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Wave turbulence and intermittency. (English) Zbl 1049.76033

From the summary: We give some new results concerning the dynamic breakdown of the weak turbulence description and discuss the fully nonlinear and intermittent behavior which follows. These results may also be important for proving or disproving the global existence of solutions for the underlying partial differential equations. Wave turbulence is a subject to which many have made important contributions. But no contributions have been more fundamental than those of Volodja Zakharov whose 60th birthday we celebrate at this meeting. He was the first to appreciate that the kinetic equations admit a far richer class of solutions than the fluxless thermodynamic solutions of equilibrium systems, and to realize the central roles that finite flux solutions play in non-equilibrium systems. It is appropriate, therefore, that we call these Kolmogorov-Zakharov spectra.

MSC:

76F55 Statistical turbulence modeling
Full Text: DOI

References:

[1] Hasselmann, K., On the nonlinear energy transfer in a gravity-wave spectrum I, J. Fluid Mech., 12, 481-500 (1962) · Zbl 0107.21402
[2] Hasselmann, K., On the nonlinear energy transfer in a gravity-wave spectrum II, J. Fluid Mech., 15, 273-281 (1963) · Zbl 0116.43401
[3] Kenyon, K., Discussion, Proc. Roy. Soc. London A, 299, 1456, 141-144 (1967)
[4] Benney, D. J.; Saffman, P. G., Nonlinear interactions of random waves in a dispersive medium, Proc. Roy. Soc. London A, 289, 301-320 (1966)
[5] Benney, D. J.; Newell, A. C., Sequential time closures of interacting random waves, J. Math. Phys., 46, 363 (1967) · Zbl 0159.59302
[6] D.J. Benney, A.C. Newell, Statistical Properties of the Sea, Physics of Fluids Special Issue, Kyoto Meeting of International Union of Theoretical and Applied Mechanics, September 1966, Vol. 10, 1967, S281 pp.; D.J. Benney, A.C. Newell, Statistical Properties of the Sea, Physics of Fluids Special Issue, Kyoto Meeting of International Union of Theoretical and Applied Mechanics, September 1966, Vol. 10, 1967, S281 pp.
[7] V.E. Zakharov, N.N. Filonenko, Energy spectrum for stochastic oscillations of the surface of a fluid, Dokl. Akad. Nauk SSSR 170 (6) (1966) 1292-1295 [English transl. in Sov. Math. Dokl.].; V.E. Zakharov, N.N. Filonenko, Energy spectrum for stochastic oscillations of the surface of a fluid, Dokl. Akad. Nauk SSSR 170 (6) (1966) 1292-1295 [English transl. in Sov. Math. Dokl.].
[8] V.E. Zakharov, N.N. Filonenko, Weak turbulence of capillary waves, Zh. Prikl. Mekh. I Tekn. Fiz. (5) (1967) 62-67 [English transl. in J. Appl. Mech. Tech. Phys.].; V.E. Zakharov, N.N. Filonenko, Weak turbulence of capillary waves, Zh. Prikl. Mekh. I Tekn. Fiz. (5) (1967) 62-67 [English transl. in J. Appl. Mech. Tech. Phys.].
[9] V.E. Zakharov, Weak turbulence in media with decay dispersion law, Zh. Prikl. Mekh I Tekhn. Fiz. 4 (1965) 35 [English transl. in J. Appl. Mech. Tech. Phys.].; V.E. Zakharov, Weak turbulence in media with decay dispersion law, Zh. Prikl. Mekh I Tekhn. Fiz. 4 (1965) 35 [English transl. in J. Appl. Mech. Tech. Phys.].
[10] V.E. Zakharov, On the spectrum of turbulence in plasma without magnetic field, Zh. Eksper. Teoret. Fiz. 51 (1966) 686-696 [English transl. in Sov. Phys. JETP 24 (1967) 455-459].; V.E. Zakharov, On the spectrum of turbulence in plasma without magnetic field, Zh. Eksper. Teoret. Fiz. 51 (1966) 686-696 [English transl. in Sov. Phys. JETP 24 (1967) 455-459].
