Three-dimensional stellarator codes P. R. Garabedian* Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012 Contributed by P. R. Garabedian, June 3, 2002 Three-dimensional computer codes have been used to develop anomalous transport can be modeled by iterating on small quasisymmetric stellarators with modular coils that are promising variations of the electric potential within the magnetic surfaces candidates for a magnetic fusion reactor. The mathematics of to achieve quasineutrality between the distributions of ions and plasma confinement raises serious questions about the numerical electrons. The method simulates complicated transport in stel- calculations. Convergence studies have been performed to assess larators remarkably well, and numerical results have been ob- the best configurations. Comparisons with recent data from large tained for the large helical device (LHD) experiment at the stellarator experiments serve to validate the theory. National Institute for Fusion Studies in Toki, Japan that are in excellent agreement with recent observations of the energy odular stellarators can be viewed as an advanced tokamak confinement time at high temperatures (6, 7). A vectorized Mhybrid appropriate for implementation as a fusion reactor version of the TRAN code runs efficiently on standard work (1). Quasisymmetry of the magnetic spectrum is predicted to stations. give good confinement at high temperatures, and adequate Convergence Studies rotational transform from the external magnetic field is expected to stabilize the plasma. New configurations have been designed Calculation of toroidal equilibrium of a plasma without two- by making imaginative use of three-dimensional computer codes. dimensional symmetry is a problem in mathematics that is not Because the mathematics of these stellarators is complicated, we well posed. In terms of the toroidal flux s and a pair of angular have performed convergence studies applicable to proof of flux coordinates ␪ and ␾, the parallel current has a Fourier principle experiments that are being planned. expansion of the form We are primarily concerned with the NSTAB equilibrium and J⅐B m Ϫ In ͑m Ϫ In͒␪ ϩ ͑n Ϫ ␫m͒␾ stability code and the TRAN Monte Carlo transport code ϭ pЈ ͸ B cosͫ ͬ , developed at New York University by Octavio Betancourt and B2 n Ϫ ␫m mn 1 Ϫ ␫I Mark Taylor (2–5). The codes are applied to a compact stel- larator called the MHH2 that has two field periods and excellent where Bmn is the magnetic spectrum of Fourier coefficients of ͞ 2 ␫ quasiaxial symmetry. For the calculations we selected a config- 1 B , is the rotational transform, and I is the net current (8). uration with realistic physical parameters that provide good A configuration is called quasisymmetric if a single row, column, convergence, enabling us to perform long runs and make or diagonal of the double array Bmn dominates that spectrum. Ϫ␫ estimates of numerical errors. Theoretical conclusions can be The small denominators n m vanish at resonant surfaces ␫ drawn that are relevant to a wider range of examples, such as the where is rational, so smooth solutions of the partial differential optimized stellarator specified in Table 1. equations describing MHD equilibrium do not in general exist in The NSTAB code is a computer implementation of the three dimensions. In numerical work this leads us to construct variational principle of ideal magnetohydrodynamics (MHD). If weak, discontinuous solutions of discrete equations that are B is the magnetic field and p is the pressure, solutions of the expressed in conservation form, but even the best methods only magnetostatics equations are found by minimizing the potential converge in an asymptotic sense. Enough spectral terms must be energy included to eliminate significant truncation error, but not so many that the results become meaningless. Our convergence studies for the NSTAB code clarify how this can be accom- E ϭ ͵͵͵͓B2͞2 Ϫ p͔dV plished in practice. For convergence studies we have selected an example of the MHH2 compact stellarator, shown in Fig. 1, that has two field in a coordinate system compatible with toroidal geometry in periods, plasma aspect ratio three, excellent quasiaxial symme- three dimensions. An accurate finite difference scheme is used try, and a limit near 5% for the average value of ␤ ϭ 2p͞B2. The in the radial direction, and dependence on the poloidal and separatrix has a smooth shape that facilitates making long runs SCIENCES toroidal angles is handled by the spectral method. It is assumed of the NSTAB code. The coordinate system for the computa- that there are nested toroidal flux surfaces, and the differential tions is chosen to rotate once in the poloidal direction over a full APPLIED PHYSICAL equations are written in a conservation form that captures