Weak Line Approximation Research Articles

Extension of the weak-line approximation and application to correlated- k methods

Global climate models require accurate and rapid computation of the radiative transfer through the atmosphere. Correlated- k methods are often used. One of the approximations used in correlated- k models is the weak-line approximation. We introduce an approximation T γ which reduces to the weak-line limit when optical depths are small, and captures the deviation from the weak-line limit as the extinction deviates from the weak-line limit. This approximation is constructed to match the first two moments of the gamma distribution to the k-distribution of the transmission. We compare the errors of the weak-line approximation with T γ in the context of a water vapor spectrum. The extension T γ is more accurate and converges more rapidly than the weak-line approximation.

On systematic errors in parametrized calculations of longwave radiation transfer

AbstractA detailed narrow band model of longwave radiation transfer in a clear‐sky atmosphere is used to study the impact on cooling rate profiles and radiative fluxes of some approximations commonly used in GCM‐type radiation codes.Systematic errors in the results arise when economies are made in the vertical and spectral resolution. A coarse discretization of the vertical profiles of temperature and humidity overestimates atmospheric absorption (by as much as +5 W m−2), as does (by as much as +5 W m−2) the use of mean values of the temperature and humidity in each model layer. Overestimation of the absorption (up to +7 W m−2) also results from the use of the strong line approximation for H2O and CO2, or of the weak line approximation for O3.Decreasing the order of the vertical quadrature used for the vertical integration tends to decrease the absorption. Effects of the water vapour continuum absorption outside the 800‐1250 cm−1 window region cannot be neglected, as this absorption accounts respectively for 21 and 8 W m−2 of the downward radiation at the surface for a tropical and a subarctic winter atmosphere, and decreases the corresponding outgoing radiation at the top of the atmosphere by 5 and 0.3 W m−2 respectively. Neglecting N2O and CH3 decreases absorption by about 5 W m−2. Restricting CO2 absorption to the 500‐800 cm−1 interval, and O3 absorption to the 970‐1110 cm−1 interval further decreases by 5 W m−2 the absorption in the tropical atmosphere.For transmission functions described by statistical models, a widening of the spectral intervals reinforces the influence of the strong lines and leads to an overestimation of the absorption, which is partly compensated by a weaker temperature dependence.

Approximate representations of absorption functions of elasser and statistical models of bands

The determinat ion of t r ansmiss ion functions on the basis of distinct model representa t ions is of great significance in the problem of t r ansmiss ion of infrared emiss ion through the a tmosphere . The s imples t model of a group of spec t ra l l ines is the E l sas se r model of equidistant lines of identical intensity. According to that model, the average absorpt ion A t in a given spectra l interval in the weak-l ine approximation (u << 1) is given by

Spectral Band Absorptance of Radiation Traversing Two or More Cells in Series

Theoretical equations are derived for the spectral band absorptance of radiation which traverses two or more cells in series. The amount of absorbing gas, pressure, and temperature of each cell may be chosen independently. Limiting expressions are obtained when the absorptance of all of the cells can be represented by either the weak line or the strong line approximation as well as when some of the cells can be represented by the weak line approximation while the remainder can be represented by the strong line approximation. Some of the results apply to any spectral band, while other results apply only to particular band models or to nonoverlapping spectral lines. The dependence of the absorptance on the various parameters is illustrated by numerous examples. The conditions for the validity of the usual interpolation procedure for obtaining the equivalent parameters for a homogeneous path are derived.

The Atmospheres of the Metallic-Line Stars in the Hyades.

view Abstract Citations (47) References (52) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS The Atmospheres of the Metallic-Line Stars in the Hyades. Conti, Peter S. Abstract A spectrophotometric study of five metallic4ine stars in the Hyades star cluster is undertaken to determine the abundances of elements with respect to a normal star of he cluster, 45 Tauri. The equivalent widths of over 400 lines in the wavelength range XX A are used. Effective temperatures are derived from photoelectric six-color observations. Model atmospheres with these temperatures, and with gravities derived from a mass-luminosity relation, are constructed by using a scaled solar T (r) and extensive opacity tables. The exact state of ionization of each ion is determined from the model atmosphere, and a weak-line approximation is used to determine the predicted shift in a curve of growth. Abundances are derived by a comparison between this and the observed line strengths. Several types of model atmospheres are investigated: normal dwarfs; those of effective gravity decreasing outward through the atmosphere; those of a higher boundary temperature than a scaled solar T (r); and those of 80 per cent helium abundance. It is concluded that none of these models or combinations of them can explain the well-known observed deficiencies of calcium in metallic-line stars in terms of atmospheric structure. Models with a slightly lower effective gravity in the atmosphere than that obtained from a massluminosity relation fit somewhat better the observed neutral-ionized line strengths. It is possible that iron is overabundant by a factor of 2 or 3 in the hotter metallic-line stars. Some elements lighter than the iron-peak elements are underabundant with respect to iron and heavier elements overabundant. Publication: The Astrophysical Journal Supplement Series Pub Date: January 1965 DOI: 10.1086/190109 Bibcode: 1965ApJS...11...47C full text sources ADS | data products SIMBAD (25) GCPD (1)