[11] Newell, A. C., The closure problem in a system of random gravity waves, Rev. Geophys., 6, 1-31 (1968)
[12] Benney, D. J.; Newell, A. C., Random wave closures, Stud. Appl. Math., 48, 1, 29-53 (1969) · Zbl 0185.55404
[13] Newell, A. C.; Aucoin, P. J., Semidispersive wave systems, J. Fluid Mech., 49, 593-609 (1971) · Zbl 0232.76021
[14] V.E. Zakharov, R.Z. Sagdeev, On spectra of acoustic turbulence, Dokl. Akad. Nauk SSSR 192 (2) (1970) 297-299 [English transl. in Sov. Phys. JETP 35 (1972) 310-314].; V.E. Zakharov, R.Z. Sagdeev, On spectra of acoustic turbulence, Dokl. Akad. Nauk SSSR 192 (2) (1970) 297-299 [English transl. in Sov. Phys. JETP 35 (1972) 310-314]. · Zbl 0223.76042
[15] E.A. Kuznetsov, On turbulence of ion sound in plasma in a magnetic field, Zh. Eksper. Teoret. Fiz. 62 (2) (1972) 584-592 [English transl. in Sov. Phys. JETP 35 (1972) 310-314].; E.A. Kuznetsov, On turbulence of ion sound in plasma in a magnetic field, Zh. Eksper. Teoret. Fiz. 62 (2) (1972) 584-592 [English transl. in Sov. Phys. JETP 35 (1972) 310-314].
[16] V.E. Zakharov, S.L. Musher, On Kolmogorov spectra in a system of nonlinear oscillators, Dokl. Akad. Nauk SSSR 209 (5) (1973) 1063-1065 [English transl. in Sov. Phys. Dokl. 18 (1973)].; V.E. Zakharov, S.L. Musher, On Kolmogorov spectra in a system of nonlinear oscillators, Dokl. Akad. Nauk SSSR 209 (5) (1973) 1063-1065 [English transl. in Sov. Phys. Dokl. 18 (1973)].
[17] Hasegawa, A.; Mima, K., Pseudo-three-dimensional turbulence in magnetized non-uniform plasma, Phys. Fluids, 21, 87 (1978) · Zbl 0374.76046
[18] E.N. Pelinovskii, Wave turbulence on a beta-plane, Okeanologia 18 (2) (1978) 192-195 (in Russian).; E.N. Pelinovskii, Wave turbulence on a beta-plane, Okeanologia 18 (2) (1978) 192-195 (in Russian).
[19] Crawford, D. R.; Saffman, P. G.; Yuen, H. C., Evolution of a random inhomogeneous field of nonlinear deep-water gravity waves, Wave Motion, 2, 1-16 (1980) · Zbl 0434.76018
[20] R. Peierls, Quantum Theory of Solids, Clarendon Press, Oxford, 1955/Pergamon Press, Oxford, 1981.; R. Peierls, Quantum Theory of Solids, Clarendon Press, Oxford, 1955/Pergamon Press, Oxford, 1981. · Zbl 0068.23207
[21] Hasselmann, S.; Hasselmann, K.; Allender, J. H.; Barnett, T. P., J. Phys. Oceanogr., 15, 1378 (1985)
[22] V.S. L’vov, G.E. Falkovich, On anisotropic spectra of weak and sound turbulence, Zh. Eksper. Teoret. Fiz. 80 (2) (1981) 593-596 [English transl. in Sov. Phys. JETP 53 (1981) 200-300].; V.S. L’vov, G.E. Falkovich, On anisotropic spectra of weak and sound turbulence, Zh. Eksper. Teoret. Fiz. 80 (2) (1981) 593-596 [English transl. in Sov. Phys. JETP 53 (1981) 200-300].
