Dissipative Structures and Weak Turbulence

Dissipative Structure and Weak Turbulence provides an understanding of the emergence and evolution of structures in macroscopic systems. This book discusses the emergence of dissipative structures.

Organized into 10 chapters, this book begins with an overview of the stability of a fluid layer with potentially unstable density stratification in the field of gravity. This text then explains the theoretical description of the dynamics of a given system at a formal level. Other chapters consider several examples of how such simplified models can be derived, complicating the picture progressively to account for other phenomena. This book discusses as well the theory and experiments on plain Rayleigh-Bénard convection by setting first the theoretical frame and deriving the analytical solution of the marginal stability problem. The final chapter deals with building a bridge between chaos as studied in weakly confined systems and more advanced turbulence in the most conventional sense.

This book is a valuable resource for physicists.

Contenu

Foreword
Chapter 1. Outlook

1. Dissipative Structures
2. Transition to Temporal Chaos
3. Spatio-Temporal Chaos and Weak Turbulence
4. Bibliographical Notes
   Chapter 2. Evolution and Stability, Basic Concepts
5. General Framework
   1.1. Discrete Versus Continuous Systems
   1.2. Autonomous Versus Time-Dependent Systems
   1.3. Deterministic Evolution
   1.4. Perturbations and Stability
6. Global Stability
   2.1. General Viewpoint and Definitions
   2.2. Energy Method
   2.3. Different Concepts of Stability
   3· Normal Modes, Linear and Nonlinear Dynamics
   3.1. Normal Mode Analysis
   3.2. Weakly Nonlinear Dynamics
7. Qualitative Dynamics
   4.1. Elements for a Phase Portrait
   4.2. Absorbing Zones and Limit Sets
   4.3. Limit Sets and Attractors
   4.4. Hyperbolicity, Structural Stability, and Bifurcations
   4.5. About More General Attractors
8. Bibliographical Notes
   Chapter 3. Instability Mechanisms
9. Rayleigh-Bénard Convection
   1.1. Qualitative Analysis
   1.2. Simplified Model
   1.3. Normal Mode Analysis
   1.4. Linear Dynamics of Unstable Modes
10. Convection in Binary Mixtures
    2.1. Stationary Mode
    2.2. Oscillatory Mode
11. Thermal Convection in Nematic Liquid Crystals
12. Electrohydrodynamic Instabilities in Nematics
    4.1. Carr-Helfrich Mechanism
    4.2. Two Instability Regimes
13. Taylor-Couette instability
    5.1. Centrifugal Instabilities
    5.2. Rayleigh Instability Mechanism
    5.3. One-Dimensional Model
14. Bénard-Marangoni Convection
    6.1. Mechanism and Simplified Model
    6.2. Marangoni Versus Buoyancy-Driven Convection
15. Bibliographical Notes
    Chapter 4. Thermal Convection
16. Boussinesq Equations and Boundary Conditions
    1.1. Evolution Equations
    1.2. Boundary Conditions
17. Normal Mode Analysis
    2.1. Stress-Free Solution, Rayleigh (1916)
    2.2. No-Slip Solution, Pellew and Southwell (1940)
    2.3. Vicinity of the Threshold (Linear Stage)
    2.4. Approximate Determination of Threshold Curves
    2.5. Treatment of the Vertical Vorticity
18. Phenomenology of Nonlinear Convection
    3.1. Introduction
    3.2. Universal Secondary Modes
    3.3. Specific Secondary Instabilities
19. Phenomenology of the Transition to Turbulence
    4.1. "Old" Results
    4.2. "Recent" Results, Concrete Examples
    4.3. Concluding Remarks
20. Bibliographical Notes
    Chapter 5· Low-Dimensional Dynamical Systems
21. Dimension Reduction: A Case Study
    1.1. The Model and Its Normal Modes
    1.2. Elimination of Slaved Modes: Heuristic Approach
22. Center Manifold and Normal Forms
    2.1. Perturbative Approach to the Center Manifold
    2.2. Normal Forms of Dynamical Equations
    2.3. Normal Forms and Symmetries
    2.4. Slightly Off the Critical Point
23. Dynamics and Bifurcations in One Dimension
    3.1. General Dynamics
