Wave Turbulence -Eric FALCON group

Non Linear Physics Group - Eric Falcon

Experimental studies on wave turbulence (papers by our group)

Wave turbulence is a domain rapidly expanding for several years. It concerns the study of the dynamical and statistical properties of a set of numerous waves in interaction. This is a ubiquitous phenomenon that occurs in various situations on very different scales: from spin waves in solids, internal or surface waves in oceanography up to plasma waves in astrophysics.

Wave turbulence theory, also called weak turbulence, assumes that the energy transfer is governed by resonant interaction between waves leading to an energy cascade from large (forcing) scales up to small (dissipative) ones. Although this theory from the end of the 60s can be applied to nearly all fields of physics involving weakly nonlinear waves, well-controlled laboratory experiments on wave turbulence remained few. The last years saw an important experimental effort, particularly in France, notably to probe the limits of validity of the theory based on very restrictive hypotheses (infinite size effects, weak nonlinearity, scale separation, constant energy flux, local interactions...). Experiments show the limitations of the current theoretical framework, which in return, arouses a theoretical and numerical renewed interest.

Our projects are funded by ANR Turbonde 2007-2011, ANR Turbulon 2012-2016, ANR Dysturb 2017-2022, and Simons Foundation 2019-2023.

Gravity-capillary wave turbulence: Laboratory experiments

For an overview: see our 2022 review in Annual Review of Fluid Mechanics.

Experimental reconstruction of a
strongly nonlinear wave field

We experimentally study and characterize gravity-capillary wave turbulence on the surface of a fluid to better understand the basic mechanisms of energy transfer between waves.

We have observed in laboratory the regime of gravity-capillary wave turbulence [1], and have reported the first observation of intermittency in wave turbulence [2]. The intermittency is shown to not come from some coherent structures at large scale (wavebreakings, capillary bursts on steep gravity waves) [10,11]. Moreover, two major experimental challenges have been faced: the measurement of injected power [3], and to measure the wave field fully resolved in space and time [12]. At the time, those quantities were not yet been measured directly for wave turbulence on the surface of a fluid. Two main results have then been obtained: (i) Energy transfer mechanisms are not restricted to purely resonant wave interactions, as assumed by the theory, but involved other mechanisms related to the presence of strong nonlinear waves (sharp crested waves, bound waves, ...) [12]; (ii) The system shows large injected power fluctuations within the fluid [3], fluctuations not taken into account by weak turbulence theory. We showed that the distribution of injected power fluctuations is well described by a simple model [7]. A book chapter has been also published on the fluctuations in wave turbulence [19], as well as a review on wave turbulence [6, 9].

We have then reported the first observation in laboratory of the direct gravity-capillary cascade when the fluid is not in a deep-water regime [14]. The study of non-stationary regime of capillary wave turbulence, when the forcing is stopped, led to the first observation of decay wave turbulence [16]. Another optical method (different form Fourier Transform Profilometry used in [12]) called Diffusing Light Videography, has been used to reconstruct the capillary wave field both in time and space. We have highlighted the role of strongly nonlinear capillary waves on the turbulent dynamics [18, 30]. The study of 3-wave interactions between gravity-capillary waves allows us to validate experimentally, for the first time for noncolinear waves, the theory of 3-wave resonant interactions [24]. We have also obtain the first indirect measurement of the energy flux at each scale of the turbulent cascade [21]. As a consequence, the highlighting of dissipation at all scales of the capillary turbulent cascade (not taken into account in the current stage of theoretical developments) allowed to understand the disagreements observed, these last years, in numerous experiences on capillary wave turbulence. The constant of the Kolmogorov-Zakharov spectrum was also inferred experimentally for the first time and compared with its theoretical value [21, 23]. Besides, we made the first direct numerical simulations of capillary wave turbulence from the two-phase Navier-Stokes equations [22]. These simulations confirm the validity of weak turbulence derivation when hypotheses are verified. Finally, we have studied for the first time wave turbulence on the interface between two immiscible fluids with free upper surface. We show that the coupling between free surface waves and interface waves modify strongly the wave turbulence regime [26].

