  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. On the other hand, Alexakis-Doering (2006) calculated upper
bound of the bulk averaged enstrophy dissipation rate of the steady-state two
dimensional turbulence. {In this paper, we show that vortex thinning induces
enhanced dissipation with strictly slower vanishing order of the enstrophy
dissipation than $Re^{-1}$.}
