A mathematical consideration of vortex thinning in 2D turbulence

Abstract

In two dimensional turbulence, vortex thinning process is one of the attractive mechanism to explain inverse energy cascade in terms of vortex dynamics. By direct numerical simulation to the two-dimensional Navier-Stokes equations with small-scale forcing and large-scale damping, Xiao-Wan-Chen-Eyink (2009) found an evidence that inverse energy cascade may proceed with the vortex thinning mechanism. The aim of this paper is to analyze the vortex-thinning mechanism mathematically (using the incompressible Euler equations), and give a mathematical evidence that large-scale vorticity gains energy from small-scale vorticity due to the vortex-thinning process.