Semiclassical Mechanics of the Wigner 6j-Symbol

The semiclassical mechanics of the Wigner 6j-symbol is examined from the standpoint of WKB theory for multidimensional, integrable systems, to explore the geometrical issues surrounding the Ponzano-Regge formula. The relations among the methods of Roberts and others for deriving the Ponzano-Regge formula are discussed, and a new approach, based on the recoupling of four angular momenta, is presented. A generalization of the Yutsis-type of spin network is developed for this purpose. Special attention is devoted to symplectic reduction, the reduced phase space of the 6j-symbol (the 2-sphere of Kapovich and Millson), and the reduction of Poisson bracket expressions for semiclassical amplitudes. General principles for the semiclassical study of arbitrary spin networks are laid down; some of these were used in our recent derivation of the asymptotic formula for the Wigner 9j-symbol.Comment: 64 pages, 50 figure

A perspective on climate model hierarchies

To understand Earth's climate, climate modelers employ a hierarchy of climate models spanning a wide spectrum of complexity and comprehensiveness. This essay, inspired by the World Climate Research Programme's recent “Model Hierarchies Workshop,” attempts to survey and synthesize some of the current thinking on climate model hierarchies, especially as presented at the workshop. We give a few formal descriptions of the hierarchy and survey the various ways it is used to generate, test, and confirm hypotheses. We also discuss some of the pitfalls of contemporary climate modeling, and how the “elegance” advocated for by Held (2005) has (and has not) been used to address them. We conclude with a survey of current activity in hierarchical modeling, and offer suggestions for its continued fruitful development

An introduction to tensors and group theory for physicists

The second edition of this highly praised textbook provides an introduction to tensors, group theory, and their applications in classical and quantum physics.  Both intuitive and rigorous, it aims to demystify tensors by giving the slightly more abstract but conceptually much clearer definition found in the math literature, and then connects this formulation to the component formalism of physics calculations.  New pedagogical features, such as new illustrations, tables, and boxed sections, as well as additional “invitation” sections that provide accessible introductions to new material, offer increased visual engagement, clarity, and motivation for students.   Part I begins with linear algebraic foundations, follows with the modern component-free definition of tensors, and concludes with applications to physics through the use of tensor products. Part II introduces group theory, including abstract groups and Lie groups and their associated Lie algebras, then intertwines this material with that of Part I by introducing representation theory.  Examples and exercises are provided in each chapter for good practice in applying the presented material and techniques.  Prerequisites for this text include the standard lower-division mathematics and physics courses, though extensive references are provided for the motivated student who has not yet had these.  Advanced undergraduate and beginning graduate students in physics and applied mathematics will find this textbook to be a clear, concise, and engaging introduction to tensors and groups. Reviews of the First Edition “[P]hysicist Nadir Jeevanjee has produced a masterly book that will help other physicists understand those subjects [tensors and groups] as mathematicians understand them… From the first pages, Jeevanjee shows amazing skill in finding fresh, compelling words to bring forward the insight that animates the modern mathematical view…[W]ith compelling force and clarity, he provides many carefully worked-out examples and well-chosen specific problems… Jeevanjee’s clear and forceful writing presents familiar cases with a freshness that will draw in and reassure even a fearful student.  [This] is a masterpiece of exposition and explanation that would win credit for even a seasoned author.” —Physics Today "Jeevanjee’s [text] is a valuable piece of work on several counts, including its express pedagogical service rendered to fledgling physicists and the fact that it does indeed give pure mathematicians a way to come to terms with what physicists are saying with the same words we use, but with an ostensibly different meaning.  The book is very easy to read, very user-friendly, full of examples...and exercises, and will do the job the author wants it to do with style.” —MAA Review

Cold Pools, Effective Buoyancy, and Atmospheric Convection

‘Cold pools’ are pools of air that have been cooled by rain evaporation, and which subsequently slump down and spread out across the Earth’s surface due to their negative buoyancy. Such cold pools, which typically arise from rain produced by convection, also feed back upon convection by kicking up new convection at their edges.This thesis studies the interaction of cold pools and convection at two levels of detail: on one end, we study the dynamics and thermodynamics of a single, idealized cold pool, and on the other, we study the interplay between a steady-state ensemble of convection and the many cold pools that accompany it. A recurring notion is that of ‘effective buoyancy’, which is the net acceleration experienced by a density anomaly such as a cold pool, including the back-reaction of the environment (i.e. the ‘virtual mass effect’) which reduces the net acceleration from its Archimedean value. We derive analytical formulae for the effective buoyancy of cold pools and other roughly cylindrical density anomalies, and use the same framework to understand the forces at play when cold pools trigger new convection. We also analyze the sizes and lifetimes of cold pools, and examine the impact of cold pools on the organization (i.e. clustering) of convection