Vorticity of a velocity field

The vorticity of a differentiable three-dimensional velocity field is

\omega=\nabla\times u.

It is a vector field obtained by applying curl to the spatial variable at each fixed time. For twice continuously differentiable velocity, $\nabla\cdot\omega=0$ .

Two-dimensional convention

For $u=(u_1(x_1,x_2),u_2(x_1,x_2),0)$ , the vorticity points in the third direction. It is commonly identified with the scalar $\omega=\partial_1u_2-\partial_2u_1$ .

Rotation and vortex terminology

Rigid rotation $u=(-\Omega x_2,\Omega x_1,0)$ has $\omega=(0,0,2\Omega)$ . A “vortex” denotes a flow structure and is not synonymous with the vorticity vector at a point; circular particle motion and local vorticity must be distinguished.

References

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

Axisymmetric cylindrical formula

For axisymmetric velocity,

\omega_r=-\partial_z u_\theta,\qquad \omega_\theta=\partial_z u_r-\partial_r u_z,\qquad \omega_z=\frac1r\partial_r(ru_\theta).

These identities follow from the cylindrical curl formula and hold directly for $r>0$ .