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10 - Robert H. Kraichnan

Published online by Cambridge University Press:  07 October 2011

By
Gregory Eyink  and
Uriel Frisch
Edited by
Peter A. Davidson ,
Yukio Kaneda ,
Keith Moffatt  and
Katepalli R. Sreenivasan
Gregory Eyink
Affiliation:
Hopkins University
Uriel Frisch
Affiliation:
UNS, CNRS, OCA, Lab. Lagrange
Peter A. Davidson
Affiliation:
University of Cambridge
Yukio Kaneda
Affiliation:
Nagoya University, Japan
Keith Moffatt
Affiliation:
University of Cambridge
Katepalli R. Sreenivasan
Affiliation:
New York University
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Summary

Introduction

Robert Harry Kraichnan (1928–2008) was one of the leaders in the theory of turbulence for a span of about forty years (mid-1950s to mid-1990s). Among his many contributions, he is perhaps best known for his work on the inverse energy cascade (i.e. from small to large scales) for forced two-dimensional turbulence. This discovery was made in 1967 at a time when two-dimensional flow was becoming increasingly important for the study of large-scale phenomena in the Earth's atmosphere and oceans. The impact of the discovery was amplified by the development of new experimental and numerical techniques that allowed full validation of the conjecture.

How did Kraichnan become interested in turbulence? His earliest scientific interest was in general relativity, which he began to study on his own at age 13. At age 18 he wrote at MIT a prescient undergraduate thesis, Quantum Theory of the Linear Gravitational Field; he received a PhD in physics from MIT in 1949 for his thesis, Relativistic Scattering of Pseudoscalar Mesons by Nucleons, supervised by Herman Feshbach. His interest in turbulence arose in 1950 while assisting Albert Einstein in search for highly nonlinear, particlelike solutions to unified field equations.

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A Voyage Through Turbulence , pp. 329 - 372
Publisher: Cambridge University Press
Print publication year: 2011

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References

Abe, R. 1973. Expansion of a critical exponent in inverse powers of spin dimensionality, Prog. Theor. Physics 49, 113–128.CrossRefGoogle Scholar
Alfvén, H. 1942. Existence of electromagnetic-hydrodynamic waves, Nature 150, 405–406.CrossRefGoogle Scholar
Batchelor, G.K. 1950. On the spontaneous magnetic field in a conducting liquid in turbulent motion, Proc. Roy. Soc. Lond. Ser. A 201, 405–416.CrossRefGoogle Scholar
Batchelor, G.K. 1953. The Theory of Homogeneous Turbulence, Cambridge University Press, Cambridge, UK.Google Scholar
Batchelor, G.K. 1959. Small-scale variation of convected quantities like temperature in turbulent fluid. Part 1. General discussion and the case of small conductivity. J. Fluid Mech. 5, 113–133.CrossRefGoogle Scholar
Batchelor, G.K. 1969. Computation of the energy spectrum in homogeneous two-dimensional turbulence, Phys. Fluids Suppl. II 12, 233–239.Google Scholar
Batchelor, G.K. & Townsend, A.A. 1949. The nature of turbulent motion at large wave-numbers. Proc. R. Soc. Lond. A 199, 238–255.CrossRefGoogle Scholar
Baudet, C., Ciliberto, S. & Pinton, J.F. 1991. Spectral analysis of the von Kármán flow using ultrasound scattering, Phys. Rev. Lett. 67, 193–195.CrossRefGoogle Scholar
Belinicher, V.I. & L'vov, V.S. 1987. The scale-invariant theory of developed hydrodynamic turbulence, Zhurn. Eksp. Teor. Fiz. 93, 533–551.Google Scholar
Boffetta, G. 2007. Energy and enstrophy fluxes in the double cascade of two-dimensional turbulence, J. Fluid Mech. 589, 253–260.CrossRefGoogle Scholar
Boffetta, G. & Musacchio, S. 2010. An update of the double cascade in two-dimensional turbulence, Phys. Rev. E 82, 016307.CrossRefGoogle Scholar
Borue, V. 1993. Spectral exponents of enstrophy cascade in stationary two-dimensional homogeneous turbulence, Phys. Rev. Lett. 71, 3967–3970.CrossRefGoogle ScholarPubMed
Bouchet, F. & Simonnet, E. 2009. Random changes of flow topology in two-dimensional and geophysical turbulence, Phys. Rev. Lett. 102, 094504.CrossRefGoogle ScholarPubMed
Boussinesq, J. V. 1870. Essai théorique sur les lois trouvées expérimentalement par M. Bazin pour l'écoulement uniforme de l'eau dans les canaux découverts, C. R. Acad. Sci. Paris 71, 389–393.Google Scholar
Bray, R.W. 1966. A study of turbulence and convection using Fourier and numerical analysis, PhD dissertation available in the library of the Department of Applied Mathematics and Theoretical Physics, Cambridge University, Cambridge [all chapters but the last on thermal convection are also available at http://www.oca.eu/etc7/bray-phd1966.pdf].
