Symmetry Induced Four-Wave Capillary Wave Turbulence

arXiv (Cornell University), 2005

We perform numerical simulations of the dynamical equations for free water surface in finite basin in presence of gravity. Wave Turbulence (WT) is a theory derived for describing statistics of weakly nonlinear waves in the infinite basin limit. Its formal applicability condition on the minimal size of the computational basin is impossible to satisfy in present numerical simulations, and the number of wave resonances is significantly depleted due to the wavenumber discreteness. The goal of this paper will be to examine which WT predictions survive in such discrete systems with depleted resonances and which properties arise specifically due to the discreteness effects. As in [1-3], our results for the wave spectrum agree with the Zakharov-Filonenko spectrum predicted within WT. We also go beyond finding the spectra and compute probability density function (PDF) of the wave amplitudes and observe an anomalously large, with respect to Gaussian, probability of strong waves which is consistent with recent theory . Using a simple model for quasi-resonances we predict an effect arising purely due to discreteness: existence of a threshold wave intensity above which turbulent cascade develops and proceeds to arbitrarily small scales. Numerically, we observe that the energy cascade is very "bursty" in time and is somewhat similar to sporadic sandpile avalanches. We explain this as a cycle: a cascade arrest due to discreteness leads to accumulation of energy near the forcing scale which, in turn, leads to widening of the nonlinear resonance and, therefore, triggering of the cascade draining the turbulence levels and returning the system to the beginning of the cycle.

Journal of Fluid Mechanics

We study the dynamics of capillary waves at the interface of a two-layer stratified turbulent channel flow. We use a combined pseudo-spectral/phase field method to solve for the turbulent flow in the two liquid layers and to track the dynamics of the liquid–liquid interface. The two liquid layers have same thickness and same density, but different viscosity. We vary the viscosity of the upper layer (two different values) to mimic a stratified oil–water flow. This allows us to study the interplay between inertial, viscous and surface tension forces in the absence of gravity. In the present set-up, waves are naturally forced by turbulence over a broad range of scales, from the larger scales, whose size is of order of the system scale, down to the smaller dissipative scales. After an initial transient, we observe the emergence of a stationary capillary wave regime, which we study by means of temporal and spatial spectra. The computed frequency and wavenumber power spectra of wave eleva...

In this paper we review recent developments in the statistical theory of weakly nonlinear dispersive waves, the subject known as Wave Turbulence (WT). We revise WT theory using a generalisation of the random phase approximation (RPA). This generalisation takes into account that not only the phases but also the amplitudes of the wave Fourier modes are random quantities and it is called the "Random Phase and Amplitude" approach. This approach allows to systematically derive the kinetic equation for the energy spectrum from the the Peierls-Brout-Prigogine (PBP) equation for the multi-mode probability density function (PDF). The PBP equation was originally derived for the three-wave systems and in the present paper we derive a similar equation for the four-wave case. Equation for the multi-mode PDF will be used to validate the statistical assumptions Yeontaek Choi et al. 2 about the phase and the amplitude randomness used for WT closures. Further, the multimode PDF contains a detailed statistical information, beyond spectra, and it finally allows to study non-Gaussianity and intermittency in WT, as it will be described in the present paper. In particular, we will show that intermittency of stochastic nonlinear waves is related to a flux of probability in the space of wave amplitudes.

Advances in Space Research, 2008

We describe a new wave mode similar to the acoustic wave in which both density and velocity fluctuate.

Advances in Space Research, 2008

We describe a new wave mode similar to the acoustic wave in which both density and velocity fluctuate. Unlike the acoustic wave in which the underlying distribution is Maxwellian, this new wave mode occurs when the underlying distribution is a ...

ABSTRACT The theory of non-linear wave interactions leading to so-called interfacial wave turbulence, where a broadband distribution of capillary wave phenomena may be induced by a monofrequency oscillator, is well known, but experimental results are rare. In particular, it is challenging to set up a physical system where both capillary wave amplitudes are easy to measure and capillary forces dominate gravitational forces.

