Advances in Wave Turbulence

The following sections are included:

• Preface

• Contents

The goals of this chapter are: To state and review the premises on which a successful asymptotic closure of the BBGKY hierarchy of moment/cumulant equations is based; to describe how and why this closure is attained; to examine the nature of solutions of the kinetic equation; to discuss obstacles which limit the theory's validity and suggest how the theory might then be modified; to compare the experimental evidence in a range of applications with the theory's predictions; and finally, and most importantly, to suggest open challenges and to encourage the reader to apply and explore wave turbulence with confidence.

The key governing parameter of wave turbulence is the energy flux that drives the waves and cascades to small scales through nonlinear interactions. In the inertial range, the energy flux is conserved across the scales, and is assumed to be constant in most theoretical approaches. It is only recently that measurements of the injected power into wave turbulence have been performed at the scale of the wave maker (integral scale). In this review, we focus on the statistical properties of the injected power fluctuations in gravity-capillary wave turbulence in a stationary regime. Fluctuations of the injected power have been found much larger than their mean value. In addition, events related to a negative injected power, i.e. an instantaneous reversed energy flux, occur with a fairly large probability. Both features are well described using a Langevin type equation. Finally, we consider the experimental dependence of the scaling law of the wave spectrum with the mean injected power and discuss possible reasons for the discrepancy with weak turbulence theory.

This chapter reviews the recent progress made mainly during the last two decades in wave turbulence in astrophysical plasmas (MHD, Hall MHD and electron MHD) in the incompressible and compressible cases. The emphasis is on homogeneous and anisotropic turbulence which usually provides the best theoretical framework to investigate space and laboratory plasmas. The interplanetary medium and the solar atmosphere are presented as two examples of media where anisotropic wave turbulence is relevant. The most important results of wave turbulence are reported and discussed in the context of space and simulated magnetized plasmas. Important issues and possible spurious interpretations are eventually discussed.

We review recent progress in optical wave turbulence with a specific focus on the fast growing field of fiber lasers. Weak irregular nonlinear interactions between a large number of resonator modes are responsible for practically important characteristics of fiber lasers such as the spectral broadening of radiation. Wave turbulence is a fundamental nonlinear phenomenon which occurs in a variety of nonlinear wave-bearing physical systems. The experimental impediments and the computationally intensive nature of simulation of hydrodynamic or plasma wave turbulence often make it rather challenging to collect a significant number of statistical data. The study of turbulent wave behavior in optical devices offers quite a unique opportunity to collect an enormous amount of data on the statistical properties of wave turbulence using high-speed, high precision optical measurements during a relatively short period of time. We present recent theoretical, numerical and experimental results in optical wave turbulence in fiber lasers ranging from weak to strong turbulences for different signs of fiber dispersion. Furthermore, we report on our studies of spectral wave condensate in fiber lasers that make interdisciplinary links with a number of other research fields.

Thin metal plates have been used for ages to simulate the thunder noise in theatres. Turbulence of flexion waves in the plate radiates acoustic waves in the air. In this way it is possible to hear the Kolmogorov spectrum (as suggested by the title of the original article on the application of the Weak Turbulence theory to this issue)! In this chapter, we present the results of the application of the theory to flexion waves in a thin elastic plate. Then we compare the experimental results to the theoretical predictions. Advanced measurements of the plates deformation have ben implemented using a high-speed Fourier transform profilometry technique. This technique provides a 2D measurement of the deformation resolved in time. In this way a precise space and time analysis of wave turbulence could be performed for the first time in a real system. This analysis shows that the observed turbulence is in agreement with the phenomenology of the Weak Turbulence Theory but that there is a quantitative disagreement between observation and theoretical predictions. We discuss the possible reasons for such discrepancy: finite size effects or dissipation.

We overview past and new experimental results on gravity wave turbulence obtained at The Deep flume facility in Hull. The relatively large size of this flume, 12m × 6m × 1.5m, allows us to study the pure gravity waves without being concerned with modifications caused by the capillary or finite depth effects. We analyze wave spectra, probability density functions of wave heights and their increments, and structure functions to characterize both random weak waves and strong singular coherent structures. We see some evidence in favor of the Zakharov–Filonenko spectrum at larger wave intensities and we see signature of the finite size effects for weaker waves. Our new experiments are devoted to evolving wave turbulence, both the stages of forming the steady state and its decay, including experiments with a tilted wall.

The existing wave turbulence paradigm assumes, in particular, a proximity to stationarity and spatial homogeneity of the wave field under consideration, which is often violated in nature. In this review, we outline a new theoretical approach free from these restrictive assumptions. We present and discuss a novel approach to the direct numerical simulation (DNS) of random wind wave fields, based on the Zakharov equation. This approach can be used in situations when the statistical theory is not applicable. It can be also used to provide an independent corroboration to the wave turbulence theory. We review the experimental evidence indicating that often wave fields evolve much faster than the existing theory predicts, we then employ the DNS to study the fast evolution of wave fields far from equilibrium. These fields evolve on the dynamic timescale typical of coherent wave processes, rather than on the kinetic timescale typical of broadband random wave fields. This phenomenon of “fast evolution”, its causes, the specifics of its mathematical description and implications are the main focus of the review. We also present a generalisation of the existing statistical theory of wave turbulence, based on the new generalised kinetic equation. This equation is free of the assumption of quasi-stationarity and can account for both the fast and kinetic-scale processes in wave fields. Throughout the review water waves are used as a typical example of weakly nonlinear wave field, however, the new approach is equally valid for any random broadband weakly nonlinear wave field evolving due to nonlinear resonant interactions, either triad or quartic; the specificity of wave field is only in the form of the interaction coefficients.