Introduction to Turbulent Dynamical Systems in Complex Systems

1. Front Matter

   Pages i-xi

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2. Introduction

   • Andrew J. Majda

   Pages 1-4

3. Prototype Examples of Complex Turbulent Dynamical Systems

   • Andrew J. Majda

   Pages 5-11

4. The Mathematical Theory of Turbulent Dynamical Systems

   • Andrew J. Majda

   Pages 13-41

5. Statistical Prediction and UQ for Turbulent Dynamical Systems

   • Andrew J. Majda

   Pages 43-63

6. State Estimation, Data Assimilation, or Filtering for Complex Turbulent Dynamical Systems

   • Andrew J. Majda

   Pages 65-83

7. Back Matter

   Pages 85-91

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This volume is a research expository article on the applied mathematics of turbulent dynamical systems through the paradigm of modern applied mathematics. It involves the blending of rigorous mathematical theory, qualitative and quantitative modeling, and novel numerical procedures driven by the goal of understanding physical phenomena which are of central importance to the field. The contents cover general framework, concrete examples, and instructive qualitative models. Accessible open problems are mentioned throughout.

Topics covered include:

·         Geophysical flows with rotation, topography, deterministic and random forcing

·         New statistical energy principles for general turbulent dynamical systems, with applications

·         Linear statistical response theory combined with information theory to cope with model errors

·         Reduced low order models

·         Recent mathematical strategies for online data assimilation of turbulent dynamical systems as well as rigorous results for finite ensemble Kalman filters

The volume will appeal to graduate students and researchers working mathematics, physics and engineering and particularly those in the climate, atmospheric and ocean sciences interested in turbulent dynamical as well as other complex systems.

• Applied mathematics
• Climate and atmospheric sciences
• Dynamical systems
• Geoscience enginering
• complex systems
• data assimilation, filtering
• fluid- and aerodynamics

“This research monograph presents recent mathematical results in the field of turbulent dynamical systems. … Numerous references are cited. The book is certainly of some interest for graduate students and researchers in the field of mathematical modeling of geophysical flows.” (Kai Schneider, zbMATH 1377.37003, 2018)

“This is a beautifully written research expository book on turbulent dynamical systems that arise in complex systems. … it contains a wealth of new information including many penetrating insights that are not available elsewhere systematically. … It belongs on the bookshelf of everybody who is seriously interested in turbulent dynamical systems in high-dimensional phase space and many related applied issues.” (Xiaoming Wang, Mathematical Reviews, July, 2017)

• New York University, New York, USA

  Andrew J. Majda

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