  We analyze the tunneling of vortex states from elliptically shaped traps.
Using the hydrodynamic representation of the Gross-Pitaevskii (Nonlinear
Schr\"odinger) equation, we derive analytically and demonstrate numerically a
novel type of quantum fluid flow: a jet-like singularity formed by the
interaction between the vortex and the nonhomogenous field. For strongly
elongated traps, the ellipticity overwhelms the circular rotation, resulting in
the ejection of field in narrow, well-defined directions. These jets can also
be understood as a formation of caustics since they correspond to a convergence
of trajectories starting from the top of the potential barrier and meeting at a
certain point on the exit line. They will appear in any coherent wave system
with angular momentum and non-circular symmetry, such as superfluids,
Bose-Einstein condensates, and light.
