mathematical and Physical Sciences Award

Allen H. Boozer, Professor of Applied Physics at Columbia University, is part of a stellarator group that has received a Simons Mathematical and Physical Sciences Award of $2M per year for four years. The Simons Foundation is particularly well known for funding applied mathematics and its applications to the fundamental sciences. Support for work as applied as the mathematical foundations of stellarators is noteworthy. The Foundation wants the title of the grant to clearly state the practical application, *Hidden Symmetries and Fusion Energy.*

The primary institutions participating in this project are Princeton University, Australian National University (ANU), Cornell University, Max-Planck Institute for Plasma Physics-Greifswald (IPP--Greifswald), New York University (NYU), Columbia University, University of Colorado-Boulder, University of Maryland (UMD), University of Texas at Austin (UT Austin), and University of Warwick (Warwick). **Chris Hegna** (APAM alumnus, Ph.D, 1989, Plasma Physics), the Harry D. Spangler Professor at the University of Wisconsin, Madison, is a collaborator on this grant. **Amitava Bhattacharjee** (APAM faculty 1984-1993), who is the head of the Theory Department at the Princeton Plasma Physics Laboratory (PPPL) and a professor at Princeton, will serve as the director of the overall project.

Steady-state fusion of ionized deuterium (D) and tritium (T) for power production occurs at a temperature of 10^{8} K a density of 10^{20} ions/m^{3} with an energy loss time of a few seconds. The simultaneous achievement of all three conditions is the objective of the ITER tokamak, which is under construction by a unique international collaboration in France. In tokamaks, the plasma has an axisymmetric toroidal (doughnut) shape with the pressure of the DT plasma balanced by a magnetic field. In stellarators, the plasma is confined in a non-axisymmetric torus, which allows many issues of tokamaks to be circumvented. These issues include sudden, possibly destructive, terminations of the plasma, called disruptions, steady-state maintenance of the magnetic field, and the reliability of computations of large changes in the plasma design.

The non-axisymmetry of the stellarator makes mathematical descriptions far subtler but paradoxically greatly enhances the reliability of computational design. In tokamaks, the properties of the magnetic field are in large part determined by a current carried by the plasma itself, which makes the system strongly non-linear and difficult to computationally predict. In stellarators the properties of the magnetic field are dominated by external coils and approximately ten times as many spatial distributions of magnetic field are available for plasma control than in axisymmetry.

The trajectories both of the charged particles that form a fusion plasma and of the magnetic field lines are given by Hamilton's equations of classical mechanics. Exact toroidal symmetry gives invariants for both the particles and the field lines that allow a simple but rigorous treatment. The practical constraints associated with these invariants can be lost when the symmetry breaking is less than a part in a thousand; stellarators break axisymmetry by tens of percent.

Prof. Boozer, together with a German colleague, Jürgen Nührenberg, received the 2010 Alfvén prize of the European Physical Society "*for the formulation and practical application of criteria allowing stellarators to have good fast-particle and neoclassical energy confinement*." This work, which was based on the adiabatic invariants of classical mechanics and the hidden symmetries they imply, led to the design of a radically different billion-Euro-class stellarator in Germany, W7-X. W7-X recently began operations, but has already demonstrated that non-axisymmetric design can give highly reliable results despite its subtleties.

Although W7-X can demonstrate that much of the physics required for a fusion power plant can be achieved, it and the much smaller stellarator HSX at the University of Wisconsin need not be end points of computational design. Far more can be done with mathematics and simulations. It is this realization that led to the funding by the Simons Foundation.

Areas in which major advances are illustrated by recent work associated with Columbia University: (1) The coils for stellarators can be challenging, but Prof. Boozer has shown that all possible externally-produced spatial distributions of magnetic field in a torus can be ordered by their exponentially increasing difficulty of production. This allows stellarators to be optimized for less challenging coils by imposing a penalty for the use of fields that are difficult to produce. (2) The magnetic field lines should lie in toroidal surfaces across most of the plasma volume. Nevertheless, these surfaces should not be allowed to extend to the walls because their first point of contact would have an excessive heat load and would be inconsistent with the pumping of the plasma exhaust. Tubes of magnetic flux should carry the plasma from the plasma edge to appropriately localized locations on the walls, where divertor structures are located. A 2017 paper on which Prof. Boozer is a co-author shows that stellarator optimization for properties distinct from divertor requirements can give natural divertor solutions. With a colleague, Prof. Alkesh Punjabi of Hampton University, Prof. Boozer has developed a numerically efficient method based on concepts from Hamiltonian mechanics, cantori and turnstiles, for determining what types of divertor solutions are available and how they can be controlled. These and many other areas involving mathematics and simulation will be advanced by the Simons Foundation grant, *Hidden Symmetries and Fusion Energy*, which will appear on the website https://www.simonsfoundation.org/collaborations/#mathematics-physical-sciences

Image: The W7-X Stellarator

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