Small disturbances of steady vortices

Abstract

A new approach to the description of the disturbed motion in the cores of localized steady vortices is developed within the context of an ideal incompressible fluid. For this purpose the linearized Helmholtz equation for the vorticity perturbations is reformulated in the language of the deformation field directly characterizing the evolution of the shape of each vortex line. The description of the disturbed motion of steady vortices in the language of the deformation field is used to investigate the vibrations of a vortex ring. A unified method of describing all its types of natural vibration, based on the expansion of the solution in a special basis of the disturbances representing the response to given vibrations of the vortex boundary, is developed. This method differs from most previous approaches mainly in that the structure of the disturbed field over the cross section of the ring is not assumed to be known in advance, but is determined dynamically in the form of successive approximations with respect to μ. The method makes it possible, in principle, to classify the natural vibrations of the vortex ring in the same way as for a Kelvin vortex. By means of the procedure proposed a family of slow vibrations of the vortex ring with frequencies ω/Ω₀=O(μ) is found. It is shown that these vibrations possess a complex structure and in the leading approximation constitute a linear combination of a barrel-shaped and a flexural perturbation. In this case the displacement of the center line of the ring is determined by the internal dynamics of the disturbances in the vortex and cannot be obtained within the framework of the thin vortex filament theory.