Chandrasekhar, S. 1955. A theory of turbulence, Proc. R. Soc. Lond. A 229, 1–19.CrossRefGoogle Scholar
Charney, J.G. 1947. The dynamics of long waves in a baroclinic westerly current, J. Meteor. 4, 135–163.2.0.CO;2>CrossRefGoogle Scholar
Chavanne, X., Chillà, F., Chabaud, B., Castaing, B. & Hébral, B. 2001. Turbulent Rayleigh–Bénard convection in gaseous and liquid He, Phys. Fluids 13, 1300–1320.CrossRefGoogle Scholar
Chen, S., Ecke, R.E., Eyink, G.L., Rivera, M., Wan, M.-P., & Xiao, Z. 2006. Physical mechanism of the two-dimensional inverse energy cascade, Phys. Rev. Lett. 96, 084502CrossRefGoogle ScholarPubMed
Chertkov, M., Connaughton, C, Kolokolov, I. & Lebedev, V. 2007. Dynamics of energy condensation in two-dimensional turbulence, Phys. Rev. Lett. 99, 084501.CrossRefGoogle ScholarPubMed
Chertkov, M., Falkovich, G., Kolokolov, I. & Lebedev, V. 1995. Normal and anomalous scaling of the fourth-order correlation function of a randomly advected passive scalar, Phys.Rev.E 52, 4924–4941.CrossRefGoogle ScholarPubMed
Cho, J.Y.N. & Lindborg, E. 2001. Horizontal velocity structure functions in the upper troposphere and lower stratosphere 1. Observations, J. Geophys. Res. 106, 10223–32.CrossRefGoogle Scholar
Cichowlas, C., Bonaïti, C., Debbasch, F. & Brachet, M. 2005. Effective dissipation and turbulence in spectrally truncated Euler flows, Phys. Rev. Lett. 95, 264502.CrossRefGoogle ScholarPubMed
Corrsin, S. 1951. On the spectrum of isotropic temperature fluctuations in an isotropic turbulence, J. Appl. Phys. 22, 469–473.CrossRefGoogle Scholar
Donzis, D.A. & Sreenivasan, K.R. 2010. The bottleneck effect and the Kolmogorov constant in isotropic turbulence, J. Fluid Mech. 657, 171–188.CrossRefGoogle Scholar
Falkovich, G., Gawȩdzki, K. & Vergassola, M. 2001. Particles and fields in fluid turbulence, Rev. Mod. Phys. 73, 913–975.CrossRefGoogle Scholar
Fjørtoft, R. 1953. On the changes in the spectral distribution of kinetic energy for two-dimensional nondivergent flow, Tellus 5, 225–230.CrossRefGoogle Scholar
Forster, D., Nelson, D.R. & Stephen, M.J. 1977. Large-distance and long-time properties of a randomly stirred fluid, Phys. Rev. A 16, 732–749.CrossRefGoogle Scholar
di Francesco, P., Ginsparg, P. & Zinn-Justin, J. 1995. 2D gravity and random matrices, Phys. Rep. 254, 1–133.CrossRefGoogle Scholar
Frisch, U. 1995. Turbulence: The Legacy of A.N. Kolmogorov, Cambridge University Press.Google Scholar
Frisch, U. & Bourret, R. 1970. Parastochastics, J. Math. Phys. 11, 364–390.CrossRefGoogle Scholar
Frisch, U., Lesieur, M. & Brissaud, A. 1974. A Markovian random coupling model for turbulence, J. Fluid Mech. 65, 145–152.CrossRefGoogle Scholar
Frisch, U., Mazzino, A. & Vergassola, M. 1998. Intermittency in passive scalar advection, Phys. Rev. Lett. 80, 5532–5537.CrossRefGoogle Scholar
Frisch, U., Mazzino, A., Noullez, A. & Vergassola, M. 1999. Lagrangian method for multiple correlations in passive scalar advection, Phys. Fluids 11, 2178–2186.CrossRefGoogle Scholar
Frisch, U. & Morf, R. 1981. Intermittency in nonlinear dynamics and singularitites at complex times, Phys. Rev. A 23, 2673–2705.CrossRefGoogle Scholar
Frisch, U., Pouquet, A., Leorat, J. & Mazure, A. 1975. Possibility of an inverse cascade of magnetic helicity in magnetohydrodynamic turbulence, J. Fluid Mech. 68, 769–778.CrossRefGoogle Scholar
