Criticality and the fractal structure of −5/3 turbulent cascades

Here we show a procedure to generate an analytical structure producing a cascade that scales as the energy spectrum in isotropic homogeneous turbulence. We obtain a function that unveils a non-self-similar fractal at the origin of the cascade. It reveals that the backbone underlying cascades is form...

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Autores:: Cabrera, Juan Luis
Gutiérrez, Esther
Rodríguez Márquez, Miguel

Tipo de recurso:: Article of journal

Fecha de publicación:: 2021

Institución:: Corporación Universidad de la Costa

Repositorio:: REDICUC - Repositorio CUC

Idioma:: eng

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dc.title.spa.fl_str_mv	Criticality and the fractal structure of −5/3 turbulent cascades
dc.title.translated.spa.fl_str_mv	Criticidad y estructura fractal de −5/3 cascadas turbulentas
title	Criticality and the fractal structure of −5/3 turbulent cascades
spellingShingle	Criticality and the fractal structure of −5/3 turbulent cascades Cascade Criticality Fractals Navier-stokes equation Nonlinear Stochastic Turbulence Maps Complex Cascada Criticidad Fractales Ecuación de navier-stokes No lineal Estocástico Turbulencia Mapas Complejo
title_short	Criticality and the fractal structure of −5/3 turbulent cascades
title_full	Criticality and the fractal structure of −5/3 turbulent cascades
title_fullStr	Criticality and the fractal structure of −5/3 turbulent cascades
title_full_unstemmed	Criticality and the fractal structure of −5/3 turbulent cascades
title_sort	Criticality and the fractal structure of −5/3 turbulent cascades
dc.creator.fl_str_mv	Cabrera, Juan Luis Gutiérrez, Esther Rodríguez Márquez, Miguel
dc.contributor.author.spa.fl_str_mv	Cabrera, Juan Luis Gutiérrez, Esther Rodríguez Márquez, Miguel
dc.subject.spa.fl_str_mv	Cascade Criticality Fractals Navier-stokes equation Nonlinear Stochastic Turbulence Maps Complex Cascada Criticidad Fractales Ecuación de navier-stokes No lineal Estocástico Turbulencia Mapas Complejo
topic	Cascade Criticality Fractals Navier-stokes equation Nonlinear Stochastic Turbulence Maps Complex Cascada Criticidad Fractales Ecuación de navier-stokes No lineal Estocástico Turbulencia Mapas Complejo
description	Here we show a procedure to generate an analytical structure producing a cascade that scales as the energy spectrum in isotropic homogeneous turbulence. We obtain a function that unveils a non-self-similar fractal at the origin of the cascade. It reveals that the backbone underlying cascades is formed by deterministic nested polynomials with parameters tuned in a Hopf bifurcation critical point. The cascade scaling is exactly obtainable (not by numerical simulations) from deterministic low dimensional nonlinear dynamics. Consequently, it should not be exclusive for fluids but also present in other complex phenomena. The scaling is obtainable both in deterministic and stochastic situations.
publishDate	2021
dc.date.accessioned.none.fl_str_mv	2021-06-04T22:38:09Z
dc.date.available.none.fl_str_mv	2021-06-04T22:38:09Z
dc.date.issued.none.fl_str_mv	2021-03-12
dc.date.embargoEnd.none.fl_str_mv	2023-03-12
dc.type.spa.fl_str_mv	Artículo de revista
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dc.relation.references.spa.fl_str_mv	[1] Eckmann JP. Roads to turbulence in dissipative dynamical systems. Rev Mod Phys 1981;53:643–54. doi:10.1103/RevModPhys.53.643. [2] Kadanoff LP. Roads to chaos. Physics Today 1983;36(12):46. doi:10.1063/1. 