    3.2. Normal/Inverse Bifurcations
    3.3. Conditional Stability, Hysteresis, and Turning Points
    3.4. Transcritical Bifurcations
    3.5. Imperfect Bifurcations
    3.6. Mathematical Context: Unfolding of Singularities
24. Introduction to Higher Dimensional Problems
25. Dynamics and Bifurcations in Two Dimensions
    5.1. Linear Dynamics
    5.2. Nonlinear Dynamics, an Example: The Pendulum
    5.3. General Dynamics in Two Dimensions
    5.4. Bifurcations in Two Dimensions
26. Conclusion
27. Bibliographical Notes
    Chapter 6. Beyond Periodic Behavior
28. Poincaré Maps
    1.1. Surface of Section and First Return Map
    1.2. Application to the Lorenz Model
29. Stability of a Limit Cycle
30. Bifurcations of a Limit Cycle
    3.1. Normal Form in the Complex Case
    3.2. Hopf Bifurcation for Maps
    3.3. Bifurcations at Strong Resonances 1/1 and 1/2
31. Nature of Turbulence and Transition Scenarios
32. Sub-Harmonic Route to Turbulence
    5.1. The Modeling Issue
    5.2. The Sub-Harmonic Cascade
    5.3. Universality and Renormalization
    5.4. Beyond the Accumulation Point
    5.5. Concluding Remark
33. Temporal Intermittency
    6.1. Modeling of Intermittent Behavior
    6.2. Type I Intermittency
    6.3. Type III Intermittency
    6.4. Type II Intermittency
34. Quasi-Periodicity
    7.1. Introduction to the Locking Phenomenon
    7.2. The Winding Number and the Structure of Lockings
    7.3. The Breakdown of a Two-Torus
    7.4. Ruelle-Takens Scenario and n-Periodicity
35. Beyond "Classical" Scenarios
36. Bibliographical Notes
    Chapter 7. Characterization of Temporal Chaos
37. Divergence of Trajectories and Lyapunov Exponents
    1.1. One-Dimensional Iterations
    1.2. Generalization to d-Dimensional Maps
    1.3. Generalization to Differential Systems
    1.4. Lyapunov Signature of Temporal Behavior
38. Probabilistic Approach
    2.1. Entropy
    2.2. Invariant Measures
    2.3. Invariant Measures for One-Dimensional Maps
    2.4. Natural Measures
39. Chaos and Dimensions
    3.1. Introduction
    3.2. Fractal Geometry
    3.3. Probabilistic Viewpoint
    3.4. Dimensions and Lyapunov Exponents
40. Experimental Approach
    4.1. The Method of Time Delays
    4.2. Embedding
    4.3. Practical Problems
    4.4. Dimension Estimates
    4.5. Determination of Lyapunov Exponents
    4.6. Final Remark
41. Bibliographical Notes
    Chapter 8. Basics of Pattern Formation in Weakly Confined Systems
42. Instabilities, Confinement, and Aspect Ratios
43. Pattern Formation
    2.1. Modeling of Weakly Confined Systems
    2.2. Pattern Formation in Two Dimensions
44. Uniform Nonlinear Solutions
    3.1. General Setting
    3.2. Steady Solutions of Two-Dimensional Models
    3.3. Rayleigh-Bénard Convection
45. Modulated Structures
    4.1. Systematic Expansion
    4.2. Roll Modulations in a Rotationally Invariant System
    4.3. Extension to Two-Dimensional Patterns
    4.4. Short-Term Stability and Early Nonlinear Selection
    4.5. Phenomenological Extension: The Example of Waves
46. Bibliographical Notes
    Chapter 9. Applications of the Envelope Formalism
47. Phase Winding Solutions
48. Long Wavelength Instabilities
    2.1. General Formulation
    2.2. Longitudinal Perturbations and Eckhaus Instability
    2.3. Transverse Perturbations and Zigzag Instability
    2.4. Stability of Waves
49. Lateral Boundary Effects
    3.1. Rolls Parallel to a Lateral Wall
    3.2. Rolls Perpendicular to a Lateral Wall
50. Structural Defects
    4.1. Grain Boundaries
    4.2. Dislocations
51. Pattern Selection at Lowest Order
52. Bibliographical Notes
    Chapter 10. Dynamics of Textures and Turbulence
53. Phase Diffusion in Steady Roll Patterns
    1.1. Phase Diffusion Formalism
    1.2. Phase Instabilities
    1.3. Rotationally Invariant Formulation of Phase Dynamics
54. Nonlinear Selection and Weak Turbulence
    2.1. Nonlinear Wavelength Selection Criteria…