Skip to main content Accessibility help

Turbulence of generalised flows in two dimensions

  • Simon Thalabard (a1) and Jérémie Bec (a2)


This paper discusses the generalised least-action principle and the associated concept of generalised flow introduced by Brenier (J. Am. Math. Soc., vol. 2, 1989, pp. 225–255), from the perspective of turbulence modelling. In essence, Brenier’s generalised least-action principle is a probabilistic generalisation of Arnold’s geometric interpretation of ideal fluid mechanics, whereby strong solutions to the Euler equations are deduced from minimising an action over Lagrangian maps. While Arnold’s framework relies on the deterministic concept of Lagrangian flow, Brenier’s least-action principle describes solutions to the Euler equations in terms of non-deterministic generalised flows, namely probability measures over sets of Lagrangian trajectories. In concept, generalised flows seem naturally fit to describe turbulent Lagrangian trajectories in terms of stochastic processes, an approach that originates from Richardson’s seminal work on turbulent dispersion. Still, Brenier’s generalised least-action principle has so far hardly found any practical application in the realm of fluid mechanics, let alone for turbulence modelling. The purpose of the present paper is to provide a hydrodynamical perspective on Brenier’s principle, and to assess its skills at coarse graining the Lagrangian motion of fluid particles, in order to reconstruct the space–time dynamics of the underlying Eulerian velocity fields. In practice, we rely on a statistical-mechanics interpretation of the concept of generalised flows, whereby the latter become akin to statistical ensembles of suitably defined ‘permutation flows’. We then employ Monte Carlo techniques to numerically solve the generalised least-action principle, and analyse the Eulerian and Lagrangian statistical features of the associated generalised flows. For simplicity, we restrict ourselves to two dimensions of space and consider situations of increasing complexity, ranging from solid rotation and cellular flows to freely decaying two-dimensional turbulence. Our analysis highlights a major caveat of the generalised variational principle. When used over long time lags, e.g. longer than a well-defined hydrodynamic turnover time, it generates artificial generalised flows, with non-physical statistical features. We argue that this limitation is not specifically inherent to Brenier’s formulation, but rather to the variational framework being formulated as a two-end boundary-value problem. When appropriately used over sufficiently short times, the generalised least-action principle is however relevant, and generalised flows can be explicitly constructed, that represent a space–time coarse graining of the underlying dynamics. We show numerical evidence that Brenier’s principle may then even accommodate irreversible Eulerian behaviours. This suggests that, if carefully used, generalised variational formulations could provide new tools to coarse grain genuine multi-scale hydrodynamics.


Corresponding author

Email address for correspondence:


