Scaling laws for turbulent relative dispersion in two-dimensional energy inverse-cascade turbulence

[1] ABRAMOWITZ, MILTON & STEGUN, IRENE A. 1964 Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, ninth dover printing, tenth gpo printing edn. New York: Dover.

[2] ALBERS, TONY & RADONS, GU¨NTER 2018 Exact results for the nonergodicity of d-dimensional generalized L´evy walks. Physical Review Letters 120 (10).

[3] ARTALE, VINCENZO, BOFFETTA, GUIDO, CELANI, ANTONIO, CENCINI, MAS- SIMO & VULPIANI, ANGELO 1997 Dispersion of passive tracers in closed basins: Beyond the diffusion coefficient. Physics of Fluids 9 (11), 3162–3171.

[4] BALKOVSKY, EUGENE & LEBEDEV, V 1998 Instanton for the Kraichnan passive scalar problem. Physical Review E 58 (5), 5776–5795.

[5] BARENBLATT, GRIGORY ISAAKOVICH 1996 Scaling, Self-similarity, and Inter-mediate Asymptotics: Dimensional Analysis and Intermediate Asymptotics. Cam- bridge Texts in Applied Mathematics . Cambridge University Press.

[6] BARENBLATT, GRIGORY ISAAKOVICH 2003 Scaling . Cambridge Texts in Applied Mathematics . Cambridge University Press.

[7] BARENBLATT, GRIGORY ISAAKOVICH 2014 Flow, Deformation and Fracture: Lectures on Fluid Mechanics and the Mechanics of Deformable Solids for Mathe- maticians and Physicists. Cambridge Texts in Applied Mathematics . Cambridge University Press.

[8] BATCHELOR, GEORGE KEITH 1950 The application of the similarity theory of turbulence to atmospheric diffusion. Quarterly Journal of the Royal Meteorological Society 76 (328), 133–146.

[9] BATCHELOR, GEORGE KEITH 1952 Diffusion in a field of homogeneous tur-bulence: II. The relative motion of particles. Mathematical Proceedings of the Cambridge Philosophical Society 48 (2), 345–362.

[10] BATCHELOR, GEORGE KEITH 1953 The theory of homogeneous turbulence. Cam- bridge University Press. BIBLIOGRAPHY

[11] BATCHELOR, GEORGE KEITH 1969 Computation of the energy spectrum in ho- mogeneous two-dimensional turbulence. Physics of Fluids 12 (12), II–233.

[12] BENNETT, Press. ANDREW 2006 Lagrangian Fluid Dynamics. Cambridge University

[13] BERG, JACOB, LU¨THI, BEAT, MANN, JAKOB & OTT, SøREN 2006 Backwards and forwards relative dispersion in turbulent flow: An experimental investigation. Physical Review E 74 (1), 016304.

[14] BERNARD, DENIS 1999 Three-point velocity correlation functions in two-dimensional forced turbulence. Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics 60 (5 Pt B), 6184–7.

[15] BIFERALE, LUCA, BOFFETTA, GUIDO, CELANI, ANTONIO, DEVENISH, BEN-JAMIN, LANOTTE, ALESSANDRA S. & TOSCHI, FEDERICO 2005 Lagrangian statistics of particle pairs in homogeneous isotropic turbulence. Physics of Fluids 17 (11), 1–9.

[16] BIFERALE, LUCA, LANOTTE, ALESSANDRA S., SCATAMACCHIA, R & TOSCHI, FEDERICO 2014 Intermittency in the relative separations of tracers and of heavy particles in turbulent flows. Journal of Fluid Mechanics 757 (6), 550–572.

[17] BIRKHOFF, GARRETT 1960 Hydrodynamics. Princeton University Press.

[18] BITANE, REHAB, HOMANN, HOLGER & BEC, JE´RE´MIE 2012 Time scales of turbulent relative dispersion. Physical Review E 86 (4), 045302.

