In a stationary vortex, the typical streamline (a line that is everywhere tangent to the flow velocity vector) is a closed loop surrounding the axis; and each vortex line (a line that is everywhere tangent to the vorticity vector) is roughly parallel to the axis. A surface that is everywhere tangent to both flow velocity and vorticity is called a vortex tube. In general, vortex tubes are nested around the axis of rotation.  The axis itself is one of the vortex lines, a limiting case of a vortex tube with zero diameter.
According to Helmholtz's theorems, a vortex line cannot start or end in the fluid – except momentarily, in non-steady flow, while the vortex is forming or dissipating. In general, vortex lines (in particular, the axis line) are either closed loops or end at the boundary of the fluid.  A whirlpool is an example of the latter, namely a vortex in a body of water whose axis ends at the free surface.  A vortex tube whose vortex lines are all closed will be a closed torus-like surface.
A newly created vortex will promptly extend and bend so as to eliminate any open-ended vortex lines. For example, when an airplane engine is started, a vortex usually forms ahead of each propeller, or the turbofan of each jet engine. One end of the vortex line is attached to the engine, while the other end usually stretches out and bends until it reaches the ground.
When vortices are made visible by smoke or ink trails, they may seem to have spiral pathlines or streamlines.  However, this appearance is often an illusion and the fluid particles are moving in closed paths.  The spiral streaks that are taken to be streamlines are in fact clouds of the marker fluid that originally spanned several vortex tubes and were stretched into spiral shapes by the non-uniform flow velocity distribution.