Hide All
Abarbanel, H. D., Holm, D. D., Marsden, J. E. & Ratiu, T. S. 1986 Nonlinear stability analysis of stratified fluid equilibria. Phil. Trans. R. Soc. Lond. A 318 (1543), 349409.
Antonia, R. A. & Burattini, P. 2006 Approach to the 4/5 law in homogeneous isotropic turbulence. J. Fluid Mech. 550, 175184.
Arnold, V. I. 1966 Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits. Ann. Inst. Fourier 16, 319361.
Arnold, V. I. 2013 Mathematical Methods of Classical Mechanics, Graduate Texts in Mathematics, vol. 60. Springer.
Batchelor, G. K. 1969 Computation of the energy spectrum in homogeneous two-dimensional turbulence. Phys. Fluids 12, II-233.
Baxter, R. J. 2016 Exactly Solved Models in Statistical Mechanics. Elsevier.
Benamou, J.-D., Carlier, G., Cuturi, M., Nenna, L. & Peyré, G. 2015 Iterative Bregman projections for regularized transportation problems. SIAM J. Sci. Comput. 37, A1111A1138.
Benamou, J.-D., Carlier, G. & Nenna, L. 2019 Generalized incompressible flows, multi-marginal transport and Sinkhorn algorithm. Numer. Math. 142, 3354.
Bernard, D.2000 Turbulence for (and by) amateurs. Preprint, arXiv:cond-mat/0007106.
Binder, K. 1986 Introduction: theory and ‘technical’ aspects of Monte Carlo simulations. In Monte Carlo Methods in Statistical Physics, pp. 145. Springer.
Bouchet, F. & Corvellec, M. 2010 Invariant measures of the 2D Euler and Vlasov equations. J. Stat. Mech. Theor. Exp. 2010, P08021.
Bouchet, F. & Venaille, A. 2012 Statistical mechanics of two-dimensional and geophysical flows. Phys. Rep. 515, 227295.
Brenier, Y. 1989 The least action principle and the related concept of generalized flows for incompressible perfect fluids. J. Am. Math. Soc. 2, 225255.
Brenier, Y. 1999 Minimal geodesics on groups of volume-preserving maps and generalized solutions of the Euler equations. Commun. Pure Appl. Maths 52, 411452.
Brenier, Y. 2008 Generalized solutions and hydrostatic approximation of the Euler equations. Physica D 237, 19821988.
Brenier, Y., De Lellis, C. & Székelyhidi, L. 2011 Weak-strong uniqueness for measure-valued solutions. Commun. Math. Phys. 305, 351361.
Buckmaster, T. 2015 Onsager’s conjecture almost everywhere in time. Commun. Math. Phys. 333, 11751198.
Buckmaster, T., Lellis, C. & Székelyhidi, L. 2016 Dissipative Euler flows with Onsager-critical spatial regularity. Commun. Pure Appl. Maths 69, 16131670.
Dematteis, G., Grafke, T., Onorato, M. & Vanden-Eijnden, E.2019 Experimental evidence of hydrodynamic instantons: the universal route to rogue waves. Preprint, arXiv:1907.01320.
DiPerna, R. J. & Majda, A. J. 1987 Oscillations and concentrations in weak solutions of the incompressible fluid equations. Commun. Math. Phys. 108, 667689.
Dmitruk, P. & Montgomery, D. C. 2005 Numerical study of the decay of enstrophy in a two-dimensional Navier–Stokes fluid in the limit of very small viscosities. Phys. Fluids 17, 035114.
Duchon, J. & Robert, R. 2000 Inertial energy dissipation for weak solutions of incompressible Euler and Navier–Stokes equations. Nonlinearity 13, 249255.
Ebin, D. G. & Marsden, J. 1970 Groups of diffeomorphisms and the motion of an incompressible fluid. Ann. Maths 92, 102163.
Eyink, G. L. 2001 Dissipation in turbulent solutions of 2D Euler equations. Nonlinearity 14, 787802.
Eyink, G. L. 2002 Local 4/5-law and energy dissipation anomaly in turbulence. Nonlinearity 16, 137145.
Eyink, G. L. & Sreenivasan, K. R. 2006 Onsager and the theory of hydrodynamic turbulence. Rev. Mod. Phys. 78, 87135.
Falkovich, G., Gawȩdzki, K. & Vergassola, M. 2001 Particles and fields in fluid turbulence. Rev. Mod. Phys. 73, 913975.
Farazmand, M. & Serra, M.2018 Variational Lagrangian formulation of the Euler equations for incompressible flow: a simple derivation. Preprint, arXiv:1807.02726.
Gallouët, T. & Mérigot, Q. 2018 A Lagrangian scheme à la Brenier for the incompressible Euler equations. Found. Comput. Maths 18, 835865.
Gawȩdzki, K. 2001 Turbulent advection and breakdown of the Lagrangian flow. In Intermittency in Turbulent Flows (ed. Vassilicos, J. C.), pp. 86104. Cambridge University Press.
Holm, D. D. & Kupershmidt, B. A. 1983 Noncanonical hamiltonian formulation of ideal magnetohydrodynamics. Physica D 7, 330333.
