Wave Turbulence Regime Research Articles

SIMULATION OF CAPILLARY WAVE TURBULENCE ON THE BASIS OF FULLY NONLINEAR PLANE-SYMMETRIC MODEL

A new model for the direct numerical simulation of capillary wave turbulence arising at a free surface of deep incompressible fluid is proposed in the work. The plane-symmetric model based on the time-dependent conformal transform is fully nonlinear and takes into account the effects of surface tension, external random forcing and dissipation of energy. The simulation results show that the system of nonlinear capillary waves can go into a quasi-stationary state (wave turbulence regime), when the action of an external force is compensated by the viscosity. In this regime, the fluid motion demonstrates quite complex and irregular behavior. The spatial and frequency spectra of surface perturbations acquire a power-law dependence in the quasi-stationary state. The exponents of the spectra do not coincide with the classical Zakharov-Filonenko spectrum obtained for isotropic capillary turbulence. In the case of anisotropic quasi-1D geometry, five-wave resonant interactions become the dominant process. The numerical results agree with high accuracy with the corresponding analytical spectra obtained on the basis of dimensional analysis of weak turbulence spectra.

Transition from wave turbulence to acousticlike shock-wave regime

An experiment is performed to examine the previously unobserved transition from a dispersive wave-turbulence regime to a nondispersive shock-wave regime on the surface of a fluid. Using a magnetic liquid subjected to a high enough external magnetic field, a set of random shock waves is indeed observed on the free surface. Their spectrum is then found to be well modeled by a spectrum of singularities (Kuznetsov-like spectrum), and their statistics are close to the predictions from the one-dimensional random-forced Burgers' equation.

Path Large Deviations for the Kinetic Theory of Weak Turbulence

We consider a generic Hamiltonian system of nonlinear interacting waves with 3-wave interactions. In the kinetic regime of wave turbulence, which assumes weak nonlinearity and large system size, the relevant observable associated with the wave amplitude is the empirical spectral density that appears as the natural precursor of the spectral density, or spectrum, for finite system size. Following classical derivations of the Peierls equation for the moment generating function of the wave amplitudes in the kinetic regime, we propose a large deviation estimate for the dynamics of the empirical spectral density, where the number of admissible wavenumbers, which is proportional to the volume of the system, appears as the natural large deviation parameter. The large deviation stochastic Hamiltonian that quantifies the minus of the log probability of a trajectory is computed within the kinetic regime which assumes the Random Phase approximation for weak nonlinearity. We compare this Hamiltonian with the one for a system of modes interacting in a mean-field way with the empirical spectrum. Its relationship with the Random Phase and Amplitude approximation is discussed. Moreover, for the specific case when no forces and dissipation are present, a few fundamental properties of the large deviation dynamics are investigated. We show that the latter conserves total energy and momentum, as expected for a 3-wave interacting systems. In addition, we compute the equilibrium quasipotential and check that global detailed balance is satisfied at the large deviation level. Finally, we discuss briefly some physical applications of the theory.

Dynamical Fractional and Multifractal Fields

Motivated by the modeling of three-dimensional fluid turbulence, we define and study a class of stochastic partial differential equations (SPDEs) that are randomly stirred by a spatially smooth and uncorrelated in time forcing term. To reproduce the fractional, and more specifically multifractal, regularity nature of fully developed turbulence, these dynamical evolutions incorporate an homogenous pseudo-differential linear operator of degree 0 that takes care of transferring energy that is injected at large scales in the system, towards smaller scales according to a cascading mechanism. In the simplest situation which concerns the development of fractional regularity in a linear and Gaussian framework, we derive explicit predictions for the statistical behaviors of the solution at finite and infinite time. Doing so, we realize a cascading transfer of energy using linear, although non local, interactions. These evolutions can be seen as a stochastic version of recently proposed systems of forced waves intended to model the regime of weak wave turbulence in stratified and rotational flows. To include multifractal, i.e. intermittent, corrections to this picture, we get some inspiration from the Gaussian multiplicative chaos, which is known to be multifractal, to motivate the introduction of an additional quadratic interaction in these dynamical evolutions. Because the theoretical analysis of the obtained class of nonlinear SPDEs is much more demanding, we perform numerical simulations and observe the non-Gaussian and in particular skewed nature of their solution.

