Determining the Validity of Tokamak Perturbed Equilibrium Modeling Using Nonlinear Equilibria

TL;DR

This paper evaluates the validity of perturbed equilibrium models in tokamaks with small 3D fields, identifying the correct reference frame for nonlinear equilibria and validating linear theory within certain tolerances.

Contribution

It determines the appropriate reference frame for nonlinear equilibria in tokamaks and validates the use of linearized MHD theory within typical coil tolerances.

Findings

The TF coil centroid approximates the correct reference frame for SPARC.

The radial location of the TF coil inboard legs approximates the frame for NSTX-U.

Linear theory remains valid within existing coil tolerances, with margins for future devices.

Abstract

The prediction of perturbed equilibrium models for tokamaks with small 3D fields is strongly dependent on which reference frame for axisymmetry is assumed - for example, the toroidal field (TF) coil vs the poloidal field (PF) coil centroid. We use fully 3D equilibria generated by VMEC to determine the correct reference frame in these models when subject to n=1 coil asymmetries. We analyze a case for SPARC and find that the appropriate frame can be well approximated by the centroid of the TF coil set. We also consider the case of NSTX-U and find that the frame can be approximated by the radial location of the TF coil inboard legs at the midplane. We determine the correct frame by analyzing the shifted magnetic axis in the nonlinear equilibria. We also analyze the magnetic field line displacement to identify where the linearized MHD theory begins to break down, and compare that to typical coil tolerances in existing tokamaks. We find that linear theory is valid for existing tolerances, validating the use of perturbative codes to set these tolerances, but with a sufficiently small margin to be of note for future devices with relative tolerances larger than approximately $1\%$ of the minor radius. This study enables engineers to confidently use 3D perturbative models for determining assembly tolerances by providing insight into the correct applications of the theory.