Explicit spectrally optimized Fourier series for nested magnetic surfaces

The nonuniqueness of the poloidal angle θ in the parametric representation of a space curve x(θ)=[R(θ),Z(θ)] can be exploited to condense the Fourier spectra of R and Z. The nonlinear equation describing this spectral condensation was previously derived and solved numerically using Lagrange multipliers. Here a special case of the condensation equation is shown to be exactly solvable, leading to an explicit representation for x. A family of such representations is generated that possesses increasingly condensed spectra as a parameter is varied. Applications to a variety of curves are considered as models for three-dimensional magnetohydrodynamic (MHD) equilibria with nested flux surfaces. A substantial improvement occurs in spectral convergence compared with a polar representation, while retaining the numerical simplicity of the polar constraint. The asymptotic behavior for the R and Z spectral coefficients near a magnetic axis is analyzed. Implications for improvements of MHD equilibrium calculations are discussed.