Frisch, U., Sulem, P.-L. & Nelkin, M. 1978. A simple dynamical model of intermittent fully developed turbulence, J. Fluid Mech. 87, 719–736.CrossRefGoogle Scholar
Frisch, U. & Wirth, A. 1996. Inertial-diffusive range for a passive scalar advected by a white-in-time velocity field, Europhys. Lett. 35, 683–687.CrossRefGoogle Scholar
Fyfe, D. & Montgomery, D. 1976. High-beta turbulence in two-dimensional magneto-hydrodynamics, J. Plasma Phys. 16, 181–191.CrossRefGoogle Scholar
Galtier, S., Nazarenko, S.V., Newell, A.C. & Pouquet, A. 2000. A weak turbulence theory for incompressible magnetohydrodynamics, J. Plasma Phys. 63, 447–488.CrossRefGoogle Scholar
Gat, O., Procaccia, I. & Zeitak, R. 1998. Anomalous scaling in passive scalar advection: Monte Carlo Lagrangian trajectories, Phys. Rev. Lett. 80, 5536–5539.CrossRefGoogle Scholar
Gawȩdzki, K. & Kupiainen, A. 1995. Anomalous scaling of the passive scalar, Phys. Rev. Lett. 75, 3834–3837.CrossRefGoogle ScholarPubMed
Goldreich, P. & Sridhar, S. 1997. Magnetohydrodynamic turbulence revisited, Astrophys. J. 485, 680–688.CrossRefGoogle Scholar
Gotoh, T. 1998. Energy spectrum in the inertial and dissipation ranges of two-dimensional steady turbulence, Phys. Rev. E 57, 2984–2991.CrossRefGoogle Scholar
Gotoh, T. & Kraichnan, R.H. 2004. Turbulence and Tsallis statistics, Physica D 193, 231–244.CrossRefGoogle Scholar
Grant, H.L., Stewart, R.W. & Moilliet, A. 1962. Turbulent spectra from a tidal channel, J. Fluid Mech. 12, 241–268.CrossRefGoogle Scholar
Hashminskii, R.Z. 1966. A limit theorem for the solutions of differential equations with random righthand sides, Theor. Prob. Appl. 11, 390–406.CrossRefGoogle Scholar
Herring, J.R. & Kraichnan, R.H. 1972. Comparison of some approximations for isotropic turbulence. In Statistical Models and Turbulence, Rosenblatt, M. & Van Atta, C., eds., Springer-Verlag, New York, 148–194.CrossRefGoogle Scholar
Hopf, E. 1952. Statistical hydromechanics and functional calculus, J. Rat. Mech. Anal. 1, 87–123.Google Scholar
't Hooft, G. 1974. A planar diagram theory for strong interactions, Nucl. Phys. B 72, 461–473.CrossRefGoogle Scholar
Iroshnikov, P.S. 1963. Turbulence of a conducting fluid in a strong magnetic field, Astronomicheskii Zhurnal 40, No. 4, 742–750 (in Russian); translated in Soviet Astronomy 7, 566–571 (1964). [The Russian version has the correct initial ‘R’ for his given name ‘Ruslan’; this was incorrectly translated as a ‘P’.]Google Scholar
Kawasaki, K. 1970. Kinetic equations and time correlation functions of critical fluctuations, Ann. Phys. (N.Y.) 61, 1–56.CrossRefGoogle Scholar
Kimura, Y. and Kraichnan, R.H. 1993. Statistics of an advected passive scalar, Phys. Fluids A 5, 2264–2277.CrossRefGoogle Scholar
Kit, L.G. & Tsinober, A.B. 1971. Possibility of creating and investigating two-dimensional turbulence in a strong magnetic field, Magnitnaya Gidrodinamika, 7, 27–34.Google Scholar
Kolmogorov, A.N. 1941. The local structure of turbulence in incompressible viscous fluid for very large Reynolds number, Dokl. Akad. Nauk SSSR 30, 299–303.Google Scholar
Kolmogorov, A.N. 1962. A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number, J. Fluid Mech. 13, 82–85.CrossRefGoogle Scholar
