Explore the vast range of titles available.

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.

The search for statistical equilibria at very low positive temperatures, using a Monte Carlo algorithm, can locate dynamical equilibria of the N-vortex problem on a sphere. Numerical results show this algorithm accurately and efficiently locates the lowest energy equilibrium. Using an essential tool called the radial distribution function, superficially similar configurations can be easily distinguished and classified. The problem of N bodies on the surface of the sphere interacting by a logarithmic potential is examined for selected N from 4 to 40,962, comparing the energies found by placing points at vertices of polyhedrons to the lowest energies found by a Monte Carlo algorithm. The polyhedron families are generated from simple polyhedrons through two triangular face splitting operations applied iteratively. The closest energy of these polyhedron configurations to the Monte Carlo minimum energy is identified and the energies agree well. The energy per particle pair asymptotically approaches a mean field theory limit of −½(log(2) − 1), approximately 0.153426, for both the polyhedron and the Monte Carlo-generated energies. On a mesh of N points distributed over the plane through methods like those employed above, the logarithmic potential is studied with the circulation is set to zero and the enstrophy held to a fixed value. Using a Monte Carlo Metropolis Rule-based algorithm to maximize the Gibbs factor exp(−β H), with β an inverse temperature and H the Hamiltonian, the arrangement of site vorticities is found to depend on the inverse temperature and number of points. From examining the effects the mesh size has by using the mean nearest neighbor parity introduced here, conclusions about the continuum limit can be drawn, and match analytical predictions that there are no phase transitions except at β = 0. Finally, meshes of N points are distributed over the unit sphere, and the logarithmic potential for this problem studied again with zero and fixed enstrophy. Through a similar Monte Carlo Metropolis algorithm the best arrangements of vorticities for mesh sizes and inverse temperatures are studied. Based on these results conclusions are drawn about the continuum limit, specifically that there are no phase transitions except at β = 0.

The following sections are included: Venus Titan The Great Red Spot Polar Vortices and Other Curiosities Outline

The following sections are included: Introduction Markov Chains Detailed Balance The Metropolis Rule Multiple Canonical Constraints Ensemble Averages Metropolis-Hastings Monte Carlo Algorithm

The following sections are included: Introduction Equilibrium Statistical Mechanics Mean-Field Theory Setting Up Coupled Barotropic Flows Proofs for a Non-Rotating Planet Mean-Field Theory on a Rotating Sphere Positive Temperatures Negative Temperatures

The following sections are included: The Physical Model Voronoi Cells and the Spin-Lattice Approximation The Solid Sphere Model The Shallow-Water Equations on the Rotating Sphere The Spin-Lattice Shallow-Water Model Circulation Constraints Enstrophy Constraints Gibbs Ensemble

The following sections are included: Energy-Relative Enstrophy Variational Theory The Augmented Energy Functional Extremals: Existence and Properties

The following sections are included: Introduction Statistical Mechanics of Macroscopic Flows Bragg-Williams Approximation Internal Energy Entropy Helmholtz Free Energy Polar State Criteria The Non-Rotating Case The Rotating Case Summary of Main Results The Infinite-Dimensional Non-Extensive Limit

The following sections are included: Introduction The Rotating Sphere Model Solution of the Spherical Model

The following sections are included: Introduction First Order Transitions Antipodal Symmetry Monte Carlo Results Phase Transitions in Latent Heat Conclusion