The plasma energy W p = j [\B 2-h p)dV is minimized over a toroidal domain ft p using an inverse representation for the cylindrical coordinates R = ^ R mn (s) cos(m6-nf) and Z = VJ Z mn (s) sin(m#-n<;) , where (.s,0,f) are radial, poloidal, and toroidal flux coordinates, respectively. The radial resolution of the MHD equations is significantly improved by separating R and Z into contributions from even and odd poloidal harmonics which are individually analytic near the magnetic axis. A free boundary equilibrium results when tt p is varied to make the total pressure-B 2 + p continuous at the plasma surface E p and when tie vacuum magnetic field B^ satisfies the Neumann condition B w • dS p = 0. The vacuum field is decomposed as B v = Bo I V$, where Bo is the field arising from plasma currents and external coils and $ is a single-valued potential necessary to satisfy B v • d*£, p = 0 when p == 0. A Green's function method is used to obtain an integral equation over S p for the scalar magnetic potential $ = YJ$ mrl sin(mfl-n<;). A linear matrix equation is solved for $ mn to determine ^B% on the boundary. Real experimental conditions are simulated by keeping the external and net plasma currents constant during the iteration. Applications to I-2 stellarator equilibria are presented.

The theoretical and experimental development of stellarators has removed some of the specific deficiencies of this configuration, viz., the limitations in β, the high neoclassical transport, and the low collisionless confinement of α particles. These optimized stellarators can best be realized with a modular coil system. The W7-AS experiment [Plasma Phys. Controlled Fusion 31, 1579 (1989)] has successfully demonstrated two aspects of advanced stellarators, the improved equilibrium and the modular coil concept. Stellarator optimization will much more viably be demonstrated by W7-X [Plasma Physics and Controlled Fusion Research, Proceedings of the 12th International Conference, Nice, 1988 (IAEA, Vienna, 1989), Vol. 2, p. 369], the successor experiment presently under design. Optimized stellarators seem to offer an independent reactor option. In addition, they supplement, in a unique form, the toroidal confinement fusion program, e.g., energy transport is anomalous in stellarators too, but possibly more easily understandable in the frame of existing theoretical concepts than in tokamaks.

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Neumann boundary value problems are solved for stellarator fields generated by external currents. The method can be applied to the construction of external coil configurations of stellarators and to studies of the properties of vacuum fields. A computer code, NESCOIL, has been developed and applied to the Helias and ATF stellarator configurations.