Determining the validity of tokamak perturbed equilibrium modeling using nonlinear equilibria

The prediction of perturbed equilibrium models for tokamaks with small non-axisymmetric fields is strongly dependent on which reference frame for axisymmetry is assumed. This assumption directly affects the applied field spectrum on the plasma, with the incorrect choice resulting in an incorrect prediction of the plasma response. We use fully 3D equilibria generated by VMEC to determine the correct reference frame when calculating error fields in perturbed equilibrium codes subject to n = 1 misalignments in several coil sets. We analyze a case of independently offset toroidal field (TF) coils and central solenoid (CS)/poloidal field (PF) coils in the SPARC tokamak and find that the appropriate reference frame can be well approximated by the centroid of the TF coil set. We also consider the case of NSTX-U with an independent centerpost consisting of the inner legs of the TF coils and CS, and find that the reference frame can be well 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, and use our findings to generalize to 3D fields from misalignments which modify the reference frame and those which impact the plasma response. 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.