Dissipative Structures and Weak Turbulence

by Paul Manneville (Editor)

Overview

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.

Product Details

ISBN-13:	9780080924458
Publisher:	Elsevier Science & Technology Books
Publication date:	06/28/2014
Sold by:	Barnes & Noble
Format:	eBook
Pages:	448
File size:	12 MB
Note:	This product may take a few minutes to download.

Table of Contents

ForewordChapter 1. Outlook 1. Dissipative Structures 2. Transition to Temporal Chaos 3. Spatio-Temporal Chaos and Weak Turbulence 4. Bibliographical NotesChapter 2. Evolution and Stability, Basic Concepts 1. General Framework 1.1. Discrete Versus Continuous Systems 1.2. Autonomous Versus Time-Dependent Systems 1.3. Deterministic Evolution 1.4. Perturbations and Stability 2. 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 4. 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 5. Bibliographical NotesChapter 3. Instability Mechanisms 1. Rayleigh-Bénard Convection 1.1. Qualitative Analysis 1.2. Simplified Model 1.3. Normal Mode Analysis 1.4. Linear Dynamics of Unstable Modes 2. Convection in Binary Mixtures 2.1. Stationary Mode 2.2. Oscillatory Mode 3. Thermal Convection in Nematic Liquid Crystals 4. Electrohydrodynamic Instabilities in Nematics 4.1. Carr-Helfrich Mechanism 4.2. Two Instability Regimes 5. Taylor-Couette instability 5.1. Centrifugal Instabilities 5.2. Rayleigh Instability Mechanism 5.3. One-Dimensional Model 6. Bénard-Marangoni Convection 6.1. Mechanism and Simplified Model 6.2. Marangoni Versus Buoyancy-Driven Convection 7. Bibliographical NotesChapter 4. Thermal Convection 1. Boussinesq Equations and Boundary Conditions 1.1. Evolution Equations 1.2. Boundary Conditions 2. 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 3. Phenomenology of Nonlinear Convection 3.1. Introduction 3.2. Universal Secondary Modes 3.3. Specific Secondary Instabilities 4. Phenomenology of the Transition to Turbulence 4.1. "Old" Results 4.2. "Recent" Results, Concrete Examples 4.3. Concluding Remarks 5. Bibliographical NotesChapter 5· Low-Dimensional Dynamical Systems 1. Dimension Reduction: A Case Study 1.1. The Model and Its Normal Modes 1.2. Elimination of Slaved Modes: Heuristic Approach 2. 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 3. 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 4. Introduction to Higher Dimensional Problems 5. 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 6. Conclusion 7. Bibliographical NotesChapter 6. Beyond Periodic Behavior 1. Poincaré Maps 1.1. Surface of Section and First Return Map 1.2. Application to the Lorenz Model 2. Stability of a Limit Cycle 3. Bifurcations of a Limit Cycle 3.1.

Editorial Reviews

Contributors to the chaos literature have frequently drawn research motivation and expository inspiration from the phenomenon of turbulence in fluids. Here the tables are turned; the author's primary focus is turbulence, and he has produced for the benefit of graduate students and specialists a detailed survey of recent developments as they related to complexly rich phenomenology of his field. Nicely produced volume derives from lectures presented at various French institutions during the past decade. (NW) Annotation c. Book News, Inc., Portland, OR (booknews.com)

Booknews