Abstract
Scaling is an important measure of multi-scale fluctuation systems. Turbulence as the most remarkable multi-scale system possesses scaling over a wide range of scales. She-Leveque (SL) hierarchical symmetry, since its publication in 1994, has received wide attention. A number of experimental, numerical and theoretical work have been devoted to its verification, extension, and modification. Application to the understanding of magnetohydrodynamic turbulence, motions of cosmic baryon fluids, cosmological supersonic turbulence, natural image, spiral turbulent patterns, DNA anomalous composition, human heart variability are just a few among the most successful examples. A number of modified scaling laws have been derived in the framework of the hierarchical symmetry, and the SL model parameters are found to reveal both the organizational order of the whole system and the properties of the most significant fluctuation structures. A partial set of work related to these studies are reviewed. Particular emphasis is placed on the nature of the hierarchical symmetry. It is suggested that the SL hierarchical symmetry is a new form of the self-organization principle for multi-scale fluctuation systems, and can be employed as a standard analysis tool in the general multi-scale methodology. It is further suggested that the SL hierarchical symmetry implies the existence of a turbulence ensemble. It is speculated that the search for defining the turbulence ensemble might open a new way for deriving statistical closure equations for turbulence and other multi-scale fluctuation systems.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
She Z.S., Leveque E.: Universal scaling laws in fully-developed turbulence. Phys. Rev. Lett. 72(3), 336–339 (1994)
Dubrulle B.: Intermittency in fully-developed turbulence: log-poisson statistics and generalized scale covariance. Phys. Rev. Lett. 73(7), 959–962 (1994)
She Z.S., Waymire E.C.: Quantized energy cascade and log-poisson statistics in fully-developed turbulence. Phys. Rev. Lett. 74(2), 262–265 (1995)
Kolmogorov A.N., Acad C.R.: Local structure of turbulence in an incompressible fluid for very large Reynolds numbers. Dokl. Akad. Nauk. SSSR. 30, 301–305 (1941)
She Z.S., Ren K., Lewis G.S., Swinney H.L.: Scalings and structures in turbulent Couette-Taylor flow. Phys. Rev. E 64, 016308 (2001)
Liu L., She Z.S.: Hierarchical structure description of intermittent structures of turbulence. Fluid. Dyn. Res. 33(3), 261–286 (2003)
Liu J., She Z.S., Guo H.Y., Li L., Ouyang Q.Y.: Hierarchical structure description of spatiotemporal chaos. Phys. Rev. E 70, 2036215 (2004)
Frisch U.P., Sul P.L., Nelkin M.: A simple dynamical model of intermittent fully developed turbulence. J. Fluid Mech. 87(4), 719–736 (2006)
Sinai Y.G.: Statistics of shocks in solutions of inviscid Burgers equation. Commun. Math. Phys. 148(3), 601–621 (1992)
She Z.S., Aurell E., Frisch U.: The inviscid Burgers equation with initial data of Brownian type. Commun. Math. Phys. 148(3), 623–641 (1992)
Benzi R., Ciliberto S., Tripiccione R., Baudet C., Massaioli F., Succi S.: Extended self-similarity in turbulent flows. Phys. Rev. E 48(1), 29–32 (1993)
Kadanoff L.P.: Imports and exports. J. Stat. Phys. 111(5-6), 1391–1396 (2003)
Sreenivasan K.R., Antonia R.A.: The phenomenology of small-scale turbulence. Annu. Rev. Fluid. Mech. 29, 435–472 (1997)
