In polar coordinates, define the axis of the torus to be r=r₀, z=0. Next take the vorticity to be

$$\omega_\vartheta = f(\rho^2), \qquad r \omega_r = - g(\rho^2) z, \qquad r \omega_z = g(\rho^2) (r - r_0),$$

where $\rho$ is the distance from the axis of the torus, i.e. $\rho = \sqrt{(r-r_0)^2 + z^2}$, and f and g are arbitrary smooth functions of compact support that stay away from the axis r=0 to avoid singularies in the velocity.

Seen in the r,z, plane, the vortex lines are found to circle around the point r=r₀, z=0. If $\phi$ indicates the angle around this point, $d\phi/d\vartheta = g/f$, hence if g/f is irrational, the vortex line never hits itself.

Note that vortex tubes do not close, even though they are constrained within a finite space and do hit themselves. Hitting is not the same as closing, as some references appear to think.

Chorin appears to have constructed this example earlier.