Perturbed Toroidal Vortices Display Internal Simply Connected Topology

TL;DR

The paper investigates the topology of zero-helicity vortices under odd-parity perturbations, showing that simply connected flux surfaces can emerge inside a crescent-shaped boundary, contrary to the common toroidal assumption. Using a Soloviev/Hill-based axisymmetric vortex model with a slowly varying, zero-vorticity perturbation, the authors develop a modified flux labeling $(\psi,\varphi)$ and perform a rigorous 3D topological classification, identifying an inner separatrix $S_{in}$ and an outer separatrix $S_{out}$. The results establish that for $0<\alpha<\alpha_c$, flux surfaces split into three regimes: open ($\psi<0$), toroidal ($0<\psi<\psi_-$), and simply connected ($\psi_-<\psi<\psi_+$), with field lines uniquely labeled by $(\psi,\varphi)$ and proven closure in 3D. Complementary particle-motion simulations in perturbed magnetic-field configurations demonstrate crescent-shaped drift surfaces that align with, yet differ from, the flux surfaces, supporting the topological findings and highlighting implications for plasma confinement and vortex dynamics. The work thus generalizes the expected topology of Hill-like vortices and RMF-driven FRC systems, with broad relevance across fluid and plasma contexts.

Abstract

This work shows that the interiors of perturbed zero-helicity vortices display simply connected topology with a crescent-shaped boundary. Flux surfaces in fluid and magnetic vortices were explored analytically, while particle trajectories in the context of plasma confinement were examined numerically, demonstrating the existence of both toroidal and simply connected topologies. This new topology appears for perturbations in a broad class, with amplitudes and spatial variance allowed to be arbitrarily small. This work proves the closedness of field lines under odd-parity perturbations of zero-helicity vortices.

Figures (4)

• Figure 1: The flux surfaces (blue) and field lines (purple) transition from toroidal to simply connected as $\psi$ increases. The red circle is the O-point null line on which the critical points $\textbf{r}_c$, defined by $\textbf{B}(\textbf{r}_c)=0$, lie. In (a), there is no intersection between the flux surface and the red circle; in (b), there is one intersection; and in (c), there are two intersections. Here, $\alpha = 0.2,\ k = 0.25\ \text{m}^{-1},\ B_0 = 2\ \text{T},\ r_s = z_s = 1\ \text{m}$.
• Figure 2: (a) shows the $1$-dimensional $\psi(0,y,0)$$y$ plot, with diamonds indicating the critical points. (b) and (c) are 2-dimensional plot of flux surfaces intersecting with $y-z$ and $x-y$ planes respectively. In all figures, green, blue and magenta indicates open, torus and simply connected topology respectively. The dashed black and solid black lines indicate outer and inner separatrix respectively. (b) directly shows field lines embedded on $y-z$ plane, while (c) can also be interpreted as Poincaré map of field lines on $x-y$ plane. The red circle is the $\textbf{B} = 0$ critical point circle in (c).Here, $\psi\in[-0.2,\psi_+=0.313]\ \text{Wb},\ \alpha = 0.2,\ k = 0.25\ \text{m}^{-1},\ B_0 = 2\ \text{T},\ r_s = z_s = 1\ \text{m}$.
• Figure 3: Categorization of compact flux surfaces, $0<\psi<\psi_+$, for all $\alpha<\alpha_c$. Surfaces in the blue region, magenta region, and solid black line are toroidal, simply connected, and transitional. Here $r_s=z_s$.
• Figure 4: Trajectories of electrons in perturbed FRC projected onto the $x$--$y$ plane. Unit in each axis is cm. Time is denoted by color. $\alpha = 0.1$, $I = 0.1$, $r_s = 25$ cm, $z_s = 75$ cm, $s\sim 800$, $T=2\times10^5\tau_{ce}$