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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
We remark that Frishman and Falkovich (2014) argued on theoretical grounds that unlike Eq. (6), the short-time expansion of the difference of forward/backward dispersion appearing in (9) for incompressible Navier–Stokes should have a finite radius of convergence at a fine-grained level (\(\ell \equiv 0\)), even in the limit of \(\nu \rightarrow 0\). This remarkable property may be useful to bridge the gap between the asymptotically short time results presented here and the observations of Richardson dispersion at later times.
We remark that this is an asymptotic statement related to the cascade at arbitrarily small scales. It does not imply that the cascade rate is constant (or even positive) throughout all scales in the inertial range, although in practice this is very often observed.
More correctly, the dual cascade picture was predicted by Kraichnan to occur in a statistically steady state for a fluid with large-scale damping (such as linear friction of hyperviscosity) and viscosity. These two effects impost cutoff wave numbers; damping imposes \(k_{ir}\) is an infrared cutoff and viscosity \(k_{uv}\) is the corresponding ultraviolet. Then, the inverse energy cascade range is predicted to be confined to \( k_{ir} \ll k \ll k_f\), whereas the direct enstrophy range to \(k_{f} \ll k \ll k_{uv}\). For simplicity, in our analysis, we consider forced Euler equations, neglecting the effects of large-scale damping and viscosity. However, our conclusions can easily be modified to accommodate the presence of a damping term and for Navier–Stokes solutions in the limit where viscosity \(\nu \) is taken to zero before all others discussed in this section.
This depends, of course, on the choice of forcing scheme. For example, energy input is ensured if the forcing is chosen to be solution dependent, for example, small-scale Lundgren forcing of the form \(f=\alpha \mathbf{P}_{k_f}[{{u}}]\) with \(\alpha :=\alpha (k_f)>0\) and \(\mathbf{P}_{k_f}\) is the projection onto a shell around \(k_f\) in wave number space. Another attractive choice of force is to take f to be a homogenous Gaussian random field which is white noise correlated in time, i.e. \(\langle f_i ({{x}},t) f_j({{x}}',t')\rangle = 2F_{ij}({{x}}-{{x}}')\delta (t-t')\). This has the theoretical advantage that, after averaging over the forcing statistics, the mean injection rate of energy is solution independent; i.e. after averaging balance (21), the injection term is \(\langle {{u}}\cdot f\rangle =F_{ii}(0)>0\), ensuring input of energy on average.
We are grateful to P. Isett for pointing out an improvement of Prop 1 from an early preprint which we present here.
Where the notation \(f(\tau )=o(\tau ^3)\) denotes \(\lim _{\tau \rightarrow 0} f(\tau )/\tau ^3\rightarrow 0\).
References
Batchelor, G.K.: Computation of the energy spectrum in homogeneous two-dimensional turbulence. Phys. Fluids 12(12), II-233 (1969)
Bitane, R., Homann, H., Bec, J.: Time scales of turbulent relative dispersion. Phys. Rev. E 86(4), 045302 (2012)
Buckmaster, T., De Lellis. C., Székelyhidi Jr, L., Vicol, V.: Onsager’s conjecture for admissible weak solutions. arXiv preprint arXiv:1701.08678 (2017)
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)
De Lellis, C., Székelyhidi Jr., L.: The \(h\)-principle and the equations of fluid dynamics. Bull. Am. Math. Soc. 49, 347–375 (2012)
Drivas, T.D., Eyink, G.L.: An Onsager singularity theorem for Leray solutions of incompressible Navier–Stokes. arXiv preprint arXiv:1710.05205 (2017)
Drivas, T.D.: Anomalous Dissipation, Spontaneous Stochasticity & Onsager’s Conjecture. Doctoral Dissertation. Johns Hopkins University (2017)