Gravity wave turbulence: Large scale experiments

Gravity wave turbulence is of primordial interest in oceanography but remains still not well understood. Beyond the observation of the direct cascade of gravity wave turbulence, we showed that the wave spectrum is non-universal and depend on the forcing parameters [1]. We reported the first laboratory observations of an inverse cascade of gravity wave turbulence [15]. Moreover, we experimentally showed that a spatially homogeneous forcing leads to a good agreement with theoretical predictions [17], contrary to previous observations with a localized forcing with wavemakers.

We performed experiments in large-scale wave basin (50 m x 30 m x 5 m) at Ecole Centrale Nantes, France, involving 4 French laboratories (MSC/Univ. Paris Diderot, LPS/ENS, SPHINX/CEA Saclay, LHEEA/Ecole Centrale Nantes). The turbulent wave field is experimentally found to strongly depend on the basin boundary conditions (absorbing - beach, or reflecting - wall) although their statistical and spectral properties are close [23]. We have also studied resonant interactions between nonlinear waves that are the fundamental mechanism that transfers energy in wave turbulence. The study of 4-wave interactions between gravity waves allow us to validate experimentally, for the first time for noncolinear or perpendicular waves, the theory of 4-wave resonant interactions [25]. For stronger nonlinearities, meaningful departures from this weakly nonlinear theory are observed [28].

Wave turbulence in low-gravity environment

We studied purely capillary waves in low-gravity environment during CNES Parabolic Flight Campaigns. We have observed capillary wave turbulence on a broad range of scales usually masked by the gravity wave regime on Earth [8]. Another advantage is to have a system with no boundary, the fluid covering all the internal surface of the spherical cell in low-gravity. Various patterns (hexagons, lines) have thus been observed on the spherical fluid surface when the forcing is periodic [8]. See pictures of wave turbulence in space ; See also pictures of the team in Space

This experiment was performed with different forcing conditions and geometry onboard the International Space Station (ISS) in 2017 and 2018 during CNES Proxima mission of the French astronaut Thomas Pesquet. The experiment called FLUIDICS (FLUId DynamICs in Space) was co-funded by CNES and Airbus. Main results have been published recently [31,35].

Hydroelastic wave turbulence

Hydroelastic wave turbulence focuses on random waves propagating on the surface of a fluid covered by an elastic sheet. It has been obtained for the first time in laboratory [20]. The existence of three-wave interactions, predicted theoretically in this system, has just been highlighted experimentally [27].

Hydroelastic waves, including gravity-bending waves, are found in various domains: on the surface of lakes or oceans covered by ice, or for very large floating structures in oceanography, flapping flags, or in biomedical applications such as heart valves.

Magnetic wave turbulence

We report the first observation of magnetic wave turbulence on the surface of a ferrofluid submitted to a magnetic field, a regime that has not yet been envisaged in theoretical studies.

When wave amplitudes are high enough, the wave turbulence theory predicts a nonlinear resonant process between waves that generates smaller wavelengths. In a ferrofluid (a liquid with a suspension of nanometric magnetic particles), the dispersion relation of surface waves was known to be tuned by applying a magnetic field. This leads the authors to the first observation of a magnetic wave turbulence regime [5]. The existence domains of gravity and capillary wave turbulence are also documented as well as a triple point of coexistence of these three regimes: these new results are understood using dimensional analysis. Such an experimental system where the dispersion relation is tuned by the operator from a non-dispersive to a dispersive system is thus of primary interest to test the wave turbulence theory. The case of a magnetic field parallel to the fluid surface shows several differences with the normal case. The striking one is the meaningful broading of the inertial domain of the magnetic wave turbulence regime [13].


