Table of Contents
Preface v
1 Planets and Inspiration 1
1.1 Venus 1
1.2 Titan 4
1.3 The Great Red Spot 5
1.4 Polar Vortices and Other Curiosities 7
1.5 Outline 9
2 Barotropic and Shallow-Water Models 13
2.1 The Physical Model 13
2.2 Voronoi Cells and the Spin-Lattice Approximation 16
2.3 The Solid Sphere Model 20
2.4 The Shallow-Water Equations on the Rotating Sphere 25
2.5 The Spin-Lattice Shallow-Water Model 32
2.5.1 Circulation Constraints 36
2.5.2 Enstrophy Constraints 37
2.5.3 Gibbs Ensemble 37
3 Dynamic Equilibria of the Barotropic Model - Variational Approach 41
3.1 Energy-Relative Enstrophy Variational Theory 41
3.2 The Augmented Energy Functional 46
3.3 Extremals: Existence and Properties 52
4 Statistical Mechanics 61
4.1 Introduction 61
4.2 Microstates and Macrostates 63
4.3 Entropy 63
4.4 Partition Functions 64
4.5 Free Energies 65
4.6 Planck's Theorem in Negative Temperatures 66
4.7 Latent Heat and Orders of Phase Transitions 68
5 The Monte Carlo Approach 71
5.1 Introduction 71
5.2 Markov Chains 72
5.3 Detailed Balance 74
5.4 The Metropolis Rule 75
5.5 Multiple Canonical Constraints 77
5.6 Ensemble Averages 78
5.7 Metropolis-Hastings Monte Carlo Algorithm 82
6 Phase Transitions in Energy-Relative Enstrophy Models 85
6.1 Introduction 85
6.2 Classical and Recent Energy-Enstrophy Theories 86
6.2.1 Gaussian Model 87
6.2.2 Spherical Model for Coupled Barotropic Flows 88
6.3 Monte Carlo Simulations of the Energy-Relative Enstrophy Model 89
6.4 Free Energy 99
7 Extremal Free Energy in the Mean-Field Theory 107
7.1 Introduction 107
7.2 Equilibrium Statistical Mechanics 108
7.3 Mean-Field Theory 109
7.3.1Setting Up Coupled Barotropic Flows 111
7.3.2 Proofs for a Non-Rotating Planet 113
7.3.3 Mean-Field Theory on a Rotating Sphere 116
7.3.4 Positive Temperatures 120
7.3.5 Negative Temperatures 122
8 Phase Transitions of Barotropic Flow 129
8.1 Introduction 129
8.2 Statistical Mechanics of Macroscopic Flows 131
8.3 Bragg-Williams Approximation 133
8.3.1 Internal Energy 136
8.3.2 Entropy 140
8.3.3 Helmholtz Free Energy 141
8.4 Polar State Criteria 142
8.4.1 The Non-Rotating Case 143
8.4.2 The Rotating Case 146
8.4.3 Summary of Main Results 154
8.5 The Infinite-Dimensional Non-Extensive Limit 155
9 Phase Transitions to Super-Rotation - Exact Closed-Form Solutions 159
9.1 Introduction 159
9.2 The Rotating Sphere Model 160
9.3 Solution of the Spherical Model 162
10 The Shallow-Water Models - High Energy, Cyclonic Solutions 169
10.1 Introduction 169
10.2 First Order Transitions 171
10.3 Antipodal Symmetry 172
10.4 Monte Carlo Results 174
10.5 Phase Transitions in Latent Heat 177
10.6 Conclusion 178
11 The Shallow-Water Model - Low-Energy Solutions 183
11.1 Introduction 183
11.2 Theoretical Predictions of the Shallow-Water Model 185
11.2.1 The Energy Gap from Large Planetary Spin and Anticyclonic Spots 186
11.2.2 North-South Asymmetry and the Energy Terms 186
11.2.3 Large Relative Enstrophies and High Rim Velocities 187
11.2.4 Angular Momentum, Moment of Inertia, Entropy, and the Location of the High Spot 188
11.3 Monte Carlo Simulations and Results 189
11.3.1 Key Features of the Great Red Spot-like Structure 190
11.3.2 First-Order Phase Transition with Latent Heat 191
11.3.3 Multiple High Spots in the Same Macrostate 193
11.3.4 Belts and Zones 193
11.4 Conclusion 194
Bibliography 201
Index 207
