Vorticity, Statistical Mechanics, and Monte Carlo Simulation

This book is drawn from across many active fields of mathematics and physics, and has connections to atmospheric dynamics, spherical codes, graph theory, constrained optimization problems, Markov Chains, and Monte Carlo methods. It addresses how to access interesting, original, and publishable research in statistical modeling of large-scale flows and several related fields. The authors f this book explicitly reach around the major branches of mathematics and physics, showing how the use of a few straightforward approaches can create a cornucopia of intriguing questions and the tools to answer them. In reading this book, the reader will learn how to research a topic and how to understand statistical mechanics treatments of fluid dynamics. Of particular interest should be the application of Monte Carlo methods to problems like dispersal of points on the sphere, the phase transitions of in viscid fluid flows in models that increasingly approach the conditions of actual planetary atmospheres, and the treatment of negative absolute temperatures and the effects these extremely high-energy states have on fluid flows. Special attention is given to spherical models as well. This book is intended for the upper-level undergraduate or the beginning graduate level courses of mathematics and physics. It will also be of interest to readers interested in statistical mechanics methods applied to fluid mechanics problems. Readers will gain an understanding of how to synthesize new mathematics by applying familiar tools in new ways, and develop new tools to fit particular applications.

Zusammenfassung
This book is meant for an audience of advanced undergraduates and graduate students taking courses on the statistical mechanics approach to turbulent ?ows and on stochastic simulations. It is also suitable for the self-study of professionals involved in the research and modelling of large scale stochastic ?uid ?ows with a substantial vortical component. Several related ideas motivate the approach in this book, namely, the application of equilibrium statistical mechanics to two-dimensional and 2- dimensional ?uid ?ows in the spirit of Onsager [337], and Kraichnan [227], is taken to be a valid starting point, and the primary importance of non-linear convection e?ects combined with the gravitational and rotational properties of large scale strati?ed ?ows over the secondary e?ects of viscosity is assumed. The latter point is corroborated by the many successful studies of ?uid v- cosity which limit its e?ects to speci?c and narrow regions such as boundary layers, and to the initial and transient phases of the experiment such as in the Ekman layer and spin-up [154] [344].

Inhalt
Probability.- Statistical Mechanics.- The Monte Carlo Approach.- Spectral Methods.- Discrete Models in Fluids.- Spin-Lattice Models.- Monte Carlo Simulations of Spin-Lattice Models on the Sphere.- Polyhedra and Ground States.- Mesh Generation.- Statistical Mechanics for a Vortex Gas.- Two-Layer Quasi-Geostrophic Models.- Coupled Barotropic Vorticity Dynamics on a Rotating Sphere.