Editing Vortex ring (section)

===Spherical vortices===
{{Main|Hill's spherical vortex}}

[[Micaiah John Muller Hill|Hill]]'s spherical vortex<ref name="hill1894">{{cite journal|last1=Hill|first1=M.J.M.|title=VI. On a spherical vortex |journal=Philosophical Transactions of the Royal Society of London A |date=1894 |volume=185 |pages=213–245 |doi=10.1098/rsta.1894.0006|bibcode=1894RSPTA.185..213H|doi-access=free }}</ref> is an example of steady vortex flow and may be used to model vortex rings having a vorticity distribution extending to the centerline. More precisely, the model supposes a linearly distributed vorticity distribution in the radial direction starting from the centerline and bounded by a sphere of radius <math>a</math> as:
<math display="block"> \frac{\omega}{r}=\frac{15}{2}\frac{U}{a^2}</math>
where <math>U</math> is the constant translational speed of the vortex.

Finally, the [[Stokes stream function]] of Hill's spherical vortex can be computed and is given by:<ref name="hill1894"/><ref name="lamb1932"/>
<math display="block">\begin{align}
&\psi(r,x) = -\frac{3}{4}\frac{U}{a^2}r^2\left(a^2-r^2-x^2\right) && \text{inside the vortex} \\
&\psi(r,x) = \frac{1}{2}Ur^2\left[1-\frac{a^3}{\left(x^2+r^2\right)^{3/2}}\right] && \text{outside the vortex}
\end{align}</math>
The above expressions correspond to the stream function describing a steady flow. In a fixed frame of reference, the stream function of the bulk flow having a speed <math>U</math> should be added.

The [[circulation (physics)|circulation]], the hydrodynamic impulse and the [[kinetic energy]] can also be calculated in terms of the translational speed <math>U</math> and radius <math>a</math>:<ref name="hill1894"/><ref name="lamb1932"/>
<!-- density ρ missing in impulse and energy , see Lamb p. 244 -->
<math display="block">\begin{align}
\Gamma &= 5Ua \\
I &= 2\pi\rho Ua^3 \\
E & = \frac{10\pi}{7}\rho U^2a^3
\end{align}</math>

Such a structure or an electromagnetic equivalent has been suggested as an explanation for the internal structure of [[ball lightning]].  For example, Shafranov {{Citation needed|date=July 2010}} used a magnetohydrodynamic (MHD) analogy to Hill's stationary fluid mechanical vortex to consider the equilibrium conditions of axially symmetric MHD configurations, reducing the problem to the theory of stationary flow of an incompressible fluid.  In axial symmetry, he considered general equilibrium for distributed currents and concluded under the [[Virial Theorem]] that if there were no gravitation, a bounded equilibrium configuration could exist only in the presence of an azimuthal current.{{Citation needed|date=June 2021}}