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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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References

Bardos, C. & Frisch, U. 1974 Séminaire Leray 1974–1975. Paris: Collège de France.
Batchelor, G. K. 1969 Computation of the energy spectrum in homogeneous two-dimensional turbulence. Phys. Fluids Suppl. 12, II.233.Google Scholar
Brissaud, A., Frisch, U., LÉORAT, J., Lesieur, M., Mazure, A., Pouquet, A., Sadourny, R. & Sulem, P. L. 1973 Catastrophe énergétique et nature de la turbulence. Ann. Geophys. 29, 539.Google Scholar
Desbois, M. 1975 Large-scale kinetic energy spectra from eulerian analysis of Eole wind data. J. Atmos. Sci. 32 (in Press).Google Scholar
Dupree, H. T. 1974 Theory of two-dimensional turbulence. Phys. Fluids. 17, 100.Google Scholar
Frisch, U., Lesieur, M. & Brissaud, A. 1974 A Markovian random coupling model for turbulence. J. Fluid Mech. 65, 145.Google Scholar
Frisch, U., Lesieur, M. & Sulem, P. L. 1974 Le modèle du champ d’épreuve (test-field-model) de Kraichnan: une nouvelle méthode de fermeture des équations de la turbulence. Observatoire de Nice Preprint.
Herring, J. R. 1973 Proc. Symp. on Turbulence in Fluids and Plasmas Culham Laboratory, Berkshire.
Herring, J. R. & Kraichnan, R. H. 1972 Comparison of some approximations for isotropic turbulence. In Statistical Models and Turbulence, p. 148. Springer.
Herring, J. R., Orszag, S. A., Kraichnan, R. H. & Fox, D. G. 1974 Decay of two-dimensional homogeneous turbulence. J. Fluid Mech. 66, 417.Google Scholar
Kolesnikov, Y. B. & Tsinober, A. B. 1972 Two-dimensional turbulent flow behind a circular cylinder. Magnitnaya Gidrodinamica, 3, 23. (English trans. Magnetohydrodynamics, October 1972.)Google Scholar
Kraichnan, R. H. 1958 In 2nd Symp. on Naval Hydrodyn. (ed. R. Cooper), p. 29. Washington: Office of Naval Research, publ. ACR-38.
Kraichnan, R. H. 1961 Dynamics of non linear stochastic systems. J. Math. Phys. 2, 124.Google Scholar
Kraichnan, R. H. 1967 Inertial ranges in two-dimensional turbulence. Phys. Fluids, 10, 1417.Google Scholar
Kraichnan, R. H. 1971a An almost-Markovian Galilean-invariant turbulence model. J. Fluid Mech. 47, 513.Google Scholar
Kraichnan, R. H. 1971b Inertial-range transfer in two- and three-dimensional turbulence. J. Fluid Mech. 47, 525.Google Scholar
Kraichnan, R. H. 1975 Statistical dynamics of two-dimensional flows. J. Fluid Mech. 67, 15.Google Scholar
Leith, C. E. 1968 Diffusion approximation for two-dimensional turbulence. Phys. Fluids, 11, 671.Google Scholar
Leith, C. E. 1971 Atmospheric predictability and two-dimensional turbulence. J. Atmos. Sci. 28, 145.Google Scholar
Leith, C. E. & Kraichnan, R. H. 1972 Predictability of turbulent flows. J. Atmos. Sci. 29, 1041.Google Scholar
Lesieur, M. 1973 Ph.D. thesis, University of Nice.
Lilly, D. K. 1969 Numerical simulation of two-dimensional turbulence. Phys. Fluids, Suppl. 12, II.240.Google Scholar
Lilly, D. K. 1971 Numerical simulation of developing and decaying two-dimensional turbulence. J. Fluid Mech. 45, 395.Google Scholar
Lilly, D. K. 1972 Numerical simulation studies of two-dimensional turbulence. I. Models of statistically steady turbulence. Geophys. Fluid Dyn. 3, 289.Google Scholar
Morel, P. & Larchevêque, M. 1974 Relative dispersion of constant-level balloons in the 200-mb general circulation. J. Atmos. Sci. 31, 2189.Google Scholar
Morel, P. & Necco, G. 1973 Scale dependance of the 200-mb divergence inferred from Eole data. J. Atmos. Sci. 30, 909.Google Scholar
Onsager, L. 1949 Statistical hydrodynamics. Nuovo Cimento Suppl. 6, 279.Google Scholar
Orszag, S. A. 1970 Analytical theories of turbulence. J. Fluid Mech. 41, 363.Google Scholar
Orszag, S. A. 1974 Statistical theory of turbulence. Les Houches Summer School on Physics. Gordon & Breach.
Proudman, I. & Reid, W. H. 1954 On the decay of a normally distributed and homogeneous turbulent velocity field. Phil. Trans. A 247, 163.Google Scholar
Saffman, P. G. 1971 On the spectrum and decay of random two-dimensional vorticity distributions at large Reynolds number. Appl. Math. 50 (4), 377.Google Scholar
Wiin-Nielsen, A. 1967 On the annual variation and spectral distribution of atmospheric energy. Tellus, 19, 540.Google Scholar