circuit of the device in the toroidal direction. Zoning of the islands and current sheets. The resolution is so good that poloidal angle has been adjusted to provide good resolution on questions of stability can be settled by a mountain pass theorem crude grids and reasonable spacing of the mesh both at the asserting that when more than one solution of the problem can separatrix and the magnetic axis, which is determined by a robust be found then an unstable equilibrium must exist corresponding algorithm. However, numerical errors appear at the magnetic to a saddle point in the energy landscape. Bifurcated equilibria axis if the radial mesh is too fine. are calculated whose magnetic surfaces have Poincare´sections Applying an accelerated method of steepest descent to the displaying the structure of the most unstable modes. variational principle, we test for stability by examining runs of The TRAN code uses a split time algorithm to calculate the the NSTAB code in which some dangerous mode has been confinement time of test particles by alternately tracking guiding triggered by introducing temporarily an appropriate forcing center orbits and applying a random walk that represents colli- term. A run predicts stability if the mode decays during further sions. The magnetic field B and the flow field U of the plasma in a background obtained by using NSTAB are held fixed during iterations that impose quasineutrality. Conservation of momen- Abbreviations: LHD, large helical device; MHD, magnetohydrodynamics; W7-AS, Wendel- tum might be enforced by the selection of U, but because it is not stein 7-AS. thermal U has little effect on the collision operator. However, *E-mail: [email protected]. www.pnas.org͞cgi͞doi͞10.1073͞pnas.162330399 PNAS ͉ August 6, 2002 ͉ vol. 99 ͉ no. 16 ͉ 10257–10259 Downloaded by guest on September 27, 2021 Table 1. Fourier coefficients ⌬mn of an MHH2 stellarator defined Table 2. Relationship between the highest degree N of the spectral terms used in a run of NSTAB and the resulting critical ؍ in cylindrical and toroidal coordinates by the formula r ؉ iz iu ؊imu؉2inv e ¥ ⌬mn e value of ␤ that is calculated n Ϫ10123N 16 20 24 32 48گm ␤ 0.060 0.050 0.045 0.040 0.039 Ϫ1 0.190 0.130 Ϫ0.015 0.000 0.000 0 0.000 1.000 0.000 0.000 0.000 1 0.150 3.000 0.250 0.050 0.000 2 0.000 Ϫ0.090 Ϫ0.420 Ϫ0.070 0.000 residuals so that the test of stability would become more reliable. 3 0.000 0.000 Ϫ0.040 0.080 0.000 Runs were continued as long as possible to achieve accuracy 4 0.000 0.015 0.000 Ϫ0.015 Ϫ0.015 sufficient for a convergence study. Recent advances in computer technology have enabled us to do this economically. In Table 2 we compare the degree N of the spectral calcula- tions with the corresponding estimate of an average ␤ limit based iterations, but if it grows then the equilibrium is unstable. A more on the mountain pass theorem. If the degree is too low the convincing conclusion can be drawn from the mountain pass method does not provide meaningful results about stability. theorem if the iterations converge to a bifurcated solution whose However, for n ϭ 24 one obtains efficiently answers that are of stellarator symmetry is visibly broken by magnetic surfaces that sufficient accuracy to optimize the design of a quasisymmetric exhibit the structure of the dangerous mode. The numerical stellarator like the MHH2. The results do not change signifi- results depend on the maximum degree N of factors in each of cantly for N Ͼ 32, which is a good value to choose from the point two angular coordinates that specify the spectral terms included of view of asymptotic convergence. At N ϭ 48 a stage is reached in the computation. The purpose of our convergence study is to where the numerical method may fail in long runs because of the decide whether the prediction about stability approaches a singular behavior of the solution. meaningful limit as N increases. Table 3 shows how the convergence of the iterative scheme For the example of the MHH2 we have performed equilibrium used in the NSTAB code depends on the degree N of the spectral and stability runs of the NSTAB code with between 14 and 28 terms that are used. For crude grids it is easy to reduce the mesh intervals in the radial flux coordinate s and with a largest residuals to the level of round-off error in the computer. degree N of the spectral terms ranging as high as 48. Because of However, as N increases the effectiveness of the scheme dete- the nested surface hypothesis, islands of small width are captured riorates, and the method may only converge in the asymptotic better on crude grids, and the calculations are relatively insen- sense that at first the errors become smaller, but later they sitive to the radial mesh size because the finite difference scheme increase without limit.