A Numerical Technique for Obtaining the Complete Bifurcated Equilibrium Solution Space for a Tokamak
This paper describes a new numerical method for finding solutions to the ideal MHD equilibrium problem. The space of solutions thus found canhave more than one bifurcation branch, depending on the tokamak being modeled. We suggest that the solutions which are difficult to obtain without the use of this technique correspond to equilibria which are difficult to maintain in the tokamak being modeled. First, this paper investigates, for a tokamak design with large poloidal field-shaping coil (PFC) to plasma distance, the bifurcated numerical solution curve as a function of flux loop position and relates this curve to the practical existence of ideal magnetohydrodynamic equilibria and practical tokamak design. In previous papers, some of the problems which could arise for large PFC-plasma distance were discussed. Then, using a regularization technique, it was shown that, for large PFC-plasma distance, the flux loops should be close to the PFCs for stable control if the full information from the flux loops is used. Here it is shown that, for large PFC-plasma distance, the structure of the equilibrium solution space becomes increasingly complex and desirable solutions become more difficult to attain as the flux loops are moved farther from the plasma. In order to explore this solution space numerically, it is necessary to obtain solutions for which the usual Picard iteration method is unstable. Here an extension of this method is given. The solution space is enlarged by adding additional variables and constraints, so that the iteration to the desired solution is stable in the extended space. A modified version of this numerical technique has been used to obtain equilibrium fits to highly elongated DIII-D plasmas. The numerical equilibria are very difficult to obtain without the use of this technique and the plasmas are difficult to maintain in the tokamak.