Kraichnan, R.H. 1953. Scattering of sound in a turbulent medium, J. Acoust. Soc. Am. 25, 1096–1104.CrossRefGoogle Scholar
Kraichnan, R.H. 1955a. Statistical mechanics of an adiabatically compressible fluid, J. Acoust. Soc. Am. 27, 438–441.CrossRefGoogle Scholar
Kraichnan, R.H. 1955b. Special-relativistic derivation of generally covariant gravitation theory, Phys. Rev. 98, 1118–1122.CrossRefGoogle Scholar
Kraichnan, R.H. 1956a. Pressure field within homogeneous anisotropic turbulence, J. Acoust. Soc. Am. 28, 64–72.CrossRefGoogle Scholar
Kraichnan, R.H. 1956b. Pressure fluctuations in turbulent flow over a flat plate, J. Acoust. Soc. Am. 28, 378–390.CrossRefGoogle Scholar
Kraichnan, R.H. 1957a. Noise transmission from boundary-layer pressure fluctuations, J. Acoust. Soc. Am. 29, 65–80.CrossRefGoogle Scholar
Kraichnan, R.H. 1957b. Relation of fourth-order to second-order moments in stationary isotropic turbulence, Phys. Rev. 107, 1385–1490.CrossRefGoogle Scholar
Kraichnan, R.H. 1958a. Irreversible statistical mechanics of incompressible hydromagnetic turbulence, Phys. Rev. 109, 1407–1422.CrossRefGoogle Scholar
Kraichnan, R.H. 1958b. Higher order interactions in homogeneous turbulence theory, Phys. Fluids 1, 358–359.CrossRefGoogle Scholar
Kraichnan, R.H. 1958c. A theory of turbulence dynamics. In Second Symposium on Naval Hydrodynamics, Ref. ACR-38, Office of Naval Research, Washington, DC, pp. 29–44.Google Scholar
Kraichnan, R.H. 1959a. Structure of isotropic turbulence at very high Reynolds numbers, J. Fluid Mech. 5, 497–543.CrossRefGoogle Scholar
Kraichnan, R. H. 1959b. Classical fluctuation–relaxation theorem, Phys. Rev. 113, 1181–1182.CrossRefGoogle Scholar
Kraichnan, R.H. 1961. Dynamics of nonlinear stochastic systems, J. Math. Phys. 2, 124–148.CrossRefGoogle Scholar
Kraichnan, R.H. 1962. Mixing-length analysis of turbulent thermal convection at arbitrary Prandtl number, Phys. Fluids 5, 1374–1389.CrossRefGoogle Scholar
Kraichnan, R.H. 1964a. Direct-interaction approximation for shear and thermally driven turbulence, Phys. Fluids 7, 1048–1062.CrossRefGoogle Scholar
Kraichnan, R.H. 1964b. Kolmogorov's hypotheses and Eulerian turbulence theory, Phys. Fluids 7, 1723–1734.CrossRefGoogle Scholar
Kraichnan, R.H. 1965a. Lagrangian-history closure approximation for turbulence, Phys. Fluids 8, 575–598.CrossRefGoogle Scholar
Kraichnan, R.H. 1965b. Inertial-range spectrum of hydrodynamic turbulence, Phys. Fluids 8, 1385–1387.CrossRefGoogle Scholar
Kraichnan, R.H. 1966a. Isotropic turbulence and inertial-range structure, Phys. Fluids 9, 1728–1752.CrossRefGoogle Scholar
Kraichnan, R.H. 1966b. Dispersion of particle pairs in homogeneous turbulence, Phys. Fluids 9, 1937–1943.CrossRefGoogle Scholar
Kraichnan, R.H. 1967a. Condensate turbulence in a weakly coupled boson gas, Phys. Rev. Lett. 18, 202–206.CrossRefGoogle Scholar
Kraichnan, R.H. 1967b. Inertial ranges in two-dimensional turbulence, Phys. Fluids 10, 1417–1423.CrossRefGoogle Scholar
Kraichnan, R.H. 1967c. Intermittency in the very small scales of turbulence, Phys. Fluids 10, 2081–2082.CrossRefGoogle Scholar
Kraichnan, R.H. 1968a. Lagrangian-history statistical theory for Burgers equation, Phys. Fluids 11, 266–277.CrossRefGoogle Scholar
Kraichnan, R.H. 1968b. Small-scale structure of a scalar field convected by turbulence, Phys. Fluids 11, 945–953.CrossRefGoogle Scholar
