The dynamics of streamwise vorticity in axisymmetric jets is studied by direct numerical integration of the Navier–Stokes equations coupled with a passive scalar. Consistently with recent experiments and inviscid numerical simulations, the present viscous simulations show the appearance of pairs of axially counter‐rotating vortex filaments. After their formation the filaments move away from the jet, dragging the tracer into finger‐shaped structures. The three‐dimensional topology of the rings, filaments, and fingers is described, well beyond the time of filament formation. Finally, visualizations of the pressure gradient field are presented, which suggest that the filaments can be directly observed experimentally by seeding the flow with microbubbles.