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A Lagrangian fluctuation–dissipation relation for scalar turbulence. Part I. Flows with no bounding walls

Published online by Cambridge University Press:  14 September 2017

Theodore D. Drivas Open the ORCID record for Theodore D. Drivas [Opens in a new window]  and
Gregory L. Eyink
Theodore D. Drivas*
Affiliation:
Department of Applied Mathematics and Statistics, The Johns Hopkins University, Baltimore, MD 21218, USA
Gregory L. Eyink
Affiliation:
Department of Applied Mathematics and Statistics, The Johns Hopkins University, Baltimore, MD 21218, USA Department of Physics and Astronomy, The Johns Hopkins University, Baltimore, MD 21218, USA
*
†Email address for correspondence: [email protected]
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Abstract

An exact relation is derived between scalar dissipation due to molecular diffusivity and the randomness of stochastic Lagrangian trajectories for flows without bounding walls. This ‘Lagrangian fluctuation–dissipation relation’ equates the scalar dissipation for either passive or active scalars to the variance of scalar inputs associated with initial scalar values and internal scalar sources, as these are sampled backward in time by the stochastic Lagrangian trajectories. As an important application, we reconsider the phenomenon of ‘Lagrangian spontaneous stochasticity’ or persistent non-determinism of Lagrangian particle trajectories in the limit of vanishing viscosity and diffusivity. Previous work on the Kraichnan (Phys. Fluids, 1968, vol. 11, pp. 945–953) model of turbulent scalar advection has shown that anomalous scalar dissipation is associated in that model with Lagrangian spontaneous stochasticity. There has been controversy, however, regarding the validity of this mechanism for scalars advected by an actual turbulent flow. We here completely resolve this controversy by exploiting the fluctuation–dissipation relation. For either a passive or an active scalar advected by any divergence-free velocity field, including solutions of the incompressible Navier–Stokes equation, and away from walls, we prove that anomalous scalar dissipation requires Lagrangian spontaneous stochasticity. For passive scalars, we prove furthermore that spontaneous stochasticity yields anomalous dissipation for suitable initial scalar fields, so that the two phenomena are there completely equivalent. These points are illustrated by numerical results from a database of homogeneous isotropic turbulence, which provide both additional support to the results and physical insight into the representation of diffusive effects by stochastic Lagrangian particle trajectories.

JFM classification

Turbulent Flows: Turbulent Flows Mixing: Turbulent mixing Turbulent Flows: Turbulence theory
Type
Papers
Information
Journal of Fluid Mechanics , Volume 829 , 25 October 2017 , pp. 153 - 189
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© 2017 Cambridge University Press 

