Preface ....................................................... VII
Introduction ................................................... XV
Part I: Stochastic Structure Formations in Random
Hydrodynamic Flows
1 Equilibrium Distributions for Hydrodynamic Flows ............. 3
1.1 Two-Dimensional Hydrodynamics ........................... 4
2 Rogue Waves as an Object of Statistical Topography .......... 15
2.1 Statistical Topography of Random Field ξ(R,t) .......... 21
Part II: Density Field Diffusion and Clustering in Random
Hydrodynamic Flows
3 Main Features of the Problem and Determining Equations ...... 27
3.1 Low-Inertia Tracer ..................................... 27
3.2 Inertialess Tracer ..................................... 29
3.2.1 Relationship between the Lagrangian and
Eulerian Descriptions ........................... 30
4 Statistical Description of Inertialess Tracer Diffusion
and Clustering .............................................. 39
4.1 General Remarks ........................................ 39
4.2 Approximation of the Delta-Correlated (in Time)
Velocity Field ......................................... 42
4.2.1 Lagrangian Description (Particle Diffusion) ..... 42
4.2.2 Eulerian Description ............................ 52
4.3 Additional Factors ..................................... 60
4.3.1 Plane-Parallel Mean Shear ....................... 60
4.3.2 Effect of Molecular Diffusion ................... 62
4.3.3 Consideration of Finite Temporal Correlation
Radius .......................................... 66
4.3.4 Diffusion Approximation ......................... 67
4.4 Features of Tracer Diffusion in Fast Random Wave
Fields ................................................. 70
4.4.1 Eulerian Description ............................ 73
5 Integral One-Point Statistical Characteristics of Density
Field ....................................................... 79
5.1 Spatial Correlation Function of Density Field .......... 80
5.2 Spatial Correlation Tensor of Density Field Gradient
and Dissipation ........................................ 83
5.2.1 Extension to the Case of Inhomogeneous Initial
Conditions ...................................... 86
6 Tracer Diffusion and Clustering in Random Nondivergent
Flows ....................................................... 89
6.1 Diffusion and Clustering of the Buoyant Inertialess
Tracer ................................................. 89
6.1.1 Buoyant Tracer in Random Surface z(R,t) ......... 91
6.2 Diffusion and Clustering of Low-Inertia Tracer ......... 93
6.2.1 A Feature of Low-Inertia Particle Diffusion
(The Lagrangian Description) .................... 94
6.2.2 Low-Inertia Tracer Diffusion (The Eulerian
Description) .................................... 97
6.2.3 Spatial Correlations of Field V(r,t) ............ 99
6.2.4 Correlation Tensor of Spatial Derivatives of
Field V(r,t) ................................... 101
6.2.5 Temporal Correlation Tensor of Field V(r,t) .... 104
6.2.6 Conditions of Applicability of the Obtained
Results ........................................ 106
6.3 Diffusion and Clustering of Low-Inertia Tracer ........ 107
6.3.1 Spatial Correlations of Field V(r,t) ........... 108
6.3.2 Temporal Correlation Tensor of Field V(r,t) .... 111
7 Diffusion and Clustering of Settling Tracer in Random
Flows ...................................................... 115
7.1 State of Art and Main Equation of the Problem ......... 115
7.1.1 Particle Diffusion (Lagrangian Description) .... 116
7.1.2 Eulerian Description of the Tracer Density
Field .......................................... 118
7.2 Diffusion and Clustering of the Density Field ......... 119
7.3 Low-Inertia Settling Trace ............................ 127
7.3.2 Diffusion Approximation ........................ 130
7.3.3 Space-Time Correlation Tensor of Field
 (r,t) ......................................... 131
7.3.4 Space-Time Correlation Tensor of Field
div (r,t) ..................................... 133
Part III: Magnetic Field Diffusion and Clustering in Random
Magnetohydrodynamics Flows
8 Probabilistic Description of Magnetic Field in Random
Velocity Field ............................................. 139
8.1 General Remarks ....................................... 139
8.2 Statistical Averaging ................................. 141
9 Probabilistic Description of Magnetic Energy in Random
Velocity Field ............................................. 145
9.1 Delta-Correlated Random Velocity Field Approximation .. 145
9.2 Stochastic Dynamo in Critical Situations .............. 150
9.2.1 Features of Magnetic Field Diffusion in
Critical Situations ............................ 150
9.2.2 The Main Equations ............................. 154
9.2.3 Pseudoequilibrium Velocity Field ............... 163
9.2.4 Random Acoustic Velocity Field ................. 167
9.2.5 Equilibrium Thermal Velocity Field ............. 171
10 Integral One-Point Statistical Characteristics of
Magnetic Field ............................................. 173
10.1 Spatial Correlation Function of Magnetic Field ........ 173
10.2 On the Magnetic Field Helicity ........................ 176
10.3 On the Magnetic Field Dissipation ..................... 179
Part IV: Wave Localization in Randomly Layered Media
11 General Remarks ............................................ 185
11.1 Wave Incidence on an Inhomogeneous Layer .............. 185
11.2 Source Inside an Inhomogeneous Layer .................. 188
12 Statistics of Scattered Field at Layer Boundaries .......... 191
12.1 Reflection and Transmission Coefficients ............. 191
12.1.1 Nondissipative Medium (Normal Wave Incidence) .. 193
12.1.2 Nondissipative Medium (Oblique Wave
Incidence) ..................................... 196
12.1.3 Dissipative Medium ............................. 199
12.2 Source Inside the Medium Layer ........................ 202
12.3 Statistical Localization of Energy .................... 203
