Optimum Fourier representations for stellarator magnetic flux surfaces

A steepest descent algorithm is used to obtain a least-squares approximation for threedimensional, toroidal stellarator flux surfaces represented by a discrete set of Poincare puncture points. A stream function λ(ρ,θ,ϕ) is introduced as a renormalization parameter to improve the mode convergence properties of the double Fourier series for the inverse co-ordinates (R, Z) representing toroidally nested magnetic surfaces: R(ρ, θ,ϕ) = Σ R_mn(ρ) cos(mθ — nϕ) and Z(ρ, θ,ϕ) = Σ Z_mn(ρ) sin(mθ - nϕ), where ρ is a scaled radial flux co-ordinate and (R, ϕ, Z) are standard cylindrical co-ordinates. The variable ϕ is the geometric toroidal angle, and θ is a parametric co-ordinate representing a poloidal angle. The stream function λ(ρ, θ,ρ) = Σ λ_mn(ϕ) sin(mθ—nϕ) and the rotational transform profile ɩ(ρ) are determined by solving a system of simultaneous linear equations obtained from the MHD equilibrium condition ∇ρ = 0, together with the stellarator condition for zero net toroidal current on each flux surface, ∇ × ∇ϕ⟩ = 0. Numerically computed Fourier representations are presented for vacuum configurations of the Advanced Toroidal Facility (ATF), Uragan-3 and TJ-II stellarator devices.