[19] BITANE, REHAB, HOMANN, HOLGER & BEC, JE´RE´MIE 2013 Geometry and violent events in turbulent pair dispersion. Journal of Turbulence 14 (2), 23–45.

[20] BOFFETTA, GUIDO 2007 Energy and enstrophy fluxes in the double cascade of two-dimensional turbulence. Journal of Fluid Mechanics 589, 253–260.

[21] BOFFETTA, GUIDO, CELANI, ANTONIO, CRISANTI, ANDREA & VULPIANI, ANGELO 1999 Pair dispersion in synthetic fully developed turbulence. Euro-physics Letters 60 (6), 6734–6741.

[22] BOFFETTA, GUIDO., CELANI, ANTONIO & VERGASSOLA, MASSIMO 2000 In-verse energy cascade in two-dimensional turbulence: Deviations from Gaussian behavior. Physical Review E 61 (1), R29–R32.

[23] BOFFETTA, GUIDO & ECKE, ROBERT E. 2012 Two-dimensional turbulence. Annual Review of Fluid Mechanics 44 (1), 427–451.

[24] BOFFETTA, GUIDO. & MUSACCHIO, STEFANO 2010 Evidence for the double cas- cade scenario in two-dimensional turbulence. Physical Review E 82 (1), 016307.

[25] BOFFETTA, GUIDO & SOKOLOV, IGOR M. 2002 Relative dispersion in fully de- veloped turbulence: the Richardson’s law and intermittency corrections. Physical Review Letters 88 (9), 094501.

[26] BOFFETTA, GUIDO & SOKOLOV, IGOR M. 2002 Statistics of two-particle dis-persion in two-dimensional turbulence. Physics of Fluids 14 (9), 3224–3232.

[27] BOTHE, MARIUS, SAGUES, FRANCESC & SOKOLOV, IGOR M. 2019 Mean squared displacement in a generalized L´evy walk model. Physical Review E 100 (1), 012117.

[28] BOURGOIN, MICKAE¨L 2006 The role of pair dispersion in turbulent flow. Science 311 (5762), 835–838.

[29] BOURGOIN, MICKAE¨L 2015 Turbulent pair dispersion as a ballistic cascade phe- nomenology. Journal of Fluid Mechanics 772, 678–704.

[30] BRAGG, ANDREW D., IRELAND, PETER J. & COLLINS, LANCE R. 2016 For- ward and backward in time dispersion of fluid and inertial particles in isotropic turbulence. Physics of Fluids 28 (1).

[31] BRUNEAU, CHARLES-HENRI & KELLAY, HAMID 2005 Experiments and direct numerical simulations of two-dimensional turbulence. Physical Review E 71 (4), 046305.

[32] BUARIA, DHAWAL, SAWFORD, BRIAN L. & YEUNG, PUI-KUEN 2015 Charac- teristics of backward and forward two-particle relative dispersion in turbulence at different Reynolds numbers. Physics of Fluids 27 (10), 105101.

[33] BUARIA, DHAWAL, YEUNG, PUI-KUEN & SAWFORD, BRIAN L. 2016 A La-grangian study of turbulent mixing: forward and backward dispersion of molecu- lar trajectories in isotropic turbulence. Journal of Fluid Mechanics 799, 352–382.

[34] CARDY, JOHN, FALKOVICH, GREGORY & GAWEDZKI, KRZYSZTOF 2008 Non- equilibrium Statistical Mechanics and Turbulence. London Mathematical Society Lecture Note Series . Cambridge University Press.

[35] CASTIGLIONE, P, MAZZINO, ANDREA, MURATORE-GINANNESCHI, P & VULPIANI, ANGELO 1999 On strong anomalous diffusion. Physica D 134, 75–93.

[36] CHAVES, MARTA, GAWDZKI, KRZYSZTOF, HORVAI, PETER, KUPIAINEN, ANTTI & VERGASSOLA, MASSIMO 2003 Lagrangian dispersion in Gaussian self- similar velocity ensembles. Journal of Statistical Physics 1136 (5).