[23] R.S. Iroshnikov, Possibility of formation of a non-isotropic spectrum of wind waves by their weak nonlinear interaction, Dokl. Akad. Nauk SSSR 280 (6) (1985) 1321-1325 [English transl. in Sov. Phys. Dokl. 30 (1985) 126-128].; R.S. Iroshnikov, Possibility of formation of a non-isotropic spectrum of wind waves by their weak nonlinear interaction, Dokl. Akad. Nauk SSSR 280 (6) (1985) 1321-1325 [English transl. in Sov. Phys. Dokl. 30 (1985) 126-128]. · Zbl 0635.76017
[24] Klimontovich, Y. L.; Kremp, D.; Kraeft, W. D., Adv. Chem. Phys., 68, 175 (1987)
[25] A.M. Balk, V.E. Zakharov, On stability of weak turbulence Kolmogorov spectra, Dokl. Akad. Nauk SSSR 299 (5) (1988) 1112-1115 [English transl. in Sov. Phys. Dokl. 33 (1988) 270-273].; A.M. Balk, V.E. Zakharov, On stability of weak turbulence Kolmogorov spectra, Dokl. Akad. Nauk SSSR 299 (5) (1988) 1112-1115 [English transl. in Sov. Phys. Dokl. 33 (1988) 270-273]. · Zbl 0695.76026
[26] A.M. Balk, V.E. Zakharov, Stability of weak turbulence Kolmogorov spectra, in: Plasma Theory and Nonlinear and Turbulent Processes in Physics, Proceedings of the International Workshop, Kiev, April 1987, World Scientific, Singapore, 1988, pp. 359-376.; A.M. Balk, V.E. Zakharov, Stability of weak turbulence Kolmogorov spectra, in: Plasma Theory and Nonlinear and Turbulent Processes in Physics, Proceedings of the International Workshop, Kiev, April 1987, World Scientific, Singapore, 1988, pp. 359-376. · Zbl 0695.76026
[27] Falkovich, G. E.; Shafarenko, A. V., On the stability of Kolmogorov spectra of a weak turbulence, Physica D, 27, 311-399 (1987) · Zbl 0631.76052
[28] V.E. Zakharov, E.I. Shulman, On additional motion invariants of classical Hamiltonian wave systems, Physica D 29 (1988) 283-320.; V.E. Zakharov, E.I. Shulman, On additional motion invariants of classical Hamiltonian wave systems, Physica D 29 (1988) 283-320. · Zbl 0651.35080
[29] Balk, A. M.; Nazarenko, S. V., On the physical realizability of anisotropic Kolmogorov spectra of weak turbulence, Sov. Phys. JETP, 70, 1031 (1990)
[30] Balk, A. M.; Nazarenko, S. V.; Zakharov, V. E., New invariant for drift turbulence, Phys. Lett. A, 152, 5-6, 276-280 (1991)
[31] A.M. Balk, S.V. Nazarenko, V.E. Zakharov, Non-local turbulence of drift waves, Zh. Eksper. Teoret. Fiz. 98 (1990) 446-467 [English transl. in Sov. Phys. JETP 71 (1990) 249-260].; A.M. Balk, S.V. Nazarenko, V.E. Zakharov, Non-local turbulence of drift waves, Zh. Eksper. Teoret. Fiz. 98 (1990) 446-467 [English transl. in Sov. Phys. JETP 71 (1990) 249-260].
[32] Dyachenko, S.; Newell, A. C.; Pushkarev, A.; Zakharov, V. E., Optical turbulence: weak turbulence, condensates and collapsing filaments in the nonlinear schrodinger equation, Physica D, 57, 96-160 (1992) · Zbl 0767.35082
[33] V.E. Zakharov, V.S. L’vov, G. Falkovich, Kolmogorov Spectra of Turbulence, Springer, Berlin, 1992.; V.E. Zakharov, V.S. L’vov, G. Falkovich, Kolmogorov Spectra of Turbulence, Springer, Berlin, 1992. · Zbl 0786.76002
[34] Banner, M. L.; Young, I. R., Modeling spectral dissipation in the evolution of wind waves. I. Assessment of existing model performance, J. Phys. Oceanogr., 24, 1550-1571 (1994)
[35] A.C. Newell, V.E. Zakharov, Optical turbulence, in: P. Tabeling, O. Cardoso (Eds.), Turbulence: A Tentative Dictionary, Plenum Press, New York, 1995.; A.C. Newell, V.E. Zakharov, Optical turbulence, in: P. Tabeling, O. Cardoso (Eds.), Turbulence: A Tentative Dictionary, Plenum Press, New York, 1995.