RuizChavarria G., Baudet C., Ciliberto S.: Hierarchy of the energy-dissipation moments in fully-developed turbulence. Phys. Rev. Lett. 74(11), 1986–1989 (1995)
Chavarria G.R., Baudet C., Benzi R., Ciliberto S.: Hierarchy of the velocity structure functions in fully-developed turbulence. J. Phys. II France 5(4), 485–490 (1995)
RuizChavarria G., Baudet C., Ciliberto S.: Scaling laws and dissipation scale of a passive scalar in fully developed turbulence. Phys. D 99(2-3), 369–380 (1996)
Camussi R., Benzi R.: Hierarchy of transverse structure functions. Phys. Fluids 9(2), 257–259 (1997)
Benzi R., Ciliberto S., Baudet C., Chavarria G.R.: On the scaling of 3-dimensional homogeneous and isotropic turbulence. Phys. D 80(4), 385–398 (1995)
Benzi R., Biferale L., Ciliberto S., Struglia M.V., Tripiccione R.: Generalized scaling in fully developed turbulence. Phys. D 96(1-4), 162–181 (1996)
Leveque E., Ruiz-Chavarria G., Baudet C., Ciliberto S.: Scaling laws for the turbulent mixing of a passive scalar in the wake of a cylinder. Phys. Fluids 11(7), 1869–1879 (1999)
Zou Z.P., Zhu Y.J., Zhou M.D., She Z.S.: Hierarchical structures in a turbulent pipe flow. Fluid. Dyn. Res. 33(5-6), 493–508 (2003)
Jiang X.Q., Gong H., Liu J.K., Zhou M.D., She Z.S.: Hierarchical structures in a turbulent free shear flow. J. Fluid Mech. 569, 259–286 (2006)
Baroud C.N., Plapp B.B., Swinney H.L., She Z.S.: Scaling in three-dimensional and quasi-two-dimensional rotating turbulent flows. Phys. Fluids 15(8), 2091–2104 (2003)
Lewis G.S., Swinney H.L.: Velocity structure functions, scaling, and transitions in high-Reynolds-number Couette-Taylor flow. Phys. Rev. E 59(5), 5457–5467 (1999)
Heslot F., Castaing B., Libchaber A.: Transitions to turbulence in helium gas. Phys. Rev. A 36(12), 5870–5873 (1987)
Sano M., Wu X.Z., Libchaber A.: Turbulence in helium-gas free-convection. Phys. Rev. A 40(11), 6421–6430 (1989)
Ching E.S.C., Kwok C.Y.: Statistics of local tperature dissipation in high Rayleigh number convection. Phys. Rev. E 62(6), R7587–R7590 (2000)
Ching E.S.C.: Intermittency of tperature field in turbulent convection. Phys. Rev. E 61(1), R33–R36 (2000)
Ching E.S.C., Leung C.K., Qiu X.L., Tong P.: Intermittency of velocity fluctuations in turbulent thermal convection. Phys. Rev. E 68, 026307 (2003)
Ching E.S.C.: Intermittency and scaling in turbulent convection. Acta. Mech. Sin. 19(5), 385–393 (2003)
Sun C., Zhou Q., Xia K.Q.: Cascades of velocity and temperature fluctuations in buoyancy-driven thermal turbulence. Phys. Rev. Lett. 97(14), 144504 (2006)
Cao N.Z., Chen S.Y., She Z.S.: Scalings and relative scalings in the Navier-Stokes turbulence. Phys. Rev. Lett. 76(20), 3711–3714 (1996)
Sreenivasan K.R., Vainshtein S.I., Bhiladvala R., SanGil I., Chen S.Y., Cao N.Z.: Asymmetry of velocity incrents in fully developed turbulence and the scaling of low-order moments. Phys. Rev. Lett. 77(8), 1488–1491 (1996)
Chen S.Y., Cao N.Z.: Inertial range scaling in turbulence. Phys. Rev. E 52(6), R5757–R5759 (1995)
Grossmann S., Lohse D., Reeh A.: Different intermittency for longitudinal and transversal turbulent fluctuations. Phys. Fluids 9(12), 3817–3825 (1997)
