Preface
1 Barotropic geophysical flows and two-dimensional fluid flows: elementary introduction
1.1 Introduction
1.2 Some special exact solutions
1.3 Conserved quantities
1.4 Barotropic geophysical flows in a channel domain - an important physical model
1.5 Variational derivatives and an optimization principle for elementary geophysical solutions
1.6 More equations for geophysical flows
References
2 The response to large-scale forcing
2.1 Introduction
2.2 Non-linear stability with Kolmogorov forcing
2.3 Stability of flows with generalized Kolmogorov forcing
References
3 The selective decay principle for basic geophysical flows
3.1 Introduction
3.2 Selective decay states and their invariance
3.3 Mathematical formulation of the selective decay principle
3.4 Energy-enstrophy decay
3.5 Bounds on the Dirichlet quotient, A(t)
3.6 Rigorous theory for selective decay
3.7 Numerical experiments demonstrating facets of selective decay
References
A.1 Stronger controls on A(t)
A.2 The proof of the mathematical form of the selective decay principle in the presence of the beta-plane effect
4 Non-linear stability of steady geophysical flows
4.1 Introduction
4.2 Stability of simple steady states
4.3 Stability for more general steady states
4.4 Non-linear stability of zonal flows on the beta-plane
4.5 Variational characterization of the steady states
References
5 Topographic mean flow interaction, non-linear instability, and chaotic dynamics
5.1 Introduction
5.2 Systems with layered topography
5.3 Integrable behavior
5.4 A limit regime with chaotic solutions
5.5 Numerical experiments
References
Appendix 1
Appendix 2
6 Introduction to information theory and empirical statistical theory
6.1 Introduction
6.2 Information theory and Shannon’’s entropy
6.3 Most probable states with prior distribution
6.4 Entropy for continuous measures on the line
6.5 Maximum entropy principle for continuous fields
6.6 An application of the maximum entropy principle to geophysical flows with topography
6.7 Application of the maximum entropy principle to geophysical flows with topography and mean flow
References
7 Equilibrium statistical mechanics for systems of ordinary differential equations
7.1 Introduction
7.2 Introduction to statistical mechanics for ODEs
7.3 Statistical mechanics for the truncated Burgers-Hopf equations
7.4 The Lorenz 96 model
References
……
8 Statistical mechanics for the truncated quasi-geostrophic equations :
9 Empirical statistical theories for most probable states
10 Assessing the potential applicability of equilibrium statistical theories for geophysical flows: an overview
11 Predictions and comparison of equilibrium statistical theories
12 Equilibrium statistical theories and dynamical modeling of flows with forcing and dissipation
13 Predicting the jets and spots on Jupiter by equilibrium statistical mechanics
14 The statistical relevance of additional conserved quantities for truncated geophysical flows
15 A mathematical framework for quantifying predictability utilizing relative entropy
16 Barotropic quasi-geostrophic equations on the sphere
Index
