surface wave turbulence




zéro G flights

video (in English) by Alan Newell
given at ENS on wave turbulence

Surface waves and wave turbulence

Contrary to hydrodynamical turbulence (2D or 3D), only few experimental studies exist of surface wave turbulence (in a liquid). The studies of the capillary regime are scarce and the case of gravity waves was not studied in the laboratory (only in situ ocean studies with wind exist). One open question is the transition between gravity waves and capillary waves on the power spectrum of the amplitude of the waves.
We study wave turbulence at the surface of mercury by means of capacitive sensors. The random forcing is done with low frequency wave makers in a rectangular tank. The energy injected at low wavenumbers is transferred to smallest scale structures by non-linearities and wave interactions. The power spectrum of the capillary and gravity waves are both power laws (Kolmogorov-like spectra). The cross-over between the two regimes is characterized as a function of the amplitude and frequency of the random forcing.
By measuring the force exerted on the wave maker by the fluid together with its velocity, the power injected in the system was measured. The distribution of the injected power is observed to be strongly asymmetrical, and it is studied in the framework of the out of equilibrium fluctuation theorems of the statistical physics.  In a low gravity environment (parabolic flights), we have recently observed the capillary wave turbulence regime on more than 2 decades in frequency (which is usually hidden by the gravity regime) in good agreement with weakly non-linear theories (weak turbulence theory).
Wave turbulence is also studied on  thin elastic plate. Multipoint measurements resolved in time show the persistence on the wave structure through the observation of a dispersion relation, ie localisation of energy in the wavenumber/frequency space.

Noise and instabilities

The dynamo effect is an instability that creates magnetic energy from the kinetic energy of the flow of an electrically conducting liquid. In most cases this flow is turbulent such that the unstable field (the magnetic field) is forced by a fluctuating field (the velocity field). In a similar framework, we develop laboratory-scale experiments where instabilities are submitted to noise.

For example, we studied the effect of random vertical vibrations on the Rosensweig instability (that creates peaks at the surface of a ferrofluid submitted to a normal-to-surface magnetic field, see picture). We have shown that a small noise postpone the onset of this instability and a larger noise leads to the intermittent appearance of  peaks. In order to describe this effect, we study  theoretically   model equations and we quantified  the importance of the zero frequency component of the noise in controlling the on-off intermittency. Other experiments of interest are granular instabilities in which fluctuations are linked to the erratic motion of the grains.

In all these situations the interplay between an instability process and fluctuations leads to a rich  behavior that still remain to understand.

Fluctuations in out of equilibrium systems

Although dissipative systems driven far from equilibrium involve many degrees of freedom, there is no complete and satisfactory statistical description of the behavior of global quantities (energy, power,...), defined on the whole volume, unlike the thermo-statistic theory describing the equilibrium states. This is mainly due to the energy fluxes between the  energy input (usually at the boundaries of the systems) and the dissipation  in the bulk. These fluxes generate correlations, inhomogeneities and large fluctuations which prohibit the use of the usual tools of statistical mechanics.

To obtain a better understanding of this out-of-equilibrium systems, experimental studies (convection, wave turbulence) together with analogical (Langevin equation) and numerical (granular gas, shell model of turbulent flows, earthquake model) simulations are performed, as well as  theoretical considerations. We mainly focus on the behavior of the injected power (which can be measured experimentally) in such a system and we considered also other global quantities like the total energy or the heat flux.

These transversal studies underline some common features in dissipative systems as different as the waves turbulence or the simulated motions of brownian particles which present  remarkable similar histograms of the fluctuation of their injected power. We also show that the fluctuations of injected and dissipated power are closely related in all dissipative systems. Finally we discovered a useful limit where a granular gas tends smoothly to equilibrium. This limit is useful for instance to estimate the relevance of alternative definitions of out-of-equilibrium temperatures which can be applied to dissipative systems.