The erosion of a distributed two-dimensional vortex in a background straining flow

Herein we present a simplified theory for the behaviour of a vortex embedded in a growing external straining flow. Such a flow arises naturally as a vortex moves relative to other vortices. While the strain may generally exhibit a complex time dependence, the salient features of the vortex evolution can be understood in the simpler context, studied here, of a linearly growing strain. Then, all of the typical stages of evolution can be seen, from linear deformation, to the stripping or erosion of low-lying peripheral vorticity, and finally to the breaking or rapid elongation of the vortex into a thin filament.

When, as is often the case in practice, the strain growth is slow, the vortex adjusts itself to be in approximate equilibrium with the background flow, Then., the vortex passes through, or near, a sequence of equilibrium states until, at a critical value of the strain, it suddenly breaks. In the intermediate period before breaking, the vortex continuously sheds peripheral vorticity, thereby steepening its edge gradients. This stripping is required to keep the vortex in a near equilibrium configuration.

We show that this behaviour can be captured., quantitatively, by a reduced model, the elliptical model, which represents the vortex by a nested set of elliptical vorticity contours, each having a (slightly) different aspect ratio and orientation. Here, we have extended the original elliptical model by allowing for edge vorticity levels to be shed when appropriate (to represent stripping) and by incorporating the flow induced by the vorticity being stripped away. The success of this model proves that the essential characteristics of vortex erosion are captured simply by the leading-order, elliptical shape deformations of vorticity contours.

Finally, we discuss the role of viscosity. Then, there is a competition between gradient steepening by stripping and smoothing by viscosity. If the strain grows too slowly, the vortex is dominated by viscous decay, and the edge gradients become very smooth. On the other hand, for sufficiently rapid strain growth (which can still be slow, depending on the viscosity), the vortex edge remains steep until the final breaking.