Configurations for a proof of principle stellarator experiment P. R. Garabedian* New York University, Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, NY 10012 Contributed by P. R. Garabedian, November 15, 1999 One of the serious limitations of tokamaks as reactors is the oc- The conventional stellarators in operation 20 years ago suf- currence of disruptions. Stellarators designed by advanced com- fered from poor stability because of magnetic hills and from putational methods provide an attractive alternative for a major unsatisfactory transport because of conflicting helical and experiment in magnetic fusion research. Configurations with ap- toroidal terms in the magnetic spectrum. Progress was made proximate two-dimensional magnetic symmetry have been found when heliacs were introduced whose shape provided a good with high β limits and good transport. Specifications are given magnetic well, but large terms in the spectrum still suggested for a compact stellarator with three field periods and 18 moder- there would be poor transport and perhaps also unfavorable ately twisted modular coils that has equilibrium with robust flux island formation. The helias configuration discovered 10 years surfaces, a deep magnetic well assuring favorable stability, and ago by Nuehrenberg and Zille using the BETA equilibrium adequate confinement of hot particles at reactor conditions. Fast code was a significant breakthrough because both the well and computer codes with sufficient accuracy to resolve the mathe- the spectrum were satisfactory (6). After that quasihelically matical problems of equilibrium, stability and transport that arise symmetric (QHS) and quasiaxially symmetric (QAS) stellara- in the more complicated geometry of the stellarator have pro- tors were developed that had remarkable physical properties, duced this breakthrough. The mathematical analysis of the meth- but only those with the essential features of the helias seem to ods used is presented. meet all the requirements for a proof of principle experiment (7, 8). This is the theory justifying approval of the very promis- ing Wendelstein 7-X (W7-X) proposal in Europe (9). However, 1. The Stellarator Concept a more compact modular helias-like heliac with just two field agnetic fusion is based on the concept of fusing deuterium periods (MHH2) becomes an attractive candidate provided that M ions to form helium and release neutrons in a hot plasma the spectral terms associated with the most dangerous reso- confined by a strong magnetic field. The most common geom- nances are kept small enough to suppress corresponding islands etry of the plasma is a torus laced by nested flux surfaces. The (10). Of more interest to us here will be a quasiaxially sym- tokamak is an axially symmetric configuration using net current metric configuration with three field periods called the QAS3 to achieve equilibrium, but the current may cause disruptions of which depends for satisfactory equilibrium on the existence of the plasma. These instabilities are suppressed in stellarators by substantial bootstrap current to augment the rotational trans- exploiting three-dimensional asymmetry of the magnetic field form. Our computations show that this stellarator has adequate instead of net current to generate the poloidal field required two-dimensional symmetry for transport, robust equilibrium to confine the plasma. New methods of computational physics over a range of assumptions about the bootstrap current, and have been introduced to handle the more complicated geometry safe magnetohydrodynamic stability to both kink and ballooning of stellarator equilibria. modes (11). The complexity of the mathematical theory of stellarators with The NSTAB and TRAN computer codes that we have used to desirable physical properties calls for computational design tech- design the QAS3 stellarator have been validated both by com- niques rather than asymptotic expansion about exact solutions. parison with experimental data and by careful convergence stud- Experience has shown that the partial differential equations of ies (12, 13). We shall discuss in Section 3 results about the sta- ideal magnetohydrodynamics furnish a satisfactory model de- bility of global modes for the Compact Helical System (CHS) spite the fact that the mean free path of charged particles in a experiment in Japan for a case where ample measurements are reactor may go hundreds of circuits around the torus. Fast and available, though the performance was not very good because of accurate numerical algorithms have led to significant progress in the presence of a magnetic hill (14). Similar studies have been recent years. Most of our results come from running the NSTAB made for a variety of tokamaks. Insofar as equilibrium, stabil- equilibrium and stability code and the TRAN neoclassical trans- ity, and transport are concerned, our prediction is that the new port code developed by Octavio Betancourt and Mark Taylor at stellarators represent a significant improvement over previous New York University (1–3). experience. The QAS3 is predicted to have an average β limit A generally accepted technique to design stellarators is to op- of at least 6% because of a strong magnetic well. Because of its timize their physical properties in dependence on the shape of two-dimensional symmetry, it should have confinement permit- the plasma surface during calculations of magnetohydrodynamic ting ignition in a reactor of moderate size. equilibrium (4). Configurations have been found this way that Smooth solutions of the magnetostatic equations in fully have an approximate helical or axial symmetry of the magnetic three-dimensional geometry do not exist, and related diffi- spectrum which leads to good confinement of hot particles. The culties have arisen in the mathematical analysis of stability. precise meaning of this symmetry will be explained below, and Accordingly, all of the computer codes that are in use to de- when it is axial we shall see that the contribution to the rota- sign stellarators have limitations on their convergence, although tional transform from bootstrap current may be large and com- they do seem to provided a plausible simulation of the most pact stellarators are obtained whose aspect ratio is comparable important physics. More progress will require construction of a to that of a tokamak (5). Moreover, it is possible to choose the multiple harmonics defining the plasma geometry so that modu- lar coils with relatively moderate twist can be constructed to gen- *E-mail: [email protected]. erate an external magnetic field matching the optimal equilib- The publication costs of this article were defrayed in part by page charge payment. This rium. These coils are preferable to the interlocked helical coils article must therefore be hereby marked “advertisement” in accordance with 18 U.S.C. of conventional torsatrons. §1734 solely to indicate this fact. 972–977 | PNAS | February 1, 2000 | vol. 97 | no. 3 Downloaded by guest on September 23, 2021 proof of principle experiment to assess the merits of the new to the generation of magnetic islands. It becomes clear that, configurations that have been proposed. A relatively modest in three dimensions, one cannot expect smooth solutions of the expenditure in this direction could lead to significant progress equilibrium problem to exist with the kind of nested toroidal sur- in magnetic fusion research and provide a desirable alternative faces s D const. required for good confinement because of the to the tokamak. The resulting experimental observations would behavior of the small denominators. In practice, that motivates not only serve to verify the computational theory, but would us to use a numerical method that constructs weak solutions also provide important information about bootstrap current, β of the problem in an appropriate conservation form, avoiding limits, and transport at plasma conditions already achieved in unnecessary differentiations. The Kolmogoroff–Arnold–Moser tokamaks. The QAS3 configuration is our candidate of choice (KAM) theory of dynamical systems describes a corresponding for such a stellarator experiment. phenomenon that occurs in tracking magnetic lines of force (12). The BETA equilibrium code applied a finite element method 2. Two-Dimensional Magnetic Symmetry to the variational principle of magnetohydrodynamics to calcu- Equilibria in plasma physics are solutions of the system of partial late weak solutions of the stellarator problem and gave satis- differential equations factory estimates of the potential energy, but was otherwise in- accurate (4). Resolution was improved in the VMEC code by ∇·B D 0; J 3 B Drp introducing a spectral algorithm to handle dependence on the poloidal and toroidal angles, but the problem of nonexistence for the magnetic field B, the current density J Dr3 B, and of solutions became harder to address and the energy landscape the pressure p, which we take to be a given function of the became more difficult to compute (16). The NSTAB equilib- toroidal flux s. We compute solutions by applying the variational rium and stability code combines these approaches, capturing principle islands and current sheets remarkably well. It is fast and accu- ZZZ rate enough to be an effective tool for the systematic design of 2 B =2 − psdx1dx2dx3 D minimum configurations involving complicated combinations of multiple harmonics in three dimensions. of ideal magnetohydrodynamics, subject to appropriate flux con- In the spectral method, weak solutions of the equilibrium straints.