Non-zero Divergence Research Articles

Nonlinear Eddy-Viscosity Turbulence Parameterization in Anelastic Three-Dimensional Flow: Some Mathematical Aspects

This paper considers some mathematical aspects of the nonlinear eddy-viscosity turbulence parameterization with quasi-isotropic partitioning of turbulent kinetic energy in three-dimensional anelastic flow. The parameterization, which several investigators have used with the Boussinesq approximation in simulating boundary layer flow and shallow cumulus convection, is generalized in a reasonable way, modifying the formulations of subgrid velocity variances and mean flow deformation to take nonzero three-dimensional divergence into account. It is shown that while strict dissipation of domain-integrated mean kinetic energy due to turbulence is insured for Boussinesq flow, a source/sink term analogous to the pressure-divergence term is also formally present for anelastic flow. Also, in seeking a straightforward parameterization of turbulent kinetic energy so as to insure mathematical realizability for the turbulent velocity variance and covariances, it is found that the formulation of Lilly (1967) meets these requirements provided that his proportionality factor is suitably restricted. In turn, this places a restriction on the relative magnitudes of two universal constants that arise in a more recent formulation of Deardorff (1973). (1973).

Parametrized post-Newtonian approximation and Rastall's gravitational field equations

The parametrized post-Newtonian (PPN) approximation is generalized to accommodate Rastall's modification of Einstein's theory of gravity, which allows nonzero divergence of the energy-momentum tensor. Rastall's theory is then shown to have consistent field equations, gauge conditions, and the correct Newtonian limit of the equations of motion. The PPN parameters are obtained and shown to agree experimentally with those for the Einstein theory. In light of the nonzero divergence condition, integral conservation laws are investigated and shown to yield conserved energy-momentum and angularmomentum. We conclude that the above generalization of metric theories, within the PPN framework, is a natural extension of the concept of metric theories.

A Balanced Diagnostic System Compatible with a Barotropic Prognostic Model

A system of diagnostic equations for the velocity field, or wind laws, for a barotropic primitive-equation model of large-scale atmospheric flow is derived. Attention is given to the classical balance equation and its ellipticity condition. Numerical solutions of the diagnostic system are presented, including examples of cases of the mixed elliptic-hyperbolic type and cases with non-zero divergence. Procedures for implementing such a system are outlined, along with a review of factors in using the technique for operational numerical weather prediction.

Integral versus local conservation laws in metric theories of gravity

Previously we considered extensions of metric theories of gravitation with non-zero divergence of the energy-momentum tensor. We confirm here that lack of local conservation laws does not exclude, a priori, existence of integral conservation laws.

Gravitational theories with nonzero divergence of the energy-momentum tensor

Assuming that the divergence of the energy-momentum tensor is nonzero leads to a class of theories with consistent field equations and gauge conditions as well as compatibility with the Newtonian limit of the conservation laws. We have used both the Einstein and the Brans-Dicke theories as models, but the extension to other viable theories such as vector-metric and two-metric theories is possible. One particularly interesting theory emerges that agrees with the ordinary Brans-Dicke theory except for the post-Newtonian parameter ${\ensuremath{\zeta}}_{2}$, which predicts nonconservation of total momentum. Unfortunately no accurate experimental limits for this parameter are known. It thus remains for future experiments in lunar-laser ranging to test this theory.

Polar Boundary Conditions in Zonally Averaged Global Climate Models

Abstract We consider global climate models based on zonally averaged balance relations. Inherent boundary conditions require the meridional fluxes of non-negative properties (temperature, humidity, energy, etc.), as well as the flux of zonal momentum, to vanish at both poles. On the other hand, the meridional divergence of these fluxes does not vanish at either pole. An important exception from this general non-zero polar divergence condition of meridional fluxes is the transport of zonal momentum; the meridional divergence of zonal momentum flux vanishes at the pole because there is neither zonal surface stress nor horizontal wind. These conditions are derived from the balance equations for energy and momentum. Furthermore, they are tested with observed flux data for specific humidity and zonal wind. The closure problem in such models is often overcome by a diffusive parameterization of the fluxes in terms of meridional gradients. It is shown that, due to the above conditions, the exchange coefficient fo...

Do massive Brans-Dicke theories of gravitation imitate Brans-Dicke theories with nonzero divergence of energy-momentum tensor?

Acharya and Hogan (1973) have introduced a massive scalar field into the usual Brans-Dicke (1961) theory of gravitation. Formally they obtain certain field equations. The assumption of a nonzero divergence for T mu nu (or equivalently the introduction of sources) formally imitates a massive Brans-Dicke and satisfies the condition of Acharya and Hogan that the theory be indistinguishable with the classical test of the Einstein theory. Although it was shown elsewhere that the modified Brans-Dicke theory agrees with the classical test under certain conditions, there were no specified limits on omega, a similar circumstance discovered by Acharya and Hogan for the massive scalar field.

Comments on stubbe's deviations from the barometric law

An approximation for the height distribution of nitrogen and atomic oxygen in the thermosphere between 120 and 500 km is obtained by the assumption that these constituents are in diffusive equilibrium. The horizontal wind system in the thermosphere may cause deviations from the diffusive equilibrium distribution due to the non-zero divergence of the horizontal flow and the vertical mass motion associated with it. Stubbe has examined this problem and deduced considerable deviations from diffusive equilibrium due to this effect. The problem is re-examined here, and it is shown that there exist arguments that the deviations from diffusive equilibrium due to horizontal wind fields are not as large as suggested by Stubbe.

Magnetostatic spin wave focusing and defocusing in cylindrically symmetric non-laplacian magnetic fields

A simple analytic criterion is developed which determines the focusing or defocusing of near axis magnetostatic spin waves propagating in a cylindrically symmetric ferromagnet. The crystal is assumed to be magnetically saturated by an axial magnetic field which, due to nonzero volume divergence of the magnetization, is in general, non-Laplacian in character. In addition, both magnetic dipolar coupling and variation of the angle between the magnetization vector and the local propagation vector are included in the model, which thus extends the analytic work of previous authors. The asymptotic behavior of the spin wave trajectories is found to predict either focusing or defocusing. In analogy to under- and over-damped resonators, the former occurs either by the monotonic or damped oscillatory approach of a ray toward the axis; the condition for critical damping is also given. The theory is applied to the case of a magnetically saturated YIG cylinder of commonly employed dimensions (3 mm d. × 10 mm).

Resonant interactions between planetary waves

In contrast to surface gravity waves, planetary waves can interact resonantly at the second order. Hence triplets of planetary waves may occur which are in resonance with each other. For simplicity the situation is studied first on a β plane. The geometrical conditions for two waves to form a triplet with a given third wave are determined, and so also is the rate of energy transfer. Some conservation theorems are proved. The analysis allows for a non-zero horizontal divergence of the motion. For waves which cover a complete sphere it is shown that the resonant interactions between three different harmonic components take place, if at all, in the neighbourhoods of two latitude circles, situated symmetrically north and south of the equator.