[37] CHEN, LIN-YUAN, GOLDENFELD, NIGEL & OONO, YOSHI 1991 Renormalization-group theory for the modified porous-medium equation. Physical Review A 44 (10), 6544–6550.

[38] CHEN, SHIYI, ECKE, ROBERT E., EYINK, GREGORY L., RIVERA, MICHAEL K., WAN, MINPING & XIAO, ZUOLI 2006 Physical mechanism of the two-dimensional inverse energy cascade. Physical Review Letters 96 (8), 084502.

[39] DANILOV, SERGEI D & GURARIE, DAVID 2001 Nonuniversal features of forced two-dimensional turbulence in the energy range. Physical Review E 63 (2), 020203.

[40] EULE, SVEN, ZABURDAEV, VASILY, FRIEDRICH, R & GEISEL, THEO 2012 Langevin description of superdiffusive L´evy processes. Physical Review E - Sta- tistical, Nonlinear, and Soft Matter Physics 86 (4), 41134.

[41] EYINK, GREGORY L. & BENVENISTE, DAMIEN 2013 Diffusion approximation in turbulent two-particle dispersion. Physical Review E 88 (4), 041001.

[42] FALKOVICH, GREGORY, GAWEDZKI, KRZYSZTOF & VERGASSOLA, MASSIMO 2001 Particles and fields in fluid turbulence. Reviews of Modern Physics 73 (4), 913–975.

[43] FJøRTOFT, RAGNAR 1953 On the changes in the spectral distribution of kinetic energy for twodimensional, nondivergent flow. Tellus 5 (3), 225–230.

[44] FRANCOIS, NICOLAS, XIA, HUA, PUNZMANN, HORST & SHATS, MICHAEL 2013 Inverse energy cascade and emergence of large coherent vortices in turbu- lence driven by faraday waves. Physical Review Letters 110 (19), 194501.

[45] FRISCH, URIEL 1995 Turbulence: The Legacy of AN Kolmogorov . Cambridge University Press.

[46] FRISCH, URIEL & SULEM, PIERRE-LOUIS 1984 Numerical simulation of the inverse cascade in two-dimensional turbulence. Physics of Fluids 27 (8), 1921.

[47] GARDINER, CRISPIN 2009 Stochastic Methods A Handbook for the Natural and Social Sciences, 4th edn. Springer.

[48] GOLDENFELD, NIGEL 1992 Lectures on Phase Transitions and the Renormaliza- tion Group. Westview Press.

[49] GOTOH, TOSHIYUKI & KANEDA, YUKIO 1991 Lagrangian velocity autocorre-lation and eddy viscosity in two-dimensional anisotropic turbulence. Physics of Fluids A: Fluid Dynamics 3 (10), 2426.

[50] GOTOH, TOSHIYUKI, ROGALLO, ROBERT S., HERRING, JACKSON R. & KRAICHNAN, ROBERT H. 1993 Lagrangian velocity correlations in homogeneous isotropic turbulence. Physics of Fluids A: Fluid Dynamics 5 (11), 2846–2864.

[51] HE, GUOWEI, JIN, GUODONG & YANG, YUE 2017 Space-time correlations and dynamic coupling in turbulent flows. Annual Review of Fluid Mechanics 49 (1), 51–70.

[52] HE, GUOWEI, JIN, GUODONG & ZHAO, XIN 2009 Scale-similarity model for Lagrangian velocity correlations in isotropic and stationary turbulence. Physical Review E 80 (6), 066313.

[53] HERRING, JACKSON R., ORSZAG, STEVEN A., KRAICHNAN, ROBERT H. & FOX, DOUGLAS G. 1974 Decay of two-dimensional homogeneous turbulence. Journal of Fluid Mechanics 66 (3), 417–444.

