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Experimental study of the two-dimensional inverse energy cascade in a square box

Published online by Cambridge University Press:  21 April 2006

J. Sommeria
J. Sommeria
Affiliation:
Gis Madylam, Institut de Mécanique de Grenoble, BP 68. No. 38402, Saint Martin D'Hères Cedex, France
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Abstract

A quantitative experimental study of the two-dimensional inverse energy cascade is presented. The flow is electrically driven in a horizontal layer of mercury and three-dimensional perturbations are suppressed by means of a uniform magnetic field, so that the flow can be well approximated by a two-dimensional Navier–Stokes equation with a steady forcing term and a linear friction due to the Hartmann layer. Turbulence is produced by the instability of a periodic square network of 36 electrically driven alternating vortices. The inverse cascade is limited at large scales, either by the linear friction or by the finite size of the domain, depending on the experimental parameters. In the first case, $k^{-\frac{5}{3}$ spectra are measured and the corresponding two-dimensional Kolmogorov constant is in the range 3–7. In the second case, a condensation of the turbulent energy in the lowest mode, corresponding to a spontaneous mean global rotation, is observed. Such a condensation was predicted by Kraichnan (1967) from statistical thermodynamics arguments, but without the symmetry breaking. Random reversals of the rotation sense, owing to turbulent fluctuations, are more and more sparse as friction is decreased. The lowest mode fluctuations and the small scales are statistically independent.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 170 , September 1986 , pp. 139 - 168
Copyright
© 1986 Cambridge University Press