Linear and nonlinear dynamics of a plate with acoustic black hole, geometric and contact nonlinearity for vibration mitigation

A rectangular plate with a wedge profile creating an Acoustic Black Hole (ABH) termination is studied numerically. A particular emphasis is put on combining two different types of nonlinearity in order to improve the passive damping capacity of the ABH by transferring energy to the high-frequency range where it is more efficient. First, the addition of contact points to create a vibro-impact black hole (VI-ABH) is taken into account, following a previous study on beams. The contact nonlinearity allows for a rapid and efficient transfer of energy. Second, the large-amplitude vibrations of the plate in the ABH region where small thickness is reached, is also considered. The geometric nonlinearity is incorporated using a von Kármán plate model, and the regime of wave turbulence is shown to be triggered thus creating an energy flux from the low to the high frequencies. The linear characteristics of the ABH plate are first analyzed. Numerical results show the appearance of overdamped modes gathered in solution branches with constant number of half-waves in the transverse direction of the ABH, seen as a waveguide. The structure of the branches is shown to be more and more prominent when increasing the width of the plate, showing a transition from beam-like to full plate structure, with a fixed value for the fundamental cut-on frequency. The combination of both contact and geometric nonlinearities to improve the ABH effect is then reported. It is shown that the coexistence of both nonlinearities provides better passive damping efficacy.

Rotating magnetohydrodynamic turbulence

Decaying turbulence in rotating Magneto-hydrodynamic systems is studied theoretically and numerically. In the linear limit, when the velocity and magnetic perturbations are small, the system supports two types of waves. When the rotation effects are stronger than the ones of the external magnetic field, one of these waves contains most of the kinetic energy (inertial wave) and the other—most of the magnetic energy (magnetostrophic wave). The weak wave turbulence (WWT) theory for decoupled inertial and magnetospheric wave systems was previously derived in Galtier (2014 J. Fluid Mech. 757 114–54). In the present paper, we suggest theory of strong turbulence for such waves based on the critical balance (CB) approach conjecturing that the linear and nonlinear timescales are of similar magnitudes in a wide range of turbulent scales. Regimes of weak and strong wave turbulence are simulated numerically. The results appear to be in good agreement with the WWT and CB predictions, particularly for the exponents of the kinetic and magnetic energy spectra.

Fibre multi-wave mixing combs reveal the broken symmetry of Fermi–Pasta–Ulam recurrence

In optical fibres, weak modulations can grow at the expense of a strong pump to form a triangular comb of sideband pairs, until the process is reversed. Repeated cycles of such conversion and back-conversion constitute a manifestation of the universal nonlinear phenomenon known as Fermi–Pasta–Ulam recurrence. However, it remains a major challenge to observe the coexistence of different types of recurrences owing to the spontaneous symmetry-breaking nature of such a phenomenon. Here, we implement a novel non-destructive technique that allows the evolution in amplitude and phase of frequency modes to be reconstructed via post-processing of the fibre backscattered light. We clearly observe how control of the input modulation seed results in different recursive behaviours emerging from the phase-space structure dictated by the spontaneously broken symmetry. The proposed technique is an important tool to characterize other mixing processes and new regimes of rogue-wave formation and wave turbulence in fibre optics.

Turbulence of Weak Gravitational Waves in the Early Universe.

We study the statistical properties of an ensemble of weak gravitational waves interacting nonlinearly in a flat space-time. We show that the resonant three-wave interactions are absent and develop a theory for four-wave interactions in the reduced case of a 2.5+1 diagonal metric tensor. In this limit, where only plus-polarized gravitational waves are present, we derive the interaction Hamiltonian and consider the asymptotic regime of weak gravitational wave turbulence. Both direct and inverse cascades are found for the energy and the wave action, respectively, and the corresponding wave spectra are derived. The inverse cascade is characterized by a finite-time propagation of the metric excitations-a process similar to an explosive nonequilibrium Bose-Einstein condensation, which provides an efficient mechanism to ironing out small-scale inhomogeneities. The direct cascade leads to an accumulation of the radiation energy in the system. These processes might be important for understanding the early Universe where a background of weak nonlinear gravitational waves is expected.

Kolmogorov Turbulence Defeated by Anderson Localization for a Bose-Einstein Condensate in a Sinai-Oscillator Trap.

We study the dynamics of a Bose-Einstein condensate in a Sinai-oscillator trap under a monochromatic driving force. Such a trap is formed by a harmonic potential and a repulsive disk located in the center vicinity corresponding to the first experiments of condensate formation by Ketterle and co-workers in 1995. We allow that the external driving allows us to model the regime of weak wave turbulence with the Kolmogorov energy flow from low to high energies. We show that in a certain regime of weak driving and weak nonlinearity such a turbulent energy flow is defeated by the Anderson localization that leads to localization of energy on low energy modes. This is in a drastic contrast to the random phase approximation leading to energy flow to high modes. A critical threshold is determined above which the turbulent flow to high energies becomes possible. We argue that this phenomenon can be studied with ultracold atoms in magneto-optical traps.

