Editing Quantum turbulence (section)

=== Properties of vortex lines ===
Vortex lines are topological line defects of the phase. Their nucleation makes the quantum fluid's region to become a multiply-connected region. As given by Fig 2, density depletion can be observed near the axis, with <math>\rho = 0 </math> on the vortex line. The size of the vortex core varies between different quantum fluids. The size of the vortex core is around <math>a_0 = 10^{-10}\text{m} </math> for helium II, <math>a_0 = 10^{-8}\text{m} </math> for <sup>3</sup>He-B and for typical atomic condensates <math>a_0 = 10^{-4}\text{m} </math>. The simplest vortex system in a quantum fluid consists of a single straight vortex line; the velocity field of such configuration is purely azimuthal given by <math>v_{\theta} = \kappa/{2\pi r}</math>. This is the same formula as for a classical vortex line solution of the Euler equation, however, classically, this model is physically unrealistic as the velocity diverges as <math>r \rightarrow 0</math>. This leads to the idea of the [[Rankine vortex]] as shown in fig 2, which combines solid body rotation for small <math>r</math> and vortex motion for large values of <math>r</math>, and is a more realistic model of ordinary classical vortices.

Many similarities can be drawn with vortices in classical fluids, for example the fact that vortex lines obey the classical [[Kelvin's circulation theorem|Kelvin circulation theorem]]: the circulation is conserved and the vortex lines must terminate at boundaries or exist in the shape of closed loops. In the zero temperature limit, a point on a vortex line will travel accordingly to the velocity field that is generated at that point by the other parts of the vortex line, provided that the vortex line is not straight (an isolated straight vortex does not move). The velocity can also be generated by any other vortex lines in the fluid, a phenomenon also present in classical fluids. A simple example of this is a [[vortex ring]] (a torus-shaped vortex) which moves at a self-induced velocity <math>v_R</math> inversely proportional to the radius of the ring <math>R</math>, where <math>R >> a_0</math>.<ref>{{Cite journal|last1=Barenghi|first1=C. F.|last2=Donnelly|first2=R J|date=October 2009|title=Vortex rings in classical and quantum systems|url=https://iopscience.iop.org/article/10.1088/0169-5983/41/5/051401|journal=Fluid Dynamics Research|volume=41|issue=5|pages=051401|doi=10.1088/0169-5983/41/5/051401|s2cid=123246632 |issn=0169-5983}}</ref> The whole ring moves at a velocity

<math>v_R = \frac{\kappa}{4\pi R}\left[\ln{\left(\frac{8R}{a_0}\right)} - \frac{1}{2}\right]</math>