Bending waves on inviscid columnar vortices

Abstract

Bending waves, perturbation modes leading to deflections of the vortex centreline, are considered for an infinitely long straight vortex embedded in an irrotational flow of unlimited extent. We first establish the general form of the dispersion relation for long waves on columnar vortices with arbitrary distributions of axial and azimuthal vorticity by a singular perturbation analysis of the Howard-Gupta equation. The asymptotic results are shown to compare favourably with numerical solutions of the Howard-Gupta equation for wavelengths comparable to the vortex core radius and longer. Dispersion relations are then found numerically for specific models of vortex core structures observed experimentally; here no restrictions are placed on wavelength. The linear dispersion relation has an infinite number of branches, falling into two families; one with infinite phase speed at zero wavenumber (which we call ‘fast’ waves), the other with zero phase speed (‘slow’ waves). In the long-wave limit, slow waves have zero group velocity, while the fast waves may have finite non-zero group speeds that depend on the form of the velocity profiles on the axis of rotation. Weakly nonlinear waves are described under most circumstances by the nonlinear Schrödinger equation. Solitons are possible in certain ‘windows’ of wavenumbers of the carrier waves. An example has already been presented by Leibovich & Ma (1983), who compute solitons and soliton windows on a fast-wave branch for a vortex with a particular core structure. Experimental data of Maxworthy, Hopfinger & Redekopp (1985) reveal solitons which appear to be associated with the slow branch, and these are computed for velocity profiles fitting their data. The nonlinear Schrödinger equation is shown to fail for long waves, and to be replaced by a nonlinear integro-differential equation.