Hostname: page-component-77c89778f8-7drxs Total loading time: 0 Render date: 2024-07-17T10:30:22.480Z Has data issue: false hasContentIssue false
  • English
  • Français

Classification of self-organized vortices in two-dimensional turbulence: the case of a bounded domain

Published online by Cambridge University Press:  26 April 2006

P. H. Chavanis  and
J. Sommeria
P. H. Chavanis
Affiliation:
Ecole Normale Supérieure de Lyon, 69364 Lyon cedex 07, France
J. Sommeria
Affiliation:
Ecole Normale Supérieure de Lyon, 69364 Lyon cedex 07, France
Get access Rights & Permissions [Opens in a new window]

Abstract

We calculate steady solutions of the Euler equations for any given value of energy and circulation (and angular momentum in the case of a circular domain). A linear relationship between vorticity and stream function is assumed. These solutions correspond to the predicted self-organization into a maximum-entropy state, in the limit of strong mixing. Vorticity mixing is then only weakly restricted by the constraint of energy conservation. While maximum-entropy solutions depend in general on the whole probability distribution of vorticity levels, these linearized results depend only on a single control parameter, yet keeping much of the general structure of the problem. A convenient classification of the maximum-entropy states is thus provided. We show furthermore how to extend these linearized results as an expansion in energy, involving successive moments of the vorticity probability distribution. They are applied to a rectangular domain and compared with existing numerical and laboratory results. We predict that the flow organizes into a single vortex in the square domain, but into a two-vortex dipolar state in a rectangle with aspect ratio greater than 1.12. The case of a circular domain is also explicitly solved, taking into account the conservation of the angular momentum.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 314 , 10 May 1996 , pp. 267 - 297
Copyright
© 1996 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bretherton, F. P. & Haidvoguel, D. B. 1976 Two-dimensional turbulence over topography. J. Fluid Mech. 78, 129–154.Google Scholar
Carnevale, G. F. & Frederiksen, J. S. 1987 Nonlinear stability and statistical mechanics of flow over topography. J. Fluid Mech. 175, 157–181.Google Scholar
Carnevale, G. F. & Vallis, G. K. 1990 Pseudo-advective relaxation to stable states of inviscid two-dimensional fluids. J. Fluid Mech. 213, 549–571.Google Scholar
Chaplygin, S. A. 1902 One case of vortex motion in fluid Proc. Phys. Sec. Natural Phil. Soc. 11, 114.Google Scholar
Chavanis, P. H. & Sommeria, J. 1995 Classification of self-organized isolated vortices in two-dimensional turbulence. In preparation.
Chen, P. & Cross, M. C. 1994 Phase diagram for coherent vortex formation in the two-dimensional inviscid fluid in circular geometries. Phys. Rev. E 50, 2022–2029.Google Scholar
Denoix, M. A., Sommeria, J. & Thess, A. 1994 Two-dimensional turbulence: the prediction of coherent structures by statistical mechanics. In Progress in Turbulence Research (ed. H. Branover & Y. Unger), pp. 88–107. AIAA.
Fofonoff, N. P. 1954 Steady flow in a frictionless homogeneous ocean. J. Mar. Res. 13, 254–262.Google Scholar
Hasegawa, A. 1985 Self-organization processes in continuous media Adv. Phys. 34, 1–42.Google Scholar
Heijst VAN, G. J. F., Davies, P. A. & Davis, R. G. 1990 Spin-up in a rectangular container Phys. Fluids A 2, 150–159.Google Scholar
Joyce, G. & Montgomery, D. 1973 Negative temperature states for the two-dimensional guiding-centre plasma. J. Plasma Phys. 10, 107–121.Google Scholar
Juttner, B., Thess, A. & Sommeria, J. 1995 On the symmetry of self-organized structures in two-dimensional turbulence, Phys. Fluids 7, 2108–2110.Google Scholar
