Lecture Notes on Turbulence and Coherent Structures in Fluids, Plasmas and Nonlinear Media

The following sections are included:

• Preface

• Contents

This is a short course on developed turbulence, weak and strong. The main emphasis is on fundamental properties like universality and symmetries. Two main notions are explained: i) fluxes of dynamical integrals of motion, ii) statistical integrals of motion.

This chapter is based on the three lectures given as Renormalization for beginners and is divided into three main sections, each corresponding to one of the lectures. After a brief introduction in the first section, we give an overview of renormalization in physics, concluding with a short discussion of its potential for application to turbulence. The second main section then discusses renormalized perturbation theories and two-point turbulence closures. In the interests of completeness, this is supplemented by material from the Workshop lecture Two-point closures revisited. Lastly we briefly discuss renormalization group as used in particle theory and the theory of critical phenomena, and then concentrate on its application to macroscopic fluid turbulence.

An introduction is given to the multi-scale processes that provide major contributions to the ocean's energy balance. Motivating problem is the issue to determine how the large-scale ocean will respond to variations in momentum and buoyancy exchange with the atmosphere. It appears that to obtain a clear answer to this problem, knowledge of turbulent flows at many different spatial and temporal scales is essential.

An overview of the current state of fundamental plasma turbulence theory as applied to magnetic confinement is presented. Various topics, both historical and contemporary, in the analytical description of plasma turbulence are discussed. Modern gyrokinetic formalism is described heuris-tically, and various important nonlinear equations derived therefrom are highlighted. Several examples of the transition to turbulence are mentioned. The venerable quasilinear theory is discussed as an important example of statistical closure. Resonance-broadening theory is motivated, and some plasma-physics uses of the direct-interaction approximation and its Markovian relatives are described. The Martin–Siggia–Rose formalism is introduced. The theory of zonal-flow generation is shown to provide interesting examples of field-theoretic functional techniques, Markovian closure theory, and adiabatic and wave-kinetic methods. As an example of intermittent phenomena, the formation and propagation of coherent plasma “blobs” are briefly discussed. Some outstanding issues and challenging research areas are enumerated.

This chapter is based on two lectures given by M. Shats at the Summer School describing studies of turbulence in toroidal plasma confinement experiments. Some plasma diagnostics relevant for turbulence studies are reviewed. The data analysis techniques and methods are described in the context of the turbulence studies performed in the low-temperature plasma of the H-1 toroidal heliac, with particular emphasis on the analysis of spectral transfer in turbulent spectra. Experimental results on self-organization of the two-dimensional fluid turbulence are presented to illustrate some similarity with processes in quasi-two-dimensional plasma turbulence. Experimental signatures of zonal flows in plasma are illustrated.

This chapter deals primarily with the numerical computation of three-dimensional Navier–Stokes incompressible turbulence. As such, it leaves a lot of ground uncovered. It will be made clear early in the chapter that turbulence is not restricted to such flows, or even only to fluids, but this is the most common case in industrial applications, and most engineering computational schemes have been developed for it. Even when compressibility is important for the flow as a whole, such as in transonic flight, it does not affect turbulence strongly, and most of what is said here can be used with minor modifications. More extreme departures, such as hypersonics, plasmas, or the strong stratification of many geophysical flows, require changes that would make much of the material in this chapter irrelevant, except perhaps for the direct numerical simulations discussed in §6.5. It would make little sense, for example, to use current RANS or LES models for two-dimensional turbulence.

This chapter has two parts. Sections §6.1. to §6.3. are a quick review of classical turbulence theory, with the goal of establishing the tools to be used later, and specially of defining the relevant turbulent length scales. The second part, formed by sections §6.4. to §6.7., describe the three levels of detail at which turbulence is nowadays computed. It is difficult in such a limited space to do more than to familiarize the reader with the requirements of each technique, and with the results that can be expected from them. The final result should at least be to allow him or her to determine which method to use in a particular situation.

A subject such as this cannot be discussed without some reference to numerical analysis, which is the tool of simulations. This is specially true in the case of turbulence, where the flow field is not smooth, and therefore not easy to represent numerically. Any reader seriously interested in computing turbulence should, of course, familiarize her or himself as much as possible with numerical techniques.

In this chapter one of the most widely used techniques in experimental studies of fluid turbulence, particle image velocimetry (PIV), is reviewed. Principles and main characteristics of the PIV method are complemented by examples of its implementation for studies of turbulent jet.

We present an introduction into the basic physics and an overview of recent developments in Singular Optics, the novel discipline of modern optics studying electro–magnetic vortices. We focus on vortex solitons and related topics of nonlinear singular optics as well as discuss the future perspectives and applications of twisted light.