Energy transfer in wave turbulence
Wave turbulence is a domain rapidly expanding for several years. It concerns the study of the dynamical and statistical properties of a group of stochastic waves in interaction. This phenomenon, very common, meets in situations varied on very different scales: astrophysics, geophysics, mathematics and various domains of physics. The main purpose is to understand the transfers of energy between nonlinear waves by means of generic laws, independently of the particular environment in which these waves propagate. Wave turbulence theory (or weak turbulence) allowed, from the end of the 60s, to calculate analytically, the out-of-equilibrium stationary solutions for the wave spectrum in almost all the domains of physics. In spite of the existence of numerous exact theoretical predictions, 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 binding hypotheses. Our experiments show the limitations of the current theoretical framework, which in return, arouses a theoretical and numerical renewed interest. <br /> <br />The objective of this program is to probe experimentally and numerically the limits of validity of the weak turbulence theory (finite size effects of the system, role of the strong nonlinearities, and of the dissipation). The second objective consists in understanding better the flows in which two mechanisms of transfer of energy are in competition (for example waves and vortices). <br /> <br />This project of fundamental research should so allow a better understanding of the mechanisms of energy transfers in wave turbulence with implications in oceanography, geophysics and astrophysics.
To reach these goals, different systems will be studied experimentally and numerically: hydrodynamic waves (of gravity and/or capillarity), and elastic waves on the surface of a thin plate. Two other systems, in which the relation of dispersal is tunable, will then be considered to understand better the various limits of the wave turbulence regime: hydro-elastic waves on the surface of an elastic membrane floating on a fluid, and waves at the interface of a liquid with its vapor close to the critical point liquid-vapor. Besides, we shall study experimentally the interaction of surface waves either with a discreet set of vortices, or with an underlying flow generated by a magnetohydrodynamic forcing.
We made the first direct numerical simulations of capillary wave turbulence from the fundamental equations of hydrodynamics. This study, published in PRL, was commented in the scientific press. These simulations validate so for the first time the wave turbulence theory applied to the capillary waves. These results generalized to the oceanic case could allow an improvement of our understanding of the role of the capillary waves during their damping by wave breaking, or during the gaseous exchanges between ocean and atmosphere that participate in the regulation of the climate.
Besides, the highlighting of the dissipation at all scales of the capillary turbulent cascade, not taken into account at the current stage of the theoretical developments, allowed to understand the disagreements observed, these last years, in a lot of experiences on wave turbulence. For that purpose, the indirect measure of the energy flux at every scale of the turbulent cascade was obtained for the first time. The constant of the Kolmogorov-Zakharov spectrum was also inferred experimentally for the first time and compared with its theoretical value.
Concerning elastic wave turbulence on the surface of a thin plate, a transition of regime of wave turbulence towards a dynamics dominated by the singularities was numerically observed. The existence of singular structures, such as angular folds leads to the breaking of the weak turbulence regime to the detriment of a regime of turbulence of singularities.
Finally, we reported, for the first time in laboratory, the existence of interactions between hydro-elastic waves - waves propagating on a floating elastic membrane. This type of waves usually meets in oceanography on the surface of an ice-covered ocean and this study allows a better understanding of their dynamics.
The project continues according to the timetable announced initially, and should have strong implications in oceanography, geophysics and astrophysics.
1. B. Miquel, A. Alexakis, C. Josserand & N. Mordant, Phys. Rev. Lett. 111, 054302 (2013) Transition from wave turbulence to dynamical crumpling in vibrated elastic plates
2. B. Issenmann & E. Falcon, Phys. Rev. E 87, 011001(R) (2013) Gravity wave t
Coupling between the atmosphere and the oceans is of major importance for climate modeling and long term weather forecast. Part of the energy transfer occurs through waves generated on the sea surface by the wind. It induces a state of wave turbulence in which energy is transferred by non-linear interacting waves from the forcing scales down to small scales at which energy is dissipated into heat. A statistical theory of wave turbulence was developed in the 60’s, the so-called weak turbulence theory which indeed exhibits such an energy transfer in out-of-equilibrium situations: the Kolmogorov-Zakharov spectrum. This theory has been applied far beyond oceanography in almost any context involving non linear waves (astrophysical or tokamak plasmas, internal gravity waves in the oceans, Rossby waves in the atmosphere, spin waves in magnets, bending waves of elastic plates, Kelvin waves in superfluid turbulence, non linear optics, etc). As any theory, the weak turbulence theory has to be validated or invalidated by experiments. So far, only few experiments have been designed to specifically test the theory and its limitations and they mostly concern surface waves on a liquid. The theory is based on quite strong hypothesis such as asymptotically small wave amplitudes and infinite systems, which may strongly limit its applicability to real systems. Our project is aimed at testing the theory in various situations: first in systems involving only waves (capillary and/or gravity waves, elastic waves) by putting the emphasis on finite size effects, strong magnitude of the waves or existence of correlations. This first axis thus consists of experimentally and numerically probing the validity domain of the weak turbulence theory and investigating the statistics of wave turbulence in finite systems involving strong nonlinearities. A second axis of the project is to develop new specific systems in which the dispersion relation will be tunable: gravity-elastic waves on the surface of an elastic membrane floating on a fluid, and waves at a liquid/vapor interface near the critical point. We will study in this way how the dispersion relation affects the statistical properties of wave turbulence. Finally, in many systems the waves are coupled to other degrees of freedom as for instance the flow vorticity field. These other degrees of freedom are likely to be present in real systems and are expected to affect wave properties. We aim at quantifying these processes. To do that, we will experimentally study the interaction of surface waves either with a discrete set of vortices, or with an underlying flow generated by a magnetohydrodynamic forcing. This project of fundamental research aims to a better understanding of wave turbulence. This project will thus leads to potential applications in geo/astrophysics, condensed matter, fluid mechanics or non-linear optics.
Laboratoire Matière et Systèmes Complexes, UMR 7057 - Université Paris Diderot (Laboratoire public)
The author of this summary is the project coordinator, who is responsible for the content of this summary. The ANR declines any responsibility as for its contents.
Laboratoire Matière et Systèmes Complexes, UMR 7057 - Université Paris Diderot
Laboratoire de Physique Statistique, UMR 8550 - Ecole Normale Supérieure
Groupe Instabilités et Turbulence, URA 2464 - CEA Saclay
Laboratoire en Hydrodynamique Energétique et Environnement Atmosphérique, UMR 6598 - Ecole Centrale de Nantes (ex-LMF)
Help of the ANR 580,992 euros
Beginning and duration of the scientific project: October 2012 - 48 Months