Periodic Norbury Vortex Rings

Norbury (1973) showed the existence of a one-parameter family of axisymmetric vortex rings ranging from the infinitesimally thin circular vortex rings of Kelvin and Helmholtz to Hill's spherical vortex, using a parameter $\alpha$ that measures the ``thickness" of the ring. These vortex rings move under their own induction in the axial direction at a constant speed that depends on $\alpha$. On the other hand, Levy and Forsdyke (1927) --- as well as Vasilev (1916) before them --- showed that an infinite coaxial array of thin vortex rings is also a solution to the Euler equations. Masroor and Stremler (2022) showed how the self-induced speed of such an array depends on two non-dimensional parameters: the $\alpha$ of Norbury as well as $\lambda$, the ratio between the inter-vortex spacing and the rings' common radii, in the realm of small $\alpha$. In this work, we adapt a modern technique for calculating the cross-sectional contours of Norbury's vortex rings (Protas 2019) to numerically determine a two-parameter family of ``periodic Norbury rings" that move in the axial direction under their mutual induction without change of form. This family of rings spans the full range from infinite arrays of thin vortex rings in the regime $\alpha \ll 1$ to infinite arrays of Hill's spherical vortex in the limit $\alpha \rightarrow \sqrt{2}$. The stability of these novel stationary solutions of the axisymmetric Euler equations will be examined in a future study.