Abstract
We study Onsager's theory of large, coherent vortices in turbulent flows in the approximation of the point-vortex model for two-dimensional Euler hydrodynamics. In the limit of a large number of point vortices with the energy perpair of vortices held fixed, we prove that the entropy defined from the microcanonical distribution as a function of the (pair-specific) energy has its maximum at a finite value and thereafter decreases, yielding the negative-temperature states predicted by Onsager. We furthermore show that the equilibrium vorticity distribution maximizes an appropriate entropy functional subject to the constraint of fixed energy, and, under regularity assumptions, obeys the Joyce-Montgomery mean-field equation. We also prove that, under appropriate conditions, the vorticity distribution is the same as that for the canonical distribution, a form of equivalence of ensembles. We establish a large-fluctuation theory for the microcanonical distributions, which is based on a level-3 large-deviations theory for exchangeable distributions. We discuss some implications of that property for the ergodicity requirements to justify Onsager's theory, and also the theoretical foundations of a recent extension to continuous vorticity fields by R. Robert and J. Miller. Although the theory of two-dimensional vortices is of primary interest, our proofs actually apply to a very general class of mean-field models with long-range interactions in arbitrary dimensions.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
L. Onsager, Statistical hydrodynamics,Nuovo Cimento Suppl. 6:279–289 (1949).
A. S. Monin and A. M. Yaglom,Statistical Fluid Mechanics, Vols. I, II (MIT Press, Cambridge, Massachusetts, 1975).
J. O. Hinze,Turbulence, 2nd ed. (McGraw-Hill, New York, 1975).
A. J. Chorin, Equilibrium statistics of a vortex filament with applications,Commun. Math. Phys. 141:619–631 (1991).
R. H. Kraichnan and D. Montgomery, Two-dimensional turbulence,Rep. Prog. Phys. 43:547–619 (1980).
G. Kirchhoff,Vorlesungen über mathematische Physik (B. G. Teubener, Leipzig, 1877).
C. Marchioro and M. Pulvirenti,Vortex Methods in Two-Dimensional Fluid Dynamics (Springer, Berlin, 1984).
E. A. Overman and N. J. Zabusky, Evolution and merger of isolated vortex structures,Phys. Fluids 25:1297–1305 (1982).
D. Dürr and M. Pulvirenti, On the vortex flow in bounded domains,Commun. Math. Phys. 85:265–273 (1982).
R. H. Kraichnan, Inertial ranges in two-dimensional turbulence,Phys. Fluids 10:1417–1423 (1967).
J. McWilliams, The emergence of isolated coherent vortices in turbulent flow,J. Fluid Mech. 146:21 (1984).
D. S. Deem and N. J. Zabusky, Ergodic boundary in numerical simulations of two-dimensional turbulence,Phys. Rev. Lett. 27:397–381 (1971).
K. M. Khanin, Quasi-periodic motions of vortex systems,Physica D 4:261–269 (1982).
T. S. Lundgren and Y. B. Pointin, Statistical mechanics of two-dimensional vortices,J. Stat. Phys. 17:323–355 (1977).
A. J. Chorin, Turbulence and vortex stretching on a lattice,Commun. Pure Appl. Math. XXXIX:547–565 (1986).
L. van Hove, Quelques propriétés générais de l'intégrale d'un système de particules avec interaction,Physica 15:951–961 (1949).
J. Fröhlich and D. Ruelle, Statistical mechanics of vortices in an in viscid two-dimensional fluid,Commun. Math. Phys. 87:1–36 (1982).
J. Messer and H. Spohn, Statistical mechanics of the isothermal Lane-Emden equation,J. Stat. Phys. 29:561–578 (1982).
E. Caglioti, P. L. Lions, C. Marchioro, and M. Pulvirenti, A special class of stationary flows for two-dimensional Euler equations: A statistical mechanics approach,Commun. Math. Phys. 143:501–525 (1992).
M. K.-H. Kiessling, Statistical mechanics of classical particles with logarithmic interactions,Commun. Pure Appl. Math. (1992).
D. Montgomery and G. Joyce, Statistical mechanics of “negative temperature” states,Phys. Fluids 17:1139–1145 (1974).
G. Joyce and D. Montgomery, Negative temperature states for the two-dimensional guiding center plasma,J. Plasma Phys. 10:107–121 (1973).
