# The Relationship between Flux Coordinates and Equilibrium-based Frames of Reference in Fusion Theory

## TL;DR

This paper explores the relationships between flux coordinate frames and equilibrium-based reference frames in fusion plasma theory, introducing new formal tools and a novel annihilation operator to simplify derivations.

## Contribution

It introduces a new annihilation operator based on current density frames and elucidates the relationships between different local reference frames in fusion plasma theory.

## Key findings

- Relationship between flux coordinates and equilibrium frames clarified.
- New annihilation operator simplifies derivation of inner layer equations.
- Formal connections between magnetic field-based and current density-based frames established.

## Abstract

The properties of two local reference frames based on the magnetic field and the current density are investigated for magnetized plasmas in toroidal geometry with symmetric angle. The magnetic field-based local frame of reference has been well-studied for example by Dewar and colleagues [Phys. Fluids 27, 1723 (1984)] An analogous frame based on the current density vector is possible because it is also divergence free and perpendicular to the gradient of the poloidal flux. The concept of straightness of a vector is introduced and used to elucidate the Boozer and Hamada coordinate systems. The relationship of these local frames to the more well-known Frenet frame of reference, which specifies a curve in terms of curvature and torsion, is given. As an example of the usefulness of the these formal relationships, we briefly review the ideal MHD theory and their use. We also present a new annihilation operator, useful for eliminating shorter time scales than the time scale of interest, for deriving the inner layer equations of Glasser, Greene, and Johnson [Phys. Fluids 18, 875 (1975)]. Compared to the original derivation that is based on the local frame of reference in terms of the magnetic field, the new annihilation operator that is based on the local frame of reference in terms of the current density simplifies the derivation.