  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 $n, \alpha \to \infty$ 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 $0 < q \ll 1$.