Kraichnan, R.H. 1968c. Convergents to infinite series in turbulence theory. Phys. Rev. 174, 240–246.CrossRefGoogle Scholar
Kraichnan, R.H. 1970a. Turbulent diffusion: evaluation of primitive and renormalized perturbation series by Padé approximants and by expansion of Stieltjes transforms into contributions from continuous orthogonal functions. In The Padé Approximant in Theoretical Physics, Baker, G.A. & Gammel, J.L., eds., Academic Press, New York, pp. 129–170.CrossRefGoogle Scholar
Kraichnan, R.H. 1970b. Instability in fully-developed turbulence, Phys. Fluids 13, 569–575.CrossRefGoogle Scholar
Kraichnan, R.H. 1970c. Convergents to turbulence functions, J. Fluid Mech. 41, 189–218.CrossRefGoogle Scholar
Kraichnan, R.H. 1971a. An almost-Markovian Galilean-invariant turbulence model, J. Fluid Mech. 47, 513–524.CrossRefGoogle Scholar
Kraichnan, R.H. 1971b. Inertial-range transfer in two- and three-dimensional turbulence, J. Fluid Mech. 47, 525–535.CrossRefGoogle Scholar
Kraichnan, R.H. 1972. Some modern developments in the statistical theory of turbulence. In Statistical Mechanics: New Concepts, New Problems, New Applications, Rice, S.A., Freed, K.F. and Light, J.C., eds., University of Chicago Press, Chicago, pp. 201–228.Google Scholar
Kraichnan, R.H. 1974a. On Kolmogorov's inertial-range theories, J. Fluid Mech. 62, 305–330.CrossRefGoogle Scholar
Kraichnan, R.H. 1974b. Passive-scalar convection by a quasi-uniform random straining field, J. Fluid Mech. 64, 737–762.CrossRefGoogle Scholar
Kraichnan, R.H. 1975a. Remarks on turbulence theory, Adv. Math. 16, 305–331.CrossRefGoogle Scholar
Kraichnan, R.H. 1975b. Statistical dynamics of two-dimensional flow, J. Fluid Mech. 67, 155–175.CrossRefGoogle Scholar
Kraichnan, R.H. 1976a. Diffusion of weak magnetic fields by isotropic turbulence. J. Fluid Mech. 75, 657–676.CrossRefGoogle Scholar
Kraichnan, R.H. 1976b. Eddy viscosity in two and three dimensions, J. Atmos. Sci. 33, 1521–1536.2.0.CO;2>CrossRefGoogle Scholar
Kraichnan, R.H. 1976c. Diffusion of passive-scalar and magnetic fields by helical turbulence, J. Fluid Mech. 77, 753–768.CrossRefGoogle Scholar
Kraichnan, R.H. 1979a. Variational method in turbulence theory, Phys. Rev. Lett. 42, 1263–1266.CrossRefGoogle Scholar
Kraichnan, R.H. 1979b. Consistency of the alpha-effect turbulent dynamo, Phys. Rev. Lett. 42 1677–1680.CrossRefGoogle Scholar
Kraichnan, R.H. 1985. Decimated amplitude equations in turbulence dynamics. In Theoretical Approaches to Turbulence, Dwoyer, D.L., Hussaini, M.Y. and Voigt, R.G., eds., Springer-Verlag, New York, pp. 91–135.Google Scholar
Kraichnan, R.H. 1988. Reduced description of hydrodynamic turbulence, J. Stat. Phys. 51, 949–963.CrossRefGoogle Scholar
Kraichnan, R.H. 1991. Turbulent cascade and intermittency growth, Proc. Roy. Soc. Lond. A 434, 65–78.CrossRefGoogle Scholar
Kraichnan, R.H. 1994. Anomalous scaling of a randomly advected passive scalar, Phys. Rev. Lett. 72, 1016–1019.CrossRefGoogle ScholarPubMed
Kraichnan, R.H. 1999. Note on forced Burgers turbulence, Phys. Fluids 11, 3738–3742.CrossRefGoogle Scholar
Kraichnan, R.H. & Chen, S. 1989. Is there a statistical mechanics of turbulence?Physica D 37, 160–172.CrossRefGoogle Scholar
