Three-dimensional free boundary calculations using a spectral Green's function method

Abstract

The plasma energy W p = ∫ ω p ( 1 2 B 2 +p dV is minimized over a toroidal domain ω p using an inverse representation for the cylindrical coordinates R = ΣR mn ( S) cos( mθ − nζ) and Z = ΣZ mn ( s) sin( mθ − nζ), where ( s, θ, ζ) 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 ω p is varied to make the total pressure 1 2 B 2 + p continuous at the plasma surface Σ p and when the vacuum magnetic field B v satisfies the Neumann condition B v ·d Σ p = 0 . The vacuum field is decomposed as B v = B 0 + ∇φ , where B 0 is the field arising from plasma currents and external coils and φ is an single-valued potential necessary to satisfy B v d Σ p = 0 when p ≠ 0. A Green's function method is used to obtain an integral equation over Σ p for the scalar magnetic potential φ = Σφ mn sin( mθ − nζ). A linear matrix equation is solved for φ mn to determine 1 2 B 2 v on the boundary. Real experimental conditions are simulated by keeping the external and net plasma currents constant during the iteration. Applications to l = 2 stellarator equilibria are presented.