Semi-infinite Atmosphere Research Articles

Rotating Stratified Flow over a Mountain Ridge as an Initial Value Problem

Abstract Solutions for steady inviscid quasi- and semi-geostrophic flow over a mountain ridge on the f-plane are degenerate in the sense that an arbitrary constant mountain-parallel flow can be added to the solution. It is shown that consideration of the problem as an initial value one removes this degeneracy. The quasi-geostrophic results presented here for a semi-infinite atmosphere vary for different initial conditions according to whether the flow is Boussinesq, anelastic, or deep. We enumerate conditions for which a mountain drag and an upstream influence exists.

Time-dependentH-,X- andY-functions for an anisotropically-scattering atmosphere

We have considered a homogeneous atmosphere scattering anisotropically with Dirac δ-function type time-dependent incidence. We used the method of integral operator developed by Ambartsumian and the theory ofN-solutions developed by Busbridge to find the correspondingH-function (in semi-infinite atmosphere) andX- andY-functions (in finite atmosphere).

Anelastic Semigeostrophic Flow over a Mountain Ridge

Abstract Scale analysis indicates that five nondimensional parameters (R02 ϵ, μ λ and kλ) characterize the disturbance generated by the steady flow of a uniform wind (U0, V0) incident on a mountain ridge of width a in an isothermal, uniformly rotating, uniformly stratified, vertically semi-infinite atmosphere. Here μ = h0/HR is the ratio of the mountain height h0 to the deformation depth HR = fa/N where f is the Coriolis parameter and N is the static buoyancy frequency. The parameters λ = HR/H and kλ are the ratios of HR to the density scale height H and the potential temperature scale height H/k respectively. There are two Rossby numbers: One based on the incident flow that is parallel to the mountain. ϵ = V0/fa, and one normal to the mountain, R0 = U0/fa. If R02 ≪1, then the mountain-parallel flow is in approximate geostrophic balance and the flow is semigeostrophic. The semigeostrophic case reduces to the quasi-geostrophic one in the limit as μ and ϵ tend to zero. If the flow is Boussinesq (λ = 0), the...

Analytic solution of transfer problem linearized with respect to a small parameter in the problem of an almost gray atmosphere

The problem of radiative transfer in a semi-infinite plane-parallel atmosphere that is almost gray is considered. The transfer equation is solved in the second approximation obtained in accordance with the iterative scheme proposed by Chandrasekhar, the case of a gray atmosphere being taken as the first approximation. An exact solution of this equation for arbitrary profiles of the absorption coefficient that do not depend on the optical depth is found.

The Neumann solution of the multiple scattering problem in a plane-parallel medium—IIc. The specific intensity in a semi-infinite space

Using a new formulation of the resolvent function [Rutily and Bergeat JQSRT 38, 61 (1987)], we solve the Milne integral equation for the mean intensity in a semi-infinite plane-parallel atmosphere. The specific intensity is then derived at any optical depth. By expanding the intensity in power series in the albedo, we derive the corresponding intensities of a n-times scattered radiation field propagating in a half-space ( n ⩾/ 1).

Orthogonal rational functions on a semi-infinite interval

By applying a mapping to the Chebyshev polynomials, we define a new spectral basis: the “rational Chebyshev functions on the semi-infinite interval,” denoted by TL n ( y). Continuing earlier work by the author and by Grosch and Orszag, we show that these rational functions inherit most of the good numerical characteristics of the Chebyshev polynomials: orthogonality, completeness, exponential or “infinite order” convergence, matrix sparsity for equations with polynomial coefficients, and simplicity. Seven numerical examples illustrate their versatility. The “Charney” stability problem of meteorology, for example, is solved to show the feasibility of applying spectral methods to a semi-infinite atmosphere. For functions that are singular at both endpoints, such as K 1( y), one may combine rational Chebyshev functions with a preliminary mapping to obtain a single, exponentially convergent expansion on y ϵ [0, ∞ ]. Finally, we successfully generalize the WKB method to obtain, for the J 0 Bessel function, an amplitude-phase approximation which is convergent rather than asymptotic and is accurate not merely for large y but for all y, even the origin.