[54] ISHIHARA, TAKASHI, GOTOH, TOSHIYUKI & KANEDA, YUKIO 2009 Study of high–Reynolds number isotropic turbulence by direct numerical simulation. An- nual Review of Fluid Mechanics 41 (1), 165–180.

[55] ISHIHARA, TAKASHI & KANEDA, YUKIO 2002 Relative diffusion of a pair of fluid particles in the inertial subrange of turbulence. Physics of Fluids 14 (11), L69–L72.

[56] JULLIEN, MARIE-CAROLINE, PARET, JE´ROˆME & TABELING, PATRICK 1999 Richardson pair dispersion in two-dimensional turbulence. Physical Review Let- ters 82, 2872.

[57] KANATANI, KENTARO, OGASAWARA, TAKESHI & TOH, SADAYOSHI 2009 Telegraph-type versus diffusion-type models of turbulent relative dispersion. Jour- nal of the Physical Society of Japan 78 (2), 024401.

[58] KANEDA, YUKIO 1981 Renormalized expansions in the theory of turbulence with the use of the Lagrangian position function. Journal of Fluid Mechanics 107 (-1), 131.

[59] KANEDA, YUKIO, GOTOH, KOJI & ISHIHARA, TAKASHI 1998 Taylor Expan-sions and Pad´e Approximations of Lagrangian and Eulerian Two-Time Velocity Correlations in Turbulence. Journal of the Physical Society of Japan 67 (4), 1075– 1078.

[60] KANEDA, YUKIO & GOTOH, TOSHIYUKI 1991 Lagrangian velocity autocorre-lation in isotropic turbulence. Physics of Fluids A: Fluid Dynamics 3 (8), 1924– 1933.

[61] KANEDA, YUKIO, ISHIHARA, TAKASHI & GOTOH, KOJI 1999 Taylor expan-sions in powers of time of Lagrangian and Eulerian two-point two-time velocity correlations in turbulence. Physics of Fluids 11 (8), 2154–2166.

[62] DE KA´RMA´N, THEODORE & HOWARTH, LESLIE 1938 On the statistical theory of isotropic turbulence. Proceedings of the Royal Society of London. Series A - Mathematical and Physical Sciences 164 (917), 192–215.

[63] KELLAY, HAMID & GOLDBURG, WALTER I 2002 Two-dimensional turbulence: A review of some recent experiments. Rep. Prog. Phys. 65, 845–894.

[64] KIDA, SHIGEO & GOTO, SUSUMU 1997 A Lagrangian direct-interaction ap-proximation for homogeneous isotropic turbulence. Journal of Fluid Mechanics 345 (September 2000), 307–345.

[65] KISHI, TATSURO, MATSUMOTO, TAKESHI & TOH, SADAYOSHI 2020 Non-Kolmogorov scaling for two-particle relative velocity in two-dimensional inverse energy-cascade turbulence. Physical Review Fluids 5 (5), 054601.

[66] KISTLER, A. L. & VREBALOVICH, T. 1966 Grid turbulence at large Reynolds numbers. Journal of Fluid Mechanics 26 (01), 37.

[67] KLAFTER, JOSEPH, SHLESINGER, MICHAEL F. & ZUMOFEN, GERT 1996 Be-yond Brownian motion. Physics Today 49 (2), 33–39.

[68] KLAFTER, JOSEPH. & SOKOLOV, IGOR M. 2011 First Steps in Random Walks. Oxford University Press.

[69] KOLMOGOROV, ANDREY N. 1941 Dissipation of energy in the locally isotropic turbulence. Dokl. Akad. Nauk SSSR A 32, 16–18.

[70] KOLMOGOROV, ANDREY N. 1941 The local structure of turbulence in incom-pressible viscous fluid for very large reynolds numbers. Dokl. Akad. Nauk SSSR 30, 301–305.

[71] KRAICHNAN, ROBERT H 1959 The structure of isotropic turbulence at very high Reynolds numbers. Journal of Fluid Mechanics 5 (04), 497.

