Stochastic Equations: Theory and Applications in Acoustics, Hydrodynamics, Magnetohydrodynamics, and Radiophysics, Volume 2: Coherent Phenomena in Stochastic Dynamic Systems

489 Pages·2015·8.97 MB·English

by  Valery I. Klyatskin (auth.)

Understanding Complex Systems
Valery I. Klyatskin
Stochastic Equations:
Theory and Applications in
Acoustics, Hydrodynamics,
Magnetohydrodynamics,
and Radiophysics, Volume 2
Coherent Phenomena in
Stochastic Dynamic Systems
Understanding Complex Systems
FoundingEditor
Prof.Dr.J.A.ScottKelso
CenterforComplexSystems&BrainSciences
FloridaAtlanticUniversity
BocaRatonFL,USA
E-mail:[email protected]
EditorialandProgrammeAdvisoryBoard
DanBraha
NewEnglandComplexSystems,InstituteandUniversityofMassachusetts,Dartmouth
PéterÉrdi
CenterforComplexSystemsStudies,KalamazooCollege,USAandHungarianAcademyof
Sciences,Budapest,Hungary
KarlFriston
InstituteofCognitiveNeuroscience,UniversityCollegeLondon,London,UK
HermannHaken
CenterofSynergetics,UniversityofStuttgart,Stuttgart,Germany
ViktorJirsa
CentreNationaldelaRechercheScientifique(CNRS),UniversitédelaMéditerranée,Marseille,
France
JanuszKacprzyk
SystemResearch,PolishAcademyofSciences,Warsaw,Poland
KunihikoKaneko
ResearchCenterforComplexSystemsBiology,TheUniversityofTokyo,Tokyo,Japan
ScottKelso
CenterforComplexSystemsandBrainSciences,FloridaAtlanticUniversity,BocaRaton,USA
MarkusKirkilionis
MathematicsInstituteandCentreforComplexSystems,UniversityofWarwick,Coventry,UK
JürgenKurths
PotsdamInstituteforClimateImpactResearch(PIK),Potsdam,Germany
AndrzejNowak
DepartmentofPsychology,WarsawUniversity,Poland
LindaReichl
CenterforComplexQuantumSystems,UniversityofTexas,Austin,USA
PeterSchuster
TheoreticalChemistryandStructuralBiology,UniversityofVienna,Vienna,Austria
FrankSchweitzer
SystemDesign,ETHZürich,Zürich,Switzerland
DidierSornette
EntrepreneurialRisk,ETHZürich,Zürich,Switzerland
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Understanding Complex Systems
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Valery I. Klyatskin
Stochastic Equations:
Theory and Applications in
Acoustics, Hydrodynamics,
Magnetohydrodynamics,
and Radiophysics, Volume 2
Coherent Phenomena in
Stochastic Dynamic Systems
Translated from Russian
by A. Vinogradov
ABC
ValeryI.Klyatskin
A.M.ObukhovInstituteof
AtmosphericPhysics
RussianAcademyofSciences
Moscow
Russia
ISSN1860-0832 ISSN1860-0840 (electronic)
ISBN978-3-319-07589-1 ISBN978-3-319-07590-7 (eBook)
DOI10.1007/978-3-319-07590-7
SpringerChamHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2014941082
(cid:2)c SpringerInternationalPublishingSwitzerland2015
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To Sonya Klyatskina
Preface
This monograph is revised and more comprehensive translation of my Rus-
sian monographs [154] and in 2 volumes [155,156]. For reasons of usability,
the material is divided into two practically independent volumes. In Volume
2, general methods developed in Volume 1 are used to consider the the-
ory of coherent phenomena occurring in stochastic dynamic systems with a
probability one, i.e., almost in all realizations of a random process. Among
the examinedphenomena areclusteringofparticlesandpassivescalartracer
field (density field) in randomvelocity field, passivevectortracer field (mag-
netic field) in a random velocity field, dynamic localization of plane waves
in layered random media, propagation of monochromatic waves in random
media,andformationofcausticstructuresindifferent-naturewavefieldsprop-
agatinginmultidimensionalrandommedia(under theassumptionthatwave
propagationis described in terms of the scalar parabolic equation).
Diffusion and clustering of both inertialess and low-inertia passive scalar
tracer (density field) in different media and under different conditions are
consideredhereinsignificantlygreaterdetailonthebasisofmonograph[154].
Inaddition,Iaddedconsiderationofdiffusionandclusteringofmagneticfield
energy in random media.
Workingatthisedition,Itriedtotakeintoaccountremarksandwishesof
readers about both style of the text and choice of specific problems. Differ-
ent mistakes and misprints are corrected. The book is destined for scientists
dealing with stochastic dynamic systems in different areas, such as hydro-
dynamics, magnetohydrodynamics, acoustics, radiophysics, theoretical and
mathematicalphysics,andappliedmathematics,andcanbeusefulforsenior
and postgraduate students.
Volume 2 consists of five Parts and one Appendix.
PartIofthisvolumedealswiththeproblemofstochasticstructureforma-
tion in random hydrodynamic flows. In particular, starting from an analysis
ofthesteady-stateprobabilitydensity,itconsiderscoherentstructuresofvor-
texformation(vortexgenesis)instochasticquasi-geostrophicflows,whichare
relatedtorotationandrandomtopographyofthebottom.ThisPartincludes
VIII Preface
alsochallenges ofanalyzingthe possibility offormationof anomalouslylarge
waves (rogue waves) on the sea surface in terms of stochastic topography.
Part II deals with diffusion and clustering of particles and passive scalar
tracer density (density field) in random hydrodynamic velocity field.
Part III deals with diffusion and clustering of passive vector field (mag-
netic) in random magnetohydrodynamic velocity field.
Part IV considers the phenomenon of dynamic localization accompanying
plane wave propagation in layered random media, and
Part V considers statistical description of wavefields propagating in ran-
dom multidimensional media including the problem of caustic structure for-
mation.Theseproblemsareformulatedintermsofbothordinaryandpartial
differential equations, and every of them can be divided into the number of
separate problems of independent physical interest.
Part VI – Appendices discusses in detail the derivation of equations of
the imbedding method that offers a possibility of reformulating boundary-
value wave problems (linear and nonlinear, stationary and nonstationary)
into initial-value problems with respect to auxiliary variables.
It is worth noting that purely mathematical and physical papers devoted
to considered issues run into thousands. It would be physically impossible
to give an exhaustive bibliography. Therefore, in this book I confine myself
to referencing the papers used or discussed in this book, recent reviews and
papers with extensive bibliography on the subject.
Moscow Valery I. Klyatskin
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 LagrangianDescription (Particle Diffusion) ......... 42
4.2.2 Eulerian Description............................. 52
4.3 Additional Factors..................................... 60
4.3.1 Plane-Parallel Mean Shear........................ 60
X Contents
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 BuoyantInertialess
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
LagrangianDescription).......................... 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 CorrelationTensor 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 CorrelationTensor 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 (LagrangianDescription) ......... 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 Tracer ............................. 127
7.3.1 General Remarks................................ 127
7.3.2 Diffusion Approximation ......................... 130