Fisher R.T., Kadanoff L.P., Lamb D.Q., Dubey A., Plewa T., Calder A., Cattaneo F., Constantin P., Foster I., Papka M.E., Abarzhi S.I., Asida S.M., Rich P.M., Glendening C.C., Antypas K., Sheeler D.J., Reid L.B., Gallagher B., Needham S.G.: Terascale turbulence computation using the FLASH3 application framework on the IBM Blue Gene/L syst. IBM J. Res. Dev. 52(1-2), 127–136 (2008)
Boratav O.N., Pelz R.B.: Structures and structure functions in the inertial range of turbulence. Phys. Fluids 9(5), 1400–1415 (1997)
Boratav O.N.: On recent intermittency models of turbulence. Phys. Fluids 9(5), 1206–1208 (1997)
Kida S.: 3-dimensional periodic flows with high-symmetry. J. Phys. Soc. Jpn. 54(6), 2132–2136 (1985)
Haugen N.E.L., Brandenburg A.: Inertial range scaling in numerical turbulence with hyperviscosity. Phys. Rev. E 70, 026405 (2004)
Pumir A.: A numerical study of the mixing of a passive scalar in three dimensions in the presence of a mean gradient. Phys. Fluids 6(6), 2118–2132 (1994)
Chen S.Y., Cao N.Z.: Anomalous scaling and structure instability in three-dimensional passive scalar turbulence. Phys. Rev. Lett. 78(18), 3459–3462 (1997)
Watanabe T., Gotoh T.: Statistics of a passive scalar in homogeneous turbulence. New J. Phys. 6(1), 40 (2004)
Cao N.Z., Chen S.Y.: An intermittency model for passive-scalar turbulence. Phys. Fluids 9(5), 1203–1205 (1997)
He G.W., Chen S.Y., Doolen G.: Hierarchy of structure functions for passive scalars advected by turbulent flows. Phys. Lett. A 246(1-2), 135–138 (1998)
Iroshnikov P.: Turbulence of a conducting fluid in a strong magnetic field. Astron. Zh. 40, 742 (1963)
Kraichnan R.: Inertial-range spectrum of hydromagnetic turbulence. Phys. Fluids 8, 1385 (1965)
Sridhar H., Goldreich P.: Toward a theory of interstellar turbulence I. Weak Alfvenic turbulence. Astrophy. J. 432, 612 (1994)
Goldreich P., Sridhar H.: Toward a theory of interstellar turbulence. 2: Strong alfvenic turbulence. Astrophy. J. 438, 763 (1995)
Grauer R., Krug J., Marliani C.: Scaling of high-order structure functions in magnetohydrodynamic turbulence. Phys. Lett. A 195(5-6), 335–338 (1994)
Burlaga L.F.: Intermittent turbulence in the solar wind. J. Geophys. Res. 96(A4), 5847–5851 (1991)
Politano H., Pouquet A.: Model of intermittency in magnetohydrodynamic turbulence. Phys. Rev. E 52(1), 636–641 (1995)
Müller W.C., Biskamp D.: Scaling properties of three-dimensional magnetohydrodynamic turbulence. Phys. Rev. Lett. 84(3), 475–478 (2000)
Biskamp D., Müller W.C.: Scaling properties of three-dimensional isotropic magnetohydrodynamic turbulence. Phys. Plasmas 7(12), 4889–4900 (2000)
Müller, W.C., Biskamp, D.: The evolving phenomenological view on magnetohydrodynamic turbulence. In: Physics Lecture Notes-New York Then Berlin, vol. 614, pp. 3–27 (2003)
Müller W.C., Biskamp D., Grappin R.: Statistical anisotropy of magnetohydrodynamic turbulence. Phys. Rev. E 67(6), 0306045 (2003)
Merrifield J.A, Müller W.C., Chapman S.C., Dendy R.O.: The scaling properties of dissipation in incompressible isotropic three-dimensional magnetohydrodynamic turbulence. Phys. Plasmas 12(2), 4889–4900 (2005)
Cho J.Y., Lazarian A., Vishniac E.T.: Simulations of magnetohydrodynamic turbulence in a strongly magnetized medium. Astrophys. J. 564(1), 291–301 (2002)
Cho J.Y., Lazarian A.: Compressible magnetohydrodynamic turbulence: mode coupling, scaling relations, anisotropy, viscosity- damped regime and astrophysical implications. Monthly Not. Roy. Astron. Soc. 345(1), 325–339 (2003)
