Advertisement

Journal of Nonlinear Science

, Volume 29, Issue 1, pp 65–88 | Cite as

Turbulent Cascade Direction and Lagrangian Time-Asymmetry

  • Theodore D. DrivasEmail author
Article

Abstract

We establish Lagrangian formulae for energy conservation anomalies involving the discrepancy between short-time two-particle dispersion forward and backward in time. These results are facilitated by a rigorous version of the Ott–Mann–Gawȩdzki relation, sometimes described as a “Lagrangian analogue of the 4 / 5-law.” In particular, we prove that for weak solutions of the Euler equations, the Lagrangian forward/backward dispersion measure matches onto the energy defect (Onsager in Nuovo Cimento (Supplemento) 6:279–287, 1949; Duchon and Robert in Nonlinearity 13(1):249–255, 2000) in the sense of distributions. For strong limits of \(d\ge 3\)-dimensional Navier–Stokes solutions, the defect distribution coincides with the viscous dissipation anomaly. The Lagrangian formula shows that particles released into a 3d turbulent flow will initially disperse faster backward in time than forward, in agreement with recent theoretical predictions of Jucha et al. (Phys Rev Lett 113(5):054501, 2014). In two dimensions, we consider strong limits of solutions of the forced Euler equations with increasingly high-wave number forcing as a model of an ideal inverse cascade regime. We show that the same Lagrangian dispersion measure matches onto the anomalous input from the infinite-frequency force. As forcing typically acts as an energy source, this leads to the prediction that particles in 2d typically disperse faster forward in time than backward, which is opposite to that which occurs in 3d. Time asymmetry of the Lagrangian dispersion is thereby closely tied to the direction of the turbulent cascade, downscale in \(d\ge 3\) and upscale in \(d=2\). These conclusions lend support to the conjecture of Eyink and Drivas (J Stat Phys 158(2):386–432, 2015) that a similar connection holds for time asymmetry of Richardson two-particle dispersion and cascade direction.

Keywords

Turbulence Inviscid limit Time irreversibility Navier-Stokes 

Mathematics Subject Classification

35Q30 35Q31 76F02 

Notes

Acknowledgements

I am grateful to G. Eyink for numerous helpful suggestions and discussions. I would also like to thank P. Constantin, N. Constantinou, A. Frishman, P. Isett, H.Q. Nguyen, V. Vicol, and M. Wilczek for their comments. I would also like to thank the anonymous referees for comments that greatly improved the paper. Research of the author is supported by NSF-DMS grant 1703997.

