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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. [ABSTRACT FROM AUTHOR]

Beltrami flows with crystallographic cubic symmetry were demonstrated numerically to have knots or links of invariant tori, which are not only stream tubes but also vortex tubes. The cubic symmetry is associated with the ABC flow with its three parameters equalised. The knots formed by invariant tori are the trefoil knot, the granny knot, true-lovers' knot, etc. The linked invariant tori are shown to form a variety of chainlike structures. • Invariant tori in Beltrami flows with cubic symmetry are considered. • Knotted invariant tori of the trefoil knot type etc. are demonstrated. • A variety of chainlike structures of linked invariant tori are shown. [ABSTRACT FROM AUTHOR]

One kind of tensor denotation of nth Beltrami axisymmetric and nonaxisymmetric spherical vortices, their classification and symmetries were discussed. Chaotic phenomena will occur in the dynamic system of the nonaxisymmetric Beltrami spherical vortices. From these aspects, it is shown that the tensor denotation has more meaningful characters and nonaxisymmetric Beltrami shperical vortices are various and very complex. [ABSTRACT FROM AUTHOR]