Kazantsev, E., Sommeria, J. & Verron, J. 1995 Subgridscale eddy parametrisation by statistical mechanics in a barotropic ocean model. J. Phys. Oceanogr. (submitted).Google Scholar
Kraichnan, R. H. & Montgomery, D. 1980 Two-dimensional turbulence Rep. Prog. Phys. 43, 547–617.Google Scholar
Kuz'min, G. A. 1982 Statistical mechanics of the organisation into two-dimensional coherent structures. In Structural Turbulence (ed. M. A. Goldshtik), pp. 103–114. Acad. Naouk CCCP Novosibirsk, Institute of Thermophysics.
Lynden-Bell, D. 1967 Statistical mechanics of violent relaxation in stellar systems. Mon. Not. R. Astron. Soc. 136, 101–121Google Scholar
Marteau, D., Cardoso, O. & Tabeling, P. 1995 Equilibrium states of 2D turbulence: an experimental study. Phys. Rev. E 51, 5124–5127.Google Scholar
Michel, J. & Robert, R. 1994 Large deviations for Young measures and statistical mechanics of infinite dimensional dynamical systems with conservation law. Commun. Math. Phys. 159, 195–215.Google Scholar
Miller, J. 1990 Statistical mechanics of Euler equations in two dimensions. Phys. Rev. Lett. 65, 2137–2140.Google Scholar
Miller, J., Weichman, P. B. & Cross, M. C. 1992 Statistical mechanics, Euler's equation and Jupiter's Red Spot. Phys. Rev. A 45, 2328–2359.Google Scholar
Montgomery, D. & Joyce, G. 1974 Statistical mechanics of negative temperature states. Phys. Fluids 17, 1139–1145.Google Scholar
Montgomery, D., Matthaeus, W., Stribling, W., Martinez, D. & Oughton, S. 1992 Relaxation in two dimensions and the sinh-Poisson equation. Phys. Fluids A 4, 3–6.Google Scholar
Onsager, L. 1949 Statistical hydrodynamics. Nuovo Cimento Suppl. 6, 279–287.Google Scholar
Pointin, Y. B. & Lundgren, T. S. 1976 Statistical mechanics of two-dimensional vortices in a bounded container. Phys. Fluids 19, 1459–1470.Google Scholar
Robert, R. 1990 Etat d’équilibre statistique pour l’écoulement bidimensionnel d'un fluide parfait. C. R. Acad. Sci. Paris 311 (I), 575–578.Google Scholar
Robert, R. 1991 Maximum entropy principle for two-dimensional Euler equations. J. Statist. Phys. 65, 531–551.Google Scholar
Robert, R. & Rosier, 1996 The modelling of small scales in 2D turbulent flows: a statistical mechanical approach. J. Fluid Mech. (submitted).Google Scholar
Robert, R. & Sommeria, J. 1991 Statistical equilibrium states for two-dimensional flows. J. Fluid Mech. 229, 291–310.Google Scholar
Smith, A. R. & O'Neil, T. M. 1990 Nonaxisymmetric thermal equilibria of a cylindrically bounded guiding-center plasma or discrete vortex system. Phys. Fluids B 2, 2961–2975.Google Scholar
Sommeria, J. 1986 Experimental study of the two-dimensional inverse energy cascade in a square box. J. Fluid Mech. 170, 139–168.Google Scholar
Sommeria, J. 1988 Electrically driven vortices in a strong magnetic field. J. Fluid Mech. 189, 139–168.Google Scholar
Sommeria, J. 1994 Organized vortices as maximum entropy structures. In Modelling of Oceanic Vortices (ed. G. J. F. van Heijst), pp. 37–50. North-Holland.
Stern, M. 1975 Minimal properties of planetary eddies. J. Mar. Res. 33, 1–13.Google Scholar
Thess, A., Sommeria, J. & Juttner, B. 1994 Inertial organization of a two-dimensional turbulent vortex street Phys. Fluids 6, 2417–2429.Google Scholar
Turkington, B. & Whitaker, N. 1995 Statistical equilibrium computations of coherent structures in turbulent shear layers. SIAM J. Sci. Comput. (to appear).Google Scholar
Verron, J. & Sommeria, J. 1987 Numerical simulation of a two-dimensional turbulence experiment in magnetohydrodynamics. Phys. Fluids 30, 732–739.Google Scholar
Whitaker, N. & Turkington, B. 1994 Maximum entropy states for rotating vortex patches. Phys. Fluids 6, 3963–3973.Google Scholar