Curvature instability of a vortex ring

A global stability analysis of Kelvin's vortex ring to three-dimensional disturbances of infinitesimal amplitude is made. The basic state is a steady asymptotic solution of the Euler equations, in powers of the ratio ε of the core radius to the ring radius, for an axisymmetric vortex ring with vorticity proportional to the distance from the symmetric axis. The effect of ring curvature appears at first order, in the form of a dipole field, and a local straining field, which is a quadrupole field, follows at second order. The eigenvalue problem of the Euler equations, retaining the terms to first order, is solved in closed form, in terms of the Bessel and the modified Bessel functions. We show that the dipole field causes a parametric resonance instability between a pair of Kelvin waves whose azimuthal wavenumbers are separated by 1. The most unstable mode occurs in the short-wavelength limit, under the constraint that the radial and the azimuthal wavenumbers are of the same magnitude, and the limiting value of maximum growth rate coincides with the value 165/256ε obtained by Hattori & Fukumoto (Phys. Fluids, vol. 15, 2003, p. 3151) by means of the geometric optics method. The instability mechanism is traced to stretching of disturbance vorticity in the toroidal direction. In the absence of viscosity, the dipole effect outweighs the straining field effect of O(ε ²) known as the Moore-Saffman-Tsai-Widnall instability. The viscosity acts to damp the former preferentially and these effects compete with each other.