Transfer radiative

The radiation field in the atmosphere is determined from the equation of radiative transfer (Chandrasekhar, 1950 Kourganoff, 1952 Sobolev, 1963 Lenoble, 1977), which is an expression of the energy balance in each unit volume of the atmosphere, including absorption, scattering, and emission. In the case of a horizontally stratified medium, the following expression can be used to describe the radiative transfer in 

When the atmosphere is assumed to be stratified and plane-parallel, it is common to introduce the radiance hv ] and hv describing upward and downward propagating photons, as was presented previously for the flux (see Eqs. 4.39 and 4.40). In this case, integrating (Eq. 4.51), we obtain  

Note that the Earth s surface infrared and microwave emissivity is a function of vegetation, snow cover, soil moisture, sea conditions, etc. 

2 Solution of the Equation of Radiative transfer for Wavelengths Less than 3.5 /im Multiple Scattering 

As indicated previously, shortwave radiation below 3.5 /rm originates from the Sun. When the direct flux penetrates into the atmosphere, it is progressively attenuated by absorption (ra) and scattering (rs). In this case, the source function must account for the diffuse radiation scattered into the beam from other directions as well as the direct sunlight 

Molecular observations are almost always concerned with specific discrete transitions. These are generally observed at millimeter or centimeter wavelengths. The intensity of a source is determined by the rate of collisional versus radiative transitions between levels. Because of the extremely low densities usually associated with molecular environments, whether in a circumstellar envelope or a molecular cloud, pressure broadening is unimportant. Instead, the molecule radiates at its local velocity into the line of sight. This dispersion of velocity may be due strictly to the thermal motions of the particles, or it may be due to the presence of turbulence or large-scale chaotic motions within the medium. Either way, the local profile, (v) is a Gaussian with a finite width in frequency. 

Like their atomic counterparts, molecular lines saturate when the populations have reached the values associated with strict equilibrium with the incoming radiation. This occurs first at the line center. Any motion in the medium, ordered or random, will broaden the line and thus the molecules will see radiation at other wavelengths against which they can absorb, or into which they can emit. If the medium is optically thick at line center but the velocity dispersion is large, the overall optical depth can be considerably reduced by spreading out the line in frequency. 

The most abundant molecules, because of the low velocities observed in the clouds and high column densities, cannot be interpreted by simple optically thin models. For CO (any isotope), the ratio of the (2 1) to (1 0) transitions should be 3 1 in strength, if completely optically thin, because of the ratio of the statistical weights and transition probabilities and the temperature known to exist in the clouds. However, (1 0) is often observed to 

The abundance of a species is related to the observed line intensity by the antenna temperature, the temperature which an equivalent blackbody radiator would have to have at the line frequency to equal the observed line intensity  

Masers also serve as a warning that the intensity of a spectral line is not necessarily a direct measure of its abundance. Population inversions enhance the brightness temperatures, leading to overestimates of excitation and abundance in those species in which masing occurs. Because not all molecules undergo maser amphfication, the assumption of thermal equilibrium is usually not bad, but should be employed with caution. 

Various physical processes modify a radiation field as it propagates through an atmosphere. The rate at which the atmosphere emits depends on its composition and thermal structure, while its absorption and scattering properties are defined by the prevailing molecular opacity and cloud structure. 

Independently of whether the radiation field is generated internally or is imposed externally, the study of how it interacts with the atmosphere is embodied in the theory of radiative transfer. Many authors have dealt with this theory in various contexts. Monographs include those by Kouiganoff (1952), Woolley Stibbs (1953), Goody (1964), and Goody Yung (1989). A standard text is by Chandrasekhar (1950), which treats the subject as a branch of mathematical physics. The emphasis is on scattered sunlight in planetary atmospheres and on various problems of astrophysical interest. 

Our own approach is somewhat different and emphasizes spectra produced by thermal emission from planetary atmospheres, especially as observed from space platforms. In order to demonstrate the connection between the thermal radiation giving rise to these spectra and the physical state of the atmosphere under consideration, it is necessary to examine how the transport of this radiation is effected. Only then is it possible to have a clear understanding of how the structure of an atmosphere leads to its spectral appearance, a topic considered at length in Chapter 4. Once this is accomplished a reversal of the procedure is feasible, and in Chapters 6 through 9 we demonstrate how the observed characteristics of the radiation field imply the underlying physical structure and the state of the interacting atmosphere. 

Our aim in this chapter is to develop the mathematical formalism that serves as the foundation for all our analyses involving the radiation field in sufficient depth to be essentially autonomous, though our indebtedness to some of the procedures developed by Chandrasekhar (1950) is obvious. The equation of transfer is derived in Section 2.1, and formal solutions are found in Section 2.2. Very general techniques for solving the transfer equation numerically are developed in Section 2.3. Though 

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Additional sources are the Journal of Applied Optics and the Journal of the Optical Society of America, particularly for surface properties the Jour nal of Quantitative Spectroscopy and Radiative Transfer for gas properties the Jour -nal of Heat Tr ansfer andthe Inter national Journal of Heat and Mass Tr ansfer lor broad coverage and the Jour nal of the Institute of Ener gy for applications to industrial furnaces. 

NOTE Figures in this table are taken from plots in Hottel and Sarofim, Radiative Transfer, McGraw-HiU, New York, 1967, chap. 6. Values in parentheses are extrapolated. To convert degrees Ranldne to kelvins, multiply hy (5.556)(10 ). To convert atmosphere-feet to Idlopascal-meters, multiply hy 30.89. 

Departure of gas from grayness has so marked an effect on radiative transfer that the subject will be presented prior to discussion of the systems covered by Table 5-10. 

Treatment of radiative transfer in combustion chambers is available at varying levels of complexity, including allowance for temperature variation in both gas and refractory walls (Hottel and Sarofim,... 

