infinitesimal compared to the radius of the ring , i.e. . As a consequence, inside and in the vicinity of the core ring, one may write: , and , and, in the limit of , the elliptic integrals can be approximated by and . For a uniform vorticity distribution in the disk, the Stokes stream function can therefore be approximated by The resulting circulation , hydrodynamic impulse and kinetic energy are It is also possible to find the translational ring speed (which is finite) of such isolated thin-core vortex ring: which finally results in the well-known expression found by Kelvin and published in the English translation by Tait of von Helmholtz's paper: Spherical vortices Hill's spherical vortex 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 as: where 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: 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 should be added. The circulation, the hydrodynamic impulse and the kinetic energy can also be calculated in terms of the translational speed and radius : Such a structure or an electromagnetic equivalent has been suggested as an explanation for the internal structure of ball lightning. For example, Shafranov 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. Fraenkel-Norbury model The Fraenkel-Norbury model of isolated vortex ring, sometimes referred as the standard model, refers to the class of steady vortex rings having a linear distribution of vorticity in the core and parametrised by the mean core radius , where is the area of the vortex core and is the radius of the ring. Approximate solutions were found for thin-core rings, i.e.