Spreading of two-dimensional axisymmetric vortices exposed to a rotating strain field

Abstract

This paper examines the evolution of an axisymmetric two-dimensional vortex in a steadily rotating strain field and the dynamical interactions that can enhance vortex spreading through resonant behaviour. Starting with a point vortex localized at the origin, the applied strain field generates a cat's eye topology in the co-rotating streamfunction, localized around a radius rext. Now the vortex is allowed to spread viscously: initially rext lies outside the vortex, but as it spreads, vorticity is advected into the cat's eyes, leading to a local flattening of the mean profile of the vortex and so to enhanced mixing and spreading of the vortex. Together with this is a feedback: the response of the vortex to the external strain depends on the modified profile. The feedback is particularly strong when rext coincides with the radius rcat at which the vortex can support cat's eyes of infinitesimal width. There is a particular time at which this occurs, as these radii change with the viscous spread of the vortex: rext moves inwards and rcat outwards. This resonance behaviour leads to increased mixing of vorticity, along with a rapid stretching of vorticity contours and a sharp increase in the amplitude of the non-axisymmetric components. The dynamical feedback and enhanced diffusion are studied for viscously spreading vortices by means of numerical simulations of their time evolution, parameterized only by the Reynolds number R and the dimensionless strength A of the external strain field.