Stochastic Tools in Turbulence

Preface1. Probability Distributions and Densities 1.1 Definition of Probability 1.2 Formalizing the Definition 1.3 Measure and Lebesgue Integration 1.4 Distribution Function 1.5 The Probability Density Function 1.6 A Simple Example of the Distribution and Density Functions 1.7 Expected Values 1.8 Joint Distribution Functions 1.9 The Joint Density Function 1.10 Expected Values 1.11 Conditional Probabilities 1.12 Conditional Expectations 1.13 Statistical Independence and Lack of Correlation 1.14 Change of Variable2. Moments, Characteristic Functions, and the Gaussian Distribution 2.1 Moments Defined 2.2 Significance of the Moments 2.3 The Correlation Matrix and Principal Axes 2.4 The Characteristic Function 2.5 Properties of the Characteristic Function 2.6 Two Simple Examples 2.7 The Central-Limit Theorem for Independent Variables 2.8 The Gaussian Distribution from a More Physical Point of View 2.9 Moments of a Gaussian Distribution 2.10 The Jointly Normal Distribution 2.11 Cumulants 2.12 The Gram-Charlier Expansion3. Random Functions 3.1 Generalities: Multipoint Characteristic Functions 3.2 Statistics for Derivatives and Integrals 3.3 Processes and Characteristic Functionals: The Gaussian Process 3.4 Limit Processes of Random Functions 3.5 The Representation Problem 3.6 Finite Total Energy and Characteristic Eddies 3.7 Calculation of the Characteristic Eddies 3.8 Rate of Convergence of the Series of Eigenfunctions 3.9 Stationarity and the Ergodic Problem 3.10 Autocorrelations of Stationary Processes and Their Properties 3.11 Estimation by Time Averages 3.12 The Representation Problem for Stationary Processes: Spectra 3.13 Estimation by Time Averages with Zero Integral Scale 3.14 Another Type of Representation Theorem for Stationary Processes: Characteristic Eddies 3.15 Alternate Approaches to Harmonic Decomposition for Stationary Processes 3.16 A Central-Limit Theorem for Random Functions4. Random Processes in More Dimensions 4.1 Multidimensional Vector Fields of Finite Energy 4.2 Homogeneity, Averaging, and Ergodicity in Several Dimensions 4.3 The Homogeneous Scalar Field: One-Dimensional Spectra 4.4 The Homogeneous Scalar Field: The Three-Dimensional Spectrum 4.5 The Homogeneous Scalar Field: Consequences of Isotropy 4.6 The Homogeneous Scalar Field: General Form of the Spectra 4.7 The Solenoidal Homogeneous Vector Field: Implications of Incompressibility 4.8 The Solenoidal Homogeneous Vector Field: One-Dimensional Spectra 4.9 The Solenoidal Homogeneous Vector Field: The Three-Dimensional Spectrum 4.10 The Solenoidal Homogenous Vector Field: Consequences of Isotropy 4.11 The Solenoidal Homogeneous Vector Field: General Form of the Spectra 4.12 Characteristic Eddies for a Homogeneous Vector Field 4.13 Incompletely Homogeneous Fields: Co- and Quadrature Spectra and Coherence 4.14 Characteristic Eddies for an Incompletely Homogeneous Field 4.15 Multiple-Valued Functions 4.16 Distribution of Solutions for an Algebraic EquationAppendix 1. Fourier Transforms A1.1 Fourier Transforms of Well-Behaved Functions A1.2 The Inverse Transform A1.3 The Convolution A1.4 Symmetry Properties A1.5 Parseval's Relation A1.6 Relations Among Derivatives A1.7 Shift of Variables A1.8 Multiple VariablesAppendix 2. Tensors A2.1 Transformation Properties: Co- and Contravariant Indices A2.2 The Metric Tensor: Changing Indices A2.3 Cartesian Systems, Numerical Tensors, and Tensor Densities A2.4 Differentiation A2.5 Eigenvalues and Eigenvectors: Representations A2.6 Principal Invariants, The Cayley-Hamilton Theorem, and InversesAppendix 3. Theory of Generalized Functions A3.1 Generalities A3.2 Linear Continuous Functionals A3.3 Addition and Multiplication by a Constant and by a Function A3.4 Convergence of Sequences of Generalized Functions A3.5 Differentiation and Integration of Generalized Functions A3.6 Support of a Generalized Function A3.7 Direct Product of Generalized Functions A3.8 Convolutions of Generalized Functions A3.9 Fourier Transforms of Generalized Functions A3.10 Several Variables A3.11 Effect of a Shift of Variables A3.12 Asymptotic Behavior of Generalized Functions A3.13 Fourier Transforms of Generalized Functions Defined on the Space of Bounded, Infinitely Differentiable Functions A3.14 The Fourier Transform of the Convolution A3.15 Behavior of the Fourier Transform A3.16 The Kernel Theorem A3.17 Representations of Generalized Functions of Finite Total EnergyAppendix 4. Invariant Theory, Isotropy, and Axisymmetry A4.1 Invariance under Transformation Groups A4.2 Independent Invariants of Tensors of Various Orders A4.3 Representations of Tensor Functions—A General Method A4.4 Tensor Constants and Functions of a Scalar A4.5 Tensor Functions of a Vector A4.6 Tensor Functions of a Tensor of Second RankReferencesIndex