The Influence of Numerous Low-Intensity Spectral Lines on Band Absorptance

The characteristics of band absorption are investigated when numerous low-intensity spectral lines occur together with any number of stronger lines. For these conditions it is shown that there is always a region of validity for the weak line and nonoverlapping line approximations, although the shape of the limiting curve in the latter case may not be simple. There is also a region of validity for the strong line approximation in most practical situations. The only exception occurs when the number of spectral lines increases very rapidly as the intensity approaches zero. The principal effects of the low-intensity lines are to introduce points of inflection in the absorptance curves and to increase the intermediate range of values of pressure and absorber concentration over which neither the strong nor weak line approximation is valid. These effects are illustrated by examples. From a study of the shape of experimentally determined absorptance curves it is possible to deduce the number of low-intensity lines in the frequency interval of interest.

Emissivity of Carbon Dioxide at 43 μ

The emissivity of carbon dioxide has been measured for temperatures from 1500° to 3000°K over the wavelength range from 4.40 to 5.30 μ. Optical densities varied from 0.04 to 4.00 atm · cm; all measurements were made at a total gas pressure of 1 atm. The test gas was heated to the desired temperature by shock compression. The temperature and pressure of the test gas were determined by measuring initial concentrations and shock wave velocities. The emissivity of the hot gas was obtained in two ways: (1) the spectral radiance from the CO2 was compared to that from a globar, which was calibrated with a blackbody source and (2) the transmission through the hot gas was measured, and emissivity obtained from Kirchhoff’s law. The emissivity obtained with these two methods is in agreement within experimental error. The observed emissivities are reasonably well represented by values calculated with the weak line approximation, except that the measured values are consistently higher than the calculated emissivities at the longer wavelengths.

Test of Theoretical Absorption Band Model Approximations

Three approximations to theoretical absorption band models have recently been proposed by Plass. Experimental studies of total band absorptances for N2O, CO2, CH4, and CO show that only the “weak line approximation” is actually completely fulfilled; the “strong line approximation” is realized only for large values of total absorptance corresponding to near saturation. The validity of the “nonoverlapping approximation” could not be determined easily from the data except under conditions for which the strong line approximation would also be expected to apply. Considerable deviation from this approximation was found and has been attributed to the presence of lines that remain weak even for large absorber concentrations. Quantitative tests of this hypothesis were made on the 2224 cm−1 N2O band and the CO fundamental at 2143 cm−1. Absorptance near the R branch maximum of the CO fundamental represented the only close approach to complete fulfillment of the strong line approximation. Since the R branch of CO is nearly free of secondary lines, it is concluded that the strong line approximation can be realized experimentally only for “clean” spectral regions.

Infrared Emissivity of Carbon Dioxide (43-μ Band)*

Simplified equations are developed for computing the emissivity of the carbon dioxide 4.3-μ band. The harmonic oscillator model is assumed in computing band intensities, but anharmonicity is considered in computing the spectral distribution of emitted radiation. Calculated values of S¯/d and 2α012S¯12/d are presented for temperatures ranging from 300° to 3000°K, from which emissivities may be readily computed for a given pressure and optical path in both the weak-line and strong-line approximations. In the absence of significant Doppler broadening, both approximations provide upper limits to the emissivity. Emissivities are also computed for the case of a pure Doppler line shape for temperatures ranging from 300° to 1500°K. Doppler broadening is shown to be an important factor in setting a lower limit to the possible values of emissivity obtained at low pressures and long path lengths. Comparisons are made with published experimental data using the weak-line approximation; in several cases, the strong-line approximation as well as a mixed strong-line–weak-line approximation is also shown. The results of the present work are shown to be in good agreement with published experimental data and in definite disagreement with other theoretical calculations.

High-Temperature Spectral Absorption of the 4.3-Micron CO2 Band

The absorption coefficients of the ν3, 4.3-μ CO2 band were measured over a temperature range of about 1200° to 2400°K at various pathlengths. The hot CO2 sample was investigated in the exhaust jet of a small supersonic burner, while the jet composition was obtained using a radiation temperature measurement and a simple thermochemical calculation. The present experimental results are compared with existing experimental values and theoretical calculations. The results indicate that above 2000°K a weak line approximation can be used to describe the absorption or emission of the band. The apparent integrated band intensity of the CO2 fundamental as a function of temperature is also reported and discussed in relation to other measurements.

Useful Representations for Measurements of Spectral Band Absorption*

Three different methods are discussed for the representation of laboratory measurements of spectral band absorption. Each method is valid over a considerable range of pressure and amount of absorbing gas; together they accurately represent the absorption for virtually all values of these variables by means of reasonably simple mathematical expressions. Since these approximations have a valid theoretical basis, they can be used to obtain reliable values of the absorption for pressures and amounts of absorbing gas outside of the original experimental range. The three methods are (1) strong line approximation; (2) weak line approximation; (3) nonoverlapping line approximation. Methods (1) and (2) are valid regardless of whether or not the spectral lines overlap. Method (3) is valid regardless of whether or not the absorption is weak or strong at the line centers. The regions of validity of each of these approximations are given for the Elsasser and statistical models. An analysis is made by these methods of the published data on the absorption of the 9.6-μ band of ozone.