Journal of Fluid Mechanics


Vanishing enstrophy dissipation in two-dimensional Navier–Stokes turbulence in the inviscid limit

a1 School of Mathematics and Statistics, University of St Andrews, St Andrews KY16 9SS, UK

Article author query
tran cv   [Google Scholar] 
dritschel dg   [Google Scholar] 


Batchelor (Phys. Fluids, vol. 12, 1969, p. 233) developed a theory of two-dimensional turbulence based on the assumption that the dissipation of enstrophy (mean-square vorticity) tends to a finite non-zero constant in the limit of infinite Reynolds number Re. Here, by assuming power-law spectra, including the one predicted by Batchelor's theory, we prove that the maximum dissipation of enstrophy is in fact zero in this limit. Specifically, as $\mbox{\it Re} \to \infty$, the dissipation approaches zero no slower than $(\ln\mbox{\it Re})^{-1/2}$. The physical reason behind this result is that the decrease of viscosity enhances the production of both palinstrophy (mean-square vorticity gradients) and its dissipation – but in such a way that the net growth of palinstrophy is less rapid than the decrease of viscosity, resulting in vanishing enstrophy dissipation. This result generalizes to a rich class of quasi-geostrophic models as well as to the case of a passive tracer in layerwise-two-dimensional turbulent flows having bounded enstrophy.

(Published Online July 19 2006)
(Received December 12 2005)
(Revised April 20 2006)