On the stability and instability of Kelvin-Stuart cat's-eye flows

Abstract

Kelvin-Stuart vortices are classical mixing layer flows with many applications in fluid mechanics, plasma physics and astrophysics. We prove that the whole family of Kelvin-Stuart vortices is nonlinearly orbitally stable for co-periodic perturbations, and linearly unstable for multi-periodic and modulational perturbations. This verifies a long-standing conjecture since the discovery of the Kelvin-Stuart cat's-eye flows in the 1960s. Kelvin-Stuart cat's eyes also appear as magnetic islands which are magnetostatic equilibria for the planar ideal MHD equations in plasmas. We prove nonlinear orbital stability of Kelvin-Stuart magnetic islands for co-periodic perturbations, and give the first rigorous proof of coalescence instability for the whole family, which is important for magnetic reconnection.