Editing Vortex ring (section)

===Circular vortex lines===
For a single zero-thickness vortex ring, the vorticity is represented by a [[Dirac delta function]] as <math> \omega\left(r,x\right)=\kappa\delta\left(r-r'\right)\delta\left(x-x'\right)</math> where <math> \left(r',x'\right)</math> denotes the coordinates of the vortex filament of strength <math>\kappa</math> in a constant <math>\theta</math> half-plane. The [[Stokes stream function]] is:<ref name="lamb1932">{{cite book|last1=Lamb|first1=H.|title=Hydrodynamics|publisher=Cambridge University Press|date=1932|pages=236–241|url=https://archive.org/details/hydrodynamics00lamb}}
<!-- minus sign missing, see Lamb p. 237 --></ref>
<math display="block"> \psi(r,x)=-\frac{\kappa}{2\pi}\left(r_1+r_2\right)\left[K(\lambda)-E(\lambda)\right] </math>
with <math> r_1^2 = \left(x-x'\right)^2+\left(r-r'\right)^2 \qquad r_2^2 = \left(x-x'\right)^2+\left(r+r'\right)^2 \qquad \lambda = \frac{r_2-r_1}{r_2+r_1} </math>
where <math>r_1</math> and <math>r_2</math> are respectively the least and the greatest distance from the point <math>P(r,x)</math> to the vortex line, and where <math>K</math> is the [[complete elliptic integral of the first kind]] and <math>E</math> is the [[complete elliptic integral of the second kind]].

A circular vortex line is the limiting case of a thin vortex ring. Because there is no core thickness, the speed of the ring is infinite, as well as the [[kinetic energy]]. The hydrodynamic impulse can be expressed in term of the strength, or 'circulation' <math>\kappa</math>, of the vortex ring as <math>I = \rho \pi \kappa R^2 </math>.