Gilbert N. Plass

Models for Spectral Band Absorption

An exhaustive, theoretical study is made of the various models that have been proposed to represent band absorption. These models are compared with each other; a derivation is given of the regions where they predict the same absorption. The regions of validity for various useful approximations to these models are also given. The statistical model is extended to include the random superposition of a finite number of Elsasser bands. Thus a continuous spectrum of absorption curves is obtained between the results for the Elsasser and the pure statistical models. The absorption predicted by the statistical model when there is a specific number of spectral lines in the frequency interval under consideration is compared with the limit as the number of lines approaches infinity. It is shown that the shape of the absorption curve obtained from the statistical model is independent of the distribution of line intensities in the band for most cases of interest. The absorption from the statistical model including the effects of overlapping is shown to depend only on an average equivalent width for a single line. This result is used to derive the band absorption for the statistical model with Lorentz, Doppler, and other line shapes.

Monte Carlo Calculations of the Polarization of Radiation in the Earth's Atmosphere-Ocean System

Abstract The degree and direction of polarization, the ellipticity, and the radiance of the radiation at various levels in the atmosphere and ocean are calculated by a Monte Carlo method which includes all orders of multiple scattering. Both Rayleigh scattering by the molecules and Mie scattering by the aerosols, as well as molecular and aerosol absorption, are included in the model of the atmosphere. Similarly, in the ocean, both Rayleigh scattering by the water molecules and Mie scattering by the hydrosols, as well as absorption by the water molecules and hydrosols, are considered. Separate single-scattering phase matrices are calculated from Mie theory for the aerosols and hydrosols. Both the reflected and refracted rays, as well as the rays that undergo total internal reflection, are followed from the ocean surface which is assumed to be flat. The degree and direction of polarization, the ellipticity, the radiance and the flux are given as functions of the turbidity of the ocean, the solar zenith angl...

Effect of Carbon Dioxide Variations on Climate

Variations in the amount of atmospheric carbon dioxide cause temperature changes sufficiently large to influence the climate. If the atmospheric carbon dioxide doubles, the surface temperature rises 3.6°C; if it is cut in half, the surface temperature falls 3.8°C. Some of the factors that can be explained by the carbon dioxide theory are: during a single glacial epoch, the climate continually oscillates between a glacial and an interglacial stage with a period of tens of thousands of years with no stable state possible, when the carbon dioxide amount is below a certain critical value; the increased precipitation at the beginning of a glacial period; the time lag between the period of mountain building and the onset of glaciation; periods of glaciation occur at the same time in both hemispheres; the general warming of the climate in the last fifty years. The various factors that enter into the carbon dioxide balance and the influence of the oceans on the atmospheric carbon dioxide amount are discussed in d...

Spectral Emissivity of Carbon Dioxide from 1800–2500 cm −1 *

The spectral emissivity of carbon dioxide from 1800–2500 cm−1 is calculated as a function of temperature, pressure, and amount of radiating gas. Careful consideration is given to the choice of a proper model to represent the emission when the overlapping of the spectral lines is taken into account. At elevated temperatures the random Elsasser model should be used since many vibrational bands overlap. The emissivities were calculated from the usual intensity and energy expressions on an electronic computer taking into account up to 890 000 spectral lines at 2400°K. The electric moment matrix element was evaluated by using the harmonic oscillator approximation to represent the vibrational states. The emission from C13O2 and from the transitions near 2100 and 1900 cm−1 was included in the calculation. The shift of the emission to lower frequencies as the temperature increases is quantitatively explained. The emission in this frequency region from any flame can readily be obtained from the results given here.