Journal of Geophysical Research, 1981

The linear coupling model of Miles and Phillips for the air-sea interaction is generalized to include the average effect of the nonlinear interactions in the dynamic equations for the gravity-capillary waves. The linearized dynamic equations are stochastic with solutions that have stable moments. In particular, a steady state power spectral density for the water wave field is calculated exactly in the context of the model for various wind speeds.

Physica D: Nonlinear Phenomena, 2012

We consider a general model of Hamiltonian wave systems with triple resonances, in the standard kinetic limit of a continuum of weakly interacting dispersive waves with random phases. We show in this limit that the leadingorder, asymptotically valid dynamical equation for multimode amplitude distributions is not the well-known equation of Peierls (also, Brout & Prigogine and Zaslavskii & Sagdeev), but is instead a reduced equation containing only a subset of the terms in that equation. Our equations are consistent with the Peierls equation in that the additional terms in the latter vanish as inverse powers of volume in the large-box limit. The equations that we derive are the direct analogue of the Boltzmann hierarchy obtained from the BBGKY hierarchy in the low-density limit for gases. We show that the asymptotic multimode equations possess factorized solutions for factorized initial data, which correspond to preservation in time of the property of "random phases & amplitudes". The factors satisfy the equations for the 1-mode probability density functions (PDF's) previously derived by Choi et al. and Jakobsen & Newell. Analogous to the Klimontovich density in the kinetic theory of gases, we introduce the concepts of the "empirical spectrum" and the "empirical 1-mode PDF". We show that the factorization of the hierarchy equations implies that these quantities are self-averaging: they satisfy the wave-kinetic closure equations of the spectrum and 1-mode PDF for almost any selection of phases and amplitudes from the initial ensemble. We show that both of these closure equations satisfy an H-theorem for an entropy defined by Boltzmann's prescription S = k B log W. We also characterize the general solutions of our multimode distribution equations, for initial conditions with random phases but with no statistical assumptions on the amplitudes. Analogous to a result of Spohn for the Boltzmann hierarchy, these are "super-statistical solutions" that correspond to ensembles of solutions of the wave-kinetic closure equations with random initial conditions or random forces. On the basis of our results, we discuss possible kinetic explanations of intermittency and non-Gaussian statistics in wave turbulence. In particular, we advance the explanation of a "superturbulence" produced by stochastic or turbulent solutions of the wave kinetic equations themselves.

2017

The wave turbulence equation is an effective kinetic equation that describes the dynamics of wave spectrum in weakly nonlinear and dispersive media. Such a kinetic model has been derived by physicists in the sixties, though the well-posedness theory remains open, due to the complexity of resonant interaction kernels. In this paper, we provide a global unique radial strong solution, the first such a result, to the wave turbulence equation for capillary waves.

In the early 1960s, it was established that the stochastic initial value problem for weakly coupled wave systems has a natural asymptotic closure induced by the dispersive properties of the waves and the large separation of linear and nonlinear time scales. One is thereby led to kinetic equations for the redistribution of spectral densities via three-and four-wave resonances together with a nonlinear renormalization of the frequency. The kinetic equations have equilibrium solutions which are much richer than the familiar thermodynamic, Fermi-Dirac or Bose-Einstein spectra and admit in addition finite flux (Kolmogorov-Zakharov) solutions which describe the transfer of conserved densities (e.g. energy) between sources and sinks. There is much one can learn from the kinetic equations about the behavior of particular systems of interest including insights in connection with the phenomenon of intermittency. What we would like to convince you is that what we call weak or wave turbulence is every bit as rich as the macho turbulence of 3D hydrodynamics at high Reynolds numbers and, moreover, is analytically more tractable. It is an excellent paradigm for the study of many-body Hamiltonian systems which are driven far from equilibrium by the presence of external forcing and damping. In almost all cases, it contains within its solutions behavior which invalidates the premises on which the theory is based in some spectral range. We give some new results concerning the dynamic breakdown of the weak turbulence description and discuss the fully nonlinear and intermittent behavior which follows. These results may also be important for proving or disproving the global existence of solutions for the underlying partial differential equations. Wave turbulence is a subject to which many have made important contributions. But no contributions have been more fundamental than those of Volodja Zakharov whose 60th birthday we celebrate at this meeting. He was the first to appreciate that the kinetic equations admit a far richer class of solutions than the fluxless thermodynamic solutions of equilibrium systems and to realize the central roles that finite flux solutions play in non-equilibrium systems. It is appropriate, therefore, that we call these Kolmogorov-Zakharov (KZ) spectra. : S 0 1 6 7 -2 7 8 9 ( 0 1 ) 0 0 1 9 2 -0 1. The set up: the equation for the Fourier amplitudes. 2. Moments, cumulants.

