The Geometry of Flux Surfaces with Quasi-Poloidal Symmetry

TL;DR

This work develops a local, 2D reduction for identifying quasi-poloidal (QP) flux surfaces in magnetized plasmas by introducing local QP (LQP) surfaces and a moving-frame representation. The central result is a decoupled set of surface equations $κ_α= -2ρ_l τ - ρ τ_l$, $τ_α= ρ_l κ + ( (ρ_{ll}-ρ τ^2)/κ )_l$, and $(κ/ρ)_α=0$, which determine a flux surface geometry and the accompanying field-line structure using only surface-local data. The authors classify two analytic solution families—generalised helicoids and Hasimoto surfaces—and show Hasimoto surfaces in vacuum act as flat mirrors with a separable field strength $B(φ,ψ)=B_0(ψ)+B_1(φ)$. Numerical verification against optimised QP fields demonstrates the surface equations capture local QP behavior, explain cusps and high-mirror regions, and point to an efficient 2D optimisation route for designing QP or quasi-poloidal devices with potential stability and reduced transport benefits. Overall, this framework offers a practical, theory-backed path to exploring QP equilibria and informs ongoing efforts to leverage QP-like confinement in stellarator concepts.

Abstract

Quasi-poloidal (QP) magnetic fields have desirable properties for confining plasma: no radial drift of guiding centres (with positive implications for neoclassical transport), zero Pfirsch-Schlüter current, a lower level of damping for poloidal flows, leading to reduced anomalous transport, and possible stability benefits. Despite their attractive properties, QP fields are not amenable to the near-axis expansion, a major theoretical tool for understanding toroidal fields. In this paper, we provide a novel framework for defining and understanding QP flux surfaces. This framework relies on a simplification that transforms the task of finding a quasi-poloidal flux surface from a 3D problem to a 2D problem. This simplification also applies to asymmetric magnetic mirrors with desirable properties. We sketch how this 2D problem can form the basis of an efficient optimisation problem for finding QP flux surfaces. We leverage this 2D problem for theoretical understanding: for instance, we identify one class of QP flux surfaces that are naturally flat mirrors (Velasco et al. 2023). The reduced model is validated against numerically optimised QP equilibria. We further utilise the reduced model to explain the prevalence of cusps, high mirror ratios, and narrow pinch points in these numerical equilibria.

Figures (11)

• Figure 1: Illustration of the Darboux frame on a flux surface (constant $\psi$). $\boldsymbol{\hat{t}}$ is parallel to field lines (shown in dark blue) and lies in the tangent plane to the surface, $\boldsymbol{\hat{n}}$ is normal to the surface, and $\boldsymbol{\hat{b}}=\boldsymbol{\hat{t}}\times\boldsymbol{\hat{n}}$ also lies in the tangent plane to the surface.
• Figure 2: An example of a generalised helicoid, with $\boldsymbol{r}= \left(-0.6 u^2 \cos b,-0.6 u^2 \sin b, u+ b \right)$ for $u \in [-2,2]$ and $b \in [0,2\pi]$. The magnetic field lines (shown in dark blue) are geodesics orthogonal to the helices given by $u=\text{const}$.
• Figure 3: An example of a surface of revolution, with $\boldsymbol{r}= \left((1+ 0.9 \sin u) \cos b,(1+ 0.9 \sin u) \sin b, u \right)$ for $u \in [-2 \pi,2 \pi]$ and $b \in [0,2\pi]$. The field lines are the meridians, highlighted in dark blue.
• Figure 4: The QP error (defined in equation \ref{['eq: QP error def']}) for six different magnetic field configurations as a function of the flux surface. Configurations 1-3 were produced using DESC and configurations 4 and 5 were produced using SPEC (QP error plotted only on boundary where a flux surface necessarily exists). 'QPS' corresponds to the magnetic field design for the Quasi-Poloidal Stellarator NELSON2003205Spong_2005. Configurations 1-3 show an improvement over QPS. Despite QP being only targeted on the boundary, there is a lower level of QP towards the axis.
• Figure 5: A schematic example of what it looks like for the surface equations to be verified or falsified on a flux surface. For $\phi < \pi/2$, there is agreement with the surface equations, since the geodesic curvature is zero and $\kappa/\rho$ only depends on $\phi$. For $\phi > \pi/2$, there is disagreement as the geodesic curvature is non-zero and $\kappa/\rho$ does not have vertical contours.