[72] KRAICHNAN, ROBERT H 1964 Kolmogorov’s hypotheses and eulerian turbulence theory. Physics of Fluids 7 (11), 1723.

[73] KRAICHNAN, ROBERT H 1965 Lagrangian-history closure approximation for tur- bulence. Physics of Fluids 8 (4), 575.

[74] KRAICHNAN, ROBERT H 1966 Dispersion of particle pairs in homogeneous tur- bulence. Physics of Fluids 9 (10), 1937.

[75] KRAICHNAN, ROBERT H 1967 Inertial ranges in two-dimensional turbulence. Physics of Fluids 10 (7), 1417.

[76] KRAICHNAN, ROBERT H 1968 Small-scale structure of a scalar field convected by turbulence. Physics of Fluids 11 (5), 945–953.

[77] LANDAU, LEV D. & LIFSHITZ, EVGENY M. 1987 Fluid Mechanics, Second Edition: Volume 6 (Course of Theoretical Physics), 2nd edn. Course of theoretical physics / by L. D. Landau and E. M. Lifshitz, Vol. 6 . Butterworth-Heinemann.

[78] LEITH, C E 1968 Diffusion approximation for two-dimensional turbulence. Physics of Fluids 11 (3), 671.

[79] LESIEUR, MARCEL. 1987 Turbulence in Fluids, Mechanics of Fluids and Trans- port Processes, vol. 6. Dordrecht: Springer Netherlands.

[80] LINDBORG, ERIK 1999 Can the atmospheric kinetic energy spectrum be explained by two-dimensional turbulence? Journal of Fluid Mechanics 388, 259–288.

[81] MAGDZIARZ, MARCIN, SZCZOTKA, W-LADYSL- AW & ZEBROWSKI, PIOTR 2012 Langevin Picture of L´evy Walks and Their Extensions. Journal of Statistical Physics 147 (1), 74–96.

[82] MAGDZIARZ, MARCIN & ZORAWIK, TOMASZ 2017 Aging ballistic L´evy walks. Physical Review E 95 (2), 022126.

[83] MANDELBROT, BENOIT B. & VAN NESS, JOHN W. 1968 Fractional Brownian motions, fractional noises and applications. SIAM Review 10 (4), 422–437.

[84] METZLER, RALF & KLAFTER, JOSEPH 2000 The random walk’s guide to anoma- lous diffusion. Physics Reports 339 (339), 1–77.

[85] MIZUTA, ATSUSHI, MATSUMOTO, TAKESHI & TOH, SADAYOSHI 2013 Tran-sition of the scaling law in inverse energy cascade range caused by a nonlocal excitation of coherent structures observed in two-dimensional turbulent fields. Physical Review E 88 (5), 53009.

[86] MONIN, ANDREI S. & YAGLOM, AKIVA M. 1971 Statistical Fluid Mechanics, , vol. 1. MIT Press.

[87] OBUKHOV, ALEXANDER M. 1941 On the distribution of energy in the spectrum of turbulent flow. Izv. Akad. Nauk SSSR, Ser. Geogr. Geofi 5, 453–466.

[88] OGASAWARA, TAKESHI & TOH, SADAYOSHI 2006 Model of turbulent relative dispersion: a self-similar telegraph equation. Journal of the Physical Society of Japan 75 (8), 083401.

[89] OGASAWARA, TAKESHI & TOH, SADAYOSHI 2006 Turbulent relative dispersion in two-dimensional free convection turbulence. Journal of the Physical Society of Japan 75 (10), 104402.

[90] OTT, SøREN & MANN, JAKOB 2000 An experimental investigation of the rela- tive diffusion of particle pairs in three-dimensional turbulent flow. J. Fluid Mech 422, 207–223.

[91] OUELLETTE, NICHOLAS T., XU, HAITAO, BOURGOIN, MICKAE¨L & BODEN-SCHATZ, EBERHARD 2006 An experimental study of turbulent relative dispersion models. New Journal of Physics 8 (6), 109–109.

