Neoclassical conductivity and bootstrap current formulas for general axisymmetric equilibria and arbitrary collisionality regime

TLDR

Expressions for neoclassical resistivity and bootstrap current coefficients in toroidal axisymmetric equilibria are widely used, yet existing formulas stem from older work limited to large aspect ratio or extreme collisionality, with the most accurate current expressions coming from Hirshman (arbitrary aspect ratio, banana regime) and Hinton–Hazeltine (large aspect ratio, arbitrary collisionality). A code solving the Fokker–Planck equation with the full collision operator and field‑line variation, employing the adjoint function formalism, was used to compute these coefficients for arbitrary equilibria and collisionality regimes. The resulting coefficients, fitted to the code results within 5 %, provide comprehensive formulas for neoclassical conductivity and bootstrap current, extending prior work and revealing inaccuracies in earlier expressions across a wide plasma parameter space.

Abstract

Expressions for the neoclassical resistivity and the bootstrap current coefficients in terms of aspect ratio and collisionality are widely used in simulating toroidal axisymmetric equilibria and transport evolution. The formulas used are in most cases based on works done 15–20 years ago, where the results have been obtained for large aspect ratio, small or very large collisionality, or with a reduced collision operator. The best expressions to date and to our knowledge are due to Hirshman [S. P. Hirshman, Phys. Fluids 31, 3150 (1988)] for arbitrary aspect ratio in the banana regime and Hinton–Hazeltine [F. L. Hinton and R. D. Hazeltine, Rev. Mod. Phys. 48, 239 (1976)] for large aspect ratio and arbitrary collisionality regime. A code solving the Fokker–Planck equation with the full collision operator and including the variation along the magnetic field line, coupled with the adjoint function formalism, has been used to calculate these coefficients in arbitrary equilibrium and collisionality regimes. The coefficients have been obtained for a wide variety of plasma and equilibrium parameters and a comprehensive set of formulas, which have been fitted to the code results within 5%, is proposed for evaluating the neoclassical conductivity and the bootstrap current coefficients. This extends previous works and also highlights inaccuracies in the previous formulas in this wide plasma parameter space.