Abstract: In fluid turbulence, energy is transferred from one scale to another by an energy cascade that depends only on the energy-dissipation 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 energy-dissipation rate over the scales. Finally, we discuss the role of intermittency using the correlation functions that exhibit anomalous exponents.

Abstract: This article presents the complete study of the long-time 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 non-equilibrium evolution to an equilibrium wave spectrum, which happens to be the well known Rayleigh-Jeans distribution. Moreover we show the existence of an energy cascade, often called the Kolmogorov-Zakharov 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 self-similar wave action inverse cascade which happens to blow-up 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.

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 singularity-mediated turbulence based on the self-focusing 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 Obukhov-Kolmogorov spectrum with a power-law dependence together with an intermittent behavior using structure function correlations, in close correspondence with the one measured in fluid turbulence.

Abstract: We study in 2-dimensions the superfluid density of periodically modulated states in the framework of the mean-field Gross-Pitaevskii model of a quantum solid. We obtain a full agreement for the superfluid fraction between a semi-theoretical approach and direct numerical simulations. As in 1-dimension, 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.