Liouville spectral gap and bifurcation-driven Lagrangian–Eulerian decoupling with nondiffusive turbulence closures
Abstract
In fully developed homogeneous and isotropic turbulence, the Lagrangian and Eulerian descriptions of motion, although formally equivalent, become statistically decoupled. In this work, by invoking Liouville theorem, we show that the joint probability density function (PDF) of the Eulerian and Lagrangian fields, evolving from arbitrary initial conditions, relaxes exponentially toward a factorized form given by the product of the corresponding marginal PDFs. This relaxation is governed by a genuine spectral gap of the Liouville operator, whose magnitude is primarily set by the bifurcation rate of the velocity-gradient dynamics, whereas the contribution of Lyapunov exponents is shown to be significantly smaller. As a consequence, Eulerian–Lagrangian correlations decay rapidly, and if the joint PDF is initially factorized, its factorized structure is preserved at all subsequent times, with each marginal evolving independently under the corresponding dynamics. We further show that the formal equivalence between the two descriptions implies the invariance of the relative kinetic energy between arbitrarily chosen points. When combined with the asymmetric statistics of instantaneous finite-scale Lyapunov exponents in incompressible turbulence, this property provides a quantitative interpretation of particle-pair separation and of the turbulent energy cascade. Finally, these results naturally lead to nondiffusive closure relations for the von Kármán–Howarth and Corrsin equations, which coincide with those previously proposed by the author, thereby providing an independent theoretical validation of those closures.
Acknowledgments
This work was partially supported by the Italian Ministry for the Universities and Scientific and Technological Research (MIUR).
-
Research ethics: Not applicable.
-
Informed consent: Not applicable.
-
Author contributions: The author has accepted responsibility for the entire content of this manuscript and approved its submission.
-
Use of Large Language Models, AI and Machine Learning Tools: None declared.
-
Conflict of interest: The author states no conflict of interest.
-
Research funding: None declared.
-
Data availability: Not applicable.
References
[1] T. von Kármán and L. Howarth, “On the statistical theory of isotropic turbulence,” Proc. R. Soc. A, vol. 164, no. 14, p. 192, 1938.10.1098/rspa.1938.0013Search in Google Scholar
[2] S. Corrsin, “The decay of isotropic temperature fluctuations in an isotropic turbulence,” J. Aeronaut. Sci., vol. 18, no. 12, pp. 417–423, 1951.10.2514/8.1982Search in Google Scholar
[3] S. Corrsin, “On the spectrum of isotropic temperature fluctuations in an isotropic turbulence,” J. Appl. Phys., vol. 22, no. 4, pp. 469–473, 1951, https://doi.org/10.1063/1.1699986.Search in Google Scholar
[4] K. Hasselmann, “Zur Deutung der dreifachen Geschwindigkeitskorrelationen der isotropen Turbulenz,” Dtsch. Hydrogr. Z., vol. 11, no. 5, pp. 207–217, 1958.10.1007/BF02020016Search in Google Scholar
[5] M. Millionshtchikov, “Isotropic turbulence in the field of turbulent viscosity,” JETP Lett., vol. 8, pp. 406–411, 1969.Search in Google Scholar
[6] M. Oberlack and N. Peters, “Closure of the two-point correlation equation as a basis for Reynolds stress models,” Appl. Sci. Res., vol. 51, nos. 1–2, pp. 533–539, 1993.10.1007/BF01082587Search in Google Scholar
[7] M. K. Baev and G. G. Chernykh, “On Corrsin equation closure,” J. Eng. Thermophys., vol. 19, no. 3, pp. 154–169, 2010, https://doi.org/10.1134/S1810232810030069.Search in Google Scholar
[8] J. A. Domaradzki and G. L. Mellor, “A simple turbulence closure hypothesis for the triple-velocity correlation functions in homogeneous isotropic turbulence,” J. Fluid Mech., vol. 140, pp. 45–61, 1984.10.1017/S0022112084000501Search in Google Scholar
[9] C. Truesdell, A First Course in Rational Continuum Mechanics, New York, Academic, 1977.Search in Google Scholar
[10] W. K. George, “A theory for the self-preservation of temperature fluctuations in isotropic turbulence,” Turbulence Research Laboratory, Technical Report 117, 1988.Search in Google Scholar
[11] W. K. George, “Self-preservation of temperature fluctuations in isotropic turbulence,” in Studies in Turbulence, Berlin, Springer, 1992.10.1007/978-1-4612-2792-2_38Search in Google Scholar