Lazarian A., Cho J.Y.: Scaling, intermittency and decay of MHD turbulence. Phys. Scripta T 116, 32–37 (2005)
Cho J.Y., Lazarian A., Vishniac E.T.: Ordinary and viscosity-damped magnetohydrodynamic turbulence. Astrophys. J. 595(2), 812–823 (2003)
Gomez T., Politano H., Pouquet A.: On the validity of a nonlocal approach for MHD turbulence. Phys. Fluids 11(8), 2298–2306 (1999)
Kowal G., Lazarian A.: Scaling relations of compressible MHD turbulence. Astrophys. J. 666(2), L69–L72 (2007)
Fang L.Z.: Intermittency of cosmic baryon fluid. AIP Conf. Proc. 1059, 129–140 (2008)
He P., Liu J.R., Feng L.L., Shu C.W., Fang L.Z.: Low-redshift cosmic baryon fluid on large scales and She-Leveque universal scaling. Phys. Rev. Lett. 96(5), 051302 (2006)
Liu J.R., Fang L.Z.: Non-Gaussianity of the cosmic baryon fluid: Log-Poisson hierarchy model. Astrophys. J. 672(1), 11–18 (2008)
Lu Y., Chu Y.Q., Fang L.Z.: Log-poisson hierarchical clustering of cosmic neutral hydrogen and ly alpha transmitted flux of qso absorption spectrum. Astrophys. J. 691(1), 43–53 (2009)
Boldyrev S.: Kolmogorov-Burgers model for star-forming turbulence. Astrophys. J. 569(2), 841–845 (2002)
Padoan P., Boldyrev S., Langer W., Nordlund A.: Structure function scaling in the Taurus and Perseus molecular cloud complexes. Astrophys. J. 583(1), 308–313 (2003)
Gustafsson M., Brandenburg A., Laire J.L., Field D.: The nature of turbulence in OMC1 at the scale of star formation: observations and simulations. Astron. Astrophys. 454(3), 815–825 (2006)
Avillez De.M.A., Breitschwerdt D.: The generation and dissipation of interstellar turbulence: results from large-scale high- resolution simulations. Astrophys. J. 665(1), L35–L38 (2007)
Uritsky V.M., Paczuski M., Davila J.M., Jones S.I.: Coexistence of self-organized criticality and intermittent turbulence in the solar corona. Phys. Rev. Lett. 99, 025001 (2007)
Kissmann R., Kleimann J., Fichtner H., Grauer R.: Local turbulence simulations for the multiphase ISM. Monthly Not. Roy. Astron. Soc. 391(4), 1577–1588 (2008)
Boldyrev S., Nordlund A., Padoan P.: Scaling relations of supersonic turbulence in star-forming molecular clouds. Astrophys. J. 573(2), 678–684 (2002)
Padoan P., Jimenez R., Nordlund A., Boldyrev S.: Structure function scaling in compressible super-Alfvénic MHD turbulence. Phys. Rev. Lett. 92(19), 191102–191102 (2004)
Pan L., Padoan P., Kritsuk A.G.: Dissipative structures in supersonic turbulence. Phys. Rev. Lett. 102(3), 034501 (2009)
Kritsuk A.G., Norman M.L.: Scaling relations for turbulence in the multiphase interstellar medium. Astrophys. J. 601(1), L55–L58 (2004)
Efrov Y.N., Elmegreen B.G., Hodge P.W.: Giant shells and stellar arcs as relics of gamma-ray burst explosions. Astrophys. J. Lett. 501(2), 163–165 (1998)
Frisch U.: Turbulence. Cambridge University Press, Cambridge (1995)
Frisch U., Sulem P.L., Nelkin M.: A simple dynamical model of intermittent fully developed turbulence. J. Fluid Mech. 87(4), 719–736 (1978)
Meneveau C., Sreenivasan K.R.: Simple multifractal cascade model for fully developed turbulence. J. Fluid Mech. Phys. Rev. Lett. 59(4), 1424–1427 (1974)
Benzi R., Paladin G., Parisi G., Vulpiani A.: On the multifractal nature of fully developed turbulence and chaotic systems. J. Phys. A 17, 3521–3531 (1984)
Guo H.Y., Li L., Qi Q.Y., Liu J., She Z.S.: A systatic study of spirals and spiral turbulence in a reaction-diffusion syst. J. Chem. Phys. 118(11), 5038–5044 (2003)