Kraichnan, R.H., Chen, H. & Chen, S. 1989. Probability distribution of a stochastically advected scalar field, Phys. Rev. Lett. 63, 2657–2657.Google Scholar
Kraichnan, R.H. & Gotoh, T. 1993. Statistics of decaying Burgers turbulence, Phys. Fluids A 5, 445–457.Google Scholar
Kraichnan, R.H. & Leith, C.E. 1972. Predictability of turbulent flows, J. Atmos. Sci. 29, 1041–1058.Google Scholar
Kraichnan, R.H. & Montgomery, D. 1980. Two-dimensional turbulence, Rep. Prog. Phys. 43, 547–618.CrossRefGoogle Scholar
Kraichnan, R.H. & Nagarajan, S. 1967. Growth of turbulent magnetic fields, Phys. Fluids 10, 859–870.CrossRefGoogle Scholar
Kraichnan, R.H. & Orszag, S.A. 1967. Model equations for strong turbulence in a Vlasov plasma, Phys. Fluids 10, 1720–1736.Google Scholar
Landau, L.D. & Lifshitz, E.M. 1987. Fluid Mechanics, 2nd ed., Pergamon Press, Oxford.Google Scholar
Lee, T.D. 1951. Difference between turbulence in a two-dimensional fluid and in a three-dimensional fluid, J. Appl. Phys. 22, 524–524.CrossRefGoogle Scholar
Lee, T.D. 1952. On some statistical properties of hydrodynamic and hydromagnetic fields, Quarterly Appl. Math. 10, 69–72.CrossRefGoogle Scholar
Leith, C.E. 1967. Diffusion approximation to inertial energy transfer in isotropic turbulence, Phys. Fluids 10, 1409–1416.CrossRefGoogle Scholar
Lesieur, M. 2008. Turbulence in Fluids, 4th ed., Springer, Heidelberg.CrossRefGoogle Scholar
Leslie, D.C. 1973. Developments in the Theory of Turbulence, Oxford University Press, Clarendon.Google Scholar
Lesieur, M., Frisch, U. & Brissaud, A. 1971. ThéoriedeKraichnandelaturbulence.Application àl'étude d'une turbulence possédant de l'hélicité, Ann. Géophys. (Paris) 27, 151–165.Google Scholar
Lewis, R.M. & Kraichnan, R.H. 1962. A space-time functional formalism for turbulence, Commun. Pure Appl. Math. 15, 397–411.CrossRefGoogle Scholar
Lighthill, M.J. 1952. On sound generated aerodynamically. I. General theory, Proc. R. Soc. Lond. A 211, 564–587.CrossRefGoogle Scholar
Lighthill, M.J. 1953. On the energy scattered from the interaction of turbulence with sound or shock waves, Proc. Camb. Phil. Soc. 49, 531–551.CrossRefGoogle Scholar
Lund, F. & Rojas, C. 1989. Ultrasound as a probe of turbulence, Physica D 37, 508–514.CrossRefGoogle Scholar
Makeenko, Y.M. & Migdal, A.A. 1979. Exact equation for the loop average in multi-color QCD, Phys. Lett. B 88, 135–137.CrossRefGoogle Scholar
Mandelbrot, B.B. 1968. On intermittent free turbulence. In Turbulence of Fluids and Plasmas. New York, April 16–18, Brooklyn Polytechnic Inst., New York (abstract).Google Scholar
Mandelbrot, B.B. 1974. Intermittent turbulence in self-similar cascades: divergence of high moments and dimension of the carrier, J. Fluid Mech. 62, 331–358.CrossRefGoogle Scholar
Martin, P.C., Siggia, E.D. & Rose, H.A. 1973. Statistical dynamics of classical systems, Phys. Rev. A 8, 423–437.CrossRefGoogle Scholar
Migdal, A.A. 1983. Loop equations and 1/N expansion, Phys. Rep. 102, 199–290.CrossRefGoogle Scholar
Millionschikov, M. 1941. On the theory of homogeneous isotropic turbulence, Dokl. Akad. Nauk SSSR 32, 615–618.Google Scholar
Monin, A.S. & Yaglom, A.M. 1975. Statistical Fluid Mechanics, Vol. 2, MIT Press, Cambridge, MA.Google Scholar
Nazarenko, S.V. 2011. Wave Turbulence, Lect. Notes in Phys., Vol. 825 Springer, Heidelberg.CrossRefGoogle Scholar