References

  1. Batchelor, G.K.: Computation of the energy spectrum in homogeneous two-dimensional turbulence. Phys. Fluids 12(12), II-233 (1969)CrossRefzbMATHGoogle Scholar
  2. Bitane, R., Homann, H., Bec, J.: Time scales of turbulent relative dispersion. Phys. Rev. E 86(4), 045302 (2012)CrossRefGoogle Scholar
  3. Buckmaster, T., De Lellis. C., Székelyhidi Jr, L., Vicol, V.: Onsager’s conjecture for admissible weak solutions. arXiv preprint arXiv:1701.08678 (2017)
  4. Constantin, P., W, E., Titi, E.: Onsager’s conjecture on the energy conservation for solutions of Euler’s equation. Commun. Math. Phys. 165, 207–209 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  5. De Lellis, C., Székelyhidi Jr., L.: The \(h\)-principle and the equations of fluid dynamics. Bull. Am. Math. Soc. 49, 347–375 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  6. Drivas, T.D., Eyink, G.L.: An Onsager singularity theorem for Leray solutions of incompressible Navier–Stokes. arXiv preprint arXiv:1710.05205 (2017)
  7. Drivas, T.D.: Anomalous Dissipation, Spontaneous Stochasticity & Onsager’s Conjecture. Doctoral Dissertation. Johns Hopkins University (2017)Google Scholar
  8. Drivas, T.D., Eyink, G.L.: An Onsager singularity theorem for turbulent solutions of compressible Euler equations. Commun. Math. Phys. 359, 733 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  9. Duchon, J., Robert, R.: Inertial energy dissipation for weak solutions of incompressible Euler and Navier–Stokes equations. Nonlinearity 13(1), 249–255 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  10. Eyink, G.L.: Turbulence Theory. Course Notes. http://www.ams.jhu.edu/~eyink/Turbulence/ (2015)
  11. Eyink, G.L.: Energy dissipation without viscosity in ideal hydrodynamics I. Fourier analysis and local energy transfer. Phys. D 78, 222–240 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  12. Eyink, G.L.: Exact results on stationary turbulence in 2d: consequences of vorticity conservation. Physica D 91(1–2), 97–142 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  13. Eyink, G.L.: Local 4/5-law and energy dissipation anomaly in turbulence. Nonlinearity 16(1), 137 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  14. Eyink, G.L., Drivas, T.D.: Spontaneous stochasticity and anomalous dissipation for Burgers equation. J. Stat. Phys. 158(2), 386–432 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  15. Faber, T., Vassilicos, J.C.: Turbulent pair separation due to multiscale stagnation point structure and its time asymmetry in two-dimensional turbulence. Phys. Fluids 21(1), 015106 (2009)CrossRefzbMATHGoogle Scholar
  16. Falkovich, G., Frishman, A.: Single flow snapshot reveals the future and the past of pairs of particles in turbulence. Phys. Rev. Lett. 110(21), 214502 (2013)CrossRefGoogle Scholar
  17. Falkovich, G., Gawȩdzki, K., Vergassola, M.: Particles and fields in fluid turbulence. Rev. Mod. Phys. 73, 913–975 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  18. Fjørtoft, R.: On the changes in the spectral distribution of kinetic energy for two-dimensional, non-divergent flow. Tellus 5(3), 225–230 (1953)MathSciNetCrossRefGoogle Scholar
  19. Frishman, A., Falkovich, G.: New type of anomaly in turbulence. Phys. Rev. Lett. 113(2), 024501 (2014)CrossRefGoogle Scholar
  20. Isett, P.: A proof of Onsager’s conjecture. arXiv preprint arXiv:1608.08301 (2016)
  21. Jucha, J., Xu, H., Pumir, A., Bodenschatz, E.: Time-reversal-symmetry breaking in turbulence. Phys. Rev. Lett. 113(5), 054501 (2014)CrossRefGoogle Scholar
  22. Kaneda, Y., Ishihara, T., Yokokawa, M., Itakura, K., Uno, A.: Energy dissipation rate and energy spectrum in high resolution direct numerical simulations of turbulence in a periodic box. Phys. Fluids 15, L21–L24 (2003)CrossRefzbMATHGoogle Scholar
  23. Kolmogorov, A.N.: Dissipation of energy in locally isotropic turbulence. Dokl. Akad. Nauk SSSR 32(1), 16–18 (1941)MathSciNetzbMATHGoogle Scholar
  24. Kraichnan, R.H.: Inertial ranges in two-dimensional turbulence. Phys. Fluids 10(7), 1417–1423 (1967)MathSciNetCrossRefGoogle Scholar
  25. Lee, T.D.: Difference between turbulence in a two-dimensional fluid and in a three- dimensional fluid. J. Appl. Phys. 22(4), 524–524 (1951)MathSciNetCrossRefzbMATHGoogle Scholar
  26. Leith, C.E.: Diffusion approximation for two-dimensional turbulence. Phys. Fluids 11(3), 671–672 (1968)CrossRefGoogle Scholar
  27. Lions, P.L.: Mathematical Topics in Fluid Mechanics. Vol. 1, Volume 3 of Oxford Lecture Series in Mathematics and its Applications (1996)Google Scholar
  28. Onsager, L.: Statistical hydrodynamics. Il Nuovo Cimento (Supplemento) 6, 279–287 (1949)MathSciNetCrossRefGoogle Scholar
  29. Ott, S., Mann, J.: An experimental investigation of the relative diffusion of particle pairs in three-dimensional turbulent flow. J. Fluid Mech. 422, 207–223 (2000)CrossRefzbMATHGoogle Scholar
  30. Pearson, B.R., Krogstad, P.A., van de Water, W.: Measurements of the turbulent energy dissipation rate. Phys. Fluids 14, 1288–1290 (2002)CrossRefGoogle Scholar
  31. Richardson, L.F.: Atmospheric diffusion shown on a distance-neighbor graph. Proc. Roy. Soc. Lond. A 110, 709–737 (1926)CrossRefGoogle Scholar
  32. Sawford, B.L., Yeung, P.K., Borgas, M.S.: Comparison of backwards and forwards relative dispersion in turbulence. Phys. Fluids 17, 095109 (2005)CrossRefzbMATHGoogle Scholar
  33. Shnirelman, A.: On the nonuniqueness of weak solution of the Euler equation. Commun. Pure Appl. Math. 50(12), 1261–1286 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  34. Sreenivasan, K.R.: On the scaling of the turbulence energy dissipation rate. Phys. Fluids 27, 1048–1051 (1984)CrossRefGoogle Scholar
  35. Sreenivasan, K.R.: An update on the energy dissipation rate in isotropic turbulence. Phys. Fluids 10, 528–529 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  36. Taylor, G.I.: Eddy motion in the atmosphere. Proc. Royal Soc. A 215, 1–26 (1915)CrossRefGoogle Scholar
  37. Taylor, G.I.: Motion of solids in fluids when the flow is not irrotational. Proc. Royal Soc. A 93(648), 99–113 (1917)CrossRefzbMATHGoogle Scholar
  38. Xu, H., Pumir, A., Bodenschatz, E.: Lagrangian view of time irreversibility of fluid turbulence. Sci. China Phys. Mech. Astron. 59(1), 1–9 (2016)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsPrinceton UniversityPrincetonUSA

Personalised recommendations