Linear and nonlinear instability of vortex columns

Abstract

We consider vortex column solutions $v=V(r)e_{\theta }+W(r)e_z$ to the $3$D Euler equations. We give a mathematically rigorous construction of the countable family of unstable modes discovered by Liebovich and Stewartson (J. Fluid Mech. 126 (1983)) via formal asymptotic analysis. The unstable modes exhibit $O(1)$ growth rates and concentrate on a ring $r=r_0$ asymptotically as the azimuthal and axial wavenumbers $n,\alpha$ tend to $\infty$ with a fixed ratio. We construct these so-called ring modes with an inner-outer gluing procedure. Finally, we prove that each linear instability implies nonlinear instability for vortex columns. In particular, our analysis yields nonlinear instability for the Batchelor trailing line vortex $V(r):=\frac{q}{r}(1-{\mathrm{e}}^{-r^2})$ and $W(r):={\mathrm{e}}^{-r^2}$ when $0<q<\log 2/\sqrt{1-\log 2}\approx 1.251$.