Journal of Fluid Mechanics



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


CHUONG V. TRAN a1 and DAVID G. DRITSCHEL a1
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] 
 

Abstract

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)



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