Journal of Statistical Physics

, Volume 167, Issue 3–4, pp 596–625 | Cite as

Turbulence: Does Energy Cascade Exist?

  • Christophe Josserand
  • Martine Le Berre
  • Thierry Lehner
  • Yves Pomeau


To answer the question whether a cascade of energy exists or not in turbulence, we propose a set of correlation functions able to test if there is an irreversible transfert of energy, step by step, from large to small structures. These tests are applied to real Eulerian data of a turbulent velocity flow, taken in the wind grid tunnel of Modane, and also to a prototype model equation for wave turbulence. First we demonstrate the irreversible character of the flow by using multi-time correlation function at a given point of space. Moreover the unexpected behavior of the test function leads us to connect irreversibility and finite time singularities (intermittency). Secondly we show that turbulent cascade exists, and is a dynamical process, by using a test function depending on time and frequency. The cascade shows up only in the inertial domain where the kinetic energy is transferred more rapidly (on average) from the wavenumber \(k_{1}\) to \(k_{2}\) than from \(k_{1}\) to \(k'_{2}\) larger than \(k_{2}\).


Turbulence Irreversibility Out of equilibrium 



We thank Dr. Yves Gagne for having provided us the Eulerian velocity data of Modane’s experiment, which were taken by him et his collaborators, and we thank ONERA for facility.


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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Christophe Josserand
    • 1
  • Martine Le Berre
    • 2
  • Thierry Lehner
    • 3
  • Yves Pomeau
    • 4
  1. 1.Institut Jean Le Rond d’AlembertSorbonne Universités, CNRS & UPMC Univ Paris 06, UMR 7190ParisFrance
  2. 2.Institut des Sciences Moléculaires d’Orsay ISMO-CNRSUniversité Paris-Sud BatOrsay CedexFrance
  3. 3.Laboratoire LUTH, UMR CNRS n°8102Observatoire de Paris-MeudonMeudon cedexFrance
  4. 4.Department of MathematicsUniversity of ArizonaTucsonUSA

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