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Statistical equilibrium states for two-dimensional flows

Published online by Cambridge University Press:  26 April 2006

R. Robert  and
J. Sommeria
R. Robert
Affiliation:
21. Avenue Plaine Fleurie, 38240 Meylan, France
J. Sommeria
Affiliation:
CNRS, Laboratoire de Physique, Ecole Normale Supérieure de Lyon, 46 al. d'Italie, 69 364 Lyon Cedex 07, France
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Abstract

We explain the emergence of organized structures in two-dimensional turbulent flows by a theory of equilibrium statistical mechanics. This theory takes into account all the known constants of the motion for the Euler equations. The microscopic states are all the possible vorticity fields, while a macroscopic state is defined as a probability distribution of vorticity at each point of the domain, which describes in a statistical sense the fine-scale vorticity fluctuations. The organized structure appears as a state of maximal entropy, with the constraints of all the constants of the motion. The vorticity field obtained as the local average of this optimal macrostate is a steady solution of the Euler equation. The variational problem provides an explicit relationship between stream function and vorticity, which characterizes this steady state. Inertial structures in geophysical fluid dynamics can be predicted, using a generalization of the theory to potential vorticity.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 229 , August 1991 , pp. 291 - 310
Copyright
© 1991 Cambridge University Press

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References

Arms, R. J. & Hama, F. R. 1965 Localized-induction concept on a curved vortex and motion of an elliptic vortex ring. Phys. Fluids 8, 553–559.Google Scholar
Benjamin, T. B. 1962 Theory of the vortex breakdown phenomenon. J. Fluid Mech. 14, 593–629.Google Scholar
Benjamin, T. B. 1967 Some developments in the theory of vortex breakdown. J. Fluid Mech. 28, 65–84.Google Scholar
Betchov, R. 1965 On the curvature and torsion of an isolated vortex filament. J. Fluid Mech. 22, 471–479.Google Scholar
Caulk, D. A. & Naghdi, P. M. 1979 The influence of twist on the motion of straight elliptical jets. Arch. Rat. Mech. Anal. 69, 1–30.Google Scholar
Cruickshank, J. O. & Munson, B. R. 1981 Viscous fluid buckling of plane and axisymmetric jets. J. Fluid Mech. 113, 221–239.Google Scholar
Dym, C. L. 1974 Stability Theory and its Applications to Structural Mechanics. Noordhoff.
Garg, A. K. & Leibovich, S. 1979 Spectral characteristics of vortex breakdown flow-fields. Phys. Fluids 22, 2053–2064.Google Scholar
Green, A. E. & Naghdi, P. M. 1976 Directed fluid sheets.. Proc. R. Soc. Lond. A 347, 447–473.Google Scholar
Green, A. E., Naghdi, P. M. & Wenner, M. L. 1974 On the theory of rods. II. Developments by direct approach.. Proc. R. Soc. Lond. A 337, 485–507.Google Scholar
Kelvin, Lord 1880 Vibrations of a columnar vortex. Phil. Mag. 10, 155–168.Google Scholar
Lundgren, T. S. & Ashurst, W. T. 1989 Area-varying waves on curved vortex tubes with application to vortex breakdown. J. Fluid Mech. 200, 283–307 (referred to herein as LA).Google Scholar
Maxworthy, T., Hopfinger, E. J. & Redekopp, L. G. 1985 Wave motions on vortex cores. J. Fluid Mech. 151, 141–165.Google Scholar
Miles, J. & Salmon, R. 1985 Weakly dispersive nonlinear gravity waves. J. Fluid Mech. 157, 519–531.Google Scholar
Moore, D. W. & Saffman, P. G. 1971 Structure of a line vortex in an imposed strain In Aircraft Wake Turbulence and its Detection, pp. 339. Plenum.
Moore, D. W. & Saffman, P. G. 1972 The motion of a vortex filament with axial flow.. Phil. Trans. R. Soc. Lond. A 272, 403–429 (referred to herein as MS).Google Scholar
Naghdi, P. M. 1980 Finite deformation of elastic rods and shells. Proc. IUTAM Symp. on Finite Elasticity, pp. 47–103. Martinus Nijhoff.
Sarpkaya, T. 1971 On stationary and travelling vortex breakdowns. J. Fluid Mech. 45, 545–559.Google Scholar
Serrin, J. 1959 Mathematical principles of classical fluid mechanics. In Handbuch der Physik (ed. S. Flügge), Vol. VIII/1, pp. 125–263. Springer.
Taylor, G. I. 1968 Instability of jets, threads and sheets of viscous fluids. Proc. Intl Congr. Appl. Mech. Springer.
Truesdell, C. & Toupin, R. 1960 The classical field theories. In Handbuch der Physik (ed. S. Flügge), Vol. III/1, pp. 226–793. Springer.
Wehausen, J. V. & Laitone, E. V. 1960 Surface waves. In Handbuch der Physik (ed. S. Flügge), Vol. IX, pp. 446–778.
Widnall, S. E. & Bliss, D. B. 1971 Slender-body analysis of the motion and stability of a vortex filament containing an axial flow. J. Fluid Mech. 50, 335–353 (referred to herein as WB).Google Scholar