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 $6j$-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

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

Observing convective aggregation

Convective self-aggregation, the spontaneous organization of initially scattered convection into isolated convective clusters despite spatially homogeneous boundary conditions and forcing, was first recognized and studied in idealized numerical simulations. While there is a rich history of observational work on convective clustering and organization, there have been only a few studies that have analyzed observations to look specifically for processes related to self-aggregation in models. Here we review observational work in both of these categories and motivate the need for more of this work. We acknowledge that self-aggregation may appear to be far-removed from observed convective organization in terms of time scales, initial conditions, initiation processes, and mean state extremes, but we argue that these differences vary greatly across the diverse range of model simulations in the literature and that these comparisons are already offering important insights into real tropical phenomena. Some preliminary new findings are presented, including results showing that a self-aggregation simulation with square geometry has too broad a distribution of humidity and is too dry in the driest regions when compared with radiosonde records from Nauru, while an elongated channel simulation has realistic representations of atmospheric humidity and its variability. We discuss recent work increasing our understanding of how organized convection and climate change may interact, and how model discrepancies related to this question are prompting interest in observational comparisons. We also propose possible future directions for observational work related to convective aggregation, including novel satellite approaches and a ground-based observational network

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

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

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

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