Rod-packing arrangements of invariant tori in solenoidal vector fields with cubic symmetries

Abstract

The arrangements of invariant tori that resemble rod packings with cubic symmetries are considered in three-dimensional solenoidal vector fields. To find them systematically, vector fields whose components are represented in the form of multiple Fourier series with finite terms are classified using magnetic groups. The maximal magnetic group compatible with each arrangement is specified on the assumption that the cores of the nested invariant tori are straight and located on the lines corresponding to the central axes of the rods packed. Desired rod-packing arrangements are demonstrated by selecting vector fields whose magnetic groups are the maximal ones and by drawing their integral curves that twine around invariant tori. In the demonstration of chiral arrangements, Beltrami flows (or force-free fields in plasma physics), which have the strongest chirality of all solenoidal vector fields satisfying the same vector Helmholtz equation, are used. As by-products, several chain-like arrangements of closed invariant tori were found. One of the chains consists of knotted invariant tori. In all vector fields (chiral or achiral) selected for the demonstration, the volume percentages of ordered regions formed by invariant tori in a unit cell were roughly measured with the aid of a supervised machine learning technique.