PUBLICATIONS on wave turbulence and wave interactions:

39. E. Falcon and N. Mordant, 2022
      Annual Review of Fluid Mechanics 54, 1-25 (2022)
      Experiments in gravity-capillary wave turbulence

38. G. Ricard and E. Falcon, 2021
      Europhysics Letters (EPL) 135, 64001 (2021)
      Experimental quasi-1D capillary-wave turbulence on the surface of a fluid

37. E. Kochurin, G. Ricard, N. Zubarev & E. Falcon 2020

JETP Letters 112, 757 - 763 (2020)
Numerical Simulation of Collinear Capillary-Wave Turbulence

36. E. Falcon, G. Michel, G. Prabhudesai, A. Cazaubiel, M. Berhanu, N. Mordant, S. Aumaître & F. Bonnefoy 2020

Physical Review Letters 125, 134501 (2020)
Saturation of the Inverse Cascade in Surface Gravity-Wave Turbulence

35. M. Berhanu, E. Falcon, G. Michel, C. Gissinger & S. Fauve 2019
EPL (Europhysics Letters) 128, 34001 (2019)
Capillary wave turbulence experiments in microgravity

34. A. Cazaubiel, S. Mawet, A. Darras, G. Grojean, J. J. W. A van Loon, S. Dorbolo & E. Falcon 2019

Physical Review Letters 123, 244501 (2019)
Wave turbulence on the surface of a fluid in a high-gravity environment

33. A. Cazaubiel, F. Haudin, E. Falcon & M. Berhanu 2019
       Physical Review Fluids 4, 074803 (2019)
       Forced three-wave interactions of gravity-capillary surface waves

32. E. Falcon 2019
       Applications in Nonlinear Dynamics (V. In, P. Longhini, A. Palacios, Eds.), Springer Nature, Chap. 25, pp. 259 – 266 (2019)
       Wave Turbulence: A Set of Stochastic Nonlinear Waves in Interaction

31. M. Berhanu, E. Falcon & S. Fauve 2018,
Wave turbulence in microgravity
Report to COSPAR (World Committee for Space Research), 42th Scientific Assembly, 14-22 July 2018, Pasasena, USA, CNES Ed., p. 66 - 67 (2018)

30. M. Berhanu, E. Falcon & L. Deike 2018
       Journal of Fluid Mechanics 850, 803 (2018)
       Turbulence of capillary waves forced by steep gravity waves

29. G. Michel, B. Semin, A. Cazaubiel, F. Haudin, T. Humbert, S. Lepot, F. Bonnefoy, M. Berhanu & E. Falcon 2018
       Physical Review Fluids 3, 054801 (2018)
       Self-similar gravity wave spectra resulting from the modulation of bound waves

28. F. Bonnefoy, F. Haudin, G. Michel, B. Semin, T. Humbert, S. Aumaître, M. Berhanu & E. Falcon 2017
      La Houille Blanche 5, 56 (2017)
      Experimental observation of four-wave resonant interactions in a wave basin

27. L. Deike, M. Berhanu & E. Falcon 2017
       Physical Review Fluids 2, 064803 (2017)
       Observation of hydroelastic three-wave interactions

26. B. Issenmann, C. Laroche & E. Falcon 2016
    EPL 116, 64005 (2016)
      Wave turbulence in a two-layer fluid: coupling between free surface and interface waves

25. F. Bonnefoy, F. Haudin, G. Michel, B. Semin, T. Humbert, S. Aumaître, M. Berhanu & E. Falcon 2016
       Journal of Fluid Mechanics (Rapids) 805, R3 (2016)
       Observation of resonant interactions among surface gravity waves

24. F. Haudin, A. Cazaubiel, L. Deike, T. Jamin, E. Falcon and M. Berhanu 2016
       Phys. Rev. E 93, 043110 (2016)
       Experimental study of three-wave interactions among capillary-gravity surface waves

23. L. Deike, B. Miquel, P. Gutiérrez, T. Jamin, B. Semin, M. Berhanu, E. Falcon & F. Bonnefoy 2015
       Journal of Fluid Mechanics 781, 196 (2015)
       Role of the basin boudary conditions in gravity wave turbulence

22. L. Deike, D. Fuster, M. Berhanu, E. Falcon 2014
       Physical Review Letters 112, 234501 (2014)
       Direct numerical simulations of capillary wave turbulence

