Nadir Jeevanjee

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.

Effective Buoyancy, Inertial Pressure, and the Mechanical Generation of Boundary Layer Mass Flux by Cold Pools

AbstractThe 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.

Mean precipitation change from a deepening troposphere

Significance Global climate models robustly predict that global mean precipitation should increase at roughly 2–3% K−1, but the origin of these values is not well understood. Here we develop a simple theory to help explain these values. This theory suggests that global mean precipitation is closely tied to the depth of the troposphere, when measured in temperature coordinates. When surface temperatures increase, this “temperature depth” of the troposphere also increases, causing an increase in global mean precipitation. Global climate models robustly predict that global mean precipitation should increase at roughly 2–3% K−1, but the origin of these values is not well understood. Here we develop a simple theory to help explain these values. This theory combines the well-known radiative constraint on precipitation, which says that condensation heating from precipitation is balanced by the net radiative cooling of the free troposphere, with an invariance of radiative cooling profiles when expressed in temperature coordinates. These two constraints yield a picture in which mean precipitation is controlled primarily by the depth of the troposphere, when measured in temperature coordinates. We develop this theory in idealized simulations of radiative–convective equilibrium and also demonstrate its applicability to global climate models.

The Representation Operator and Its Applications

This chapter applies the material of the previous chapters to some particular topics, specifically the Wigner–Eckart theorem, selection rules, and gamma matrices and Dirac bilinears. We begin by discussing the perennially confusing concepts of vector operators and spherical tensors, and use these to give a quick overview of the Wigner–Eckart theorem and selection rules. These latter subjects are then made precise using the notion of a representation operator. We conclude by showing that Dirac’s famous gamma matrices can be understood in terms of representation operators, which then immediately gives the transformation properties of the “Dirac bilinears” of QED.