Coordinates based on a magnetic mirror field

TL;DR

The paper develops a coordinate system adapted to an axisymmetric magnetic field by introducing a field-line coordinate $s$ along $\mathbf{B}$ and an azimuthal coordinate $\alpha$ in the $r$-$z$ plane. It derives the necessary conditions for the existence of $(\alpha,s)$ as coordinates via a Jacobian framework and a commutation constraint, showing that $\Delta = \partial\alpha/\partial r\ partial s/\partial z - \partial\alpha/\partial z\partial s/\partial r$ must satisfy $\Delta' = 0$ and can be written as $\Delta = r B G(\alpha)$ (taken as $G=1$), along with $\mathbf{b}\cdot\nabla\alpha = 0$. A Clebsch (flux) representation $\mathbf{B}=\nabla\alpha\times\nabla\theta$ is established, together with flux relations $\mathbf{B}\mathbf{a}=\nabla\theta\times\nabla s$ and $\mathbf{B}\mathbf{b}=\nabla\alpha\times\nabla\theta$, linking the field to the coordinates. An explicit mirror-field example demonstrates the construction, yielding $\alpha = B_0\frac{r^2}{2}(1+ q\cos kz)$ and a computable $s(r,z)$. Overall, the work provides a structured, Clebsch-based coordinate framework for analyzing axisymmetric magnetic fields and their field-line geometry, with potential implications for magnetic confinement and flux-based descriptions.

Abstract

We construct a coordinate system fitting the geometry of a given, cylindrically symmetric, magnetic field.

Figures (1)

• Figure 1: Two paths, each beginning at $P_{0}$ and following the vector fields $\bb$ and $\ba$ in opposite order.