Vortex line - Knowlpedia

At a fixed time $t$ , a vortex line is a curve tangent to the vorticity field. On a region where $\omega(t,x)\ne0$ , one parametrization solves

\frac{d\gamma}{ds}=\omega(t,\gamma(s)).

Reparametrizing by a nonzero scalar factor gives the same unparametrized tangent curves.

Trajectories and structures

The time $t$ is held fixed in this definition. A vortex line is therefore different from a fluid particle trajectory, whose time derivative is the velocity $u$ . Vortex tubes and other vortex structures are organized using vorticity geometry, but the word “vortex” by itself does not select a universal quantitative criterion.

References

Lenya Ryzhik, Lecture Notes for Math 256B (2024), Sections 1–3.