# Random Functions and Turbulence

## 1st Edition

### International Series of Monographs in Natural Philosophy

**Authors:**S. Panchev

**Editors:**D. ter Haar

**eBook ISBN:**9781483145594

**Imprint:**Pergamon

**Published Date:**1st January 1971

**Page Count:**458

## Description

International Series of Monographs in Natural Philosophy, Volume 32: Random Functions and Turbulence focuses on the use of random functions as mathematical methods. The manuscript first offers information on the elements of the theory of random functions. Topics include determination of statistical moments by characteristic functions; functional transformations of random variables; multidimensional random variables with spherical symmetry; and random variables and distribution functions. The book then discusses random processes and random fields, including stationarity and ergodicity of random processes; influence of finiteness of the interval of averaging; scalar and vector random fields; and statistical moments. The text takes a look at the statistical theory of turbulence. Topics include turbulence with very large Reynolds numbers; emergence of turbulent motion; and energy spectrum in isothermal turbulent shear flow. The book also discusses small-scale and large-scale atmospheric turbulence and applications to numerical weather analysis and prediction. The manuscript is a vital source of data for readers interested in random theory.

## Table of Contents

Contents

Foreword to the English Edition

Introduction

Part I. Elements of the Theory of Random Functions

Chapter 1. Certain Data on the Theory of Probability

§ 1. Random Variables and Distribution Functions

§ 2. Numerical Characteristics of Random Variables

§ 3. Multidimensional Random Variables with Spherical Symmetry

§ 4. Functional Transformations of Random Variables

§ 5. Certain Generalizations

§ 6. The Characteristic Function

§ 7. The Determination of Statistical Moments by Means of Characteristic Functions

Chapter 2. Random Processes

§ 1. The Definition of the Random Function of One Variable. The Probability Distribution of a Random Function

§ 2. Statistical Moments. The Autocorrelation Function

§ 3. The Two-dimensional Random Process. The Cross Correlation Function

§ 4. The Stationarity and Ergodicity of Random Processes

§ 5. Fundamental Characteristics of the Autocorrelation and Cross Correlation Functions with Stationary Random Processes

§ 6. The Differentiation of Random Functions

§ 7. The Integration of Random Functions

§ 8. The Normally Distributed Random Processes

§ 9. The Harmonic Analysis of Random Processes

§ 10. Generalized Harmonic Analysis. Spectral Expansions

§ 11. Random Processes with Stationary Increment. Structure Functions

§ 12. The Determination of the Correlation Function with Experimental Data

§ 13. The Influence of Finiteness of the Interval of Averaging

Chapter 3. Random Fields

§ 1. Supplementary Information

§ 2. Scalar and Vector Random Fields. The Random Functions of Several Variables

§ 3. Statistical Moments

§ 4. Homogeneous and Isotropic Random Fields

§ 5. Normal Random Fields

§ 6. The General Form of Tensor Statistical Moments

§ 7. The Structure and Certain General Characteristics of Tensor Moments

§ 8. Spectral Expansions

§ 9. The Correlation of Random Solenoidal Vector Fields

§ 10. The Correlation of Random Potential Vector Fields

§ 11. The Joint Correlation of Solenoidal and Potential Random Vector Fields

§ 12. The Correlation of Certain Derived Fields

§ 13. Locally Homogeneous and Isotropie Random Fields. Structure Functions

§ 14. Some Additional Problems Concerning the Theory of Random Fields

Part II hydrodynamic Turbulence

Chapter 4. The Statistical Theory of Turbulence—The Method of Similarity and Dimensionality

§ 1. Some Data from the Theory of Dimensionality

§ 2. The Emergence of Turbulent Motion

§ 3. Turbulence with Very Large Reynolds Numbers

§ 4. Locally Isotropie Turbulence. The Theory of Kolmogorov

§ 5. The Microstructure of a Temperature Field in a Locally Isotropie Turbulent Flow. The Theory of Obukhov

Chapter 5. The Statistical Theory of Turbulence—The Correlation Method

§ 1. Isotropie Turbulence. The Equation of Kärmän-Howarth

§ 2. The Invariant of Loitzianskii

§ 3. Fundamental Laws of Decay of Isotropie Turbulence

§ 4. On the Hypothesis of Millionshchikov and its Generalizations

§5. Locally Isotropie Turbulence—Kolmogorov's Equation

§ 6. The Spatial Correlation of Pressure

§ 7. The Spatial Correlation of Acceleration

§ 8. The Spatial Correlation of Temperature

§ 9. Correlation of the Vorticity

Chapter 6. The Statistical Theory of Turbulence— The Spectral Method

§ 1. The Turbulent Energy Balance Equation

§ 2. The Formulation of Fundamental Concepts and Laws in Terms of the Spectral Theory

