20 research outputs found

    Uniform Approximation from Symbol Calculus on a Spherical Phase Space

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    We use symbol correspondence and quantum normal form theory to develop a more general method for finding uniform asymptotic approximations. We then apply this method to derive a result we announced in an earlier paper, namely, the uniform approximation of the 6j6j-symbol in terms of the rotation matrices. The derivation is based on the Stratonovich-Weyl symbol correspondence between matrix operators and functions on a spherical phase space. The resulting approximation depends on a canonical, or area preserving, map between two pairs of intersecting level sets on the spherical phase space.Comment: 18 pages, 5 figure

    Semiclassical Mechanics of the Wigner 6j-Symbol

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    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

    Convective self-aggregation, cold pools, and domain size

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    On the sizes and lifetimes of cold pools

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    Cold pools of air, which are formed by evaporating precipitation, play a critical role in the triggering of new precipitation. Despite their recognized importance, little effort has been devoted to building simple models of their dynamics. Here, analytical equations are derived for the radius, height, and buoyancy of a cylindrical cold pool as a function of time, and a scale analysis reveals that entrainment is a dominant influence. These governing equations yield simple expressions for the maximum sizes and lifetimes of cold pools. The terminal radius of a cold pool is relatively insensitive to its initial conditions, with a typical maximum radius of about 14 times the initial radius, give or take a factor of 2. The terminal time of a cold pool, on the other hand, can vary over orders of magnitude depending on its initial potential and kinetic energies. These predictions are validated against large-eddy simulations
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