Contents -- Preface -- 1. Planets and Inspiration -- 1.1 Venus -- 1.2 Titan -- 1.3 The Great Red Spot -- 1.4 Polar Vortices and Other Curiosities -- 1.5 Outline -- 2. Barotropic and Shallow-Water Models -- 2.1 The Physical Model -- 2.2 Voronoi Cells and the Spin-Lattice Approximation -- 2.3 The Solid Sphere Model -- 2.4 The Shallow-Water Equations on the Rotating Sphere -- 2.5 The Spin-Lattice Shallow-Water Model -- 2.5.1 Circulation Constraints -- 2.5.2 Enstrophy Constraints -- 2.5.3 Gibbs Ensemble -- 3. Dynamic Equilibria of the Barotropic Model - Variational Approach -- 3.1 Energy-Relative Enstrophy Variational Theory -- 3.2 The Augmented Energy Functional -- 3.3 Extremals: Existence and Properties -- 4. Statistical Mechanics -- 4.1 Introduction -- 4.2 Microstates and Macrostates -- 4.3 Entropy -- 4.4 Partition Functions -- 4.5 Free Energies -- 4.6 Planck's Theorem in Negative Temperatures -- 4.7 Latent Heat and Orders of Phase Transitions -- 5. The Monte Carlo Approach -- 5.1 Introduction -- 5.2 Markov Chains -- 5.3 Detailed Balance -- 5.4 The Metropolis Rule -- 5.5 Multiple Canonical Constraints -- 5.6 Ensemble Averages -- 5.7 Metropolis-Hastings Monte Carlo Algorithm -- 6. Phase Transitions in Energy-Relative Enstrophy Models -- 6.1 Introduction -- 6.2 Classical and Recent Energy-Enstrophy Theories -- 6.2.1 Gaussian Model -- 6.2.2 Spherical Model for Coupled Barotropic Flows -- 6.3 Monte Carlo Simulations of the Energy-Relative Enstrophy Model -- 6.4 Free Energy -- 7. Extremal Free Energy in the Mean-Field Theory -- 7.1 Introduction -- 7.2 Equilibrium Statistical Mechanics -- 7.3 Mean-Field Theory -- 7.3.1 Setting Up Coupled Barotropic Flows -- 7.3.2 Proofs for a Non-Rotating Planet -- 7.3.3 Mean-Field Theory on a Rotating Sphere -- 7.3.4 Positive Temperatures -- 7.3.5 Negative Temperatures -- 8. Phase Transitions of Barotropic Flow.

8.1 Introduction -- 8.2 Statistical Mechanics of Macroscopic Flows -- 8.3 Bragg-Williams Approximation -- 8.3.1 Internal Energy -- 8.3.2 Entropy -- 8.3.3 Helmholtz Free Energy -- 8.4 Polar State Criteria -- 8.4.1 The Non-Rotating Case -- 8.4.2 The Rotating Case -- 8.4.3 Summary of Main Results -- 8.5 The Infinite-Dimensional Non-Extensive Limit -- 9. Phase Transitions to Super-Rotation - Exact Closed-Form Solutions -- 9.1 Introduction -- 9.2 The Rotating Sphere Model -- 9.3 Solution of the Spherical Model -- 10. The Shallow-Water Models - High Energy, Cyclonic Solutions -- 10.1 Introduction -- 10.2 First Order Transitions -- 10.3 Antipodal Symmetry -- 10.4 Monte Carlo Results -- 10.5 Phase Transitions in Latent Heat -- 10.6 Conclusion -- 11. The Shallow-Water Model - Low-Energy Solutions -- 11.1 Introduction -- 11.2 Theoretical Predictions of the Shallow-Water Model -- 11.2.1 The Energy Gap from Large Planetary Spin and Anticyclonic Spots -- 11.2.2 North-South Asymmetry and the Energy Terms -- 11.2.3 Large Relative Enstrophies and High Rim Velocities -- 11.2.4 Angular Momentum, Moment of Inertia, Entropy, and the Location of the High Spot -- 11.3 Monte Carlo Simulations and Results -- 11.3.1 Key Features of the Great Red Spot-like Structure -- 11.3.2 First-Order Phase Transition with Latent Heat -- 11.3.3 Multiple High Spots in the Same Macrostate -- 11.3.4 Belts and Zones -- 11.4 Conclusion -- Bibliography -- Index.