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ОбложкаKlyatskin V.I. Stochastic equations: theory and applications in acoustics, hydrodynamics, magnetohydrodynamics, and radiophysics. Vol.2: Coherent phenomena in stochastic dynamic systems / transl. from russ. by A.Vinogradov. - Cham: Springer, 2015. - xvii, 491 p.: ill. - Bibliogr.: p.471-488. - Ind.: p.489-491. - (Understanding complex systems) - ISBN 978-3-319-07589-1; ISSN 1860-0832
Шифр: (И/В17-K51/2) 02
 

Место хранения: 02 | Отделение ГПНТБ СО РАН | Новосибирск

Оглавление / Contents
 
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
               fig.3(r,t) ......................................... 131
        7.3.4  Space-Time Correlation Tensor of Field
               div fig.3(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 fig.2 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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