INCAS BULLETIN

The paper shows the importance of the dispersion relation in characterizing the capillary waves seen on liquid jets. Several theoretical models are given to better understand the stability of cylindrical interfaces when various parameters are considered, such as confinement, bulk elasticity, or the viscosity ratio between the two liquid phases. Theoretical predictions are compared with experimental data in terms of the fastest-growing mode for several liquid-in-air systems. Capillary-wave decay factors are also investigated, for stationary wave trains created at the impact of a liquid jet on a horizontal liquid bath, via the dispersion relation.

Physical Review Letters, 2008

We report that the power driving gravity and capillary wave turbulence in a statistically stationary regime displays fluctuations much stronger than its mean value. We show that its probability density function (PDF) has a most probable value close to zero and involves two asymmetric roughly exponential tails. We understand the qualitative features of the PDF using a simple Langevin type model. PACS numbers: 47.35.+i, 92.10.Hm, 47.20.Ky, 68.03.Cd When a dissipative system is driven in a statistically stationary regime by an external forcing, a given amount of power per unit mass, ǫ, is transfered from the driving device to the system and is ultimately dissipated. In fully developed turbulence, a flow is driven at large scales and nonlinear interactions transfer kinetic energy toward small scales where viscous dissipation takes place. In the intermediate range of scales (the inertial range) the key role of the energy flux ǫ has been first understood by Kolmogorov [1]. Using dimensional arguments, he derived the law E(k) ∝ ǫ 2/3 k −5/3 for the energy density E(k) as a function of the wavenumber k. Kolmogorov type spectra have been derived analytically in wave turbulence, i.e. in various systems involving an ensemble of weakly interacting nonlinear waves (see for instance [2] for a review). In all cases, it has been assumed that ǫ is a given constant parameter. However, it should be kept in mind that ǫ is not an input parameter in most experiments or simulations of dissipative systems. Its value is not externally controlled but determined by the impedance of the system. In addition, as we have already shown for a variety of different dissipative systems , the energy flux or related global quantities, strongly fluctuate in time although being averaged in space on the whole system or on its boundaries. These fluctuations should not be confused with small scale intermittency which occurs in fully developed turbulence. The later is related to the spotness of dissipation in space [6] and its description does not involve a time dependent ǫ.

Phys. Fluids}, 2010

We study the discrete wave turbulent regime of capillary water waves with constant non-zero vorticity. The explicit Hamiltonian formulation and the corresponding coupling coefficient are obtained. We also present the construction and investigation of resonance clustering. Some physical implications of the obtained results are discussed.

ERCOFTAC Series, 2019

We investigate experimentally turbulence of surface gravity waves in the Coriolis facility in Grenoble by using both high sensitivity local probes and a time and space resolved stereoscopic reconstruction of the water surface. We show that the water deformation is made of the superposition of weakly nonlinear waves following the linear dispersion relation and of bound waves resulting from non resonant triadic interaction. Although the theory predicts a 4-wave resonant coupling supporting the presence of an inverse cascade of wave action, we do not observe such inverse cascade. We investigate 4-wave coupling by computing the tricoherence i.e. 4-wave correlations. We observed very weak values of the tricoherence at the frequencies excited on the linear dispersion relation that are consistent with the hypothesis of weak coupling underlying the weak turbulence theory.

2021

Two coupled time-dependent two dimensional nonlinear Schrödinger equations have been derived using multiscale expansion for two nonlinearly interacting capillary-gravity waves over an infinite depth of water. These equations are then utilised to discuss the modulational (Benjamin-Feir) instability of two Stokes wavetrains due to unidirectional and bidirectional perturbations. It is found from the graphs and the three-dimensional contour plots that the rate of growth of instability for two wave packets interacting obliquely is higher than the instance of modulation of one wave packet. We have likewise examined the influence of capillarity on modulational instability.