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

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Tran2006-JFluidMech2006-Vanishing.pdf (120.9Kb)
Date
25/07/2006
Author
Tran, Chuong Van
Dritschel, David Gerard
Keywords
Quasi-geostrophic turbulence
Spectral distribution
Energy
Decay
Equations
QA Mathematics
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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 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.
Citation
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 . https://doi.org/10.1017/S0022112006000577
Publication
Journal of Fluid Mechanics
Status
Peer reviewed
DOI
https://doi.org/10.1017/S0022112006000577
ISSN
0022-1120
Type
Journal article
Rights
(c)2006 Cambridge University Press
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  • University of St Andrews Research
URL
http://www.scopus.com/inward/record.url?scp=33746335656&partnerID=8YFLogxK
http://journals.cambridge.org/action/displayIssue?iid=454619
URI
http://hdl.handle.net/10023/1564

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