Though this is a quartic equation, it is capable of explicit solution because of the absence of second and third degree terms. Trial-and-error enters, however, because (GSi)r and are mild functions of Tg and related Te, respectively, and aprehminary guess of Tg is necessaiy. An ambiguity can exist in interpretation of terms. If part of the enclosure surface consists of screen tubes over the chamber-gas exit to a convection section, radiative transfer to those tubes is included in the chamber energy balance, but convection is not, because it has no effect on chamber gas temperature. 

The mechanism of radiative transfer in flares was found to depend on compn, flare diameter and pressure (Ref 69). The flare efficiency calcn is complicated by the drop-off in intensity at increasing altitudes and at very large diameters owing to the lower reaction temps (Ref 11, p 13) and the narrowing of the spectral emittance band (Ref 35). The prediction of the light output in terms of compn and pressure (ie, altitude) is now possible using a computer program which computes the equilibrium thermodynamic properties and the luminance (Ref 104) Flare Formulations... 

Propagation of Gasless Reactions in Solids , Combustion Flame 21, No 1 (1973), 91-97 69) B.E. Douda, Radiative Transfer Model of a Pyrotechnic Flare , NAD-RDTR No 258... 

Hottel, H, C. and Sarofim, A. F. Radiative Transfer (McGraw-Hill, New York, 1967)... 

Heat transfer processes besides pure radiative transfer are involved in control of the temperature of the air, especially below the effective emission height of 6 km. Referring back to Chapter 7, we see that vertical motions of air in the troposphere are a main factor dictating that temperature decreases as altitude increases - air loses internal energy... 

Atmospheric aerosols have a direct impact on earth s radiation balance, fog formation and cloud physics, and visibility degradation as well as human health effect[l]. Both natural and anthropogenic sources contribute to the formation of ambient aerosol, which are composed mostly of sulfates, nitrates and ammoniums in either pure or mixed forms[2]. These inorganic salt aerosols are hygroscopic by nature and exhibit the properties of deliquescence and efflorescence in humid air. That is, relative humidity(RH) history and chemical composition determine whether atmospheric aerosols are liquid or solid. Aerosol physical state affects climate and environmental phenomena such as radiative transfer, visibility, and heterogeneous chemistry. Here we present a mathematical model that considers the relative humidity history and chemical composition dependence of deliquescence and efflorescence for describing the dynamic and transport behavior of ambient aerosols[3]. 

The standard approach to modeling PDRs is to use a one-dimensional approach in which the radiation strikes perpendicularly. The region is divided into slabs, so that the equations of radiative transfer and chemical kinetics can be solved conveniently. The slabs can be homogeneous, or can have different gas densities. The radiation is scattered and absorbed by dust particles, but, in addition, both H2 and... 

Size makes a difference propagating and nonpropagating energy near- and far-field zones radiative transfer... 

Optical designers and specialists in heat transfer calculations in the chemical engineering and mechanical engineering sciences are familiar with the mathematical construct known as The Equation of Radiative Transfer, although most chemists and spectroscopists are not. The Equation of Radiative Transfer states that, disregarding absorbance and scattering, in a lossless optical system... 

Fig. 2. VLA detection of 3He+ in the PN J 320. We have modeled the radio continuum and line emission using the radiative transfer code NEBULA [1], assuming an expanding shell of ionized gas. The dashed line is the model including the H171 7 and 3He+ transitions. The solid line shows the observed spectrum and only includes the 3He+ transition. The model fits the data reasonably well even though the morphology is bipolar as indicated by the HST image [6]...

Our multi-level carbon model atom is adapted from D. Kiselman (private communication), with improved atomic data and better sampling of some absorption lines. The statistical equilibrium code MULTI (Carlsson 1986), together with ID MARCS stellar model atmospheres for a grid of 168 late-type stars with varying Tefj, log g, [Fe/H] and [C/Fe], were used in all Cl non-LTE spectral line formation calculations, to solve radiative-transfer and rate equations and to find the non-LTE solution for the multi-level atom. We put particular attention in the study of the permitted Cl lines around 9100 A, used by Akerman et al. (2004). 

M. Carlsson, A Computer Program for Solving Multi-Level Non-LTE Radiative Transfer Problems in Moving or Static Atmospheres . In Uppsala Astronomical Report No. 33 (1986)... 

Fig. 1. Model Spectra re-binned to CRIRES Resolution To demonstrate the potential for precise isotopic abundance determination two representative sample absorption spectra, normalized to unity, are shown. They result from a radiative transfer calculation using a hydrostatic MARCS model atmosphere for 3400 K. MARCS stands for Model Atmosphere in a Radiative Convective Scheme the methodology is described in detail e.g. in [1] and references therein. The models are calculated with a spectral bin size corresponding to a Doppler velocity of 1 They are re-binned to the nominal CRIRES resolution (3 p), which even for the slowest rotators is sufficient to resolve absorption lines. The spectral range covers ss of the CRIRES detector-array and has been centered at the band-head of a 29 Si16 O overtone transition at 4029 nm. In both spectra the band-head is clearly visible between the forest of well-separated low- and high-j transitions of the common isotope. The lower spectrum is based on the telluric ratio of the isotopes 28Si/29Si/30Si (92.23 4.67 3.10) whereas the upper spectrum, offset by 0.4 in y-direction, has been calculated for a ratio of 96.00 2.00 2.00.

Integrate the equation of radiative transfer to produce a synthetic spectrum as a function of assumed abundances of each element of interest. 

The reversing-layer picture is sometimes referred to as the Schuster-Schwarzschild model , but Arthur Schuster and Karl Schwarzschild considered more sophisticated models of radiative transfer (including scattering) than the one used here. 

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