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During, G., Josserand, C., Krstulovic, G., & Rica, S. (2019). Strong turbulence for vibrating plates: Emergence of a Kolmogorov spectrum. Phys. Rev. Fluids, 4(6), 12 pp.
Abstract: In fluid turbulence, energy is transferred from one scale to another by an energy cascade that depends only on the energydissipation rate. It leads by dimensional arguments to the Kolmogorov 1941 (K41) spectrum. Here we show that the normal modes of vibrations in elastic plates also manifest an energy cascade with the same K41 spectrum in the fully nonlinear regime. In particular, we observe different patterns in the elastic deformations such as folds, developable cones, and even more complex stretching structures, in analogy with spots, swirls, vortices, and other structures in hydrodynamic turbulence. We show that the energy cascade is dominated by the kinetic contribution and that the stretching energy is at thermodynamical equilibrium. We characterize this energy cascade, the validity of the constant energydissipation rate over the scales. Finally, we discuss the role of intermittency using the correlation functions that exhibit anomalous exponents.

During, G., Josserand, C., & Rica, S. (2017). Wave turbulence theory of elastic plates. Physica D, 347, 42–73.
Abstract: This article presents the complete study of the longtime evolution of random waves of a vibrating thin elastic plate in the limit of small plate deformation so that modes of oscillations interact weakly. According to the wave turbulence theory a nonlinear wave system evolves in longtime creating a slow redistribution of the spectral energy from one mode to another. We derive step by step, following the method of cumulants expansion and multiscale asymptotic perturbations, the kinetic equation for the second order cumulants as well as the second and fourth order renormalization of the dispersion relation of the waves. We characterize the nonequilibrium evolution to an equilibrium wave spectrum, which happens to be the well known RayleighJeans distribution. Moreover we show the existence of an energy cascade, often called the KolmogorovZakharov spectrum, which happens to be not simply a power law, but a logarithmic correction to the Rayleigh Jeans distribution. We perform numerical simulations confirming these scenarii, namely the equilibrium relaxation for closed systems and the existence of an energy cascade wave spectrum. Both show a good agreement between theoretical predictions and numerics. We show also some other relevant features of vibrating elastic plates, such as the existence of a selfsimilar wave action inverse cascade which happens to blowup in finite time. We discuss the mechanism of the wave breakdown phenomena in elastic plates as well as the limit of strong turbulence which arises as the thickness of the plate vanishes. Finally, we discuss the role of dissipation and the connection with experiments, and the generalization of the wave turbulence theory to elastic shells. (C) 2017 Elsevier B.V. All rights reserved.

During, G., Josserand, C., & Rica, S. (2015). Selfsimilar formation of an inverse cascade in vibrating elastic plates. Phys. Rev. E, 91(5), 10 pp.
Abstract: The dynamics of random weakly nonlinear waves is studied in the framework of vibrating thin elastic plates. Although it has been previously predicted that no stationary inverse cascade of constant wave action flux could exist in the framework of wave turbulence for elastic plates, we present substantial evidence of the existence of a timedependent inverse cascade, opening up the possibility of selforganization for a larger class of systems. This inverse cascade transports the spectral density of the amplitude of the waves from short up to large scales, increasing the distribution of long waves despite the shortwave fluctuations. This dynamics appears to be selfsimilar and possesses a powerlaw behavior in the shortwavelength limit which significantly differs from the exponent obtained via a Kolmogorov dimensional analysis argument. Finally, we show explicitly a tendency to build a longwave coherent structure in finite time.

During, G., Picozzi, A., & Rica, S. (2009). Breakdown of weakturbulence and nonlinear wave condensation. Physica D, 238(16), 1524–1549.
Abstract: The formation of a largescale coherent structure (a condensate) as a result of the long time evolution of the initial value problem of a classical partial differential nonlinear wave equation is considered. We consider the nonintegrable and unforced defocusing NonLinear Schrodinger (NLS) equation as a representative model. In spite of the formal reversibility of the NLS equation, the nonlinear wave exhibits an irreversible evolution towards a thermodynamic equilibrium state. The equilibrium state is characterized by a homogeneous solution (condensate), with smallscale fluctuations superposed (uncondensed particles), which store the information necessary for “time reversal”. We analyze the evolution Of the cumulants of the random wave as originally formulated by DJ. Benney and P.G. Saffman [D.J. Bentley, P.G. Saffman, Proc. Roy. Soc. London A 289 (1966) 301] and A.C. Newell [A.C. Newell, Rev. Geophys. 6 (1968) 1]. This allows us to provide a selfconsistent weakturbulence theory of the condensation process, in which the nonequilibrium formation of the condensate is a natural consequence of the spontaneous regeneration of a nonvanishing firstorder cumulant in the hierarchy of the cumulants' equations. More precisely, we show that in the presence of a small condensate amplitude, all relevant statistical information is contained in the offdiagonal second order cumulant, as described by the usual weakturbulence theory. Conversely, in the presence of a highamplitude condensate, the diagonal secondorder cumulants no longer vanish in the long time limit, which signals a breakdown of the weakturbulence theory. However, we show that all asymptotic closure of the hierarchy of the cumulants' equations is still possible provided one considers the Bogoliubov's basis rather than the standard Fourier's (free particle) basis. The nonequilibrium dynamics turns out to be governed by the Bogoliubov's offdiagonal second order cumulant, while the corresponding diagonal cumulants, as well as the higher order cumulants, are shown to vanish asymptotically. The numerical discretization of the NLS equation implicitly introduces an ultraviolet frequency cutoff. The simulations are in quantitative agreement with the weak turbulence theory without adjustable parameters, despite the fact that the theory is expected to breakdown nearby the transition to condensation. The fraction of condensed particles vs energy is characterized by two distinct regimes: For small energies (H << Hc) the Bogoliubov's regime is established, whereas for H less than or similar to Hc the smallamplitude condensate regime is described by the weakturbulence theory. In both regimes we derive coupled kinetic equations that describe the coupled evolution of the condensate amplitude and the incoherent field component. The influence of finite size effects and of the dimensionality of the system are also considered. It is shown that, beyond the thermodynamic limit, wave condensation is reestablished in two spatial dimensions, in complete analogy with uniform and ideal 2D Bose gases. (C) 2009 Elsevier B.V. All rights reserved.

Humbert, T., Cadot, O., During, G., Josserand, C., Rica, S., & Touze, C. (2013). Wave turbulence in vibrating plates: The effect of damping. Epl, 102(3), 6 pp.
Abstract: The effect of damping in the wave turbulence regime for thin vibrating plates is studied. An experimental method, allowing measurements of dissipation in the system at all scales, is first introduced. Practical experimental devices for increasing the dissipation are used. The main observable consequence of increasing the damping is a significant modification in the slope of the power spectral density, so that the observed power laws are not in a pure inertial regime. However, the system still displays a turbulent behavior with a cutoff frequency that is determined by the injected power which does not depend on damping. By using the measured damping powerlaw in numerical simulations, similar conclusions are drawn out. Copyright (C) EPLA, 2013