[92] PARET & TABELING 1998 Intermittency in the two-dimensional inverse cascade of energy: Experimental observations. Physics of Fluids 10 (12), 3126.

[93] PARET, JE´ROˆME, JULLIEN, MARIE-CAROLINE & TABELING, PATRICK 1999 Vorticity Statistics in the two-dimensional enstrophy cascade. Physical Review Letters 83 (17), 3418–3421.

[94] PARET, JE´ROˆME & TABELING, PATRICK 1997 Experimental observation of the two-dimensional inverse energy cascade. Physical Review Letters 79 (21), 4162– 4165.

[95] PORRA`, JOSEP M, WANG, KE GANG & MASOLIVER, JAUME 1996 Generalized Langevin equations: Anomalous diffusion and probability distributions. Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics 53 (6), 5872–5881.

[96] REBENSHTOK, ADI, DENISOV, SERGEY, HA¨NGGI, PETER & BARKAI, ELI 2014 Infinite densities for L´evy walks. Physical Review E - Statistical, Nonlinear, and Soft Matter Physics 90 (6), 062135.

[97] REBENSHTOK, ADI, DENISOV, SERGEY, HA¨NGGI, PETER & BARKAI, ELI 2014 Non-normalizable densities in strong anomalous diffusion: Beyond the central limit theorem. Physical Review Letters 112 (11), 110601.

[98] RICHARDSON, LEWIS F 1926 Atmospheric diffusion shown on a distance-neighbour graph. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 110 (756), 709–737.

[99] RIVERA, MICHAEL K. & ECKE, ROBERT E 2005 Pair dispersion and doubling time statistics in two-dimensional turbulence. Physical Review Letters 95 (19), 194503.

[100] RIVERA, MICHAEL K. & ECKE, ROBERT E. 2016 Lagrangian statistics in weakly forced two-dimensional turbulence. Chaos: An Interdisciplinary Journal of Nonlinear Science 26 (1), 013103.

[101] RIVERA, MICHAEL K., VOROBIEFF, PETER & ECKE, ROBERT E 1998 Tur-bulence in flowing soap films: velocity, vorticity, and thickness fields. Physical Review Letters 81 (7), 1417–1420.

[102] RUTGERS, MAARTEN A. 1998 Forced 2D turbulence: experimental evidence of simultaneous inverse energy and forward enstrophy cascades. Physical Review Letters 81 (11), 2244–2247.

[103] SALAZAR, JUAN P.L.C. & COLLINS, LANCE R 2009 Two-particle dispersion in isotropic turbulent flows. Annual Review of Fluid Mechanics 41 (1), 405–432.

[104] SAWFORD, BRIAN L., YEUNG, PUI-KUEN & HACKL, JASON F. 2008 Reynolds number dependence of relative dispersion statistics in isotropic turbulence. Physics of Fluids 20 (6), 065111.

[105] SCATAMACCHIA, R., BIFERALE, LUCA & TOSCHI, FEDERICO 2012 Extreme events in the dispersions of two neighboring particles under the influence of fluid turbulence. Physical Review Letters 109 (14), 144501.

[106] SCHULZ, JOHANNES H.P., BARKAI, ELI & METZLER, RALF 2014 Aging re-newal theory and application to random walks. Physical Review X 4 (1), 011028.

[107] SHLESINGER, MICHAEL F., ZASLAVSKY, GEORGE M. & KLAFTER, JOSEPH 1993 Strange kinetics. Nature 363 (6424), 31–37.

[108] SREENIVASAN, KATEPALLI R & ANTONIA, ROBERT A 1997 The phenomenol- ogy of small-scale turbulence. Annual Review of Fluid Mechanics 29 (1), 435–472.

[109] TABELING, PATRICK 2002 Two-dimensional turbulence: A physicist approach. Physics Report 362 (1), 1–62.