Wave turbulence in a two-layer fluid: Coupling between free surface and interface waves

We experimentally study gravity-capillary wave turbulence on the interface between two immiscible fluids of close density with free upper surface. We locally measure the wave height at the interface between both fluids by means of a highly sensitive laser Doppler vibrometer. We show that the inertial range of the capillary wave turbulence regime is significantly extended when the upper fluid depth is increased: The crossover frequency between the gravity and capillary wave turbulence regimes is found to decrease whereas the dissipative cut-off frequency of the spectrum is found to increase. We explain these observations by the progressive decoupling between waves propagating at the interface and the ones at the free surface, using the full dispersion relation of gravity-capillary waves in a two-layer fluid of finite depths. The cut-off evolution is due to the disappearance of parasitic capillaries responsible for the main wave dissipation for a single fluid.

Variations of characteristic time scales in rotating stratified turbulence using a large parametric numerical study.

We study rotating stratified turbulence (RST) making use of numerical data stemming from a large parametric study varying the Reynolds, Froude and Rossby numbers, Re, Fr and Ro in a broad range of values. The computations are performed using periodic boundary conditions on grids of 1024(3) points, with no modeling of the small scales, no forcing and with large-scale random initial conditions for the velocity field only, and there are altogether 65 runs analyzed in this paper. The buoyancy Reynolds number defined as R(B) = ReFr2 varies from negligible values to ≈ 10(5), approaching atmospheric or oceanic regimes. This preliminary analysis deals with the variation of characteristic time scales of RST with dimensionless parameters, focusing on the role played by the partition of energy between the kinetic and potential modes, as a key ingredient for modeling the dynamics of such flows. We find that neither rotation nor the ratio of the Brunt-Väisälä frequency to the inertial frequency seem to play a major role in the absence of forcing in the global dynamics of the small-scale kinetic and potential modes. Specifically, in these computations, mostly in regimes of wave turbulence, characteristic times based on the ratio of energy to dissipation of the velocity and temperature fluctuations, T(V) and T(P), vary substantially with parameters. Their ratio γ=T(V)/T(P) follows roughly a bell-shaped curve in terms of Richardson number Ri. It reaches a plateau - on which time scales become comparable, γ≈0.6 - when the turbulence has significantly strengthened, leading to numerous destabilization events together with a tendency towards an isotropization of the flow.

Phenomenological model for predicting stationary and non-stationary spectra of wave turbulence in vibrating plates

A phenomenological model describing the time-frequency dependence of the power spectrum of thin plates vibrating in a wave turbulence regime, is introduced. The model equation contains as basic solutions the Rayleigh–Jeans equipartition of energy, as well as the Kolmogorov–Zakharov spectrum of wave turbulence. In the Wave Turbulence Theory framework, the model is used to investigate the self-similar, non-stationary solutions of forced and free turbulent vibrations. Frequency-dependent damping laws can easily be accounted for. Their effects on the characteristics of the stationary spectra of turbulence are then investigated. Thanks to this analysis, self-similar universal solutions are given, relating the power spectrum to both the injected power and the damping law.

Modal approach for nonlinear vibrations of damped impacted plates: Application to sound synthesis of gongs and cymbals

This paper presents a modal, time-domain scheme for the nonlinear vibrations of perfect and imperfect plates. The scheme can take into account a large number of degrees-of-freedom and is energy-conserving. The targeted application is the sound synthesis of cymbals and gong-like musical instruments, which are known for displaying a strongly nonlinear vibrating behaviour. This behaviour is typical of a wave turbulence regime, in which the wide-band spectrum of excited modes is observable in the form of an energy cascade. The modal method is selected for its versatility in handling complex damping laws that can be implemented easily by selecting appropriate damping values in each one of the modal equations. In the first part of the paper, the modal method is explained in its generality, and it will be seen that the method is valid for plates with arbitrary geometry and boundary conditions as long as the eigenmodes are known. Secondly, a time-integration, energy-conserving scheme for perfect and imperfect plates is presented, and implementation comments are given in order to treat efficiently the high-dimensionality of the resulting dynamical system. The scheme is run with appropriate parameters in order to produce sound samples. A simple impact law is considered for the excitation, whereas the flexibility of the method is highlighted by showing simulations for free-edge circular plates and simply-supported rectangular plates, together with various damping laws.

