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Wave turbulence: beyond weak turbulence

Periodic Reporting for period 3 - WATU (Wave turbulence: beyond weak turbulence)

Reporting period: 2018-10-01 to 2020-03-31

"The sea surface is deformed by waves induced by the wind. These waves impact marine transportation and alter the morphological of the shore by erosion which is a major issue in Europe. Prediction of sea states (""wave meteorology"") requires an accurate knowledge of the energy exchanges between waves as well as their dissipation. On a global scale the roughness of the sea impacts the exchanges between atmosphere and ocean which are a key issue in climate modeling. The complex distribution of waves at the surface of the sea shares the same phenomenology with many other nonlinear waves systems in vibrating structures, magnetized plasmas for fusion energy or optics in communication fibers as well as other geophysical waves that propagate in the interior of the ocean (and even in planetary or star cores). It can be described generically as wave turbulence: waves exchange energy among each other so that on average, the energy is transferred to short waves and ultimately dissipated by viscosity.

Wave turbulence and fluid turbulence belong to the same class of turbulent states made of a large number of nonlinearly coupled degrees of freedom driven far from equilibrium. In wave turbulence, the degrees of freedom are waves while in fluid turbulence they can be pictured as vortices. Building a full statistical theory of turbulence is the object of strong sustained efforts for decades with the hope of building more efficient numerical models for societal applications. Such a theory has been proposed for wave turbulence in the 1960’s called Weak Turbulence. It is a statistical theory of low amplitude turbulent waves. The predicted phenomenology (energy cascade in scale) is very similar to that of fluid turbulence. Weak Turbulence is thus a promising mathematical framework as wave turbulence can be observed in a vast collection of physical systems: planetary atmospheres, astrophysical plasmas, tokomak fusion plasmas, superfluid turbulence or Bose- Einstein condensates for example. To a large extent this theory lacks experimental validation in well controlled laboratory conditions and its extension to large amplitude waves is far from achieved.

My project aims at studying several physical systems (vibrating elastic plate, 1D and 2D water surface waves, 3D internal waves in a stratified fluid) specifically chosen to highlight various features of wave turbulence both in the weakly and strongly forced regimes. The vibrating plate is used for centuries to mimic thunder noise in theaters and operas. It appears to be also a very interesting toy model for wave turbulence in which time and space resolved measurements are made possible by using high speed imaging. Having space and time information is required to really be able to investigate the structure of the waves and their coupling. My project is aimed at developing similar experimental techniques to study waves at the surface of a liquid either at small scales (centimeters) of large scale (meter size). We will also study the case of so called internal gravity waves that can develop in the interior of the ocean due to the vertical variation of the water density related to fluctuations of temperature and salinity. I will specifically use two unique large-scale facilities available in LEGI (Grenoble, France): the 30 m 1D transparent wave flume for water surface waves and the 13m- diameter Coriolis turntable for water surface waves and internal waves. I will setup advanced space-time resolved profilometry and velocimetry techniques adapted to the dimensionality and size of each one of these systems. Advanced statistical tools on massive datasets will provide a profound insight into the coupling between waves and structures in the various regimes of wave turbulence.
We are currently studying several wave configurations: elastic waves in a vibrating plate, capillary waves at the surface of water in small wave tanks (half a meter size) and gravity waves in a large pool (13m in diameter) and a linear wave tank (36m long). In all these systems we have developed an imaging system that can record the deformation of the surface with both spatial resolution in 2D and good temporal resolution so that to obtain movies of the wave field. For the elastic plate and capillary waves we use a profilometry technique and a high speed camera (250 frames/s for capillary waves, 10000 frames/s for the vibrating plate). For the gravity waves in the 13m wave tank we developed a stereoscopic technique using 3 high resolution cameras and in the 36 m 1D wave flume we use a set of 8 cameras to record the wave elevation over 16m long. A part of the work was also to develop schemes to excite the waves. Large databases are currently being built in each of these experiments. In parallel to these experiments, we are also developing an experiment devoted to the study of internal gravity waves. The density of water is changed by dissolving large quantities of salt. The Coriolis facility is filled with water whose density is decreasing with altitude up to 1m deep (about 110 tons of water and 2 tons of salt). Large amplitude waves are generated by oscillating vertical panels so that to generate a 3D nonlinear regime of interacting waves. This regime has to be compared to observations of similar states in the ocean.

We use high order statistical analysis to identify the wave coupling in these turbulent waves and compare them to the theoretical predictions. We investigate the effect of altering the physical conditions such as changing the water depth or adding stress to the vibrating plate. For instance, for capillary waves, we observed a clear transition from a wave turbulence state to a solitonic regime at low depth. In contrast, applying stress to the vibrating plate does not lead to such a change although the main effect is also to reduce the dispersion of the waves. For gravity waves we are also working on the distinction between 3-wave or 4-wave coupling and the influence of bound waves. In the 36m long wave flume we observed a promising regime of soliton gaz: solitons are localized propagating structures that interact by collisions. We were able to set up a random regime involving a large number of interacting solitons.

These results led already to publications in international journals and communications in international conferences.
The main progress beyond the state of the art is the development of time-space resolved reconstruction technique of the deformation of the water surface in the Coriolis facility. It already allowed us to performed detailed analysis of the coupling among surface waves and to make comparison with oceanographic data recorded in the Black Sea. The experimental observation of a soliton gaz in the 36m long wave flume is also a significant progress that will allow us to make bridges with non linear optics. Indeed our regime shares many similarities with the propagation of solitons in communication optical fibers. Finally the state of large scale 3D nonlinear internal waves is also an experimental première that is very promising in terms of future interaction with the oceanographic community.
time-space reconstruction of the water surface in the soliton gas regime (36m-long wave flume)
view of turbulent gravity surface waves generated in the Coriolis facility in Grenoble
global view of the Coriolis facility (13m in diameter)
view of the water surface (with turbulent gravity waves) from below
view of particles floating at the surface of water with turbulent waves