Vortex Dynamics, Statistical Mechanics, And Planetary Atmospheres

by Chjan C Lim, Xueru Ding, Joseph Nebus

Overview

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.

Product Details

ISBN-13:	9789812839138
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	04/07/2009
Pages:	224
Product dimensions:	6.00(w) x 8.90(h) x 0.50(d)

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