Vortex dynamics, statistical mechanics, and planetary atmospheres

Contents......Page 10
Preface......Page 6
1.1 Venus......Page 14
1.2 Titan......Page 17
1.3 The Great Red Spot......Page 18
1.4 Polar Vortices and Other Curiosities......Page 20
1.5 Outline......Page 22
2.1 The Physical Model......Page 26
2.2 Voronoi Cells and the Spin-Lattice Approximation......Page 29
2.3 The Solid Sphere Model......Page 33
2.4 The Shallow-Water Equations on the Rotating Sphere......Page 38
2.5 The Spin-Lattice Shallow-Water Model......Page 45
2.5.1 Circulation Constraints......Page 49
2.5.3 Gibbs Ensemble......Page 50
3.1 Energy-Relative Enstrophy Variational Theory......Page 54
3.2 The Augmented Energy Functional......Page 59
3.3 Extremals: Existence and Properties......Page 65
4.1 Introduction......Page 74
4.3 Entropy......Page 76
4.4 Partition Functions......Page 77
4.5 Free Energies......Page 78
4.6 Planck’s Theorem in Negative Temperatures......Page 79
4.7 Latent Heat and Orders of Phase Transitions......Page 81
5.1 Introduction......Page 84
5.2 Markov Chains......Page 85
5.3 Detailed Balance......Page 87
5.4 The Metropolis Rule......Page 88
5.5 Multiple Canonical Constraints......Page 90
5.6 Ensemble Averages......Page 91
5.7 Metropolis-Hastings Monte Carlo Algorithm......Page 95
6.1 Introduction......Page 98
6.2 Classical and Recent Energy-Enstrophy Theories......Page 99
6.2.1 Gaussian Model......Page 100
6.2.2 Spherical Model for Coupled Barotropic Flows......Page 101
6.3 Monte Carlo Simulations of the Energy-Relative Enstrophy Model......Page 102
6.4 Free Energy......Page 112
7.1 Introduction......Page 120
7.2 Equilibrium Statistical Mechanics......Page 121
7.3 Mean-Field Theory......Page 122
7.3.1 Setting Up Coupled Barotropic Flows......Page 124
7.3.2 Proofs for a Non-Rotating Planet......Page 126
7.3.3 Mean-Field Theory on a Rotating Sphere......Page 129
7.3.4 Positive Temperatures......Page 133
7.3.5 Negative Temperatures......Page 135
8.1 Introduction......Page 142
8.2 Statistical Mechanics of Macroscopic Flows......Page 144
8.3 Bragg-Williams Approximation......Page 146
8.3.1 Internal Energy......Page 149
8.3.2 Entropy......Page 153
8.3.3 Helmholtz Free Energy......Page 154
8.4 Polar State Criteria......Page 155
8.4.1 The Non-Rotating Case......Page 156
8.4.2 The Rotating Case......Page 159
8.4.3 Summary of Main Results......Page 167
8.5 The Infinite-Dimensional Non-Extensive Limit......Page 168
9.1 Introduction......Page 172
9.2 The Rotating Sphere Model......Page 173
9.3 Solution of the Spherical Model......Page 175
10.1 Introduction......Page 182
10.2 First Order Transitions......Page 184
10.3 Antipodal Symmetry......Page 185
10.4 Monte Carlo Results......Page 187
10.5 Phase Transitions in Latent Heat......Page 190
10.6 Conclusion......Page 191
11.1 Introduction......Page 196
11.2 Theoretical Predictions of the Shallow-Water Model......Page 198
11.2.2 North-South Asymmetry and the Energy Terms......Page 199
11.2.3 Large Relative Enstrophies and High Rim Velocities......Page 200
11.2.4 Angular Momentum, Moment of Inertia, Entropy, and the Location of the High Spot......Page 201
11.3 Monte Carlo Simulations and Results......Page 202
11.3.1 Key Features of the Great Red Spot-like Structure......Page 203
11.3.2 First-Order Phase Transition with Latent Heat......Page 204
11.3.4 Belts and Zones......Page 206
11.4 Conclusion......Page 207
Bibliography......Page 214
Index......Page 220