Thierry Gallay, Yasunori Maekawa
Three-dimensional stability of Burgers vortices
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ABSTRACT. 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 $Re = \Gamma/\nu$, where $\Gamma$
is the total circulation of the vortex and $\nu$ is the kinematic
viscosity. The purpose of this paper is to show that Burgers vortex
is asymptotically stable with respect to general three-dimensional
perturbations, for all values of the Reynolds number. This definitive
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 crucial observation 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
developped 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.