Weakly Quasisymmetric Near-Axis Solutions to all Orders

TL;DR

This paper demonstrates that weakly quasisymmetric magnetic fields near the axis can be constructed to any order in a power series, providing a systematic method without addressing global convergence.

Contribution

It offers a systematic high-order construction of solutions for weakly quasisymmetric fields near the axis, independent of force balance considerations.

Findings

Solutions can be constructed to arbitrary high order in powers of the distance from the axis.

Existence depends on choosing appropriate parameters like current or rotational transform.

The series may be asymptotic and not necessarily convergent.

Abstract

We show that the equations satisfied by weakly quasisymmetric magnetic fields can be solved to arbitrarily high order in powers of the distance from the magnetic axis. This demonstration does not consider force balance. The existence of solutions requires an appropriate choice of parameters, most notably the toroidal current or rotational transform profiles. We do not prove that the expansion converges (it is likely divergent but asymptotic), and thus the demonstration here should not be taken as definitive proof of the existence of global solutions. Instead, we provide a systematic construction of solutions to arbitrarily high order.