Advances In Wave Turbulence

Wave or weak turbulence is a branch of science concerned with the evolution of random wave fields of all kinds and on all scales, from waves in galaxies to capillary waves on water surface, from waves in nonlinear optics to quantum fluids. In spite of the enormous diversity of wave fields in nature, there is a common conceptual and mathematical core which allows us to describe the processes of random wave interactions within the same conceptual paradigm, and in the same language. The development of this core and its links with the applications is the essence of wave turbulence science (WT) which is an established integral part of nonlinear science.

We often think of our natural environment as being composed of very many interacting particles, undergoing individual chaotic motions, of which only very coarse averages are perceptible at scales natural to us. However, we could as well think of the world as being made out of individual waves. This is so not just because the distinction between waves and particles becomes rather blurred at the atomic level, but also because even phenomena at much larger scales are better describedin terms of waves rather than of particles: It is rare in both fluids and solids to observe energy being carried from one region of space to another by a given set of material particles; much more often, this transfer occurs through chains of particles, neither of them moving much, but eachcommunicating with the next, and hence creating these immaterial objects we call waves. Waves occur at many spatial and temporal scales. Many of these waves have small enough amplitude that they can be approximately described by linear theory. However, the joint effect of large sets of waves is governed by nonlinear interactions which are responsible for huge cascades of energy among very disparate scales. Understanding these energy transfers is crucial in order to determine the response oflarge systems, such as the atmosphere and the ocean, to external forcings and dissipation mechanisms which act on scales decades apart. The field of wave turbulence attempts to understand the average behavior of large ensembles of waves, subjected to forcing and dissipation at opposite ends of theirspectrum. It does so by studying individual mechanisms for energy transfer, such as resonant triads and quartets, and attempting to draw from them effects that should not survive averaging. This book presents the proceedings of the AMS-IMS-SIAM Joint Summer Research Conference on Dispersive Wave Turbulence held at Mt. Holyoke College (MA). It drew together a group of researchers from many corners of the world, in the context of a perceived renaissance of the field, driven by heated debate aboutthe fundamental mechanism of energy transfer among large sets of waves, as well as by novel applications-and old ones revisited-to the understanding of the natural world. These proceedings reflect the spirit that permeated the conference, that of friendly scientific disagreement and genuine wonderat the rich phenomenology of waves.

This book is a useful source of ideas and information for scientists whose work involves understanding and modelling turbulent flows with free surfaces. It has the following merits: (1) It provides a framework for developing the analysis of this field, which, although important, has received only limited study; (2) It recognizes the importance of the two-phase nature of strongly disturbed free surface flows, with both natural and technological applications; (3) It suggests possible lines of future research (especially experimental) to quantify the characteristics of flow regimes which are mainly known qualitatively at present.

The European Turbulence Conferences have been organized under the auspices of the European Mechanics Committee (Euromech) to provide a forum for discussion and exchange of recent and new results in the field of turbulence. The first conference was organized in Lyon in 1986 with 152 participants. The second and third conferences were held in Berlin (1988) and Stockholm (1990) with 165 and 172 participants respectively. The fourth was organized in Delft from 30 June to 3 July 1992 by the J.M. Burgers Centre. There were 214 participants from 22 countries. This steadily growing number of participants demonstrates both the success and need for this type of conference. The main topics of the Fourth European Turbulence Conference were: Dynamical Systems and Transition; Statistical Physics and Turbulence; Experiments and Novel Experimental Techniques; Particles and Bubbles in Turbulence; Simulation Methods; Coherent Structures; Turbulence Modelling and Compressibility Effects. In addition a special session was held on the subject of CeBular Automata. Each of the sessions was introduced with a survey lecture. The lecturers were: W. Eckhaus, AJ. Libchaber, L. Katgerman, F. Durst, M. Lesieur, B. Legras, D.G. Dritschel and P. Bradshaw. The contributions of the participants were subdivided into oral and poster presentations. In addition to the normal program, some Speciai Interest Groups of Ercoftac (European Research Community on Flow, Turbulence and Combustion) presented their research activities in the form of a poster.

This book is a collection of papers on dynamical and statistical theory of nonlinear wave propagation in dispersive conservative media. Emphasis is on waves on the surface of an ideal fluid and on Rossby waves in the atmosphere. Although the book deals mainly with weakly nonlinear waves, it is more than simply a description of standard perturbation techniques. The goal is to show that the theory of weakly interacting waves is naturally related to such areas of mathematics as Diophantine equations, differential geometry of waves, Poincare normal forms and the inverse scattering method.

Wave Turbulence refers to the statistical theory of weakly nonlinear dispersive waves. There is a wide and growing spectrum of physical applications, ranging from sea waves, to plasma waves, to superfluid turbulence, to nonlinear optics and Bose-Einstein condensates. Beyond the fundamentals the book thus also covers new developments such as the interaction of random waves with coherent structures (vortices, solitons, wave breaks), inverse cascades leading to condensation and the transitions between weak and strong turbulence, turbulence intermittency as well as finite system size effects, such as “frozen” turbulence, discrete wave resonances and avalanche-type energy cascades. This book is an outgrow of several lectures courses held by the author and, as a result, written and structured rather as a graduate text than a monograph, with many exercises and solutions offered along the way. The present compact description primarily addresses students and non-specialist researchers wishing to enter and work in this field.

WAVE TURBULENCE is a state of a system of many simultaneously excited and interacting waves characterized by an energy distribution which is not in any sense close to thermodynamic equilibrium. Such situations in a choppy sea, in a hot plasma, in dielectrics under arise, for example, a powerful laser beam, in magnets placed in a strong microwave field, etc. Among the great variety of physical situations in which wave turbulence arises, it is possible to select two large limiting groups which allow a detailed analysis. The first is fully developed wave turbulence arising when energy pumping and dissipation have essentially different space scales. In this case there is a wide power spectrum of turbulence. This type of turbulence is described in detail e. g. in Zakharov et al. 1 In the second limiting case the scales in which energy pumping and dissipation occur are the same. As a rule, in this case a narrow, almost singular spectrum of turbulence appears which is concentrated near surfaces, curves or even points in k-space. One of the most important, widely investigated and instructive examples of this kind of turbulence is parametric wave turbulence appearing as a result of the evolution of a parametric instability of waves in media under strong external periodic modulation (laser beam, microwave electromagnetic field, etc. ). The present book deals with parametric wave turbulence.