[12] R. A. Antonia, R. J. Smalley, T. Zhou, F. Anselmet, and L. Danaila, “Similarity solution of temperature structure functions in decaying homogeneous isotropic turbulence,” Phys. Rev. E, vol. 69, no. 1, p. 016305, 2004. https://doi.org/10.1103/PhysRevE.69.016305.Search in Google Scholar PubMed
[13] A. Onufriev, “On a model equation for probability density in semi-empirical turbulence transfer theory,” in The Notes on Turbulence, Moscow, Nauka, 1994.Search in Google Scholar
[14] V. N. Grebenev and M. Oberlack, “A chorin-type formula for solutions to a closure model for the von Kármán–Howarth equation,” J. Nonlinear Math. Phys., vol. 12, no. 1, pp. 1–9, 2005.10.2991/jnmp.2005.12.1.1Search in Google Scholar
[15] V. N. Grebenev and M. Oberlack, “A geometric interpretation of the second-order structure function arising in turbulence,” Math. Phys. Anal. Geom., vol. 12, no. 1, pp. 1–18, 2009, https://doi.org/10.1007/s11040-008-9049-4.Search in Google Scholar
[16] F. Thiesset, R. A. Antonia, L. Danaila, and L. Djenidi, “Kármán–Howarth closure equation on the basis of a universal eddy viscosity,” Phys. Rev. E, vol. 88, no. 1, p. 011003(R), 2013. https://doi.org/10.1103/PhysRevE.88.011003.Search in Google Scholar PubMed
[17] N. de Divitiis, “Lyapunov analysis for fully developed homogeneous isotropic turbulence,” Theor. Comput. Fluid Dyn., vol. 25, no. 6, pp. 421–445, 2011. https://doi.org/10.1007/s00162-010-0211-9.Search in Google Scholar
[18] N. de Divitiis, “Finite scale Lyapunov analysis of temperature fluctuations in homogeneous isotropic turbulence,” Appl. Math. Model., vol. 38, nos. 21–22, pp. 5279–5297, 2014. https://doi.org/10.1016/j.apm.2014.04.016.Search in Google Scholar
[19] N. de Divitiis, “von Kármán–Howarth and Corrsin equations closure based on Lagrangian description of the fluid motion,” Ann. Phys., vol. 368, pp. 296–309, 2016, https://doi.org/10.1016/j.aop.2016.02.010.Search in Google Scholar
[20] N. de Divitiis, “Statistical Lyapunov theory based on bifurcation analysis of energy cascade in isotropic homogeneous turbulence: A physical-mathematical review,” Entropy, vol. 21, no. 5, p. 520, 2019, https://doi.org/10.3390/e21050520.Search in Google Scholar PubMed PubMed Central
[21] N. de Divitiis, “von Kármán–Howarth and Corrsin equations closures through Liouville theorem,” Results Phys., vol. 16, 2020, https://doi.org/10.1016/j.rinp.2020.102979.Search in Google Scholar
[22] N. de Divitiis, “Turbulent energy cascade through equivalence of Euler and Lagrange motion descriptions and bifurcation rates,” J. Appl. Comput. Mech., vol. 7, no. 4, pp. 2221–2237, 2022. https://doi.org/10.22055/jacm.2021.38082.3149.Search in Google Scholar
[23] M. Pollicott, “On the rate of mixing of Axiom A flows,” Invent. Math., vol. 81, no. 3, pp. 413–426, 1985. https://doi.org/10.1007/BF01388579.Search in Google Scholar
[24] D. Ruelle, “Resonances of chaotic dynamical systems,” Phys. Rev. Lett., vol. 56, no. 5, pp. 405–407, 1986. https://doi.org/10.1103/PhysRevLett.56.405.Search in Google Scholar PubMed
[25] D. Ruelle, “Resonances for Axiom A flows,” J. Differ. Geom., vol. 25, no. 1, pp. 99–116, 1987. https://doi.org/10.4310/jdg/1214440726.Search in Google Scholar
[26] A. Tsinober, An Informal Conceptual Introduction to Turbulence: Second Edition of an Informal Introduction to Turbulence, Dordrecht, Springer Science & Business Media, 2009.10.1007/978-90-481-3174-7Search in Google Scholar
[27] D. Ruelle and F. Takens, “On the nature of turbulence,” Commun. Math. Phys., vol. 20, no. 3, p. 167, 1971.10.1007/BF01646553Search in Google Scholar
[28] J. P. Eckmann, “Roads to turbulence in dissipative dynamical systems,” Rev. Mod. Phys., vol. 53, no. 4, pp. 643–654, 1981. https://doi.org/10.1103/revmodphys.53.643.Search in Google Scholar
[29] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, New York, Springer, 1990.Search in Google Scholar
[30] D. Ruelle, “Microscopic fluctuations and turbulence,” Phys. Lett., vol. 72A, no. 2, p. 81, 1979.10.1016/0375-9601(79)90653-4Search in Google Scholar
[31] A. N. Kolmogorov, “Dissipation of energy in locally isotropic turbulence,” Dokl. Akad. Nauk, vol. 32, no. 1, pp. 19–21, 1941.Search in Google Scholar
[32] M. J. Feigenbaum, “Quantitative universality for a class of nonlinear transformations,” J. Stat. Phys., vol. 19, no. 1, 1978.10.1007/BF01020332Search in Google Scholar