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References

Albeverio, S. & Belopolskaya, Ya. 2010 Generalized solutions of the Cauchy problem for the Navier–Stokes system and diffusion processes. Cubo (Temuco) 12 (2), 77–96.Google Scholar
Batchelor, G. K. 1951 Pressure fluctuations in isotropic turbulence. Math. Proc. Camb. Phil. Soc. 47 (2), 359–374.CrossRefGoogle Scholar
Berg, J., Lüthi, B., Mann, J. & Ott, S. 2006 Backwards and forwards relative dispersion in turbulent flow: an experimental investigation. Phys. Rev. E 74, 016304.Google Scholar
Berkovitz, L. D. 1974 Lower semicontinuity of integral functionals. Trans. Am. Math. Soc. 192, 51–57.Google Scholar
Bernard, D., Gawȩdzki, K. & Kupiainen, A. 1998 Slow modes in passive advection. J. Stat. Phys. 90, 519–569.Google Scholar
Biferale, L., Boffetta, G., Celani, A., Lanotte, A. & Toschi, F. 2005 Particle trapping in three-dimensional fully developed turbulence. Phys. Fluids 17 (2), 021701.Google Scholar
Biferale, L., Lanotte, A. S., Scatamacchia, R. & Toschi, F. 2014 Intermittency in the relative separations of tracers and of heavy particles in turbulent flows. J. Fluid Mech. 757, 550–572.CrossRefGoogle Scholar
Bitane, R., Homann, H. & Bec, J. 2013 Geometry and violent events in turbulent pair dispersion. J. Turbul. 14 (2), 23–45.Google Scholar
Boffetta, G. & Sokolov, I. M. 2002 Relative dispersion in fully developed turbulence: the Richardson’s law and intermittency corrections. Phys. Rev. Lett. 88, 094501.Google Scholar
Braides, A. 2002 Gamma-Convergence for Beginners. Oxford University Press.Google Scholar
Buaria, D., Yeung, P. K. & Sawford, B. L. 2016 A Lagrangian study of turbulent mixing: forward and backward dispersion of molecular trajectories in isotropic turbulence. J. Fluid Mech. 799, 352–382.Google Scholar
Chaves, M., Gawȩdzki, K., Horvai, P., Kupiainen, A. & Vergassola, M. 2003 Lagrangian dispersion in Gaussian self-similar velocity ensembles. J. Stat. Phys. 113 (5–6), 643–692.CrossRefGoogle Scholar
Constantin, P. & Iyer, G. 2008 A stochastic Lagrangian representation of the three-dimensional incompressible Navier–Stokes equations. Commun. Pure Appl. Maths 61, 330–345.Google Scholar
Constantin, P. & Iyer, G. 2011 A stochastic-Lagrangian approach to the Navier–Stokes equations in domains with boundary. Ann. Appl. Probab. 21, 1466–1492.Google Scholar
Corrsin, S. 1951 On the spectrum of isotropic temperature fluctuations in an isotropic turbulence. J. Appl. Phys. 22 (4), 469–473.Google Scholar
Davila, J. & Vassilicos, J. C. 2003 Richardson’s pair diffusion and the stagnation point structure of turbulence. Phys. Rev. Lett. 91 (14), 144501.Google Scholar
Donzis, D. A., Sreenivasan, K. R. & Yeung, P. K. 2005 Scalar dissipation rate and dissipative anomaly in isotropic turbulence. J. Fluid Mech. 532, 199–216.CrossRefGoogle Scholar
Drivas, T. D. & Eyink, G. L. 2017 A Lagrangian fluctuation–dissipation relation for scalar turbulence. Part II. Wall-bounded flows. J. Fluid Mech. (submitted). arXiv:1703.08133.Google Scholar
Eyink, G., Vishniac, E., Lalescu, C., Aluie, H., Kanov, K., Bürger, K., Burns, R., Meneveau, C. & Szalay, A. 2013 Flux-freezing breakdown in high-conductivity magnetohydrodynamic turbulence. Nature 497 (7450), 466–469.Google Scholar
Eyink, G. L. 2001 Dissipation in turbulent solutions of 2D Euler equations. Nonlinearity 14 (4), 787–802.Google Scholar
Eyink, G. L. 2009 Stochastic line motion and stochastic flux conservation for nonideal hydromagnetic models. J. Math. Phys. 50 (8), 083102.Google Scholar
Eyink, G. L. 2010 Stochastic least-action principle for the incompressible Navier–Stokes equation. Physica D 239, 1236–1240.Google Scholar
Eyink, G. L. 2011 Stochastic flux freezing and magnetic dynamo. Phys. Rev. E 83 (5), 056405.Google Scholar
Eyink, G. L. & Drivas, T. D.2015a Quantum spontaneous stochasticity. arXiv:1509.04941.Google Scholar
Eyink, G. L. & Drivas, T. D. 2015b Spontaneous stochasticity and anomalous dissipation for Burgers equation. J. Stat. Phys. 158 (2), 386–432.Google Scholar
Eyink, G. L. & Drivas, T. D. 2017 A Lagrangian fluctuation–dissipation relation for scalar turbulence. Part III. Turbulent Rayleigh–Bénard convection. J. Fluid Mech. (submitted). arXiv:1703.09604.Google Scholar