Holm, D. D., Marsden, J. E., Ratiu, T. & Weinstein, A. 1985 Nonlinear stability of fluid and plasma equilibria. Phys. Rep. 123, 1116.
Isett, P. 2018 A proof of Onsager’s conjecture. Ann. Maths 188, 871963.
José, J. & Saletan, E. 2000 Classical Dynamics: A Contemporary Approach. Cambridge University Press.
Khesin, B. & Arnold, V. I. 2005 Topological fluid dynamics. Not. Am. Math. Soc. 52, 919.
Kraus, M., Tassi, E. & Grasso, D. 2016 Variational integrators for reduced magnetohydrodynamics. J. Comput. Phys. 321, 435458.
Laurie, J. & Bouchet, F. 2015 Computation of rare transitions in the barotropic quasi-geostrophic equations. New J. Phys. 17, 015009.
Lecoanet, D. & Kerswell, R. R. 2018 Connection between nonlinear energy optimization and instantons. Phys. Rev. E 97, 012212.
Lopes Filho, M. C., Mazzucato, A. L. & Nussenzveig Lopes, H. J. 2006 Weak solutions, renormalized solutions and enstrophy defects in 2d turbulence. Arch. Rat. Mech. Anal. 179, 353387.
Marsden, J. E. & West, M. 2001 Discrete mechanics and variational integrators. Acta Numer. 10, 357514.
Mérigot, Q. & Mirebeau, J.-M. 2016 Minimal geodesics along volume-preserving maps, through semidiscrete optimal transport. SIAM J. Numer. Anal. 54, 34653492.
Miller, J., Weichman, P. B. & Cross, M. C. 1992 Statistical mechanics, Euler’s equation, and Jupiter’s red spot. Phys. Rev. A 45, 23282359.
Morrison, P. J. 1998 Hamiltonian description of the ideal fluid. Rev. Mod. Phys. 70, 467521.
Morrison, P. J. 2005 Hamiltonian and action principle formulations of plasma physics. Phys. Plasmas 12, 058102.
Nenna, L.2016 Numerical methods for multi-marginal optimal transportation. PhD thesis, PSL Research University.
Nocedal, J. & Wright, S. 2006 Numerical Optimization. Springer Science & Business Media.
Onsager, L. 1949 Statistical hydrodynamics. Nuovo Cimento 6, 279287.
Protas, B. 2008 Vortex dynamics models in flow control problems. Nonlinearity 21 (9), R203.
Protas, B., Noack, B. R. & Östh, J. 2015 Optimal nonlinear eddy viscosity in Galerkin models of turbulent flows. J. Fluid Mech. 766, 337367.
Robert, R. & Sommeria, J. 1991 Statistical equilibrium states for two-dimensional flows. J. Fluid Mech. 229, 291310.
Salmon, R. 1983 Practical use of Hamilton’s principle. J. Fluid Mech. 132, 431444.
Salmon, R. 1985 New equations for nearly geostrophic flow. J. Fluid Mech. 153, 461477.
Salmon, R. 1988 Hamiltonian fluid mechanics. Annu. Rev. Fluid Mech. 20, 225256.
Saw, E.-W., Debue, P., Kuzzay, D., Daviaud, F. & Dubrulle, B. 2018 On the universality of anomalous scaling exponents of structure functions in turbulent flows. J. Fluid Mech. 837, 657669.
Shepherd, T. G. 1990 Symmetries, conservation laws, and Hamiltonian structure in geophysical fluid dynamics. In Adv. Geophys., vol. 32, pp. 287338. Elsevier.
Shnirelman, A. 2000 Weak solutions with decreasing energy of incompressible euler equations. Commun. Math. Phys. 210, 541603.
Sreenivasan, K. R. 1984 On the scaling of the turbulence energy dissipation rate. Phys. Fluids 27, 10481051.
Thalabard, S. & Turkington, B. 2017 Optimal response to non-equilibrium disturbances under truncated Burgers–Hopf dynamics. J. Phys. A 50, 175502.
Tu, J. H., Griffin, J., Hart, A., Rowley, C. W., Cattafesta, L. N. & Ukeiley, L. S. 2013 Integration of non-time-resolved PIV and time-resolved velocity point sensors for dynamic estimation of velocity fields. Exp. Fluids 54, 1429.
Van Nguyen, L., Laval, J.-P. & Chainais, P. 2015 A Bayesian fusion model for space-time reconstruction of finely resolved velocities in turbulent flows from low resolution measurements. J. Stat. Mech. 2015, P10008.
Vigdorovich, I. 2018 Enstrophy spectrum in freely decaying two-dimensional self-similar turbulent flow. Phys. Rev. E 98, 033110.
Zeitlin, V. 2004 Self-consistent finite-mode approximations for the hydrodynamics of an incompressible fluid on nonrotating and rotating spheres. Phys. Rev. Lett. 93, 264501.
MathJax is a JavaScript display engine for mathematics. For more information see

JFM classification

Related content

Powered by UNSILO

Turbulence of generalised flows in two dimensions

  • Simon Thalabard (a1) and Jérémie Bec (a2)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.