BASIC HYDRODYNAMICS

By T. BESAL

CONTENTS PART I: DYNAMICAL DESCRIPTION OF STOCHASTIC SYSTEMS EXAMPLES, BASIC PROBLEMS, PECULIAR FEATURES OF SOLUTIONS ORDINARY DI?ERENTIAL EQUATIONS: INITIAL-VALUE PROBLEMS PARTICLES UNDER RANDOM VELOCITY FIELD PARTICLES UNDER RANDOM FORCES THE HOPPING PHENOMENON SYSTEMS WITH BLOW-UP SINGULARITIES OSCILLATOR WITH RANDOMLY VARYING FREQUENCY (STOCHASTIC PARAMETRIC RESONANCE) LINEAR ORDINARY DI?ERENTIAL EQUATIONS: BOUNDARY-VALUE PROBLEMS PLANE WAVES IN LAYERED MEDIA: A WAVE INCIDENT ON A MEDIUM LAYER PLANE WAVES IN LAYERED MEDIA: SOURCE INSIDE THE MEDIUM PLANE WAVES IN LAYERED MEDIA: TWO-LAYER MODEL FIRST-ORDER PARTIAL DI?ERENTIAL EQUATIONS LINEAR FIRST-ORDER PARTIAL DI?ERENTIAL EQUATIONS QUASILINEAR EQUATIONS BOUNDARY-VALUE PROBLEMS FOR NONLINEAR ORDINARY DI?ERENTIAL EQUATIONS NONLINEAR FIRST-ORDER PARTIAL DI?ERENTIAL EQUATIONS PARTIAL DI?ERENTIAL EQUATIONS OF HIGHER ORDERS FUNDAMENTAL SOLUTIONS OF WAVE PROBLEMS IN FREE SPACE AND LAYERED MEDIA STATIONARY PROBLEMS FOR MAXWELL’S EQUATIONS THE HELMHOLTZ EQUATION (BOUNDARY-VALUE PROBLEM) AND THE PARABOLIC EQUATION OF QUASI-OPTICS (WAVES IN RANDOMLY INHOMOGENEOUS MEDIA) THE NAVIER–STOKES EQUATION: RANDOM FORCES IN HYDRODYNAMIC THEORY OF TURBULENCE EQUATIONS OF GEOPHYSICAL HYDRODYNAMICS SOLUTION DEPENDENCE ON PROBLEM TYPE, MEDIUM PARAMETERS, AND INITIAL DATA FUNCTIONAL REPRESENTATION OF PROBLEM SOLUTION VARIATIONAL (FUNCTIONAL) DERIVATIVES PRINCIPLE OF DYNAMIC CAUSALITY SOLUTION DEPENDENCE ON PROBLEM’S PARAMETERS SOLUTION DEPENDENCE ON INITIAL DATA IMBEDDING METHOD FOR BOUNDARY-VALUE PROBLEMS INDICATOR FUNCTION AND LIOUVILLE EQUATION ORDINARY DI?ERENTIAL EQUATIONS LINEAR EQUATIONS GENERAL-FORM NONLINEAR EQUATIONS HIGHER-ORDER PARTIAL DI?ERENTIAL EQUATIONS PARABOLIC EQUATION OF QUASI-OPTICS RANDOM FORCES IN HYDRODYNAMIC THEORY OF TURBULENCE PART II: RANDOM QUANTITIES, PROCESSES, AND FIELDS RANDOM QUANTITIES AND THEIR CHARACTERISTICS RANDOM PROCESSES AND THEIR CHARACTERISTICS GENERAL REMARKS RANDOM PROCESS TYPICAL REALIZATION CURVE STATISTICS OF RANDOM PROCESS CROSS POINTS WITH A LINE GAUSSIAN RANDOM PROCESS LOGARITHMICALLY NORMAL RANDOM PROCESS DISCONTINUOUS RANDOM PROCESSES POISSON (IMPULSE) RANDOM PROCESS TELEGRAPHER’S RANDOM PROCESS GENERALIZED TELEGRAPHER’S RANDOM PROCESS MARKOVIAN PROCESSES GENERAL PROPERTIES CHARACTERISTIC FUNCTIONAL OF THE MARKOVIAN PROCESS RANDOM FIELDS GAUSSIAN VECTOR RANDOM FIELD STATISTICAL TOPOGRAPHY OF RANDOM PROCESSES AND FIELDS ON THE CRITERION OF STOCHASTIC STRUCTURE FORMATION IN RANDOM MEDIA CORRELATION SPLITTING GAUSSIAN PROCESS GAUSSIAN RANDOM FIELDS POISSON PROCESS GENERALIZED TELEGRAPHER’S RANDOM PROCESS GENERAL-FORM MARKOVIAN PROCESSES DELTA-CORRELATED RANDOM PROCESSES ASYMPTOTIC MEANING OF DELTA-CORRELATED PROCESSES AND FIELDS PART III: STOCHASTIC EQUATIONS GENERAL APPROACHES TO ANALYZING STOCHASTIC DYNAMIC