Spectral Properties of the Linearization at the Burgers Vortex in the High Rotation Limit

Abstract

We study a linearized operator of the equation for the axisymmetric Burgers vortex which gives a stationary solution to the three dimensional Navier–Stokes equations with an axisymmetric background straining flow. It is numerically known that the Burgers vortex obtains better stabilities as the circulation number (or the vortex Reynolds number) is increasing. Although the global stability of the axisymmetric Burgers vortex is already proved rigorously, mathematical understanding of this numerical observation has been lacking. In this paper we study a linearized operator that includes the circulation number as a parameter, and prove that if the operator is restricted on a suitable invariant subspace, then its spectrum moves to the left as the circulation number goes to infinity. As an application, we show that the axisymmetric Burgers vortex with a high rotation has a better stability, in the sense that the non-radially symmetric part of solutions to the associated evolution equation decays faster in time if the circulation number is sufficiently large.