Reflection of radiation from a semi-infinite inhomogeneous atmosphere in the case of general laws of incoherent scattering

The author discusses the reflection from a semi-infinite inhomogeneous atmosphere in the case of general laws of incoherent scattering. The question of determination of the reflection function for the atmosphere as well as the mean number of scatterings of the reflected photons is investigated. Ambartsumyan's invariance principle is used to obtain equations for these quantities. The approach developed earlier for the solution of such problems in the case of a homogeneous atmosphere is generalized to an inhomogeneous medium. The results of numerical calculations in a special case are given as an illustration.

Numerical evaluation of time-dependent reflected intensity from an anisotropically scattering semi-infinite atmosphere

A numerical scheme, designed to provide four or five digit accuracy for the reflected photon intensity from an anisotropically scattering semi-infinite medium, is presented. The atmosphere is characterized by both a mean free time between collisions and a mean time of temporal capture. Tables of reflected intensities for a test-phase function are included for use as benchmarks to which other numerical solutions can be compared.

Deep and shallow quasi-geostrophic flow over mountains

The effect of a finite density scale height on steady quasi-geostrophic flow over an isolated mountain is examined using analytic solutions for an isothermal, uniformly stratified, vertically semi-infinite atmosphere on an f -plane. A nondimensional measure of the depth of flow and hence of the importance of the vertical density variations of the basic state is the ratio of the Rossby height of the flow, H R , to the density scale height H . Here H R = fa / N where a is the mountain halfwidth and N is the buoyancy frequency. The flow is deep for Λ = H R / H ≥ O(1) and shallow for Δ ⋍ 0. It is found that the mountain-induced surface anticyclone is stronger for a deep flow than for a shallow one. Unlike the shallow case, no region of cyclonic vorticity is generated. The enhanced response for deep flow decreases the critical mountain height necessary for the formation of a stagnation point (e.g., Taylor cone). This critical height decreases with decreasing scale height and with increasing stratification. In addition, the mountain's influence decays less rapidly with height for deep flow. The far-field response for the deep case exhibits a circulation consistent with the lift force acting on the mountain. Since it is shown that the volume of vertically displaced fluid at any level equals the mountain volume, the transmission of the lift to the fluid arises from the decreasing density field, while volume expansion generates the far-field anticyclonic vorticity. These results hold for both the standard and the modified quasi-geostrophic equations. In the modified theory, however, the mountain surface is not isentropic. DOI: 10.1111/j.1600-0870.1986.tb00462.x

Numerical evaluations in radiative transfer

We compare different methods of calculating the monochromatic mean intensity and flux divergence for different dependences of the source function on the optical depth. We consider a semi-infinite plane-parallel atmosphere and a plane-parallel symmetric slab. We show that in some cases the methods commonly used are quite inaccurate, and propose another which gives consistent values for both the flux divergence and the mean intensity, even in extreme cases of high and small optical depths.

Radiative transfer of resonance lines with internal sources

The modeling of resonance lines in the aurorae and dayglow usually requires a large computer code to solve the radiative transfer problem. We present here a simple theory that gives reasonable estimates of the emergent intensities, including the angular distribution, for very optically thick lines with internal sources. The method uses an extension of the H-functions (implying a semi-infinite atmosphere) and assumes that the source is monochromatic and may be reasonably represented by a Chapman profile. The theory is compared with detailed numerical calculations for resonance lines with Voigt emission line profiles, including the cases of both partial and complete frequency redistribution. The effects of frequency redistribution on the angular distribution of the emergent radiation and on the line profiles are also shown.