Queiros-Conde D.: Geometrical extended self-similarity and intermittency in diffusion-limited aggregates. Phys. Rev. Lett. 78(23), 4426–4429 (1997)
Turiel A., Mato G., Parga N., Nadal J.P.: Self-similarity properties of natural images resble those of turbulent flows. Phys. Rev. Lett. 80(5), 1098–1101 (1998)
Turiel A., Parga N., Ruderman D.L., Cronin T.W.: Multiscaling and information content of natural color images. Phys. Rev. E 62(1), 1138–1148 (2000)
Wang J., Zhang Q.D., Ren K., She Z.S.: Multi-scaling hierarchical structure analysis on the sequence of E-coli complete genome. Chin. Sci. Bull. 46(23), 1988–1992 (2001)
Ouyang Z.Q., Wang C., She Z.S.: Scaling and hierarchical structures in DNA sequences. Phys. Rev. Lett. 93(7), 078103 (2004)
Ouyang Z.Q., Liu J.K., She Z.S.: Hierarchical structure analysis describing abnormal base composition of genomes. Phys. Rev. E 72(4), 041915 (2005)
She Z.S., Fu Z.T., Chen J., Liang S., Liu S.D.: Hierarchical structures in climate and atmospheric turbulence. Pro. Nat. Sci. 12(10), 747–752 (2002)
She Z.S., Liu L.: Measuring intermittency parameters of energy cascade in turbulence. Acta Mech. Sin. 19(5), 453–457 (2003)
Zhou L.Q., Ouyang Q.: Spiral instabilities in a reaction- diffusion syst. J. Phys. Chem. A 105(1), 112–118 (2001)
Winfree A.T.: Electrical turbulence in three-dimensional heart muscle. Science 266(5187), 1003–1006 (1994)
Witkowski F.X., Leon L.J., enkoske P.A., Giles W.R., Spano M.L., Ditto W.L., Winfree A.T.: Spatiotporal evolution of ventricular fibrillation. Nature(London) 392(6671), 78–82 (1998)
Turiel A., Parga N.: The multifractal structure of contrast changes in natural images: From sharp edges to textures. Neural Comput. 12(4), 763–793 (2000)
Grazzini J., Turiel A., Yahia H., Herlin I.: A multifractal approach for extracting relevant textural areas in satellite meteorological images. Environ. Model. Softw. 22(3), 323–334 (2007)
Ivanov P.C., Amaral L.A.N., Goldberger A.L., Havlin S., Rosenblum M.G., Struzik Z., Stanley H.E.: Multifractality in human heartbeat dynamics. Nature 399, 461–465 (1999)
Ching E.S.C., Lin D.C., Zhang C.: Hierarchical structure in healthy and diseased human heart rate variability. Phys. Rev. E 69, 051919 (2004)
Biskamp D., Hallatschek K., Schwarz E.: Scaling laws in two-dimensional turbulent convection. Phys. Rev. E 63, 045302 (2001)
Haugen N.E.L., Brandenburg A., Dobler W.: Simulations of nonhelical hydromagnetic turbulence. Phys. Rev. E 70, 016308 (2004)
Cao Y.H., Chen J., She Z.S.: The nature of near-wall convection velocity in turbulent channel flow. Acta Mech. Sin. 24, 587–590 (2008)
Acknowledgements
The authors wish to thank S.Y. Chen, C.B. Lee, J. Chen, J.J. Tao, Y.P. Shi, X.T. Shi, N. Hu, Y. Wu, X. Chen, J. Pei, K. Ding, Y.Z. Wang for helpful discussions.
Open Access
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
The project supported by the National Natural Science Foundation (90716008) and by MOST 973 project (2009CB724100).
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
She, ZS., Zhang, ZX. Universal hierarchical symmetry for turbulence and general multi-scale fluctuation systems. Acta Mech Sin 25, 279–294 (2009). https://doi.org/10.1007/s10409-009-0257-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10409-009-0257-3