Vortex Dynamics, Statistical Mechanics, and Planetary Atmospheres

by Lim, Chjan C.; Ding, Xueru; Nebus, Joseph

• ISBN13: 9789812839121
• ISBN10: 9812839127
• Format: Hardcover
• Copyright: 2009-04-15
• Publisher: World Scientific Pub Co Inc

Summary

Vortex Dynamics, Statistical Mechanics, and Planetary Atmospheres introduces the reader with a background in either fluid mechanics or statistical mechanics to the modeling of planetary atmospheres by barotropic and shallow-water models. These potent models are introduced in both analytical and numerical treatments highlighting the ways both approaches inform and enlighten the other. This book builds on Vorticity, Statistical Mechanics, and Monte Carlo Simulations by Lim and Nebus in providing a rare introduction to this intersection of research fields. While the book reaches the cutting edge of atmospheric models, the exposition requires little more than an undergraduate familiarity with the relevant fields of study, and so this book is well suited to individuals hoping to swiftly learn an exciting new field of study. With inspiration drawn from the atmospheres of Venus and of Jupiter, the physical relevance of the work is never far from consideration, and the bounty of results shows a new and fruitful perspective with which to study planetary atmospheres.

Table of Contents

Preface	p. v
Planets and Inspiration	p. 1
Venus	p. 1
Titan	p. 4
The Great Red Spot	p. 5
Polar Vortices and Other Curiosities	p. 7
Outline	p. 9
Barotropic and Shallow-Water Models	p. 13
The Physical Model	p. 13
Voronoi Cells and the Spin-Lattice Approximation	p. 16
The Solid Sphere Model	p. 20
The Shallow-Water Equations on the Rotating Sphere	p. 25
The Spin-Lattice Shallow-Water Model	p. 32
Circulation Constraints	p. 36
Enstrophy Constraints	p. 37
Gibbs Ensemble	p. 37
Dynamic Equilibria of the Barotropic Model - Variational Approach	p. 41
Energy-Relative Enstrophy Variational Theory	p. 41
The Augmented Energy Functional	p. 46
Extremals: Existence and Properties	p. 52
Statistical Mechanics	p. 61
Introduction	p. 61
Microstates and Macrostates	p. 63
Entropy	p. 63
Partition Functions	p. 64
Free Energies	p. 65
Planck's Theorem in Negative Temperatures	p. 66
Latent Heat and Orders of Phase Transitions	p. 68
The Monte Carlo Approach	p. 71
Introduction	p. 71
Markov Chains	p. 72
Detailed Balance	p. 74
The Metropolis Rule	p. 75
Multiple Canonical Constraints	p. 77
Ensemble Averages	p. 78
Metropolis-Hastings Monte Carlo Algorithm	p. 82
Phase Transitions in Energy-Relative Enstrophy Models	p. 85
Introduction	p. 85
Classical and Recent Energy-Enstrophy Theories	p. 86
Gaussian Model	p. 87
Spherical Model for Coupled Barotropic Flows	p. 88
Monte Carlo Simulations of the Energy-Relative Enstrophy Model	p. 89
Free Energy	p. 99
Extremal Free Energy in the Mean-Field Theory	p. 107
Introduction	p. 107
Equilibrium Statistical Mechanics	p. 108
Mean-Field Theory	p. 109
Setting Up Coupled Barotropic Flows	p. 111
Proofs for a Non-Rotating Planet	p. 113
Mean-Field Theory on a Rotating Sphere	p. 116
Positive Temperatures	p. 120
Negative Temperatures	p. 122
Phase Transitions of Barotropic Flow	p. 129
Introduction	p. 129
Statistical Mechanics of Macroscopic Flows	p. 131
Bragg-Williams Approximation	p. 133
Internal Energy	p. 136
Entropy	p. 140
Helmholtz Free Energy	p. 141
Polar State Criteria	p. 142
The Non-Rotating Case	p. 143
The Rotating Case	p. 146
Summary of Main Results	p. 154
The Infinite-Dimensional Non-Extensive Limit	p. 155
Phase Transitions to Super-Rotation - Exact Closed-Form Solutions	p. 159
Introduction	p. 159
The Rotating Sphere Model	p. 160
Solution of the Spherical Model	p. 162
The Shallow-Water Models - High Energy, Cyclonic Solutions	p. 169
Introduction	p. 169
First Order Transitions	p. 171
Antipodal Symmetry	p. 172
Monte Carlo Results	p. 174
Phase Transitions in Latent Heat	p. 177
Conclusion	p. 178
The Shallow-Water Model - Low-Energy Solutions	p. 183
Introduction	p. 183
Theoretical Predictions of the Shallow-Water Model	p. 185
The Energy Gap from Large Planetary Spin and Anticyclonic Spots	p. 186
North-South Asymmetry and the Energy Terms	p. 186
Large Relative Enstrophies and High Rim Velocities	p. 187
Angular Momentum, Moment of Inertia, Entropy, and the Location of the High Spot	p. 188
Monte Carlo Simulations and Results	p. 189
Key Features of the Great Red Spot-like Structure	p. 190
First-Order Phase Transition with Latent Heat	p. 191
Multiple High Spots in the Same Macrostate	p. 193
Belts and Zones	p. 193
Conclusion	p. 194
Bibliography	p. 201
Index	p. 207
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