Faber, T. & Vassilicos, J. C. 2009 Turbulent pair separation due to multiscale stagnation point structure and its time asymmetry in two-dimensional turbulence. Phys. Fluids 21 (1), 015106.Google Scholar
Falkovich, G. & Frishman, A. 2013 Single flow snapshot reveals the future and the past of pairs of particles in turbulence. Phys. Rev. Lett. 110 (21), 214502.CrossRefGoogle ScholarPubMed
Falkovich, G., Gawȩdzki, K. & Vergassola, M. 2001 Particles and fields in fluid turbulence. Rev. Mod. Phys. 73, 913–975.CrossRefGoogle Scholar
Florescu, L. C. & Godet-Thobie, C. 2012 Young Measures and Compactness in Measure Spaces. de Gruyter.CrossRefGoogle Scholar
Friedman, A. 2006 Stochastic Differential Equations and Applications. Dover.Google Scholar
Frisch, U. 1995 Turbulence: The Legacy of A. N. Kolmogorov. Cambridge University Press.Google Scholar
Gawȩdzki, K. 2008 Soluble models of turbulent transport. In Non-Equilibrium Statistical Mechanics and Turbulence (ed. Cardy, J., Falkovich, G. & Gawȩdzki, K.), London Mathematical Society Lecture Note Series, vol. 355, pp. 44–107. Cambridge University Press.CrossRefGoogle Scholar
Gawȩdzki, K.2013 Fluctuation relations in stochastic thermodynamics. arXiv:1308.1518.Google Scholar
Gawȩdzki, K. & Vergassola, M. 2000 Phase transition in the passive scalar advection. Physica D 138, 63–90.Google Scholar
Hartman, P. 2002 Ordinary Differential Equations, 2nd edn. SIAM.Google Scholar
Ikeda, N. & Watanabe, S. 1989 Stochastic Differential Equations and Diffusion Processes. North-Holland.Google Scholar
Itô, K. 1984 An Introduction to Probability Theory. Cambridge University Press.CrossRefGoogle Scholar
Jiřina, M. 1959 On regular conditional probabilities. Czech. Math. J. 9 (3), 445–451.Google Scholar
Jucha, J., Xu, H., Pumir, A. & Bodenschatz, E. 2014 Time-reversal-symmetry breaking in turbulence. Phys. Rev. Lett. 113 (5), 054501.Google Scholar
Kaneda, Y., Ishihara, T., Yokokawa, M., Itakura, K. & Uno, A. 2003 Energy dissipation rate and energy spectrum in high resolution direct numerical simulations of turbulence in a periodic box. Phys. Fluids 15, L21–L24.CrossRefGoogle Scholar
Kloeden, P. E. & Platen, E. 1992 Numerical Solutions of Stochastic Differential Equations. Springer.Google Scholar
Kolmogorov, A. N. 1941a Dissipation of energy in locally isotropic turbulence. Dokl. Akad. Nauk SSSR 32 (1), 16–18.Google Scholar
Kolmogorov, A. N. 1941b Equations of turbulent motion in an incompressible fluid. Dokl. Akad. Nauk SSSR 30, 299–303.Google Scholar
Kolmogorov, A. N. 1941c The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers. Dokl. Akad. Nauk SSSR 30 (4), 299–303.Google Scholar
Kraichnan, R. H. 1968 Small-scale structure of a scalar field convected by turbulence. Phys. Fluids 11, 945–953.CrossRefGoogle Scholar
Kunita, H. 1997 Stochastic Flows and Stochastic Differential Equations, Cambridge Studies in Advanced Mathematics, vol. 24. Cambridge University Press.Google Scholar
Kupiainen, A. 2003 Nondeterministic dynamics and turbulent transport. Ann. Henri Poincaré 4 (Suppl. 2), S713–S726.Google Scholar
Le Jan, Y. & Raimond, O. 2002 Integration of Brownian vector fields. Ann. Probab. 30, 826–873.CrossRefGoogle Scholar
Le Jan, Y. & Raimond, O. 2004 Flows, coalescence and noise. Ann. Probab. 32, 1247–1315.Google Scholar
Li, Y., Perlman, E., Wan, M., Yang, Y., Burns, R., Meneveau, C., Chen, S., Szalay, A. & Eyink, G. L. 2008 A public turbulence database cluster and applications to study Lagrangian evolution of velocity increments in turbulence. J. Turbul. 9, N31.Google Scholar
Lions, P. L. 1996 Mathematical Topics in Fluid Mechanics: Volume 1: Incompressible Models. Oxford University Press.Google Scholar
Novikov, E. A. 1965 Functionals and the random-force method in turbulence theory. Sov. Phys. JETP 20 (5), 1290–1294.Google Scholar
Obukhov, A. M. 1949 Structure of the temperature field in turbulent flow. Izv. Akad. Nauk. SSSR, Ser. Geogr. i Geofiz. 13, 58–69.Google Scholar
Oksendal, B. 2013 Stochastic Differential Equations: An Introduction with Applications. Springer Science & Business Media.Google Scholar
Onsager, L. 1945 The distribution of energy in turbulence. Phys. Rev. 68, 286.Google Scholar
Onsager, L. 1949 Nuovo Cim. Suppl. VI, 279–287.CrossRefGoogle Scholar
