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Evolution of high Reynolds number two-dimensional turbulence

Published online by Cambridge University Press:  29 March 2006

A. Fouquet ,
M. Lesieur ,
J. C. André  and
C. Basdevant
A. Fouquet
Affiliation:
National Center for Atmospheric Research
The National Center for Atmospheric Research is sponsored by the National Science Foundation.
Boulder, Colorado 80302 Permanent address: Observatoire de Nice, France.
M. Lesieur
Affiliation:
Centre National de la Recherche Scientifique, Observatoire de Nice, France
J. C. André
Affiliation:
EERM/GMD, Météorologie Nationale, Paris, France
C. Basdevant
Affiliation:
Laboratoire de Météorologie Dynamique, Paris, France
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Abstract

Kraichnan's (1967) predictions concerning a simultaneous direct enstrophy cascade and inverse energy cascade for high Reynolds number two-dimensional turbulence are tested numerically using a variant of the eddy-damped quasi-normal approximation. For the initial-value problem, an analytic study using this theory shows that, in the zero-viscosity limit, energy and enstrophy are conserved for arbitrarily long times, contrary to the three-dimensional case, where the energy is conserved for only a finite time, after which it is dissipated. Non-local effects in the enstrophy inertial range, which are difficult to treat by conventional numerical schemes (Leith 1971; Leith & Kraichnan 1972), are shown to be representable by an additional diffusion term in the spectral equation. The resulting equation, including non-local effects, is integrated numerically. When enstrophy and energy are continuously injected at a fixed wavenumber, it is shown numerically that a quasi-steady regime is obtained where enstrophy cascades to large wavenumbers across a k−3 inertial range with zero energy transfer while energy flows indefinitely to small wavenumbers across a $k^{-\frac{5}{3}}$ inertial range with zero enstrophy transfer.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 72 , Issue 2 , 25 November 1975 , pp. 305 - 319
Copyright
© 1975 Cambridge University Press

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