This book provides an introduction to the theory of turbulence in fluids based on the representation of the flow by means of its vorticity field. It has long been understood that, at least in the case of incompressible flow, the vorticity representation is natural and physically transparent, yet the development of a theory of turbulence in this representation has been slow. The pioneering work of Onsager and of Joyce and Montgomery on the statistical mechanics of two-dimensional vortex systems h

Preface
Introduction
1.The Equations of Motion
1.1 The Euler and Navier-Stokes Equations
1.2 Vorticity Form of the Equations
1.3 Discrete Vortex Representations
1.4 Magnetization Variables
1.5 Fourier Representation for Periodic Flow
2.Random Flow and Its Spectra
2.1 Introduction to Probability Theory
2.2 Random Fields
2.3 Random Solutions of the Navier-Stokes Equations
2.4 Random Fourier Transform of a Homogeneous Flow Field
2.5 Brownian Motion and Brownian Walks
3.The Kolmogorov Theory
3.1 The Goals of Turbulence Theory:Universal Equilibrium
3.2 Kolmogorov Theory:Dimensional Considerations
3.3 The Kolmlgorov Spectrum and an Energy Cascade
3.4 Fractal Dimension
3.5 A First Discussion of Intermittency
4.Equilibrium Flow in Spectral Variables and in Two Space Dimensions
4.1 Statistical Equilibrium
4.2 The“Absolute”Statistical Equilibrium in Wave Number Space
4.3 The Combinatorial Method:The Approach to Equilibrium and Negative Temperatures
4.4 The Onsager Theory and the Joyce-Montgomery Equation
4.5 The Continuum Limit and the Role of Invariants
4.6 The Approach to Equilibrium,Viscosity,and Inertial Power Laws
5.Vortex Stretching
6.Polymers,Percolation,Renormalization
7.Vortex Equilibria a in Three-Dimensional Space
Bibliography
Index