Pearson, B. R., Krogstad, P.-A. & van de Water, W. 2002 Measurements of the turbulent energy dissipation rate. Phys. Fluids 14, 1288–1290.CrossRefGoogle Scholar
Rezakhanlou, F.2014 Regular flows for diffusions with rough drifts. arXiv:1405.5856.Google Scholar
Richardson, L. F. 1926 Atmospheric diffusion shown on a distance-neighbor graph. Proc. R. Soc. Lond. A 110, 709–737.Google Scholar
Robert, R. 1991 A maximum-entropy principle for two-dimensional perfect fluid dynamics. J. Stat. Phys. 65 (3–4), 531–553.Google Scholar
Robert, R. & Sommeria, J. 1991 Statistical equilibrium states for two-dimensional flows. J. Fluid Mech. 229, 291–310.Google Scholar
Rudin, W. 2006 Functional Analysis. McGraw-Hill.Google Scholar
Saffman, P. G. 1960 On the effect of the molecular diffusivity in turbulent diffusion. J. Fluid Mech. 8 (02), 273–283.Google Scholar
Salmon, R. 1988 Hamiltonian fluid mechanics. Annu. Rev. Fluid Mech. 20 (1), 225–256.Google Scholar
Sawford, B. 2001 Turbulent relative dispersion. Annu. Rev. Fluid Mech. 33 (1), 289–317.Google Scholar
Sawford, B. L., Yeung, P. K. & Borgas, M. S. 2005 Comparison of backwards and forwards relative dispersion in turbulence. Phys. Fluids 17, 095109.Google Scholar
Schuster, H. G., Klages, R., Just, W. & Jarzynski, C. 2013 Nonequilibrium Statistical Physics of Small Systems: Fluctuation Relations and Beyond. Wiley.Google Scholar
Shlesinger, M. F., West, B. J. & Klafter, J. 1987 Lévy dynamics of enhanced diffusion: application to turbulence. Phys. Rev. Lett. 58 (11), 1100.Google Scholar
Silverman, B. W. 1986 Density Estimation for Statistics and Data Analysis. CRC Press.Google Scholar
Sommeria, J., Staquet, C. & Robert, R. 1991 Final equilibrium state of a two-dimensional shear layer. J. Fluid Mech. 233, 661–689.CrossRefGoogle Scholar
Sreenivasan, K. R. 1998 An update on the energy dissipation rate in isotropic turbulence. Phys. Fluids 10 (2), 528–529.Google Scholar
Sreenivasan, K. R. & Schumacher, J. 2010 Lagrangian views on turbulent mixing of passive scalars. Phil. Trans. R. Soc. Lond. A 368 (1916), 1561–1577.Google ScholarPubMed
Stevenson, R. M. 1981 Optimized perturbation theory. Phys. Rev. D 23 (12), 2916.Google Scholar
Taylor, G. I. 1917 Observations and speculations on the nature of turbulent motion. In Reports and Memoranda of the Advisory Committee for Aeronautics, No. 345 (ed. Batchelor, G. K.), Reprinted in The Scientific Papers of Sir Geoffrey Ingram Taylor: Volume 2, Meteorology, Oceanography and Turbulent Flow. Cambridge University Press, 1960.Google Scholar
Taylor, G. I. 1922 Diffusion by continuous movements. Proc. Lond. Math. Soc. 20 (1), 196–212.Google Scholar
Taylor, G. I. & Green, A. E. 1937 Mechanism of the production of small eddies from large ones. Proc. R. Soc. Lond. A 158 (895), 499–521.Google Scholar
Thalabard, S., Krstulovic, G. & Bec, J. 2014 Turbulent pair dispersion as a continuous-time random walk. J. Fluid Mech. 755, R4.Google Scholar
Tran, C. V. & Dritschel, D. G. 2006 Vanishing enstrophy dissipation in two-dimensional Navier–Stokes turbulence in the inviscid limit. J. Fluid Mech. 559, 107–116.Google Scholar
Tsinober, A. 2009 An Informal Conceptual Introduction to Turbulence. Springer.CrossRefGoogle Scholar
Valadier, M. 1973 Désintégration d’une mesure sur un produit. C. R. Acad. Sci. A 276, 33–35.Google Scholar
Valadier, M. 1994 A course on Young measures. Rend. Istit. Mat. Univ. Trieste, Supplemento XXVI, 349–394.Google Scholar
Vanden-Eijnden, E. W. & Vanden-Eijnden, E. 2000 Generalized flows, intrinsic stochasticity and turbulent transport. Proc. Natl Acad. Sci. USA 97, 8200–8205.Google Scholar
Vanden-Eijnden, E. W. & Vanden-Eijnden, E. 2001 Turbulent Prandtl number effect on passive scalar advection. Physica D 152–153, 636–645.Google Scholar
Yaglom, A. M. 1949 On the local structure of the temperature field in a turbulent flow. Dokl. Acad. Sci. USSR 69 (6), 743.Google Scholar
Yeung, P. K., Donzis, D. A. & Sreenivasan, K. R. 2005 High-Reynolds-number simulation of turbulent mixing. Phys. Fluids 17 (8), 081703.Google Scholar
Yu, H., Kanov, K., Perlman, E., Graham, J., Frederix, E., Burns, R., Szalay, A., Eyink, G. L. & Meneveau, C. 2012 Studying Lagrangian dynamics of turbulence using on-demand fluid particle tracking in a public turbulence database. J. Turbul. 12, N13.Google Scholar