Drivas, T.D., Eyink, G.L.: An Onsager singularity theorem for turbulent solutions of compressible Euler equations. Commun. Math. Phys. 359, 733 (2018)
Duchon, J., Robert, R.: Inertial energy dissipation for weak solutions of incompressible Euler and Navier–Stokes equations. Nonlinearity 13(1), 249–255 (2000)
Eyink, G.L.: Turbulence Theory. Course Notes. http://www.ams.jhu.edu/~eyink/Turbulence/ (2015)
Eyink, G.L.: Energy dissipation without viscosity in ideal hydrodynamics I. Fourier analysis and local energy transfer. Phys. D 78, 222–240 (1994)
Eyink, G.L.: Exact results on stationary turbulence in 2d: consequences of vorticity conservation. Physica D 91(1–2), 97–142 (1996)
Eyink, G.L.: Local 4/5-law and energy dissipation anomaly in turbulence. Nonlinearity 16(1), 137 (2002)
Eyink, G.L., Drivas, T.D.: Spontaneous stochasticity and anomalous dissipation for Burgers equation. J. Stat. Phys. 158(2), 386–432 (2015)
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)
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)
Falkovich, G., Gawȩdzki, K., Vergassola, M.: Particles and fields in fluid turbulence. Rev. Mod. Phys. 73, 913–975 (2001)
Fjørtoft, R.: On the changes in the spectral distribution of kinetic energy for two-dimensional, non-divergent flow. Tellus 5(3), 225–230 (1953)
Frishman, A., Falkovich, G.: New type of anomaly in turbulence. Phys. Rev. Lett. 113(2), 024501 (2014)
Isett, P.: A proof of Onsager’s conjecture. arXiv preprint arXiv:1608.08301 (2016)
Jucha, J., Xu, H., Pumir, A., Bodenschatz, E.: Time-reversal-symmetry breaking in turbulence. Phys. Rev. Lett. 113(5), 054501 (2014)
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)
Kolmogorov, A.N.: Dissipation of energy in locally isotropic turbulence. Dokl. Akad. Nauk SSSR 32(1), 16–18 (1941)
Kraichnan, R.H.: Inertial ranges in two-dimensional turbulence. Phys. Fluids 10(7), 1417–1423 (1967)
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)
Leith, C.E.: Diffusion approximation for two-dimensional turbulence. Phys. Fluids 11(3), 671–672 (1968)
Lions, P.L.: Mathematical Topics in Fluid Mechanics. Vol. 1, Volume 3 of Oxford Lecture Series in Mathematics and its Applications (1996)
Onsager, L.: Statistical hydrodynamics. Il Nuovo Cimento (Supplemento) 6, 279–287 (1949)
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)
Pearson, B.R., Krogstad, P.A., van de Water, W.: Measurements of the turbulent energy dissipation rate. Phys. Fluids 14, 1288–1290 (2002)
Richardson, L.F.: Atmospheric diffusion shown on a distance-neighbor graph. Proc. Roy. Soc. Lond. A 110, 709–737 (1926)
Sawford, B.L., Yeung, P.K., Borgas, M.S.: Comparison of backwards and forwards relative dispersion in turbulence. Phys. Fluids 17, 095109 (2005)
Shnirelman, A.: On the nonuniqueness of weak solution of the Euler equation. Commun. Pure Appl. Math. 50(12), 1261–1286 (1997)
Sreenivasan, K.R.: On the scaling of the turbulence energy dissipation rate. Phys. Fluids 27, 1048–1051 (1984)
Sreenivasan, K.R.: An update on the energy dissipation rate in isotropic turbulence. Phys. Fluids 10, 528–529 (1998)
Taylor, G.I.: Eddy motion in the atmosphere. Proc. Royal Soc. A 215, 1–26 (1915)
Taylor, G.I.: Motion of solids in fluids when the flow is not irrotational. Proc. Royal Soc. A 93(648), 99–113 (1917)
Xu, H., Pumir, A., Bodenschatz, E.: Lagrangian view of time irreversibility of fluid turbulence. Sci. China Phys. Mech. Astron. 59(1), 1–9 (2016)
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Charles R. Doering.
Rights and permissions
About this article
Cite this article
Drivas, T.D. Turbulent Cascade Direction and Lagrangian Time-Asymmetry. J Nonlinear Sci 29, 65–88 (2019). https://doi.org/10.1007/s00332-018-9476-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00332-018-9476-8