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This book addresses the processes of stochastic structure formation in two-dimensional geophysical fluid dynamics based on statistical analysis of Gaussian random fields, as well as stochastic structure formation in dynamic systems with parametric excitation of positive random fields f(r,t) described by partial differential equations. Further, the book considers two examples of stochastic structure formation in dynamic systems with parametric excitation in the presence of Gaussian pumping. In dynamic systems with parametric excitation in space and time, this type of structure formation either happens ? or doesn’t! However, if it occurs in space, then this almost always happens (exponentially quickly) in individual realizations with a unit probability.

In the case considered, clustering of the field f(r,t) of any nature is a general feature of dynamic fields, and one may claim that structure formation is the Law of Nature for arbitrary random fields of such type. The study clarifies the conditions under which such structure formation takes place. To make the content more accessible, these conditions are described at a comparatively elementary mathematical level by employing ideas from statistical topography.

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This book addresses the processes of stochastic structure formation in two-dimensional geophysical fluid dynamics based on statistical analysis of Gaussian random fields, as well as stochastic structure formation in dynamic systems with parametric excitation of positive random fields f(r,t) described by partial differential equations. Further, the book considers two examples of stochastic structure formation in dynamic systems with parametric excitation in the presence of Gaussian pumping. In dynamic systems with parametric excitation in space and time, this type of structure formation either happens ? or doesn’t! However, if it occurs in space, then this almost always happens (exponentially quickly) in individual realizations with a unit probability.

In the case considered, clustering of the field f(r,t) of any nature is a general feature of dynamic fields, and one may claim that structure formation is the Law of Nature for arbitrary random fields of such type. The study clarifies the conditions under which such structure formation takes place. To make the content more accessible, these conditions are described at a comparatively elementary mathematical level by employing ideas from statistical topography.

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CONTENTS
1 Two-Dimensional Geophysical Fluid Dynamics = 1
 1.1 Equilibrium Distributions for Hydrodynamic Flows = 1
 1.2 Plane Motion Under the Action of a Periodic Force = 8
2 Parametrically Excited Dynamic Systems = 15
 2.1 Lognormal Random Process = 16
 2.2 Uncorrected Error of the Past Unavoidably Results in Errors of the Present and Future = 17
 2.3 Oscillator with Randomly Varying Frequency(Stochastic Parametric Resonance) = 19
3 Examples of Stochastic Dynamic Systems = 21
 3.1 Particles Under the Random Velocity Field = 21
  3.1.1 The Simplest Numerical Example of Particle Dynamics = 22
 3.2 Plane Waves in Layered Media = 26
 3.3 Partial Differential Equations = 36
 3.4 Model of a Stochastic Velocity Field Allowing Analytical Solutions to Transport Problems = 41
  3.4.1 Model of Passive Tracer Diffusion = 42
  3.4.2 Turbulent Dynamo Model = 42
4 Statistical Characteristics of a Random Velocity Field u(r, t) = 45
5 Lognormal Processes, Intermittency, and Dynamic Localization = 49
 5.1 Typical Realization Curve of a Random Process = 51
 5.2 Dynamic Localization = 52
6 Stochastic Parametric Resonance = 55
7 Wave Localization in Randomly Layered Media = 59
 7.1 Statistics of Scattered Field at Layer Boundaries = 60
  7.1.1 Reflection and Transmission Coefficients = 60
  7.1.2 Source Inside the Layer of a Medium = 67
  7.1.3 Statistical Localization = 68
 7.2 Statistical Theory of Radiative Transfer = 69
  7.2.1 Normal Wave Incidence on the Layer of Random Media = 69
  7.2.2 Nondissipative Medium(Stochastic Wave Parametric Resonance and Dynamic Wave Localization) = 72
  7.2.3 Dissipative Medium = 80
  7.2.4 Plane Wave Source Located in Random Medium = 84
 7.3 Numerical Simulation = 87
  7.3.1 Wave Incident on the Medium Layer = 89
  7.3.2 Plane Wave Source in the Medium Layer = 90
8 Lognormal Fields, Statistical Topography, and Clustering = 95
 8.1 Lognormal Random Fields = 95
 8.2 Statistical Topography of Random Fields = 97
  8.2.1 Conditions of Cluster Structure Formation = 99
  8.2.2 Statistical Topography of Lognormal Random Fields = 104
9 Stochastic Transport Phenomena in a Random Velocity Field = 109
 9.1 Clustering of the Density Field in a Random Velocity Field = 109
 9.2 Probabilistic Description of a Magnetic Field and Its Energy in a Random Velocity Field = 116
  9.2.1 Probabilistic Description of a Magnetic Field = 116
  9.2.2 Probabilistic Description of Magnetic Field Energy = 118
  9.2.3 The Critical Case of α＝0(DP＝Dˢ) = 120
10 Parametrically Excited Dynamic Systems with Gaussian Pumping = 123
 10.1 Statistical Analysis of Simple Turbulent Dynamo Problem with Gaussian Pumping = 123
 10.2 Anomalous Sea Surface Structures = 126
  10.2.1 Problem Statement = 127
  10.2.2 Equation in Probability Density = 132
  10.2.3 Statistical Analysis of the Problem = 137
  10.2.4 Statistical Topography of the Sea Surface Elevation Field = 140
Conclusion = 143
Appendix : Elements of Mathematical Tools for Describing Coherent Phenomena = 145
Bibliography = 187