12.4 Diffusion Approximation ............................... 205
12.4.1 Unmatched Boundary ............................. 205
12.4.2 Matched Boundary ............................... 207
13 Statistical Description of a Wavefield in Random
Medium ..................................................... 213
13.1 Normal Wave Incidence on the Layer of Random Media .... 213
13.1.1 Nondissipative Medium (Stochastic Wave
Parametric Resonance and Dynamic Wave
Localization) .................................. 216
13.1.2 Dissipative Medium ............................. 225
13.2 Plane Wave Source Located in Random Medium ............ 229
13.2.1 Half-Space of Random Medium .................... 233
13.2.2 Asymptotic Case of Small Dissipation ........... 235
13.3 Peculiarity of Statistical Description of Acoustic
Field ................................................. 238
13.4 Numerical Simulation .................................. 243
13.4.1 Wave Incident on the Medium Layer .............. 245
13.4.2 Plane Wave Source in the Medium Layer .......... 246
13.4.3 Nonlinear Problem on Wave Self-action in
Random Media ................................... 248
14 Eigenvalue and Eigenfunction Statistics .................... 253
14.1 General Remarks ....................................... 253
14.2 Statistical Averaging ................................. 256
15 Multidimensional Wave Problems in Layered Random Media ..... 261
15.1 Nonstationary Problems ................................ 261
15.1.1 Formulation of Boundary-Value Wave Problems .... 261
15.1.2 Statistical Description ........................ 264
15.2 Point Source in Randomly Layered Medium ............... 268
15.2.1 Factorization of the Wave Equation in Layered
Medium ......................................... 268
15.2.2 Parabolic Equation ............................. 270
15.2.3 General Case ................................... 273
16 Two-Layer Model of the Medium .............................. 277
16.1 Formulation of Boundary-Value Problems ................ 277
16.2 Statistical Description ............................... 281
Part V: Wave Propagation in Random Media
17 Method of Stochastic Equation .............................. 289
17.1 Input Stochastic Equations and Their Implications ..... 289
17.2 Delta-Correlated Approximation for Medium Parameters .. 293
17.2.1 Estimation of Depolarization Phenomena in
Random Media ................................... 305
17.3 The Delta-Correlated Approximation and the Diffusion
Approximation for Wavefield ........................... 309
17.3.1 Perturbation Method ............................ 309
17.3.2 Diffusion Approximation for the Wavefield ...... 312
17.4 Wavefield Amplitude-Phase Fluctuations ................ 317
17.4.1 Random Phase Screen (Δx  x) .................. 321
17.4.2 Continuous Medium (Δx = x) ..................... 321
18 Geometrical Optics Approximation in Randomly
Inhomogeneous Media ........................................ 325
18.1 Ray Diffusion in Random Media (The Lagrangian
Description ........................................... 325
18.2 Formation of Caustics in Randomly Inhomogeneous
Media ................................................. 329
18.3 Wavefield Amplitude-Phase Fluctuations (The Eulerian
Description) .......................................... 336
19 Method of Path Integral .................................... 343
19.1 General Remarks ....................................... 343
19.2 Statistical Description of Wavefield .................. 347
19.3 Asymptotic Analysis of Plane Wave Intensity
Fluctuations .......................................... 351
19.3.1 Random Phase Screen ............................ 353
19.3.2 Continuous Medium .............................. 356
20 Caustic Structure of Wavefield in Random Media ............. 363
20.1 Elements of Statistical Topography of Random
Intensity Field ....................................... 364
20.2 Weak Intensity Fluctuations ........................... 365
20.3 Strong Intensity Fluctuations ......................... 369
Appendices: Appendices Imbedding Method in Boundary-Value
Wave Problems General Remarks ................................. 375
A Stationary Boundary-Value Wave Problems .................... 377
A.l One-Dimensional Stationary Boundary-Value Wave
Problems .............................................. 377
A.1.1 Helmholtz Equation With Unmatched Boundary ..... 377
A.1.2 Helmholtz Equation with Matched Boundary ....... 391
A.1.3 Acoustic Waves in Variable-Density Media and
Electromagnetic Waves in Layered Inhomogeneous
Media .......................................... 394
A.1.4 Acoustic-Gravity Waves in Layered Ocean ........ 403
A.2 Waves in Periodically Inhomogeneous Media ............. 412
A.2.1 Wave Incident on the Layer of Periodically
Inhomogeneous Medium ........................... 413
A.2.2 Bragg Resonance in Inhomogeneous Media ......... 417
А.3 Boundary-Value Stationary Nonlinear Wave Problem on
Self-Action ........................................... 419
A.3.1 General Equation ............................... 419
A.3.2 Wave Incidence on a Half-Space of Nonlinear
Medium ......................................... 426
A.3.3 Examples of Wavefield Calculations in
Nonlinear Medium ............................... 430
A.4 Stationary Multidimensional Boundary-Value Problem .... 438
A.4.1 Stationary Nonlinear Multidimensional
Boundary-Value Problem ......................... 447
В One-Dimensional Nonstationary Boundary-Value Wave Problem .. 455
B.l Nonsteady Medium ...................................... 455
B.1.1 Problem on a Wave Incident on Medium Layer ..... 457
B.2 Steady Medium ......................................... 460
B.2.1 Inverse Problem Solution ....................... 465
B.3 One-Dimensional Nonlinear Wave Problem ................ 468
References ................................................. 471
Index ......................................................... 489
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