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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. (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., 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.

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

Josserand, C., Pomeau, Y., & Rica, S. (2020). Finitetime localized singularities as a mechanism for turbulent dissipation. Phys. Rev. Fluids, 5(5), 15 pp.
Abstract: The nature of the fluctuations of the dissipation rate in fluid turbulence is still under debate. One reason may be that the observed fluctuations are strong events of dissipation, which reveal the trace of spatiotemporal singularities of the Euler equations, which are the zero viscosity limit of ordinary incompressible fluids. Viscosity regularizes these hypothetical singularities, resulting in a chaotic fluctuating state in which the strong events appear randomly in space and time, making the energy dissipation highly fluctuating. Yet, to date, it is not known if smooth initial conditions of the Euler equations with finite energy do or do not blow up in finite time. We overcome this central difficulty by providing a scenario for singularitymediated turbulence based on the selffocusing nonlinear Schrodinger equation. It avoids the intrinsic difficulty of Euler equations since it is well known that solutions of this NLS equation with smooth initial conditions do blow up in finite time. When adding viscosity, the model shows (i) that dissipation takes place near the singularities only, (ii) that such intense events are random in space and time, (iii) that the mean dissipation rate is almost constant as the viscosity varies, and (iv) the observation of an ObukhovKolmogorov spectrum with a powerlaw dependence together with an intermittent behavior using structure function correlations, in close correspondence with the one measured in fluid turbulence.

Mason, P., Josserand, C., & Rica, S. (2012). Activated Nucleation of Vortices in a DipoleBlockaded Supersolid Condensate. Phys. Rev. Lett., 109(4), 5 pp.
Abstract: We investigate theoretically and numerically a model of a supersolid in a dipoleblockaded BoseEinstein condensate. The dependence of the superfluid fraction with an imposed thermal bath and a uniform boost velocity on the condensate is considered. Specifically, we observe a critical velocity for the nucleation of vortices in our system that is strongly linked to a steplike decrease in the superfluid fraction. We are able to use a scaling argument based on the energy required to activate a vortex, relating the critical temperature to the critical velocity, and find that this relationship is in good agreement with the numerical simulations carried out on the nonlocal GrossPitaevskii equation.

Sepulveda, N., Josserand, C., & Rica, S. (2010). Superfluid density in a twodimensional model of supersolid. Eur. Phys. J. B, 78(4), 439–447.
Abstract: We study in 2dimensions the superfluid density of periodically modulated states in the framework of the meanfield GrossPitaevskii model of a quantum solid. We obtain a full agreement for the superfluid fraction between a semitheoretical approach and direct numerical simulations. As in 1dimension, the superfluid density decreases exponentially with the amplitude of the particle interaction. We discuss the case when defects are present in this modulated structure. In the case of isolated defects (e.g. dislocations) the superfluid density only shows small changes. Finally, we report an increase of the superfluid fraction up to 50% in the case of extended macroscopical defects. We show also that this excess of superfluid fraction depends on the length of the complex network of grain boundaries in the system.