Nazarenko, S.V., Newell, A.C. & Galtier, S. 2001. Non-local MHD turbulence, Physica D 152, 646–652.CrossRefGoogle Scholar
Neumann, J. von. 1949. Recent theories of turbulence. In Collected Works (1949–1963)6, 437–472, ed. A.H., Taub. Pergamon Press, New York (1963).Google Scholar
Ng, C.S. & Bhattacharjee, A. 1996. Interaction of shear-Alfven wave packets: implication for weak magnetohydrodynamic turbulence in astrophysical plasmas, Astrophys. J. 465, 845–854.CrossRefGoogle Scholar
Niemela, J.J., Skrbek, L., Sreenivasan, K.R. & Donnelly, R.J. 2000. Turbulent convection at high Rayleigh numbers, Nature 404, 837–841.CrossRefGoogle ScholarPubMed
Novikov, E.A. & Stewart, R. 1964. Intermittency of turbulence and spectrum of fluctuations in energy-dissipation, Izv. Akad. Nauk. SSSR. Ser. Geofiz 3, 408–413.Google Scholar
Obukhov, A.M. 1949. Structure of the temperature field in a turbulent flow, Izv. Akad. Nauk SSSR, Ser. Geogr. i Geofiz. 13, 58–69.Google Scholar
Obukhov, A.M. 1962. Some specific features of atmospheric turbulence, J. Fluid Mech. 13, 77–81.CrossRefGoogle Scholar
Ogura, Y. 1963. A consequence of the zero-fourth-cumulant approximation in the decay of isotropic turbulence, J. Fluid Mech. 16, 33–40.CrossRefGoogle Scholar
Onsager, L. 1949. Statistical hydrodynamics, Nuovo Cim. Suppl. 6, 279–287.CrossRefGoogle Scholar
Orszag, S.A. 1966. Dynamics of fluid turbulence, Princeton Plasma Physics Laboratory, report PPL-AF-13.
Orszag, S.A. 1977. Statistical theory of turbulence. In Fluid Dynamics, Les Houches 1973, eds. R., Balian & J.L., Peube. Gordon and Breach, New York, pp. 237–374.Google Scholar
Orszag, S.A. & Kruskal, M.D. 1966. Theory of turbulence, Phys. Rev. Lett. 16, 441–444.CrossRefGoogle Scholar
Ott, S. & Mann, J. 2000. An experimental investigation of the relative diffusion of particle pairs in three-dimensional turbulent flow, J. Fluid Mech. 422, 207–223.CrossRefGoogle Scholar
Parisi, G. & Frisch, U. 1985. On the singularity structure of fully developed turbulence. In Turbulence and Predictability in Geophysical Fluid Dynamics, Proceedings of the International School of Physics ‘E. Fermi’, 1983, Varenna, Italy. Eds. M., Ghil, R., Benzi & G., Parisi, North Holland, Amsterdam, pp. 84–87.Google Scholar
Pasquero, C. & Falkovich, G. 2002. Stationary spectrum of vorticity cascade in two-dimensional turbulence, Phys. Rev. E 65, 056305.CrossRefGoogle ScholarPubMed
Pierotti, D. 1997. Intermittency in the large-N limit of a spherical shell model for turbulence, Europhys. Lett. 37, 323–328CrossRefGoogle Scholar
Pierotti, D., L'vov, V.S., Pomyalov, A. & Procaccia, I. 2000. Anomalous scaling in a model of hydrodynamic turbulence with a small parameter, Europhys. Lett. 50, 473–479.CrossRefGoogle Scholar
Prandtl, L. 1925. Bericht über Untersuchungen zur ausgebildeten Turbulenz, Zs. angew. Math. Mech. 5, 136–139.Google Scholar
Priestley, C.H.B. 1959. Turbulent Transfer in the Lower Atmosphere, University of Chicago Press, Chicago.Google Scholar
Proudman, I. 1961. On Kraichnan's theory of turbulence. In Mécanique de la Turbulence. Colloques Internat. CNRS108, 107–112, ed. A., Favre, CNRS, Paris.Google Scholar
Reid, W.H. 1959. Tech. Rept. No. 23, Division of Applied Mathematics, Brown University. Rhines, P.B. 1979. Geostrophic turbulence, Ann. Rev. Fluid Mech. 11, 401–441.Google Scholar