[36] Pushkarev, A. N.; Zakharov, V. E., Turbulence of capillary waves, Phys. Rev. Lett., 76, 3326-3329 (1996)
[37] U. Frisch, Turbulence, Cambridge University Press, Cambridge, 1996.; U. Frisch, Turbulence, Cambridge University Press, Cambridge, 1996.
[38] L’vov, V.; L’vov, Y. V.; Newell, A. C.; Zakharov, V. E., Statistical description of acoustic turbulence, Phys. Rev. E, 56, 390, 1 (1997)
[39] L’vov, Y. V.; Newell, A. C., Semiconductor lasers and Kolmogorov spectra, Phys. Lett. A, 235, 499-503 (1997)
[40] L’vov, Y. V.; Binder, R.; Newell, A. C., Quantum weak turbulence with applications to semiconductor lasers, Physica D, 121, 317-343 (1998) · Zbl 0938.82061
[41] V.E. Zakharov (Ed.), Dispersive Nonlinear Waves and Weak Turbulence, Vol. 182, AMS Translations Series 2, American Mathematical Society, Providence, RI, 1998.; V.E. Zakharov (Ed.), Dispersive Nonlinear Waves and Weak Turbulence, Vol. 182, AMS Translations Series 2, American Mathematical Society, Providence, RI, 1998. · Zbl 0879.00029
[42] Balk, A. M.; Ferapontov, E. V., Invariants of wave system and web geometry, Am. Math. Soc. Transl., 182, 2, 1-20 (1998) · Zbl 0913.76008
[43] Balk, A. M.; Zakharov, V. E., Stability of weak-turbulence Kolmogorov spectra, Am. Math. Soc. Transl., 182, 2, 31-82 (1998) · Zbl 0898.76047
[44] Zakharov, V. E., Weakly nonlinear waves on the surface of an ideal finite depth fluid, Am. Math. Soc. Transl., 182, 2, 167-197 (1998) · Zbl 0914.76015
[45] L’vov, Y. V.; Newell, A. C., Finite flux solutions of the quantum Boltzmann equation and semiconductor lasers, Phys. Rev. Lett., 84, 18-94 (2000)
[46] S. Galtier, S.V. Nazarenko, A.C. Newell, A. Pouquet, A weak turbulence theory for incompressible magnetohydrodynamics, in: T. Passot, P.L. Sulem (Eds.), Nonlinear MHD Waves and Turbulence, Lecture Notes in Physics, Springer, Berlin, 1999, pp. 291-330.; S. Galtier, S.V. Nazarenko, A.C. Newell, A. Pouquet, A weak turbulence theory for incompressible magnetohydrodynamics, in: T. Passot, P.L. Sulem (Eds.), Nonlinear MHD Waves and Turbulence, Lecture Notes in Physics, Springer, Berlin, 1999, pp. 291-330. · Zbl 0941.76045
[47] Falkovich, G. E.; Shafarenko, A. V., Non-stationary wave turbulence, J. Nonlinear Sci., 1, 452-480 (1991) · Zbl 0796.76045
[48] Newell, A. C.; Zakharov, V. E., Rough sea foam, Phys. Rev. Lett., 63, 1149 (1992)
[49] Galtier, S.; Nazarenko, S. V.; Newell, A. C.; Pouquet, A., A weak turbulence theory for incompressible MHD, J. Plasma Phys., 63, 447-488 (2000)
[50] Gurarie, V., Probability density, diagrammatic technique and epsilon expansion in the theory of wave turbulence, Nucl. Phys. B, 441, 569-594 (1995)
[51] Frisch, U.; Fournier, J. D., Quelques resultats exacts pour l’equation de Burgers aleatoire, J. Phys., 39, C5-C19 (1978)
[52] Svistunov, B. V., Highly non-equilibrium Bose condensation in a weakly interacting gas, J. Moscow Phys. Soc., 1, 373 (1991)
[53] R. Lacaze, P. Lallemand, Y. Pomeau, S. Rica, Dynamical formation of a Bose-Einstein condensate, Physica D 152-153 (2001) 779-786.; R. Lacaze, P. Lallemand, Y. Pomeau, S. Rica, Dynamical formation of a Bose-Einstein condensate, Physica D 152-153 (2001) 779-786. · Zbl 0979.83028
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