Vanishing enstrophy dissipation in two-dimensional Navier--Stokes turbulence in the inviscid limit
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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 Re -> infinity, the dissipation approaches zero no slower than (ln 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.
Tran , C V & Dritschel , D G 2006 , ' Vanishing enstrophy dissipation in two-dimensional Navier--Stokes turbulence in the inviscid limit ' Journal of Fluid Mechanics , vol. 559 , pp. 107-116 . DOI: 10.1017/S0022112006000577
Journal of Fluid Mechanics
(c)2006 Cambridge University Press
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