  We prove that there exist Beltrami fields in Euclidean space, with sharp
decay at infinity, which have a prescribed set of invariant tori (possibly
knotted or linked) that enclose an arbitrarily large number of hyperbolic
periodic orbits. These hyperbolic orbits are cablings over the core curve of
each torus. Moreover, the domain bounded by each invariant torus is covered by
an almost full measure set of invariant tori. We show that an analogous result
holds for high-frequency Beltrami fields on the flat 3-torus.