The Influence of Ozone and Aerosols on the Brightness and Color of the Twilight Sky

Abstract The radiance and color of the twilight sky are calculated for single scattered radiation with the use of spherical symmetric models of the earths atmosphere. Spherical geometry is used throughout the calculations with no plane parallel approximations. Refraction effects are taken into account through fine subdivision of the atmosphere into spherical shells of fixed index of refraction. Snells law of refraction is used to calculate a new direction of travel each time that a photon traverses the interface between layers. Five different models of the atmosphere were used: a pure molecular scattering atmosphere; molecular atmosphere plus ozone absorption; and three models with aerosol concentrations of one, three and ten times normal together with molecular scattering and ozone absorption. The results of the calculations are shown for various observation positions and local viewing angles in the solar plane for wavelengths in the range of 0.40 to 0.75 µm.

A Method for the Determination of Atmospheric Transmission Functions from Laboratory Absorption Measurements

It is shown that the average transmission function in the atmosphere can be calculated from the fractional absorption as measured in the laboratory for frequency intervals involving either large or small optical thicknesses. These relations are very general and are independent of any special assumptions as to the shape of the pressure broadened lines, of the variation of spacing, intensity, or half-width of the lines, of the extent of the overlapping of the lines, or of the distribution of the radiating gases in the atmosphere. The optical thickness is considered large or small for a particular interval depending on the intensities and half-widths of the lines contributing to the absorption. The solutions of radiation transfer problems can be obtained from the average transmission function calculated by the above method. A number of examples are given of the application of these results to particular problems. It is shown that in general for pressure broadened lines the fractional absorption is a function only of pressure times optical thickness for large optical thicknesses, and of optical thickness times a function of the pressure for small optical thicknesses.

Carbon Dioxide and the Climate

Scientists have long been fascinated with the problem of explaining variations in the climate. For at least nine-tenths of the time since the beginning of recorded geological history, the average temperature of the Earth has been higher than it is today. Between these warm epochs there have been severe periods of glaciation which have lasted a few million years and which have occurred at intervals of roughly 250,000,000 years. Of more immediate interest to us is the general warming of the climate that has taken place in the last sixty years.

Infrared Radiation in the Atmosphere

The infrared radiation flux in the atmosphere is determined by the distribution of three gases—water vapor, carbon dioxide, and ozone. The general equations for the transfer of radiation in a gas are derived. These equations are solved for four simple model atmospheres that illustrate various features of the more complicated results for the earths atmosphere. The atmospheric infrared radiation flux can be calculated from laboratory absorption measurements. The results of such calculations are discussed in detail for the frequency ranges that are influenced by the carbon dioxide and ozone bands.

PARALLEL-BEAM AND DIFFUSE RADIATION IN THE ATMOSPHERE

Abstract The calculation of atmospheric transmission functions for infrared spectral lines having a pressure-broadened line shape is extended to include the case where the fractional concentration of the radiating gas varies with height. The radiation exchange between two atmospheric layers can now be calculated under a wide variety of conditions. A recent formulation of the radiation problem by Plass is used to show that the conversion factor for changing parallel-beam radiation to diffuse radiation varies between one and two, depending on the particular frequency interval and optical thickness considered. The value of the conversion factor is two for a frequency interval involving only very weak lines; the value is 1.78 when the centers of the lines are black, if the lines do not overlap appreciably; the value approaches unity as the lines overlap more and more. The use of an average value for the conversion factor can lead to errors of 17 per cent in the calculation of radiation exchange. Since the ang...

A METHOD FOR THE INTEGRATION OF THE RADIATIVE-TRANSFER EQUATION

Abstract The equations for radiative transfer are integrated exactly for a band of spectral lines which do not overlap and for an Elsasser band. The result of the two-fold integration of the absorption over frequency and over the atmospheric path can be expressed in terms of the Legendre functions. Here it is assumed that the mixing ratio is constant, that the line intensity is independent of temperature, and that the Lorentz line shape is valid. Asymptotic forms of the Legendre functions are used to obtain the solutions to the following problems from these exact results. The regions of validity of the single-line and strong-line approximations are precisely stated. It is shown that the strongest line in the band absorbs more radiation than any other line in an atmospheric layer, when the overlap of the lines can be neglected. For an Elsasser band, an expression is derived for the line strength that gives the maximum absorption in an atmospheric layer for radiation emitted either by a black body at anothe...