Enhanced dissipation and axisymmetrization of two-dimensional viscous vortices

Abstract

This paper is devoted to the stability analysis of the Lamb-Oseen vortex in the regime of high circulation Reynolds numbers. When strongly localized perturbations are applied, it is shown that the vortex relaxes to axisymmetry in a time proportional to $Re^{2/3}$, which is substantially shorter than the diffusion time scale given by the viscosity. This enhanced dissipation effect is due to the differential rotation inside the vortex core. Our result relies on a recent work by Li, Wei, and Zhang, where optimal resolvent estimates for the linearized operator at Oseen's vortex are established. A comparison is made with the predictions that can be found in the physical literature, and with the rigorous results that were obtained for shear flows using different techniques.