The effect of fine structure on the stability of planar vortices

Abstract

This study considers the linear, inviscid response to an external strain field of classes of planar vortices. The case of a Gaussian vortex has been considered elsewhere, and an enstrophy rebound phenomenon was noted: after the vortex is disturbed enstrophy feeds from the non-axisymmetric to mean flow. At the same time an irreversible spiral wind-up of vorticity fluctuations takes place. A top-hat or Rankine vortex, on the other hand, can support a non-decaying normal mode. In vortex dynamics processes such as stripping and collisions generate vortices with sharp edges and often with bands or rings of fine scale vorticity at their periphery, rather than smooth profiles. This paper considers the stability and response of a family of vortices that vary from a broad profile to a top-hat vortex. As the edge of the vortex becomes sharper, a quasi-mode emerges and vorticity winds up in a critical layer, at the radius where the angular velocity of the fluid matches that of a normal mode on a top-hat vortex. The decay rate of these quasi-modes is proportional to the vorticity gradient at the critical layer, in agreement with theory. As the vortex edge becomes sharper it is found that the rebound of enstrophy becomes stronger but slower. The stability and linear behaviour of coherent vortices is then studied for distributions which exhibit additional fine structure within the critical layer. In particular we consider vorticity profiles with ‘bumps’, ‘troughs’ or ‘steps’ as this fine structure. The modified evolution equation that governs the critical layer is studied using numerical simulations and asymptotic analysis. It is shown that depending on the form of the short-scale vorticity distribution, this can stabilise or destabilise quasi-modes, and it may also lead to oscillatory behaviour.