2915388. [3] Bohr T, Jensen MH, Paladin G, Vulpiani A. Dynamical Systems Approach to Turbulence. Cambridge: Cambridge University Press; 1998. [4] Frisch U. Turbulence: The Legacy of A. N. Kolmogorov. Cambridge University Press; 1995. [5] McDonough JM. Three-dimensional poor man’s navier-stokes equation: A discrete dynamical system exhibiting k−5/3 inertial subrange energy scaling. Phys Rev E 2009;79:065302. doi:10.1103/PhysRevE.79.065302. [6] May RM. Simple mathematical models with very complicated dynamics. Nature 1976;261:459–67. doi:10.1038/261459a0. [7] Feigenbaum MJ. The transition to aperiodic behavior in turbulent systems. Communications in Mathematical Physics 1980;77:65–86. doi:10.1007/ BF01205039. [8] Libchaber A, Maurer J. Une experience de rayleigh-benard de geometrie reduite; multiplication, accrochage et demultiplication de frequences. J Phys Colloques 1980;41(C3):51–6. doi:10.1051/jphyscol:1980309. [9] Richardson LF. Atmospheric Diffusion Shown on a Distance-Neighbour Graph. Proceedings of the Royal Society of London Series A 1926;110(756):709–37. doi:10.1098/rspa.1926.0043. [10] Kolmogorov AN. Local structure of turbulence in an incompressible viscous fluid at very high reynolds numbers. Soviet Physics Uspekhi 1968;10(6):734– 46. doi:10.1070/pu1968v010n06abeh003710. [11] Onsager L. The distribution of energy in turbulence. Phys Rev 1945;68:286. doi:10.1103/PhysRev.68.281. [12] Obukhov AM. On the distribution of energy in the spectrum of turbulent flow. Dok Akad Nauk SSSR 1941;32:22–4. [13] Heisenberg W. Zur statistischen theorie der turbulenz. Zeitschrift für Physik 1948;124(7):628–57. [14] Weizsäcker CFv. Das spektrum der turbulenz bei großen reynoldsschen zahlen. Zeitschrift für Physik 1948;124(7):614–27. [15] Maynard Smith J. Mathematical Ideas in Biology. Cambridge University Press; 1968. [16] Aronson DG, Chory MA, Hall GR, McGehee RP. Bifurcations from an invariant circle for two-parameter families of maps of the plane: A computer-assisted study. Communications in Mathematical Physics 1982;83(3):303–54. [17] Pounder JR, Rogers TD. The geometry of chaos: Dynamics of a nonlinear second-order difference equation. Bulletin of Mathematical Biology 1980;42(4):551–97. [18] Morimoto Y. Hopf bifurcation in the simple nonlinear recurrence equation x(t+1) = ax(t)[1-x(t-1)]. Physics Letters A 1988;134(3):179–82. [19] Cabrera JL. Contribución a la descripción estocástica de algunos sistemas dinámicos no lineales con retardo temporal. Madrid, Spain: UNED; 1997. [20] Cabrera JL, de la Rubia FJ. Numerical analysis of transient behavior in the discrete random logistic equation with delay. Physics Letters A 1995;197(1):19–24. [21] Cabrera JL, de la Rubia FJ. Analysis of the behavior of a random nonlinear delay discrete equation. International Journal of Bifurcation and Chaos 1996;06(09):1683–90. [22] Cabrera JL, Rubia FJdl. Resonance-like phenomena induced by exponentially correlated parametric noise. Europhys Lett 1997;39(2):123–8. [23] Cabrera JL, Gorroñogoitia J, de la Rubia FJ. Noise-correlation-time–mediated localization in random nonlinear dynamical systems. Phys Rev Lett 1999;82:2816–19. doi:10.1103/PhysRevLett.82.2816. [24] Trifonov V, Pasqualucci L, Dalla-Favera R, Rabadan R. Fractal-like distributions over the rational numbers in high-throughput biological and clinical data. Scientific Reports 2011;1(1):191. [25] Saddoughi SG, Veeravalli SV. Local isotropy in turbulent boundary layers at high reynolds number. J Fluid Mechanics 1994;268:333–72. [26] Eyink GL, Sreenivasan KR. Onsager and the theory of hydrodynamic turbulence. Rev Mod Phys 2006;78:87–135. doi:10.1103/RevModPhys.78.87. [27] Flierl G, Ferrari R. Turbulence in the Ocean and Atmosphere. MIT OpenCourseWare. https://ocw.mit.edu.; 2006. [28] Dotti M, Schlander R, Buchhave P, Velte CM. Experimental investigation of the turbulent cascade development by injection of single large-scale fourier modes. 2020. 10.1007/s00348-020-03041-2. [29] Buchhave P, Velte CM. Dynamic triad interactions and evolving turbulence spectra. 2021. arXiv:1906.04756. [30] Josserand C, Le Berre M, Lehner T, Pomeau Y. Turbulence: Does energy cascade exist? Journal of Statistical Physics 2017;167:596–625. doi:10.1007/ s10955-016-1642-5. [31] Cabrera JL, Milton JG. Human stick balancing: Tuning lévy flights to improve balance control. Chaos: An Interdisciplinary Journal of Nonlinear Science 2004;14(3):691–8. [32] Mantegna RN, Stanley HE. Scaling behaviour in the dynamics of an economic index. Nature 1995;376(6535):46–9. [33] González-Díaz LA, Gutiérrez ED, Varona P, Cabrera JL. Winnerless competition in coupled lotka-volterra maps. Phys Rev E 2013;88:012709. doi:10.1103/ PhysRevE.88.012709. [34] Gutiérrez ED, Cabrera JL. A neural coding scheme reproducing foraging trajectories. Scientific Reports 2015;5(1):18009. doi:10.1038/srep18009. [35] Allen LJ. Some discrete-time si, sir, and sis epidemic models. Mathematical Biosciences 1994;124(1):83–105. doi:10.1016/0025-5564(94)90025-6.
