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A vortex filament is the ideal thin-core limit of a vortex tube, concentrating a finite circulation on a curve. In an unbounded incompressible fluid with velocity tending to zero at infinity, its induced velocity field follows the vorticity version of the Biot-Savart law. The ideal field is singular on the filament; finite-core models are needed there. A closed circular filament is a thin vortex ring.

A vortex ring is a vortex tube bent into a closed loop, often approximately a circle. Its vorticity runs around the loop, while the induced velocity field passes through the centre and returns outside. For a thin ring of radius $r$ and circulation $\Gamma$, the Biot-Savart law for incompressible vorticity gives an axial velocity $u_z(z)=\Gamma r^2/[2(r^2+z^2{)}^{3/2}]$, measured relative to the ring centre, with its sign set by the circulation orientation. A vortex ring is not a loop of fluid whose velocity everywhere merely follows the circular vortex line.