The transfer of polarized radiation in spectral lines: Formalism and solutions in simple cases

A general treatment of the transfer of polarized radiation in spectral lines assuming a Rayleigh phase function and a general law of frequency redistribution is derived. It is shown how nine families of coupled integral equations for the moments of the radiation field arise which are necessary to fully describe the state of polarization of the emergent radiation from a plane-parallel, semi-infinite atmosphere. The special case of angle independent redistribution functions is derived from the general formalism, and it is shown how the nine families of integral equations reduce to the six linearly independent integral equations derived by Collins (1972). To serve as a test of the formulation, solutions for isothermal atmospheres are given.

Non-coherent scattering in subordinate line—V: Solutions of the transfer problem

We present several examples of the numerical solution of the radiative transfer in subordinate lines. Using a simplified physical model that yields the line source function analogous to the usual two-level-atom form modified by the presence of the redistribution function R v in the scattering integral, we have solved the transfer problem for isothermal, plane-parallel atmospheres, both finite and semi-infinite. For finite atmospheres, we have found substantial differences between the solutions with R v and those with complete redistribution. On the other hand, for semi-infinite atmospheres the complete redistribution appears to be a good approximation, at least for a l⋍ a u (damping parameters for the lower and upper levels, respectively). It is shown that the effect of R v becomes more pronounced with increasing ratio a u/ a l. Finally, it is demonstrated that an approximate form for R v analogous to that of Kneer for R II serves as a very good approximation for computing the line profiles, particularly in the line wings.

On the nonuniqueness of solutions for nonlinear integral equations in radiative transfer theory

The nonuniqueness problem is considered for the nonlinear integral equations satisfied by the reflection and transmission matrices of homogeneous plane-parallel atmospheres. The analysis of the problems for semi-infinite and finite atmospheres is based on a recently developed biorthogonolity concept. Explicit expressions for nonphysical solutions are derived. The structure of these solutions reveal that iterative solution procedures may easily yield nonphysical results, if no proper attention is paid to certain linear constraints.

Number of terms required in the fourier expansion of the reflection function for optically thick atmospheres

Computational results have been obtained for the separate terms in the expansion of the reflection function of an optically thick, conservatively scattering, atmosphere composed of cloud particles. The computations were obtained by successive applications of the invariant imbedding, doubling and asymptotic fitting methods to cover the range from very thin to very thick atmospheres. Results are presented which illustrate the magnitude of the separate terms in the Fourier expansion of the phase function and the Fourier expansion of the reflection function of a semi-infinite atmosphere as a function of the zenith angles of incidence and reflection. The azimuthally independent reflection function is enhanced by as much as a factor of 115 over the first-order reflection function, whereas the azimuth-dependent reflection functions generally result from less multiple scattering. These results are compared with those for an atmosphere having a Henyey-Greenstein phase function with the same asymmetry factor ( g = 0.84123) as in the cloud model. The relative difference in the escape function and azimuthally independent reflection function is generally less than a few per cent, though differences up to 70% occur in the reflection function at angles where single scattering is important. Results are also presented which show the number of terms required in the Fourier expansion of the reflection function to be assured an accuracy of 0.1%. The number of terms required depends strongly on the zenith angles of incidence and reflections as well as on details of the phase function.

Intensity and polarization line profiles in a semi-infinite Rayleigh-scattering planetary atmosphere. II. Variations of equivalent width over the disk

The study of the variation of equivalent width in a Rayleighscattering planetary atmosphere along the intensity equator and along the mirror meridian on whichμ =μ 0 shows that the equivalent widths decrease monotonically towards the poles, the limb and the terminator with the following characteristics: (i) the weakest lines exhibit the maximum change; (ii) theI e r component shows more change than theI e r component; (iii) the decrease towards the limb or the terminator is not as sharp as that towards the poles; (iv)I e r component shows more decrease towards the limb whileI e r component shows more decrease towards the terminator; and (v) the relationW (μ, φ;μ 0,φ 0)= W (μ 0,φ 0;μ, φ) holds for the total intensity. These results are qualitatively in agreement with the observations of absorption bands in the spectra of Venus, Jupiter and Saturn.