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spelling	Cabrera, Juan LuisGutiérrez, EstherRodríguez Márquez, Miguel2021-06-04T22:38:09Z2021-06-04T22:38:09Z2021-03-122023-03-120960-0779https://hdl.handle.net/11323/8361https://doi.org/10.1016/j.chaos.2021.110876Corporación Universidad de la CostaREDICUC - Repositorio CUChttps://repositorio.cuc.edu.co/Here we show a procedure to generate an analytical structure producing a cascade that scales as the energy spectrum in isotropic homogeneous turbulence. We obtain a function that unveils a non-self-similar fractal at the origin of the cascade. It reveals that the backbone underlying cascades is formed by deterministic nested polynomials with parameters tuned in a Hopf bifurcation critical point. The cascade scaling is exactly obtainable (not by numerical simulations) from deterministic low dimensional nonlinear dynamics. Consequently, it should not be exclusive for fluids but also present in other complex phenomena. The scaling is obtainable both in deterministic and stochastic situations.Aquí mostramos un procedimiento para generar una estructura analítica produciendo una cascada que escala como el espectro de energía en turbulencias isotrópicas homogéneas. Obtenemos una función que revela un fractal no auto-similar en el origen de la cascada. Revela que la columna vertebral subyacente a las cascadas está formada por polinomios anidados deterministas con parámetros ajustados en un punto crítico de bifurcación de Hopf. La escala en cascada se puede obtener exactamente (no mediante simulaciones numéricas) a partir de dinámicas no lineales deterministas de baja dimensión. En consecuencia, no debe ser exclusivo de los fluidos sino también estar presente en otros fenómenos complejos. La escala se puede obtener tanto en situaciones deterministas como estocásticas.Cabrera, Juan LuisGutiérrez, Esther-will be generated-orcid-0000-0001-7579-0711-600Rodríguez Márquez, Miguelapplication/pdfengChaos, Solitons and FractalsAttribution-NonCommercial-NoDerivatives 4.0 Internationalhttp://creativecommons.org/licenses/by-nc-nd/4.0/info:eu-repo/semantics/embargoedAccesshttp://purl.org/coar/access_right/c_f1cfCascadeCriticalityFractalsNavier-stokes equationNonlinearStochasticTurbulenceMapsComplexCascadaCriticidadFractalesEcuación de navier-stokesNo linealEstocásticoTurbulenciaMapasComplejoCriticality and the fractal structure of −5/3 turbulent cascadesCriticidad y estructura fractal de −5/3 cascadas turbulentasArtículo de revistahttp://purl.org/coar/resource_type/c_6501http://purl.org/coar/resource_type/c_2df8fbb1Textinfo:eu-repo/semantics/articlehttp://purl.org/redcol/resource_type/ARTinfo:eu-repo/semantics/acceptedVersionhttps://www.sciencedirect.com/science/article/abs/pii/S0960077921002290[1] Eckmann JP. Roads to turbulence in dissipative dynamical systems. Rev Mod Phys 1981;53:643–54. doi:10.1103/RevModPhys.53.643.[2] Kadanoff LP. Roads to chaos. Physics Today 1983;36(12):46. doi:10.1063/1. 2915388.[3] Bohr T, Jensen MH, Paladin G, Vulpiani A. Dynamical Systems Approach to Turbulence. Cambridge: Cambridge University Press; 1998.[4] Frisch U. Turbulence: The Legacy of A. N. Kolmogorov. Cambridge University Press; 1995.[5] McDonough JM. Three-dimensional poor man’s navier-stokes equation: A discrete dynamical system exhibiting k−5/3 inertial subrange energy scaling. Phys Rev E 2009;79:065302. doi:10.1103/PhysRevE.79.065302.[6] May RM. Simple mathematical models with very complicated dynamics. Nature 1976;261:459–67. doi:10.1038/261459a0.[7] Feigenbaum MJ. The transition to aperiodic behavior in turbulent systems. Communications in Mathematical Physics 1980;77:65–86. doi:10.1007/ BF01205039.[8] Libchaber A, Maurer J. Une experience de rayleigh-benard de geometrie reduite; multiplication, accrochage et demultiplication de frequences. J Phys Colloques 1980;41(C3):51–6. doi:10.1051/jphyscol:1980309.