Conefield approach to identifying regions without flux surfaces for magnetic fields

TL;DR

The paper addresses identifying regions in 3D magnetic fields where no flux surfaces (invariant $2$-tori) exist for a given class. It introduces a conefield formulation of the Converse KAM method to bound the slope of potential tori and adds a killends extension to enlarge the detected nonexistence region. Applied to toroidal fields perturbed by helical modes, the method detects magnetic islands and chaos, with killends increasing the eliminated volume and reducing computation time. The approach provides a rigorous, practical means to map nonexistence regions of flux surfaces in complex magnetic geometries, and code implementing the method is publicly available.

Abstract

The conefield variant of a Converse KAM method for 3D vector fields, identifying regions through which no invariant 2-tori pass transverse to a specified direction field, is tested on some helical perturbations of an axisymmetric magnetic field in toroidal geometry. This implementation computes bounds on the slopes of invariant tori of a given class and allows to apply a subsidiary criterion to extend the non-existence region, saving significant computation time. The method finds regions corresponding to magnetic islands and chaos for the fieldline flow.

Figures (11)

• Figure 1: A cone, consisting of all the covectors at a point whose kernels are planes, like the blue one, that lie in the sector between those for $\alpha^+$ and $\alpha^-$ that does not contain $\xi$.
• Figure 2: Typical evolution of the values of $\alpha_\tau^\pm(\eta)$ and $\alpha_\tau^\pm(\xi)$ (top), along with the associated slopes $\sigma_\tau^\pm$ (bottom) for the magnetic field derived from (\ref{['A_ex2']}), for two different initial conditions: [(a)(b)] a point in the complement of the non-existence region ($(\tilde{y}_1,\tilde{z}_1) = (0.3,0.2)$); and [(c)(d)] a point inside the non-existence region ($(\tilde{y}_2,\tilde{z}_2) = (0.35,0.45)$). The insets show blowups of the indicated regions.
• Figure 3: Diagram of a transverse section in which the conefield data allows identification of a point (blue) where any possible torus passing through it (e.g. the blue dashed curve) enters a region $\mathcal{S}$ where non-existence has been established, thus allowing us to add it to the non-existence region.
• Figure 4: Diagram of the alternative 'killends' approach to enlarge the non-existence region $\mathcal{S}$ by considering the continuation of the cones of points tangent to $\mathcal{S}$.
• Figure 5: Conefield results (shown in hues) and three zoomed-in insets (on the left) displaying the computed conefield on the complement of the detected non-existence region $\tilde{\mathcal{S}}$, on a regular grid of initial conditions in the positive quadrant of the poloidal plane $\phi = 0$ for the magnetic field derived from (\ref{['A_ex2']}) with $\varepsilon = 0.003$ in symplectic coordinates. The hues vary from fast detection (red) to no detection at all (blue) within the timeout $t_f = 80$ (which corresponds to about 15 toroidal revolutions). $q$ denotes the ratio of the time to detection to the timeout, except that $q=1$ denotes no detection within the timeout.

Theorems & Definitions (1)

• Theorem 3.1