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Wave turbulence: beyond weak turbulence

Periodic Reporting for period 4 - WATU (Wave turbulence: beyond weak turbulence)

Reporting period: 2020-04-01 to 2020-12-31

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 from long 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. To a large extent this theory lacked experimental validation in well controlled laboratory conditions and its extension to large amplitude waves is far from achieved.

My project aimed at studying several physical systems (vibrating elastic plate, 1D and 2D water surface waves, 3D internal waves in a density stratified fluid) specifically chosen to highlight various features of wave turbulence both in the weakly and strongly forced regimes. Using innovative high-speed imaging techniques I could provide a very significant advance in experimental wave turbulence that highlight various limitations of the theory as compared to real systems. I also observed original states such as soliton gas and stratified turbulence, the latter in the unique Coriolis facility dedicated to investigation of geophysical flows with density stratification and or rotation (relevant for oceans or atmospheres).
We studied several wave configurations: elastic waves in a vibrating plate, capillary waves at the surface of water in small wave tanks (half a meter size), gravity surface waves in a large pool (13m in diameter) and a linear wave tank (36m long) and internal waves in the bulk of stratified water (which density is varying vertically as in the ocean or the atmosphere). In all these systems we have developed an imaging system that can record the wave field in 1D, 2D or 3D and with good temporal resolution. 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 surface gravity waves in 2D 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 synchronized 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 have being built in each of these experiments. For the case of internal waves, we use all the unique capabilities of the Coriolis facility for the study of geophysical flows in stratified and/or rotating fluids. The density of water is varied by dissolving large quantities of salt. The 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 shares physical parameters with turbulence in the interior of the ocean.

We used 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 for water waves 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 that are also encountered in the propagation of light in optical fibers. We were able to set up a random regime involving a large number of interacting solitons.

These results led to publications in international journals and communications in international conferences.
The main progress beyond the state of the art is the development of space & time resolved reconstruction technique of the wave field for many systems that allowed us to investigate the relevance of the phenomenology predicted by the weak turbulence theory. It also allowed us to characterize deeply the departure from weak turbulence for strong wave amplitude. At strong wave forcing, we observed the emergence of various sorts of singularities: wave overturning for surface water waves and internal waves, developable cones for vibrating plates and transition to soliton gas in shallow water. In several systems, the emergence of singularities is associated to the emergence of intermittency (lack of self-similarity in scales) which is also a key ingredient of the classical fluid turbulence (with vorticies) and the object of investigation for a long time in this community.

A striking result is the observation of a random state involving solitons called soliton gaz. This state is also named integrable turbulence and was predicted theoretically decades ago and mostly observed in somewhat idealized numerical simulations. We could observed such a state in a 35-meter long linear wave flume with gentle sine forcing. The accumulation of energy leads to the destabilization of the sine wave and the generation of random solitons. Solitons are present in many fields of physics involving non linearities, the most common one being the propagation of intense light in optical fibers. The realization of a soliton gaz in our wave flume is a première for water waves and it attracted citations from many fields of physics.

Another promising result is the generation of stratified turbulence in the unique Coriolis facility (in Grenoble, France). This regime is relevant of the turbulence present in the core of the ocean or atmosphere as well as in stars or liquid planetary cores. We could reach experimentally such a regime due to the large size of the Coriolis facility. These regimes are challenging to reproduce numerically as well as massive simulations are required. This opens venues for further experimental research on this geophysical flows.
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