Application of the free-boundary SIESTA MHD equilibrium code to bootstrap control scenarios in the W7-X stellarator

摘要

SIESTA is a recently developed [1] MHD equilibrium code that has been designed to perform fast and accurate calculations of ideal MHD equilibria for three-dimensional magnetic configurations. It is an iterative code that uses the solution previously obtained by the wellknown VMEC code [2] (for the same problem) to provide an Eulerian background coordinate system and an initial guess of the equilibrium solution, from which the iteration starts. Since VMEC assumes well-defined closed magnetic surfaces, the solution that VMEC provides provides a good, non-singular, polar-like generalized coordinate system. But in contrast to VMEC, SIESTA does not assume the existence of closed magnetic surfaces. Thus, the final equilibrium solution that SIESTA converges too may include other magnetic topologies such as magnetic islands and stochastic regions. Magnetic flux and mass conservation are the only constraints imposed on the solution. Numerically, SIESTA iterates through a series of plasma displacements ￡ while it looks for a minimum of the total MHD energy, given by: {formula} The minimum energy state (i.e., equilibrium) is obtained when the displacement satisfies F(ξ)= {formula}, where the ideal MHD force isF= J×B-▽p. The displacement required is obtained iteratively, by applying a Newton method and solving the associated linear problem by a combination of preconditioning and iterative algorithms such as GMRES, among others.