21. L. Deike, M.Berhanu & E. Falcon 2014
    Energy flux measurement from the dissipated energy in capillary wave turbulence
    Physical Review E 89, 023003 (2014)

20. L. Deike, J.-C. Bacri & E. Falcon 2013
    Nonlinear waves on the surface of a fluid covered by an elastic sheet
    Journal of Fluid Mechanics 733, 394 (2013)

19. S. Aumaître, E. Falcon & S. Fauve 2013
      Fluctuations of the Energy Flux in Wave Turbulence
      Advances In Wave Turbulence (Ed. V. Shrira, S. Nazarenko, World Scientific, Chap. 2, pp. 53-72, 2013)

18. M.Berhanu & E. Falcon 2013
    Space-time-resolved capillary wave turbulence
    Physical Review E 89, 033003 (2013)

17. B. Issenmann & E. Falcon 2013
    Gravity wave turbulence revealed by horizontal vibrations of the container
      Physical Review E 87, 011001(R) (2013)

16. L. Deike, M. Berhanu & E. Falcon 2012
      Decay of capillary wave turbulence
    Physical Review E 85, 066311 (2012)

15. L. Deike, C.Laroche & E. Falcon 2011
    Experimental study of the inverse cascade in gravity wave turbulence
    EPL 96, 34004 (2011)

14. E. Falcon & C.Laroche 2011
    Observation of depth-induced properties in wave turbulence on the surface of a fluid
    EPL 94, 34003 (2011)

13. S. Dorbolo & E. Falcon 2011
     Wave turbulence on the surface of a ferrofluid in a horizontal magnetic field
       Phys. Rev. E 83, 046303 (2011)

12. E. Herbert, N. Mordant & E. Falcon 2010
      Observation of the nonlinear dispersion relation and spatial statistics of wave turbulence on the surface of a fluid
      Phys. Rev. Lett. 105, 144502 (2010)

11. E. Falcon, S.G. Roux & B. Audit 2010
    Revealing intermittency in experimental data with steep power spectra
       EPL 90, 50007 (2010)

10. E. Falcon, S.G. Roux & C. Laroche 2010
      On the origin of intermittency in wave turbulence
    EPL 90, 34005 (2010)

9. E. Falcon 2010
    Laboratory experiments on wave turbulence
    Discrete and Continuous Dynamical Systems - Series B Vol. 13, N°4, 819 - 840 (2010)

8. C. Falcón, E. Falcon, U. Bortolozzo & S. Fauve 2009
        Capillary wave turbulence on a spherical fluid surface in zero gravity
       EPL 86, 14002 (2009)

7. C. Falcón & E. Falcon 2009
       Fluctuations of energy flux in a simple dissipative out-of-equilibrium system
       Phys. Rev. E 79, 041110 (2009)

6. E. Falcon 2008
       Etudes Expérimentales en Turbulence d'Ondes
       Habilitation à Diriger les Recherches, 244 pages, Université Paris Diderot (2008) (in french)

5. F. Boyer & E. Falcon 2008
        Wave turbulence on the surface of a ferrofluid in a magnetic field
       Phys. Rev. Lett. 101, 244502 (2008)

4. S. Fauve & E. Falcon 2008,
        Gravity-capillary wave turbulence
        Report to COSPAR (World Committee for Space Research), 37th Scientific Assembly, 13-20 July 2008, Montréal, Canada, CNES Ed., p. 90 - 91 (2008)

3. E. Falcon, S. Aumaître, C. Falcón, C. Laroche & S. Fauve 2008
      Fluctuations of energy flux in wave turbulence
        Physical Review Letters 100, 064503 (2008)

2. E. Falcon, S. Fauve & C. Laroche 2007
        Observation of intermittency in wave turbulence,
          Physical Review Letters 98, 154501 (2007)
1. E. Falcon, C. Laroche & S. Fauve 2007
          Observation of gravity-capillary wave turbulence,
        Physical Review Letters 98, 094503 (2007)