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Viscosity renormalization based on direct-interaction closure

Published online by Cambridge University Press:  20 April 2006

George F. Carnevale  and
Jorgen S. Frederiksen
George F. Carnevale
Affiliation:
Center for Studies of Nonlinear Dynamics, La Jolla Institute, P.O. Box 1434, La Jolla, CA 92038, U.S.A.
Jorgen S. Frederiksen
Affiliation:
CSIRO, Division of Atmospheric Physics, Mordialloc, Victoria 3195, Australia
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Abstract

Approximations in statistical turbulence theory often rely on modelling the decay in time of velocity correlations with a simple exponential decay. The decay rate is viewed as a renormalized viscosity. The three simplest implementations of this approximation scheme were originally given independently by Kraichnan, Edwards and Leslie. Each of these investigators used a different formalism and each achieved different renormalization prescriptions. These three different results are reexamined here entirely in terms of direct-interaction theory. The difference in the prescriptions of Kraichnan and Leslie is shown to be the product of different definitions of renormalized viscosity. Edwards’ prescription is shown to result from an inconsistent identification of the non-stationary energy-spectrum relaxation rate with the viscosity. An assessment of the validity of the Markovian closure approximation, and a prescription for non-stationary renormalized viscosity are provided.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 131 , June 1983 , pp. 289 - 303
Copyright
© 1983 Cambridge University Press

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References

André, J. C. 1974 Phys. Fluids 17, 15–21.
Bell, T. L. 1980 J. Atmos. Sci. 37, 1700–1707.
Bjorken, J. D. & Drell, S. D. 1965 Relativistic Quantum Fields. McGraw-Hill.
Carnevale, G. F. 1982 Phys. Fluids 25, 1547–1549.
Carnevale, G. F. & Frederiksen, J. S. 1983 A statistical dynamical theory of strongly nonlinear internal gravity waves. Geophys. Astrophys. Fluid Dyn. (in press).Google Scholar
Carnevale, G. F. & Martin, P. C. 1982 Geophys. Astrophys. Fluid Dyn. 20, 131–164.
Chandrasekhar, S. 1943 Rev. Mod. Phys. 15, 1–89.
Common, A. K. 1970 In The Fadé Approximant in Theoretical Physics (ed. G. A. Baker & J. L. Gammel), pp. 241–256. Academic.
Deker, V. & Haake, F. 1975 Phys. Rev. A11, 2043–2056.
Dewitt, R. J. & Wright, J. 1982 J. Fluid Mech. 115, 283–302.
Dubois, D. F. & Rose, H. A. 1982 Phys. Rev. A24, 1476–1504.
Edwards, S. F. 1964 J. Fluid Mech. 18, 239–273.
Fetter, A. L. & Walecka, J. D. 1971 Quantum Theory of Many Particle Systems. McGraw-Hill.
Fournier, J. D. & Frisch, U. 1978 Phys. Rev. A17, 747–762.
Herring, J. R. 1965 Phys. Fluids 8, 2219–2225.
Herring, J. R. & Kraichnan, R. H. 1972 In Statistical Models and Turbulence (ed. M. Rosenblatt & C. Van Atta). Lecture Notes in Physics, vol. 12, pp. 148–194. Springer.
Holloway, G. & Hendershott, M. C. 1977 J. Fluid Mech. 82, 747–765.
Kadanoff, L. P. & Baym, G. 1963 Quantum Statistical Mechanics. Benjamin.
Kraichnan, R. H. 1959a J. Fluid Mech. 5, 497–543.
Kraichnan, R. H. 1959b Phys. Rev. 113, 1181–1182.
Kraichnan, R. H. 1961 J. Math Phys. 2, 124–148.
Kraichnan, R. H. 1964 Phys. Fluids 7, 1163–1168.
Kraichnan, R. H. 1970 In The Padé Approximant in Theoretical Physics (ed. G. A. Baker & J. L. Gammel), pp. 129–170. Academic.
Kraichnan, R. H. 1971 J. Fluid Mech. 47, 513–524.
Kraichnan, R. H. 1975 J. Fluid Mech. 67, 155–175.
Kraichnan, R. H. 1976 J. Atmos. Sci. 33, 1521–1536.
Langreth, D. C. 1976 Linear and nonlinear response theory with application. In Proc. 1975 NATO Advanced Study Institute on Linear and Nonlinear Electron Transport in Solids, Antwerp (ed. J. T. Devreese & V. E. van Doren), pp. 3–32. Plenum.
Lee, J. 1974 J. Math. Phys. 15, 1571–1586.
Legras, B. 1980 Geophys. Astrophys. Fluid Dyn. 15, 253–281.
Leith, C. E. 1971 J. Atmos. Sci. 28, 145–161.
Leith, C. E. 1975 J. Atmos. Sci. 32, 2022–2026.
Leslie, D. C. 1973 Developments in the Theory of Turbulence. Clarendon.
Martin, P. C., Siggia, E. D. & Rose, H. A. 1973 Phys. Rev. A8, 423–437.
Orszag, S. A. 1970 J. Fluid Mech. 41, 363–386.
Roman, P. 1969 Introduction to Quantum Field Theory. Wiley.
Thompson, P. D. 1972 J. Fluid Mech. 55, 711–717.
Thompson, P. D. 1982 Phys. Fluids 25, 1159–1161.