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References

Alemany A., Moreau R., Sulem, P. L. & Frisch U.1979 Influence of an external magnetic field on homogeneous turbulence. J. Méc. 18 (2), 277.Google Scholar
Babiano A., Basdevant, C. & Sadourny R.1985 Structure functions and dispersion laws in two-dimensional turbulence. J. Atmos. Sci. 42 (9), 941.Google Scholar
Batchelor G. K.1969 Computation of the energy spectrum in homogeneous two-dimensional turbulence. Phys. Fluids. Suppl. 2 (12), 233.Google Scholar
Brachet M. E., Meneguzzi, M. & Sulem P. L.1985 Numerical simulation of decaying two-dimensional turbulence. In Macroscopic Modelling of Turbulent Flows. Lecture Notes in Physics, vol. 230, p. 347. Springer.
Colin De VerdiÈre A.1980 Quasi geostrophic turbulence in a rotating homogeneous fluid. Geophys. Astrophys. Fluid Dyn. 15, 213.Google Scholar
Couder Y.1984 Two-dimensional turbulence in a laboratory experiment. J. Phys. Lett. 45, 353.Google Scholar
Creveling H. F., de Pas, J. F., Baladi, J. Y. & Schoenhals, R. J. 1975 Stability characteristics of a singlephase free convection loop. J. Fluid Mech. 67, 65.Google Scholar
Frisch, U. & Sulem P. L.1984 Numerical simulation of the inverse cascade in two-dimensional turbulence. Phys. Fluids 27 (8), 1921. Erratum: Phys. Fluids 1985, 28, 438.Google Scholar
Harris F. J.1978 On the use of windows for harmonic analysis with the discrete Fourier transform. Proc. IEEE 66 (1), 51.Google Scholar
Herring, J. H. & McWilliams J. C.1985 Comparison of direct numerical simulation of two-dimensional turbulence with two-point closure: the effect of intermittency. J. Fluid Mech. 153, 229.Google Scholar
Hinze J. O.1975 Turbulence. McGraw-Hill.
Hopfinger E. J., Browand, F. K. & Gagne Y.1982 Turbulence and waves in a rotating tank. J. Fluid Mech. 125, 505.Google Scholar
Hossain M., Matthaeus, W.H. & Montgomery D.1983 Long time states of inverse cascades in the presence of a maximum length scale. J. Plasma Phys. 30 (3), 479.Google Scholar
Hitchings J. W.1955 Turbulence theory applied to large scale atmospheric phenomena. J. Met. 12, 263.Google Scholar
Kit, L. G. & Tsinober A. B.1971 Possibility of creating and investigating two-dimensional turbulence in a strong magnetic field. Magnitnaya Gidrodinamika 3, 27.Google Scholar
Kolesnikov, Y. B. & Tsinober A. B.1974 Experimental investigation of two-dimensional turbulence behind a grid Isv. Akad. Nauk. SSSR Mech. Zhid. i Gaza 4, 146.Google Scholar
Kraichnan R. H.1967 Inertial ranges in two-dimensional turbulence Phys. Fluids 10, 1417.Google Scholar
Kraichnan R. H.1971 An almost-markovian Galilean-invariant turbulence model. J. Fluid Mech. 47, 513.Google Scholar
Kraichnan R. H.1975 Statistical dynamics of two-dimensional flow. J. Fluid Mech. 67, 155.Google Scholar
Kraichnan, R. H. & Montgomery D.1980 Two-dimensional turbulence. Rep. Prog. Phys. 43, 549.Google Scholar
Lesieur M.1983 Introduction à la turbulence bidimensionnelle. J. Méc. (Special issue on Two-dimensional turbulence), p. 5.Google Scholar
Lielausis O.1975 Liquid metal magnetohydrodynamics. Atomic Energy Rev. 13 (3), 527.Google Scholar
Lilly D. K.1973 Lectures in sub-synoptic scales of motions and two-dimensional turbulence. In Dynamic Meteorology (ed. P. Morel), p. 353. Reidel.
Mcwilliams J. C.1983 On the relevance of two-dimensional turbulence to geophysical motions. J. Méc. (Special issue), p. 83.Google Scholar
Mcwilliams J. C.1984 The emergence of isolated, coherent vortices in turbulent flow. J. Fluid Mech. 146, 21.Google Scholar
Mory M.1984 Turbulence dans un fluide soumis à forte rotation. Thesis at the University of Grenoble (France).
Pouquet A., Lesieur M., Andre, J. C. & Basdevant C.1975 Evolution of high Reynolds number two-dimensional turbulence. J. Fluid Mech. 72 (2), 305.Google Scholar
Rhines P. B.1979 Geostrophic turbulence. Ann. Rev. Fluid Mech. 11, 401.Google Scholar
Shercliff J. A.1965 A Textbook of Magnetohydrodynamics. Pergamon.
Selyuto S. F.1984 Effect of magnetic field on formation of turbulent structure behind arrays of different configurations. Magnitnaya Gidrodinamika 3, 55.Google Scholar
Siggia, E. D. & Aref H.1981 Point vortex simulation of the inverse cascade in two-dimensional turbulence Phys. Fluids 24 (1), 171.Google Scholar
Sivashinski, G. & Yakhot V.1985 Negative viscosity effects in large-scale flows. Phys. Fluids 28 (4), 1040.Google Scholar
Sommeria J.1983 Two-dimensional behaviour of MHD fully developed turbulence (Rm 1). J. Méc. (Special issue), p. 169.
Sommeria J.1984 Two-dimensional behaviour of electrically driven flows at high Hartmann numbers, 4th Beer Sheva Seminar on MHD flows and turbulence, Israel. Proceedings in AIAA Progress in Aeronautics and Astronautics.Google Scholar
Sommeria J.1986 Electrically driven vortices in a strong magnetic field. J. Fluid Mech. (submitted).Google Scholar
Sommeria, J. & Moreau R.1982 Why, how and when MHD turbulence becomes two-dimensional. J. Fluid Mech. 118, 507.Google Scholar
Sommeria, J. & Verron J.1984 An investigation of non linear interactions in a two-dimensional recirculating flow. Phys. Fluids 27 (8), 1918.Google Scholar
Sommeria, J. & Verkon J.1985 The two-dimensional inverse energy cascade in a confined domain: numerical and laboratory experiments. 5th Turbulent Shear Flow Symp. Cornell University, Ithaca.
Staquet, C. & Lesieur M.1985 Study of the mixing layer from the point of view of two-dimensional turbulence. 5th Turbulent Shear Flow Symp. Cornell University, Ithaca.Google Scholar
Sukoriansky S., Zilberman, L. & Branover H.1984 Experiments in duct flows with reversed turbulent energy cascades. 4th Beer Sheva Seminar on MHD flows and turbulence, Israel. Proceedings in AIAA Progress in Aeronautics and Astronautics.
Tolstov C. P.1962 Fourier Series. (Series in applied maths, trans. from Russian.) Prentice-Hall.