Stability of co-axial vortex rings with implications for hot spot formation in supernova remnants*

Perturbations along two interacting vortex cores can grow under the influence of their self- and mutually induced velocity fields in a process known as the Crow instability. While generally considered in the context of wingtip vortices (i.e., streamwise vortices in a planar geometry), other interacting vortex cores, including rings, can also demonstrate perturbation growth. In this study, we analyze the collision of two co-axial vortex rings of equal strength. We show that the zero-order motion of the flow, unlike the case of two interacting line vortices, causes the growth of different wavenumbers to vary in time, providing an explanation for the dominance of the low-frequency symmetric mode and the observed Reynolds-number dependence of the dominant wavenumber observed in experiments. Our results may have important implications related to the formation of hot spots along gaseous tori ejected from stars for which a supernova event is imminent; specifically, our analysis predicts maximum growth for a wavenumber consistent with the number of hot spots observed on the circumstellar ring of supernova 1987A.