Rose, H.A. & Sulem, P.-L. 1978. Fully developed turbulence and statistical mechanics, J. Phys. France 39, 441–484.CrossRefGoogle Scholar
Rota, G.-C. 1996. Indiscrete Thoughts, Birkhäuser, Boston.Google Scholar
Rüdiger, G. 1974. The influence of a uniform magnetic field of arbitrary strength on turbulence, Astron. Nachr. 295, 275–284.CrossRefGoogle Scholar
Salmon, R. 1980. Baroclinic instability and geostrophic turbulence, Geophys. Astrophys. Fluid Dyn. 15, 167–211.CrossRefGoogle Scholar
Salmon, R. 1998. Lectures on Geophysical Fluid Dynamics, Oxford University Press.Google Scholar
Scott, R.B. & Wang, F. 2005. Direct evidence of an oceanic inverse kinetic energy cascade from satellite altimetry, J. Phys. Ocean. 35, 1650–1666.CrossRefGoogle Scholar
Semikoz, D.V. & Tkachev, I.I. 1997. Condensation of bosons in the kinetic regime, Phys. Rev. D 55, 489–502.CrossRefGoogle Scholar
Shebalin, J.V., Matthaeus, W.H. & Montgomery, D. 1983. Anisotropy in MHD turbulence due to a mean magnetic field, J. Plasma Phys. 29, 525–547.CrossRefGoogle Scholar
Shraiman, B.I. & Siggia, E.D. 1995. Anomalous scaling and small scale anisotropy of a passive scalar in turbulent flow, C.R. Acad. Sci. 321, 279–284.Google Scholar
Smith, L.M. & Yakhot, V. 1993. Bose condensation and small-scale structure generation in a random force driven 2D turbulence, Phys. Rev. Lett. 71, 352–355.CrossRefGoogle Scholar
Spiegel, E.A. 1971. Convection in stars, Ann. Rev. Astron. Astrophys. 9, 323–352.CrossRefGoogle Scholar
Spiegel, E.A. 2011. Chandrasekhar's lecture notes on the theory of turbulence (1954). Lect. Notes in Phys. Vol. 810 Springer, Heidelberg.Google Scholar
Starr, V.P. 1968. Physics of Negative Viscosity Phenomena, McGraw-Hill, New York.Google Scholar
Tatsumi, T. 1957. The theory of decay processes of incompressible isotropic turbulence. Proc. R. Soc. Lond. A 239, 16–45.CrossRefGoogle Scholar
Taylor, G.I. 1917. Observations and speculations on the nature of turbulence motion, Reports and Memoranda of the Advisory Committee for Aeronautics, no. 345; reproduced in G.I. Taylor's Scientific Papers, Batchelor, G.K., ed., Cambridge University Press, 1960, Vol. II, paper no. 7, pp. 69–78.Google Scholar
Ting, L. & Miksis, M.J. 1990. On vortical flow and sound generation, Siam. J. Appl. Math. 50, 521–536.CrossRefGoogle Scholar
Xiao, Z., Wan, M., Chen, S. & Eyink, G.L. 2008. Physical mechanism of the inverse energy cascade of two-dimensional turbulence: a numerical investigation, J. Fluid Mech. 619, 1–44.CrossRefGoogle Scholar
Yaglom, A.M. 1949. Local structure of the temperature field in a turbulent flow, Dokl. Akad. Nauk SSSR 69, 743–746.Google Scholar
Yaglom, A.M. 1966. The influence of fluctuations in energy dissipation on the shape of turbulence characteristics in the inertial interval, Dokl. Akad. Nauk. SSSR 166, 49–52.Google Scholar
Zakharov, V.E. 1966. Some aspects of nonlinear theory of surface waves (in Russian), PhD Thesis, Budker Institute for Nuclear Physics for Nuclear Physics, Novosibirsk, USSR.Google Scholar
Zakharov, V., L'vov, V. & Falkovich, G. 1992. Kolmogorov Spectra of Turbulence, Springer-Verlag, Berlin.CrossRefGoogle Scholar

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