[33] Y. Pomeau and P. Manneville, “Intermittent transition to turbulence in dissipative dynamical systems,” Commun. Math. Phys., vol. 74, no. 2, p. 189, 1980. https://doi.org/10.1007/bf01197757.Search in Google Scholar
[34] G. K. Batchelor, The Theory of Homogeneous Turbulence, Cambridge, Cambridge University Press, 1953.Search in Google Scholar
[35] J. M. Ottino, The Kinematics of Mixing: Stretching, Chaos, and Transport, New York, Cambridge Texts in Applied Mathematics, 1989.Search in Google Scholar
[36] J. M. Ottino, “Mixing, chaotic advection, and turbulence,” Annu. Rev. Fluid. Mech., vol. 22, no. 1, pp. 207–253, 1990. https://doi.org/10.1146/annurev.fl.22.010190.001231.Search in Google Scholar
[37] G. Nicolis, Introduction to Nonlinear Science, Cambridge, Cambridge University Press, 1995.10.1017/CBO9781139170802Search in Google Scholar
[38] H. Federer, Geometric Measure Theory, Berlin Heidelberg, Springer-Verlag, 1969.Search in Google Scholar
[39] N. de Divitiis, “Self-similarity in fully developed homogeneous isotropic turbulence using the lyapunov analysis,” Theor. Comput. Fluid Dyn., vol. 26, nos. 1–4, pp. 81–92, 2012. https://doi.org/10.1007/s00162-010-0213-7.Search in Google Scholar
[40] S. Chen, G. D. Doolen, R. H. Kraichnan, and Z.-S. She, “On statistical correlations between velocity increments and locally averaged dissipation in homogeneous turbulence,” Phys. Fluids A, vol. 5, no. 2, pp. 458–463, 1992.10.1063/1.858897Search in Google Scholar
[41] S. A. Orszag and G. S. Patterson, “Numerical simulation of three-dimensional homogeneous isotropic turbulence,” Phys. Rev. Lett., vol. 28, no. 2, pp. 76–79, 1972. https://doi.org/10.1103/physrevlett.28.76.Search in Google Scholar
[42] R. Panda, V. Sonnad, E. Clementi, S. A. Orszag, and V. Yakhot, “Turbulence in a randomly stirred fluid,” Phys. Fluids A, vol. 1, no. 6, pp. 1045–1053, 1989, https://doi.org/10.1063/1.857395.Search in Google Scholar
[43] R. Anderson and C. Meneveau, “Effects of the similarity model in finite-difference LES of isotropic turbulence using a lagrangian dynamic mixed model,” Flow Turbul. Combust., vol. 62, no. 3, pp. 201–225, 1999. Available at: https://doi.org/10.1023/a:1009967228812.Search in Google Scholar
[44] D. Carati, S. Ghosal, and P. Moin, “On the representation of backscatter in dynamic localization models,” Phys. Fluids, vol. 7, no. 3, pp. 606–616, 1995, https://doi.org/10.1063/1.868585.Search in Google Scholar
[45] H. S. Kang, S. Chester, and C. Meneveau, “Decaying turbulence in an active-gridgenerated flow and comparisons with large-eddy simulation,” J. Fluid Mech., vol. 480, pp. 129–160, 2003, https://doi.org/10.1017/s0022112002003579.Search in Google Scholar
[46] G. K. Batchelor, “Small-scale variation of convected quantities like temperature in turbulent fluid. Part 1. General discussion and the case of small conductivity,” J. Fluid Mech., vol. 5, no. 1, pp. 113–133, 1959. https://doi.org/10.1017/s002211205900009x.Search in Google Scholar
[47] G. K. Batchelor, I. D. Howells, and A. A. Townsend, “Small-scale variation of convected quantities like temperature in turbulent fluid. Part 2. The case of large conductivity,” J. Fluid Mech., vol. 5, no. 1, pp. 134–139, 1959. https://doi.org/10.1017/s0022112059000106.Search in Google Scholar
[48] A. M. Obukhov, “The structure of the temperature field in a turbulent flow,” Dokl. Akad. Nauk, vol. 39, p. 391, 1949.Search in Google Scholar
[49] C. H. Gibson and W. H. Schwarz, “The universal equilibrium spectra of turbulent velocity and scalar fields,” J. Fluid Mech., vol. 16, no. 3, pp. 365–384, 1963. https://doi.org/10.1017/s0022112063000835.Search in Google Scholar
[50] L. Mydlarski and Z. Warhaft, “Passive scalar statistics in high-Péclet-number grid turbulence,” J. Fluid Mech., vol. 358, pp. 135–175, 1998.10.1017/S0022112097008161Search in Google Scholar
[51] J. Chasnov, V. M. Canuto, and R. S. Rogallo, “Turbulence spectrum of strongly conductive temperature field in a rapidly stirred fluid,” Phys. Fluids A, vol. 1, no. 10, pp. 1698–1700, 1989. https://doi.org/10.1063/1.857535.Search in Google Scholar
[52] D. A. Donzis, K. R. Sreenivasan, and P. K. Yeung, “The batchelor spectrum for mixing of passive scalars in isotropic turbulence,” Flow Turbul. Combust., vol. 85, nos. 3–4, pp. 549–566, 2010, https://doi.org/10.1007/s10494-010-9271-6.Search in Google Scholar
© 2026 Walter de Gruyter GmbH, Berlin/Boston
