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Preface; Acknowledgements; Contents; 1 Introduction; 1.1 Turbulent Dynamical Systems for Complex Systems: #x83;; 1.2 Detailed Structure and Energy Conservation Principles; 2 Prototype Examples of Complex Turbulent Dynamical Systems; 2.1 Turbulent Dynamical Systems for Complex Geophysical Flows: One-Layer Model; 2.2 The L-96 Model as a Turbulent Dynamical System; 2.3 Statistical Triad Models, the Building Blocks of Complex Turbulent Dynamical Systems; 2.4 More Rich Examples of Complex Turbulent Dynamical Systems; 2.4.1 Quantitative Models; 2.4.2 Qualitative Models
3 The Mathematical Theory of Turbulent Dynamical Systems3.1 Nontrivial Turbulent Dynamical Systems with a Gaussian Invariant Measure; 3.2 Exact Equations for the Mean and Covariance of the Fluctuations; 3.2.1 Turbulent Dynamical Systems with Non-Gaussian Statistical Steady States and Nontrivial Third-Order Moments; 3.2.2 Statistical Dynamics in the L-96 Model and Statistical Energy Conservation; 3.2.3 One-Layer Geophysical Model as a Turbulent Dynamical System; 3.3 A Statistical Energy Conservation Principle for Turbulent Dynamical Systems
3.3.1 Details About Deterministic Triad Energy Conservation Symmetry3.3.2 A Generalized Statistical Energy Identity; 3.3.3 Enhanced Dissipation of the Statistical Mean Energy, the Statistical Energy Principle, and ``Eddy Viscosity''; 3.3.4 Stochastic Lyapunov Functions for One-Layer Turbulent Geophysical Flows; 3.4 Geometric Ergodicity for Turbulent Dynamical Systems; 4 Statistical Prediction and UQ for Turbulent Dynamical Systems; 4.1 A Brief Introduction; 4.1.1 Low-Order Truncation Methods for UQ and Their Limitations; 4.1.2 The Gaussian Closure Method for Statistical Prediction
4.1.3 A Fundamental Limitation of the Gaussian Closure Method4.2 A Mathematical Strategy for Imperfect Model Selection, Calibration, and Accurate Prediction: Blending Information Theory and Statistical Response Theory; 4.2.1 Imperfect Model Selection, Empirical Information Theory, and Information Barriers; 4.2.2 Linear Statistical Response and Fluctuation-Dissipation Theorem for Turbulent Dynamical Systems; 4.2.3 The Calibration and Training Phase Combining Information Theory and Kicked Statistical Response Theory
4.2.4 Low-Order Models Illustrating Model Selection, Calibration, and Prediction with UQ4.3 Improving Statistical Prediction and UQ in Complex Turbulent Dynamical Systems by Blending Information Theory and Kicked Statistical Response Theory; 4.3.1 Models with Consistent Equilibrium Single Point Statistics and Information Barriers; 4.3.2 Models with Consistent Unperturbed Equilibrium Statistics for Each Mode; 4.3.3 Calibration and Training Phase; 4.3.4 Testing Imperfect Model Prediction Skill and UQ with Different Forced Perturbations
3 The Mathematical Theory of Turbulent Dynamical Systems3.1 Nontrivial Turbulent Dynamical Systems with a Gaussian Invariant Measure; 3.2 Exact Equations for the Mean and Covariance of the Fluctuations; 3.2.1 Turbulent Dynamical Systems with Non-Gaussian Statistical Steady States and Nontrivial Third-Order Moments; 3.2.2 Statistical Dynamics in the L-96 Model and Statistical Energy Conservation; 3.2.3 One-Layer Geophysical Model as a Turbulent Dynamical System; 3.3 A Statistical Energy Conservation Principle for Turbulent Dynamical Systems
3.3.1 Details About Deterministic Triad Energy Conservation Symmetry3.3.2 A Generalized Statistical Energy Identity; 3.3.3 Enhanced Dissipation of the Statistical Mean Energy, the Statistical Energy Principle, and ``Eddy Viscosity''; 3.3.4 Stochastic Lyapunov Functions for One-Layer Turbulent Geophysical Flows; 3.4 Geometric Ergodicity for Turbulent Dynamical Systems; 4 Statistical Prediction and UQ for Turbulent Dynamical Systems; 4.1 A Brief Introduction; 4.1.1 Low-Order Truncation Methods for UQ and Their Limitations; 4.1.2 The Gaussian Closure Method for Statistical Prediction
4.1.3 A Fundamental Limitation of the Gaussian Closure Method4.2 A Mathematical Strategy for Imperfect Model Selection, Calibration, and Accurate Prediction: Blending Information Theory and Statistical Response Theory; 4.2.1 Imperfect Model Selection, Empirical Information Theory, and Information Barriers; 4.2.2 Linear Statistical Response and Fluctuation-Dissipation Theorem for Turbulent Dynamical Systems; 4.2.3 The Calibration and Training Phase Combining Information Theory and Kicked Statistical Response Theory
4.2.4 Low-Order Models Illustrating Model Selection, Calibration, and Prediction with UQ4.3 Improving Statistical Prediction and UQ in Complex Turbulent Dynamical Systems by Blending Information Theory and Kicked Statistical Response Theory; 4.3.1 Models with Consistent Equilibrium Single Point Statistics and Information Barriers; 4.3.2 Models with Consistent Unperturbed Equilibrium Statistics for Each Mode; 4.3.3 Calibration and Training Phase; 4.3.4 Testing Imperfect Model Prediction Skill and UQ with Different Forced Perturbations