[9] Richardson LF. Atmospheric Diffusion Shown on a Distance-Neighbour Graph. Proceedings of the Royal Society of London Series A 1926;110(756):709–37. doi:10.1098/rspa.1926.0043.[10] Kolmogorov AN. Local structure of turbulence in an incompressible viscous fluid at very high reynolds numbers. Soviet Physics Uspekhi 1968;10(6):734– 46. doi:10.1070/pu1968v010n06abeh003710.[11] Onsager L. The distribution of energy in turbulence. Phys Rev 1945;68:286. doi:10.1103/PhysRev.68.281.[12] Obukhov AM. On the distribution of energy in the spectrum of turbulent flow. Dok Akad Nauk SSSR 1941;32:22–4.[13] Heisenberg W. Zur statistischen theorie der turbulenz. Zeitschrift für Physik 1948;124(7):628–57.[14] Weizsäcker CFv. Das spektrum der turbulenz bei großen reynoldsschen zahlen. Zeitschrift für Physik 1948;124(7):614–27.[15] Maynard Smith J. Mathematical Ideas in Biology. Cambridge University Press; 1968.[16] Aronson DG, Chory MA, Hall GR, McGehee RP. Bifurcations from an invariant circle for two-parameter families of maps of the plane: A computer-assisted study. Communications in Mathematical Physics 1982;83(3):303–54.[17] Pounder JR, Rogers TD. The geometry of chaos: Dynamics of a nonlinear second-order difference equation. Bulletin of Mathematical Biology 1980;42(4):551–97.[18] Morimoto Y. Hopf bifurcation in the simple nonlinear recurrence equation x(t+1) = ax(t)[1-x(t-1)]. Physics Letters A 1988;134(3):179–82.[19] Cabrera JL. Contribución a la descripción estocástica de algunos sistemas dinámicos no lineales con retardo temporal. Madrid, Spain: UNED; 1997.[20] Cabrera JL, de la Rubia FJ. Numerical analysis of transient behavior in the discrete random logistic equation with delay. Physics Letters A 1995;197(1):19–24.[21] Cabrera JL, de la Rubia FJ. Analysis of the behavior of a random nonlinear delay discrete equation. International Journal of Bifurcation and Chaos 1996;06(09):1683–90.[22] Cabrera JL, Rubia FJdl. Resonance-like phenomena induced by exponentially correlated parametric noise. Europhys Lett 1997;39(2):123–8.[23] Cabrera JL, Gorroñogoitia J, de la Rubia FJ. Noise-correlation-time–mediated localization in random nonlinear dynamical systems. Phys Rev Lett 1999;82:2816–19. doi:10.1103/PhysRevLett.82.2816.[24] Trifonov V, Pasqualucci L, Dalla-Favera R, Rabadan R. Fractal-like distributions over the rational numbers in high-throughput biological and clinical data. Scientific Reports 2011;1(1):191.[25] Saddoughi SG, Veeravalli SV. Local isotropy in turbulent boundary layers at high reynolds number. J Fluid Mechanics 1994;268:333–72.[26] Eyink GL, Sreenivasan KR. Onsager and the theory of hydrodynamic turbulence. Rev Mod Phys 2006;78:87–135. doi:10.1103/RevModPhys.78.87.[27] Flierl G, Ferrari R. Turbulence in the Ocean and Atmosphere. MIT OpenCourseWare. https://ocw.mit.edu.; 2006.[28] Dotti M, Schlander R, Buchhave P, Velte CM. Experimental investigation of the turbulent cascade development by injection of single large-scale fourier modes. 2020. 10.1007/s00348-020-03041-2.[29] Buchhave P, Velte CM. Dynamic triad interactions and evolving turbulence spectra. 2021. arXiv:1906.04756.[30] Josserand C, Le Berre M, Lehner T, Pomeau Y. Turbulence: Does energy cascade exist? Journal of Statistical Physics 2017;167:596–625. doi:10.1007/ s10955-016-1642-5.[31] Cabrera JL, Milton JG. Human stick balancing: Tuning lévy flights to improve balance control. Chaos: An Interdisciplinary Journal of Nonlinear Science 2004;14(3):691–8.[32] Mantegna RN, Stanley HE. Scaling behaviour in the dynamics of an economic index. Nature 1995;376(6535):46–9.[33] González-Díaz LA, Gutiérrez ED, Varona P, Cabrera JL. Winnerless competition in coupled lotka-volterra maps. Phys Rev E 2013;88:012709. doi:10.1103/ PhysRevE.88.012709.[34] Gutiérrez ED, Cabrera JL. A neural coding scheme reproducing foraging trajectories. Scientific Reports 2015;5(1):18009. doi:10.1038/srep18009.[35] Allen LJ. Some discrete-time si, sir, and sis epidemic models. 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Criticality and the fractal structure of −5/3 turbulent cascades

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