[110] TAYLOR, GEOFFREY INGRAM 1922 Diffusion by continuous movements. Pro-ceedings of the London Mathematical Society s2-20 (1), 196–212.

[111] TENNEKES, HENK 1975 Eulerian and Lagrangian time microscales in isotropic turbulence. Journal of Fluid Mechanics 67 (3), 561–567.

[112] TENNEKES, HENK & LUMLEY, JOHN L. 1972 A first course in turbulence. MIT Press.

[113] THALABARD, SIMON, KRSTULOVIC, GIORGIO & BEC, JE´RE´MIE 2014 Turbulent pair dispersion as a continuous-time random walk. Journal of Fluid Mechanics 755, R4.

[114] TRAN, CHUONG V. & BOWMAN, JOHN C. 2004 Robustness of the inverse cascade in two-dimensional turbulence. Physical Review E 69 (3), 036303.

[115] TRAN, CHUONG V. & SHEPHERD, THEODORE G. 2002 Constraints on the spec- tral distribution of energy and enstrophy dissipation in forced two-dimensional turbulence. Physica D: Nonlinear Phenomena 165 (3-4), 199–212.

[116] VALLGREN, ANDREAS 2011 Infrared reynolds number dependency of the two-dimensional inverse energy cascade. Journal of Fluid Mechanics 667, 463–473.

[117] VEZZANI, ALESSANDRO, BARKAI, ELI & BURIONI, RAFFAELLA 2019 Single-big-jump principle in physical modeling. Physical Review E 100 (1), 012108.

[118] VEZZANI, ALESSANDRO, BARKAI, ELI & BURIONI, RAFFAELLA 2020 Rare events in generalized L´evy Walks and the Big Jump principle. SUPPLEMEN- TARY MATERIALS. Scientific Reports 10 (1), 2732.

[119] VON KAMEKE, ALEXANDRA, HUHN, FLORIAN, FERNA´NDEZ-GARC´IA, G, MUN˜UZURI, ALBERTO P. & PE´REZ-MUN˜UZURI, VICENTE 2011 Double cas-cade turbulence and richardson dispersion in a horizontal fluid flow induced by Faraday waves. Physical Review Letters 107 (7), 074502.

[120] WALLACE, JAMES M. 2014 Space-time correlations in turbulent flow: A review. Theoretical and Applied Mechanics Letters 4 (2), 022003.

[121] WANG, XUDONG, CHEN, YAO & DENG, WEIHUA 2019 L´evy-walk-like Langevin dynamics. New Journal of Physics 21 (1), 013024.

[122] XIAO, Z., WAN, MINPING, CHEN, SHIYI & EYINK, GREGORY L. 2009 Phys- ical mechanism of the inverse energy cascade of two-dimensional turbulence: a numerical investigation. Journal of Fluid Mechanics 619, 1–44.

[123] YAGLOM, AKIVA M. 1990 Alexander Mikhailovich Obukhov, 1918-1989. Boundary-Layer Meteorology 53 (1-2), v–xi.

[124] YAKHOT, VICTOR 1999 Two-dimensional turbulence in the inverse cascade range. Physical Review E 60 (5), 5544–5551.

[125] YEUNG, PUI-KUEN 1994 Direct numerical simulation of two-particle relative diffusion in isotropic turbulence. Physics of Fluids 6 (10), 3416–3428.

[126] YEUNG, PUI-KUEN & BORGAS, MICHAEL S. 2004 Relative dispersion in isotropic turbulence. Part 1. Direct numerical simulations and Reynolds-number dependence. Journal of Fluid Mechanics 503, 93–124.

[127] ZABURDAEV, VASILY, DENISOV, SERGIY & KLAFTER, JOSEPH 2015 L´evy walks. Reviews of Modern Physics 87 (2), 483–530.

[128] 木田, 重雄 & 柳瀬, 眞一郎 1999 乱流力学. 朝倉書店, japanese script.