Three-Dimensional Stability of Burgers Vortices

Burgers vortices are explicit stationary solutions of the Navier-Stokes equations which are often used to describe the vortex tubes observed in numerical simulations of three-dimensional turbulence. In this model, the velocity field is a two-dimensional perturbation of a linear straining flow with axial symmetry. The only free parameter is the Reynolds number Re = I"/nu, where I" is the total circulation of the vortex and nu is the kinematic viscosity. The purpose of this paper is to show that Burgers vortices are asymptotically stable with respect to small three-dimensional perturbations, for all values of the Reynolds number. This general result subsumes earlier studies by various authors, which were either restricted to small Reynolds numbers or to two-dimensional perturbations. Our proof relies on the fact that the linearized operator at Burgers vortex has a simple and very specific dependence upon the axial variable. This allows to reduce the full linearized equations to a vectorial two-dimensional problem, which can be treated using an extension of the techniques developed in earlier works. Although Burgers vortices are found to be stable for all Reynolds numbers, the proof indicates that perturbations may undergo an important transient amplification if Re is large, a phenomenon that was indeed observed in numerical simulations.