  We provide a general theory for the structure of the quantum flow near 3-d
nodal lines, i.e. one-dimensional loci where the 3-d wavefunction becomes equal
to zero. In suitably defined co- ordinates (co-moving with the nodal line) the
generic structure of the flow implies the formation of 3-d quantum vortices. We
show that such vortices are accompanied by nearby invariant lines of the
co-moving quantum flow, called X-lines, which are normally hyperbolic.
Furthermore, the stable and unstable manifolds of the X-lines produce chaotic
scatterings of nearby quantum (Bohmian) trajectories, thus inducing an
intricate form of the quantum current in the neighborhood of each 3-d quantum
vortex. Generic formulas describing the structure around 3-d quantum vortices
are provided, applicable to an arbitrary choice of 3-d wavefunction. We also
give specific numerical examples, as well as a discussion of the physical
consequences of chaos near 3-d quantum vortices.