J. Sommeria, C. Staquet, and R. Robert, Final equilibrium state of a two-dimensional shear layer,J. Fluid Mech. 233:661–689 (1991).
D. Montgomery, W. H. Matthaeus, W. T. Stribling, D. Martinez, and S. Oughton, Relaxation in two dimensions and the “sinh-Poisson” equation,Phys. Fluids A 4:3–6 (1992).
J. Miller, Statistical mechanics of Euler equations in two dimensions,Phys. Rev. Lett. 65:2137–2140 (1990).
V. Arnol'd,Mathematical Methods of Classical Mechanics (Springer, New York, 1978).
J. Marsden and A. Weinstein, Coadjoint orbits, vortices and Clebsch variables for incompressible fluids,Physica 7:305–323 (1983).
J. Miller, Statistical mechanics of two-dimensional Euler equations and Jupiter's Great Red Spot, Ph.D. Thesis, California Institute of Technology, Pasadena, California.
R. Robert, A maximum-entropy principle for two-dimensional perfect fluid dynamics,J. Stat. Phys. 65:531–554 (1991).
B. de Finetti, Funzione caratteristica di un fenomeno aleatorio,Atti R. Accad. Naz. Lincei Ser. 6 Mem. Classe Sci. Fis. Mat. Nat. 4:251–300 (1931).
L. Breiman,Probability (Addison-Wesley, Reading, Massachusetts, 1968).
E. Caglioti, Misure invarianti per l'equazione di Eulero bidimensionale. Meccanica statistica per il modello a vortici, Dottorato di Ricerca in Física IV Ciclo, Université di Roma, Rome, Italy.
H.-O. Georgii, Large deviations and maximum entropy principle for interacting random fields on ℤd,Ann. Prob. (1992).
I. Csiszár, Sanov property, generalizedI-projection, and a conditional limit theorem,Ann. Prob. 12:768–793 (1984).
I. Ioffe and V. Tikhomirov, Duality of convex functions and extremum problems,Russ. Math. Surv. 23:53–124 (1968).
P. Groeneboom, J. Oosterhoff, and F. H. Ruymgaart, Large deviation theorems for empirical probability measures,Ann. Prob. 7:553–586 (1979).
P. Marcus, Vortex dynamics in a shearing zonal flow,J. Fluid Mech. 215:393–430 (1990).
J. Miller, P. B. Weichman, and M. C. Cross, Statistical mechanics, Euler's equations, and Jupiter's Red Spot,Phys. Rev. A 45:2328–2359 (1992).
V. Zeitlin, Finite-mode analogs of 2-D ideal hydrodynamics: Coadjoint orbits and local canonical structure,Physica D 49:353–362 (1991).
O. Hald, Convergence of Fourier methods for Navier-Stokes equations,J. Comp. Phys. 40:305–317 (1981).
E. Hewitt and L. J. Savage, Symmetric measures on Cartesian products,Trans. Am. Math. Soc. 80:470–501 (1955).
H.-O. Georgii,Gibbs Measures and Phase Transitions (Walter de Gruyter, Berlin, 1988).
K. R. Parthasarathy,Probability Measures on Metric Spaces (Academic Press, New York, 1967).
E. B. Dynkin, Klassy ekvivalentnyh slučainyh veličin,Uspekhi Mat. Nauk 8:125–134 (1953).
S. R. S. Varadhan,Large Deviations and Applications (Society for Industrial and Applied Mathematics, Philadelphia, 1984).
H. Föllmer, Random fields and diffusion processes, inÉcole d'Été de Probabilités de Saint-Flour XV–XVII, P. L. Hennequin, ed. (Springer, Berlin, 1988).
H. Föllmer and S. Orey, Large deviations for the empirical field of a Gibbs measure,Ann. Prob. 16:961–977 (1988).
R. S. Ellis,Entropy, Large Deviations and Statistical Mechanics (Springer, New York, 1985).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Eyink, G.L., Spohn, H. Negative-temperature states and large-scale, long-lived vortices in two-dimensional turbulence. J Stat Phys 70, 833–886 (1993). https://doi.org/10.1007/BF01053597
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01053597