The three-dimensional instability of a strained vortex tube revisited

Abstract

We revisit the Moore–Saffman–Tsai–Widnall instability, a parametric resonance between left- and right-handed bending waves of infinitesimal amplitude, on the Rankine vortex strained by a weak pure shear flow. The results of Tsai & Widnall (1976) and Eloy & Le Dizès (2001), as generalized to all pairs of Kelvin waves whose azimuthal wavenumbers $m$ are separated by 2, are simplified by providing an explicit solution of the linearized Euler equations for the disturbance flow field. Given the wavenumber $k_0$ and the frequency $\omega_0$ of an intersection point of dispersion curves, the growth rate is expressible solely in terms of the modified Bessel functions, and so is the unstable wavenumber range. Every intersection leads to instability. Most of the intersections correspond to weak instability that vanishes in the short-wave limit, and dominant modes are restricted to particular intersections. For helical waves $m=\pm 1$, the growth rate of non-rotating waves is far larger than that of rotating waves. The wavenumber width of stationary instability bands broadens linearly in $k_0$, while that of rotating instability bands is bounded. The growth rate of the stationary instability takes, in the long-wavelength limit, the value of $\varepsilon/2$ for the two-dimensional displacement instability, and, in the short-wavelength limit, the value of $9\varepsilon/16$ for the elliptical instability, being larger at large but finite values of $k_0$. Here $\varepsilon$ is the strength of shear near the core centre. For resonance between higher azimuthal wavenumbers $m$ and $m+2$, the same limiting value is approached as $k_0\,{\to}\,\infty$, along sequences of specific crossing points whose frequency rapidly converges to $m+1$, in two ways, from above for a fixed $m$ and from below for $m\,{\to}\,\infty$. The energy of the Kelvin waves is calculated by invoking Cairns' formula. The instability result is compatible with Krein's theory for Hamiltonian spectra.