Dynamics of the wave turbulence spectrum in vibrating plates: A numerical investigation using a conservative finite difference scheme

The dynamics of the local kinetic energy spectrum of an elastic plate vibrating in a wave turbulence (WT) regime is investigated with a finite difference, energy-conserving scheme. The numerical method allows the simulation of pointwise forcing together with realistic boundary conditions, a set-up which is close to experimental conditions. In the absence of damping, the framework of non-stationary wave turbulence is used. Numerical simulations show the presence of a front propagating to high frequencies, leaving a steady spectrum in its wake. Self-similar dynamics of the spectra are found with and without periodic external forcing. For the periodic forcing, the mean injected power is found to be constant, and the frequency at the cascade front evolves linearly with time resulting in a increase of the total energy. For the free turbulence, the energy contained in the cascade remains constant while the frequency front increases as t1/3. These self-similar solutions are found to be in accordance with the kinetic equation derived from the von Kármán plate equations. The effect of the pointwise forcing is observable and introduces a steeper slope at low frequencies, as compared to the unforced case. The presence of a realistic geometric imperfection of the plate is found to have no effect on the global properties of the spectra dynamics. The steeper slope brought by the external forcing is shown to be still observable in a more realistic case where damping is added.

Nonlinear waves on the surface of a fluid covered by an elastic sheet

AbstractWe experimentally study linear and nonlinear waves on the surface of a fluid covered by an elastic sheet where both tension and flexural waves occur. An optical method is used to obtain the full space–time wave field, and the dispersion relation of the waves. When the forcing is increased, a significant nonlinear shift of the dispersion relation is observed. We show that this shift is due to an additional tension of the sheet induced by the transverse motion of a fundamental mode of the sheet. When the system is subjected to a random-noise forcing at large scales, a regime of hydroelastic wave turbulence is observed with a power-law spectrum of the scale, in disagreement with the wave turbulence prediction. We show that the separation between relevant time scales is well satisfied at each scale of the turbulent cascade as expected theoretically. The wave field anisotropy, and finite size effects are also quantified and are not at the origin of the discrepancy. Finally, the dissipation is found to occur at all scales of the cascade, contrary to the theoretical hypothesis, and could thus explain this disagreement.

Transition from Wave Turbulence to Dynamical Crumpling in Vibrated Elastic Plates

We study the dynamical regime of wave turbulence of a vibrated thin elastic plate based on experimental and numerical observations. We focus our study on the strongly nonlinear regime described in a previous Letter by Yokoyama and Takaoka. At small forcing, a weakly nonlinear regime is compatible with the weak turbulence theory when the dissipation is localized at high wave number. When the forcing intensity is increased, a strongly nonlinear regime emerges: singular structures dominate the dynamics at large scales whereas at small scales the weak turbulence is still present. A turbulence of singular structures with folds and D cones develops that alters significantly the energy spectra and causes the emergence of intermittency.

Non-stationary regimes of surface gravity wave turbulence

We present experimental results about rising and decaying gravity wave turbulence in a large laboratory flume. We consider the time evolution of the wave energy spectral components in ω- and k-domains and demonstrate that emerging wave turbulence can be characterized by two time scales—a short dynamical scale due to nonlinear wave interactions and a longer kinetic time scale characterizing formation of a stationary wave energy spectrum. In the decay regime we observed the maximum of the wave energy spectrum decreasing in time initially as the power law, ∝t−1/2, as predicted by the weak turbulence theory, and then exponentially due to viscous friction.

Wave turbulence in vibrating plates: The effect of damping

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 cut-off frequency that is determined by the injected power which does not depend on damping. By using the measured damping power-law in numerical simulations, similar conclusions are drawn out.

Space-time-resolved capillary wave turbulence

We report experiments on the full space- and time-resolved statistics of capillary wave turbulence at the air-water interface. The three-dimensional shape of the free interface is measured as a function of time by using the optical method of diffusing light photography associated with a fast camera. Linear and nonlinear dispersion relations are extracted from the spatiotemporal power spectrum of wave amplitude. When wave turbulence regime is reached, we observe power-law spectra both in frequency and in wave number whose exponents are found to agree with the predictions of capillary wave turbulence theory. Finally, the temporal dynamics of the spatial energy spectrum highlight the occurrence of stochastic bursts transferring wave energy through the spatial scales.

Rogue waves in Alfvénic turbulence

Rogue waves, in the form of giant breathers, are shown to develop in the Alfvén wave (AW) turbulence regime described by the randomly driven derivative nonlinear Schrödinger equation in the presence of a weak dissipation. The distribution of the instantaneous global maxima of the AW intensity fluctuations is seen to be accurately fitted by power laws, which contrasts with the integrable regime (absence of dissipation and forcing) where the behavior is rather exponential. As the dissipation is reduced, freak waves form less frequently but reach larger amplitudes.