Intensity and polarization line profiles in a semi-infinite Rayleigh-scattering planetary atmosphere. I. Integrated flux

Absorption and polarization line profiles as well as the curves of growth in the integrated light of a planet over the whole range of phase angles have been computed assuming a semi-infinite atmosphere scattering according to Rayleigh’s phase-matrix which takes polarization into account. The relative change in line depth and equivalent widths qualitatively agree with the observations of the CO2 bands in Venus reported by Young, Schorn and Young (1980). It is pointed out that the bands might be formed in a part of the atmosphere which is different from that where continuum polarization originates.

A generalised principle of invariance and radiation field in multilayer atmospheres

A generalized principle of invariance is derived for a plane-parallel atmosphere. On the basis of this principle a method for determining the radiation field in a multilayer atmosphere is proposed. This method, the first part of which is the well-known adding method, permits the application to problems involving optically finite as well as semi-infinite atmospheres. The reflecting boundaries may be incorporated, though in that case it is not possible to use the adding method.

Neutrino emission from a hot, dense, plane-parallel atmosphere in hydrostatic equilibrium. I

view Abstract Citations (24) References (22) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Neutrino emission from a hot, dense, plane-parallel atmosphere in hydrostatic equilibrium. I Schinder, P. J. ; Shapiro, S. L. Abstract The plane-parallel, semiinfinite static neutrino atmosphere problem is solved. The atmosphere is assumed to consist of free nucleons, electrons, positrons, and photons, with electron neutrinos and antineutrinos the only means by which energy and lepton number may be transported. The atmosphere is assumed to be in hydrostatic equilibrium with a constant surface gravitational acceleration. The transport equations for neutrinos and antineutrinos and the three constraint equations which completely determine the structure of the atmosphere are solved simultaneously. It is found that in general the spectra are well approximated by Fermi distributions at high energies where opacities are large, but deviate significantly from Fermi distributions at lower energies. High values of energy flux and low values of lepton number flux produce atmospheres dominated by relativistic particles. High values of lepton number flux cause large neutrino chemical potential gradients, so that energy is carried inward by antineutrinos while neutrinos carry energy outward. Publication: The Astrophysical Journal Pub Date: August 1982 DOI: 10.1086/160168 Bibcode: 1982ApJ...259..311S Keywords: Dense Plasmas; Gravitational Collapse; High Temperature Plasmas; Hydrostatics; Neutrinos; Particle Emission; Stellar Atmospheres; Convective Flow; Flux Density; Nuclear Reactions; Astrophysics full text sources ADS |

Monte Carlo calculations of resonance radiative transfer through a semi-infinite atmosphere

The results of Monte Carlo calculations of radiative transfer through a semi-infinite plane-parallel atmosphere of resonant scatterers are presented. With a photon source at optical depth tau/sub ES/ we model the semi-infinite geometry by embedding a perfectly reflecting mirror at depth tau/sub MS/+tau/sub ES/. Although some quantities characterizing the emergent photons diverge as tau/sub MS/..-->..infinity, the mean number of scatters, N/sub ES/, and path length, L/sub ES/, accumulated between the source and the edge of the atmosphere converge. Accurate results of N/sub ES/, L/sub ES/, X/sub pk/, the most probable frequency shift of the escaping photons, and tau/sub LAST/, the mean optical depth at which they last scatter, are obtained by choosing tau/sub MS/ = 4tau/sub ES/. Approximate analytic calculations of N/sub ES/, L/sub ES/, N, the mean total number of scatters undergone by escaping photons, L, their mean total path length, and , their mean (absolute) frequency shift, are presented for a symmetric slab with ..cap alpha..tau/sub ES/>>1 and tau/sub MS/>>tau/sub ES/. Analogous calculations for an asymmetric slab are discussed. Analytic fitting formulae for N/sub ES/, L/sub ES/, X/sub pk/, and tau/sub LAST/ are given.