SYSTEMS PARTIAL DI?ERENTIAL EQUATIONS PASSIVE TRACER TRANSFER IN RANDOM FIELD OF VELOCITIES RANDOM FORCES IN THE THEORY OF HYDRODYNAMIC TURBULENCE STOCHASTIC INTEGRAL EQUATIONS (METHODS OF QUANTUM FIELD THEORY) LINEAR INTEGRAL EQUATIONS NONLINEAR INTEGRAL EQUATIONS COMPLETELY SOLVABLE STOCHASTIC DYNAMIC SYSTEMS DELTA-CORRELATED FIELDS AND PROCESSES ONE-DIMENSIONAL NONLINEAR DI?ERENTIAL EQUATION LINEAR OPERATOR EQUATION STOCHASTIC EQUATIONS WITH THE MARKOVIAN FLUCTUATIONS OF PARAMETERS TELEGRAPHER’S PROCESSES SYSTEM OF LINEAR OPERATOR EQUATIONS PARTICLE IN THE ONE-DIMENSIONAL POTENTIAL FIELD ORDINARY DI?ERENTIAL EQUATION OF THE N-TH ORDER STATISTICAL INTERPRETATION OF TELEGRAPHER’S EQUATION GENERALIZED TELEGRAPHER’S PROCESS STOCHASTIC LINEAR EQUATION ORDINAL DI?ERENTIAL EQUATION OF THE N-TH ORDER GAUSSIAN MARKOVIAN PROCESSES THE SQUARE OF THE GAUSSIAN MARKOVIAN PROCESS MARKOVIAN PROCESSES WITH FINITE-DIMENSIONAL PHASE SPACE TWO-STATE PROCESS CAUSAL STOCHASTIC INTEGRAL EQUATIONS GAUSSIAN MARKOVIAN PROCESS PART IV: ASYMPTOTIC AND APPROXIMATE METHODS FOR ANALYZING STOCHASTIC EQUATIONS GAUSSIAN RANDOM FIELD DELTA-CORRELATED IN TIME (ORDINARY DI?ERENTIAL EQUATIONS) 10 1 THE FOKKER–PLANCK EQUATION 10 2 TRANSITIONAL PROBABILITY DISTRIBUTIONS 10 3 APPLICABILITY RANGE OF THE FOKKER–PLANCK EQUATION 10 3 1 LANGEVIN EQUATION 10 4 THE SIMPLEST MARKOVIAN RANDOM PROCESSES 10 4 1 SYSTEM OF LINEAR EQUATIONS 10 4 2 WIENER RANDOM PROCESS 10 4 3 WIENER RANDOM PROCESS WITH SHEAR 10 4 4 LOGNORMAL RANDOM PROCESS, INTERMITTENCY AND DYNAMIC LOCALIZATION 10 5 LOGARITHMIC-NORMAL RANDOM FIELDS, INTERMITTENCY AND CLUSTERING 10 5 1 LOGARITHMIC-NORMAL RANDOM FIELDS 10 5 2 STATISTICAL TOPOGRAPHY OF THE LOGNORMAL RANDOM FIELDS 10 6 CAUSAL INTEGRAL EQUATIONS 10 6 1 GENERAL REMARKS 10 6 2 STATISTICAL AVERAGING METHODS FOR SOLVING AND ANALYZING THE FOKKER-PLANCK EQUATION 11 1 INTEGRAL TRANSFORMATIONS 11 2 STEADY-STATE SOLUTIONS OF THE FOKKER–PLANCK EQUATION 11 2 1 ONE-DIMENSIONAL NONLINEAR DI?ERENTIAL EQUATION 11 2 2 HAMILTONIAN SYSTEMS 11 2 3 SYSTEMS OF HYDRODYNAMIC TYPE 11 3 BOUNDARY-VALUE PROBLEMS FOR THE FOKKER-PLANCK EQUATION (HOPPING PHENOMENON) 11 3 1 HOPPING PHENOMENON IN REGULAR SYSTEMS 11 3 2 HOPPING PHENOMENA IN SINGULAR SYSTEMS 11 4 ASYMPTOTIC AND APPROXIMATE METHODS OF SOLVING THE FOKKER-PLANK EQUATION 11 4 1 ASYMPTOTIC EXPANSION 11 4 2 METHOD OF CUMULANT EXPANSIONS 11 4 3 METHOD OF FAST OSCILLATION AVERAGING DI?USION AND HIGHER APPROXIMATIONS 12 1 GENERAL REMARKS 12 2 DI?USION APPROXIMATION 12 2 1 DYNAMICS OF A PARTICLE 12 3 HIGHER APPROXIMATIONS SOME OTHER APPROXIMATE APPROACHES TO THE PROBLEMS OF STATISTICAL HYDRODYNAMICS 13 1 QUASI-ELASTIC PROPERTIES OF ISOTROPIC AND STATIONARY NONCOMPRESSIBLE TURBULENT MEDIA 13 2 SOUND RADIATION BY VORTEX MOTIONS 13 2 1 SOUND RADIATION BY VORTEX LINES 13 2 2 SOUND RADIATION BY VORTEX RINGS REFERENCES