Crystallography of three-dimensional fluid flows with chirality in hexagonal cases

Magnetic groups are applied to three-dimensional fluid flows with chirality, which are called Beltrami flows (or force-free fields in plasma physics). First, six Beltrami flows are derived so that their symmetries and antisymmetries are described by six different hexagonal magnetic groups. The general Wyckoff positions are used to derive the flows. Special Wyckoff positions are shown to be useful for finding the zero points of the flows. Tube-like surfaces called invariant tori are observed to interlace and form various crystal-like structures when streamlines winding around the surfaces are numerically plotted. Next, two simpler hexagonal Beltrami flows are derived, and their zero points and invariant tori are studied. Some families of the invariant tori have arrangements similar to those observed in materials science.