§ 3. Obukhov's Spectral Theory

§ 4. Heisenberg^ Spectral Theory

§ 5. Another Approximation for the Energy Transfer Function

§ 6. Energy Spectrum in Isothermal Turbulent Shear Flow

§ 7. Temperature Spectrum in Isotropic Turbulence

§ 8. Spectral Theory of Decaying Turbulence

§ 9. Experimental Data for Turbulent Spectra

Chapter 7. Some Additional Problems of the Statistical Theory of Turbulence

§ 1. The Space-Time Correlation of the Velocity in a Homogeneous, Isotropic and Stationary Turbulent Flow

§ 2. The Space-Time Correlation of Temperature

§ 3. The Velocity and Temperature Correlation in «-Dimensional Isotropic Turbulent Flow

§ 4. The Spatial Correlation of Local Changes of Temperature in a Homogeneous and Isotropic Turbulent Flow

§ 5. The Joint Correlation of the Pressure and Velocity in an Isotropic Turbulent Flow

§ 6. The Description of Turbulence in Lagrangian Coordinates. Turbulent Diffusion

§ 7. Certain New Directions in the Statistical Theory of Turbulence

Part III. Atmospheric Turbulence

Chapter 8. Small-scale Atmospheric Turbulence

§ 1. The General Nature of Small-scale Atmospheric Turbulence

§ 2. The Microstructure of Turbulent Fluctuations of Meteorological Elements with Neutral Stratification of the Atmosphere's Surface Layer

§ 3. The Influence of Archimedean Force

§ 4. Spectra of Fluctuations of Wind Velocity, Temperature, etc., in a Micrometeorological Region

§ 5. Inertia of Meteorological Instruments in Turbulent Atmosphere

§ 6. Turbulence in Clouds and the Coalescence of Drops

Chapter 9. Large-scale Atmospheric Turbulence

§ 1. The Empirical Structure and Correlation Functions of Fundamental Meteorological Elements with Large-scale Motions

§ 2. The Analytical Approximation of Empirical Structure and Correlation Functions

§ 3. The Application of the Similarity and Dimensionality Method in the Investigation of the Macrostructure of Fundamental Meteorological Elements

§ 4. The Transition to the Range of "Saturation" of Structure Functions

§ 5. The Spatial Structure of a Geopotential Field in the Free Atmosphere

§ 6. The Time Macrostructure of Fields of Fundamental Meteorological Elements

§ 7. Spectra of Fundamental Meteorological Elements with Large-scale Motions

§ 8. The Spatial Structure and Spectrum of the Geostrophic Vorticity

§ 9. The Space-Time Macrostructure of Meteorological Elements

§ 10. The Three-dimensional Spatial Macrostructure of Meteorological Elements

§11. The Spatial Macrostructure and Spectrum of Geostrophic Thermal Advection in the Free Atmosphere

§ 12. The Spatial Macrostructure and Spectrum of Geostrophic Advection of the Vorticity in Barotropic Atmosphere

§ 13. The Structure of Geopotential Tendency

§ 14. The Joint Correlation of the Vorticity and Absolute Geopotential

§ 15. The Problem of Nonstationarity of Atmospheric Processes and of Deviations of Wind from the Geostrophic. Isallobaric Wind

§ 16. The Statistical Structure of Macrofluctuations of Wind Direction in the Free Atmosphere

§ 17. The Coefficient of Horizontal Macroturbulent Exchange in the Atmosphere

Chapter 10. Some Applications to Numerical Weather Analysis and Prediction

§ 1. The Optimal Interpolation of Meteorological Fields

§ 2. The Calculation of Characteristic Values of Spatial Derivatives of Meteorological Elements

§ 3. The Selection of Step for the Numerical Differentiation of Meteorological Elements

§ 4. The Accuracy of Determining Characteristic Values of Finite Differences of Meteorological Elements

§ 5. The Optimal Smoothing of Meteorological Fields

§ 6. The Predictability of Synoptic Processes

§ 7. The Correlation Functions of Meteorological Elements on the Spherical Earth

§ 8. The Prognosis of Smoothed Values of the Stream Function at a Mean Level of Atmosphere by Means of Correlation Functions Following Blinova's Method

§9. The Prognosis of Characteristics of General Atmospheric Circulation with a Consideration of Macroturbulence

Appendix. Large-scale Lagrangian Turbulence in the Atmosphere

References

Author Index

Subject Index

## Details

- No. of pages:
- 458

- Language:
- English

- Copyright:
- © Pergamon 1971

- Published:
- 1st January 1971

- Imprint:
- Pergamon

- eBook ISBN:
- 9781483145594