20 research outputs found
Uniform Approximation from Symbol Calculus on a Spherical Phase Space
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 -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
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
Recommended from our members
Convective self-aggregation in numerical simulations: a review
Organized convection in the Tropics occurs across a range of spatial and temporal scales and strongly influences cloud cover and humidity. One mode of organization found is “self-aggregation”, in which moist convection spontaneously organizes into one or several isolated clusters despite spatially homogeneous boundary conditions and forcing. Self-aggregation is driven by interactions between clouds, moisture, radiation, surface fluxes, and circulation, and occurs in a wide variety of idealized simulations of radiative-convective equilibrium. Here we provide a review of convective self-aggregation in numerical simulations, including its character, causes, and effects. We describe the evolution of self-aggregation including its time and length scales and the physical mechanisms leading to its triggering and maintenance, and we also discuss possible links to climate and climate change
On the sizes and lifetimes of cold pools
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
Recommended from our members
Effective buoyancy, inertial pressure, and the mechanical generation of boundary layer mass flux by cold pools
The Davies-Jones formulation of effective buoyancy is used to define inertial and buoyant components of vertical force and to develop an intuition for these components by considering simple cases. This decomposition is applied to the triggering of new boundary layer mass flux by cold pools in a cloud-resolving simulation of radiative-convective equilibrium (RCE). The triggering is found to be dominated by inertial forces, and this is explained by estimating the ratio of the inertial forcing to the buoyancy forcing, which scales as H/h, where H is the characteristic height of the initial downdraft and h is the characteristic height of the mature cold pool's gust front. In a simulation of the transition from shallow to deep convection, the buoyancy forcing plays a dominant role in triggering mass flux in the shallow regime, but the force balance tips in favor of inertial forcing just as precipitation sets in, consistent with the RCE results
Recommended from our members
On the sizes and lifetimes of cold pools
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
Recommended from our members
Effective buoyancy at the surface and aloft
It is shown here that a wide, buoyant parcel of air at the surface accelerates far less rapidly than it does aloft. In particular, analytical formulae are derived for the effective buoyancy (i.e. the net vertical acceleration due to parcel buoyancy and environmental response) of idealized cylinders of diameter D and height H, located in free space and at the surface. These formulae quantify the decrease of effective buoyancy with increasing aspect ratio D/H, and show that this effect is more pronounced for surface cylinders, especially when D/H > 1. We gain intuition for these results by considering the pressure fields generated by these buoyant parcels, and we test our results with large-eddy simulations. Our formulae can inform parametrizations of the vertical velocity equation for clouds, and also provide a quantitative map of the 'grey zone' in numerical modelling between hydrostatic and non-hydrostatic regimes