Abstract
We use Kullback entropy for Young measures to define statistical equilibrium states for a two-dimensional incompressible flow of a perfect fluid. This approach is justified, as it gives a concentration property about the equilibrium state in the phase space. It might give a statistical understanding of the appearance of coherent structures in two-dimensional turbulence.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
S. Albeverio and A. Cruzerio, Global flows with invariant Gibbs measures for Euler and Navier-Stokes two-dimensional fluids,Commun. Math. Phys. 129:431–444 (1990).
S. Albeverio, M. Ribeiro de Faria, and R. Hoegh Krohn, Stationary measures for the periodic Euler flow in two dimensions,J. Stat. Phys. 20:585–595 (1979).
V. I. Arnold, Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits,Ann. Inst. Fourier 16:319–361 (1966).
V. I. Arnold, On ana priori estimate in the theory of hydrodynamical stability,Am. Math. Soc. Transl. 79:267–269 (1969).
R. Azencott, Grandes déviations et applications, inEcole d'été de Probabilités de Saint-Flour, VIII Lecture Notes in Mathematics, No. 774 (Springer, Berlin, 1978).
P. Baldi, Large deviations and stochastic homogenization,Ann. Mat. Pura Appl. 4(151):161–177 (1988).
C. Bardos, Existence et unicité de la solution de l'équation d'Euler en dimension deux,J. Math. Anal. Appl. 40:769–790 (1972).
C. Boldrighini and S. Frigio, Equilibrium states for a plane incompressible perfect fluid,Commun. Math. Phys. 72:55–76 (1980).
B. Castaing, Conséquences d'un principe d'extremum en turbulence,J. Phys. (Paris)50:147–156 (1989).
G. H. Cottet, Analyse numérique des méthodes particulaires pour certains problèmes non linéaires, Thèse, Université Paris VI (1987).
G. S. Deem and N. J. Zabusky, Vortex waves: Stationary V-states, interactions, recurrence, and breaking,Phys. Rev. Lett. 40:859–862 (1978).
T. Dumont and M. Schatzman, to appear.
R. S. Ellis,Entropy, Large Deviations and Statistical Mechanics (Springer-Verlag, 1985).
J. Fröhlich and D. Ruelle, Statistical mechanics of vortices in an inviscid two dimensional fluid,Commun. Math. Phys. 87:1–36 (1982).
E. J. Hopfinger, Turbulence and vortices in rotating fluids, inProceedings of the XVIIth International Congress of Theoretical and Applied Mechanics (North-Holland, Amsterdam, 1989).
E. T. Jaynes, Where do we go from here? inMaximum Entropy and Bayesian Methods in Inverse Problems, C. Ray Smith and W. T. Grandy, eds. (Reidel, 1985).
M. Jirina, On regular conditional probabilities,Czech. Math. J. 9:445 (1959).
T. Kato, On the classical solutions of the two-dimensional non stationary Euler equation,Arch. Rat. Mech. Anal. 25:302–324 (1967).
R. H. Kraichnan, Statistical dynamics of two-dimensional flow,J. Fluid Mech. 67:155–175 (1975).
R. H. Kraichnan and D. Montgomery,Rep. Prog. Phys. 43:547 (1980).
T. D. Lee,Q. Appl. Math. 10:69 (1952).
C. E. Leith, Minimum enstrophy vortices,Phys. Fluids 27:1388–1395 (1984).
J. Michel and R. Robert, To appear.
J. Miller, Statistical mechanics of Euler equations in two dimensions,Phys. Rev. Lett. 65:2137–2140 (1990).
D. Montgomery, Maximal entropy in fluid and plasma turbulence, inMaximum Entropy and Bayesian Methods in Inverse Problems, C. Ray Smith and W. T. Grandy, eds. (Reidel, 1985).
J. M. Nguyen Duc and J. Sommeria, Experimental characterization of steady two-dimensional vortex couples,J. Fluid Mech. 192:175–192 (1988).
E. A. Novikov, Dynamics and statistics of a system of vortices,Sov. Phys. JETP 41:937–943 (1976).
P. J. Olver,Applications of Lie Groups to Differential Equations (Springer-Verlag, 1986).
L. Onsager, Statistical hydrodynamics,Nuovo Cimento Suppl. 6:279 (1949).
E. A. Overman and N. J. Zabusky, Evolution and merging of isolated vortex structures,Phys. Fluids 25:1297 (1982).
Y. B. Poitin and T. S. Lundgren, Statistical mechanics of two dimensional vortices in a bounded container,Phys. Fluids 10:1459–1470 (1976).
R. Robert, Concentration et entropie pour les mesures d'Young,C. R. Acad. Sci. Paris Ser. I 309:757–760 (1989).
R. Robert, Concentration and entropy for Young measures, Preprint, Laboratoire d'analyse numérique de Lyon, no. 85 (1989).
R. Robert, Etats d'équilibre statistique pour l'écoulement bidimensionnel d'un fluide parfait,C. R. Acad. Sci. Paris Ser. I 311:575–578 (1990).
R. Robert and J. Sommeria, Statistical equilibrium states for two-dimensional flows,J. Fluid Mech., to appear.
P. G. Saffman and G. R. Baker, Vortex interactions,Annu. Rev. Fluid Mech. 11:95–122 (1979).
D. Serre, Les invariants du premier ordre de l'équation d'Euler en dimension trois,Physica 13D:105–136 (1984).
J. Sommeria and M. A. Denoix, to appear.
J. Sommeria, S. D. Meyers, and H. L. Swinney, Laboratory simulation of Jupiter's Great Red Spot,Nature 331:1 (1988).
J. Sommeria, C. Nore, T. Dumont, and R. Robert, Théorie statistique de la tache rouge de Jupiter,C. R. Acad. Sci. Paris Ser. II 312:999–1005 (1991).
J. Sommeria, C. Staquet, and R. Robert, Final equilibrium state of a two-dimensional shear layer,J. Fluid Mech., to appear.
V. I. Youdovitch, Non stationary flow of an ideal incompressible liquid,Zh. Vych. Mat. 3:1032–1066 (1963).
L. C. Young, Generalized surfaces in the calculus of variations,Ann. Math. 43:84–103 (1942).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Robert, R. A maximum-entropy principle for two-dimensional perfect fluid dynamics. J Stat Phys 65, 531–553 (1991). https://doi.org/10.1007/BF01053743
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01053743