Spatio-temporal development of the pairing instability in an infinite array of vortex rings

In this paper, we study the linear stability of an infinite vortex ring array with respect to the pairing instability, using a spectral code. The base flow solution, obtained after a short relaxation process, is composed of rings with a Gaussian azimuthal vorticity profile. The temporal stability properties are first obtained and compared to the theoretical predictions obtained by Levy and Forsedyke (1927 Proc. R. Soc. Lond. A 114 594-604). The spatio-temporal evolution of a localized perturbation is then computed. The growth rate σ ≤ft( v \right) of the perturbation in the frame moving at the speed v is obtained for all v. The variation of σ ≤ft( v \right) with respect to the parameters of the flow is provided.