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Abstract
The doubling method is described for multiple scattering of light in plane-parallel atmospheres. The polarization of the radiation and the azimuth dependence are correctly accounted for. The method is practical for application to realistic simulations of clear, hazy and cloudy planetary atmospheres.
Abstract
The doubling method is described for multiple scattering of light in plane-parallel atmospheres. The polarization of the radiation and the azimuth dependence are correctly accounted for. The method is practical for application to realistic simulations of clear, hazy and cloudy planetary atmospheres.
Abstract
The intensity and polarization of sunlight reflected by terrestrial water clouds are computed with the doubling method. The calculations illustrate that this method can be effectively used in problem involving strongly anisotropic phase matrices. The method can therefore be used to derive information about planetary clouds, including those of the earth, from polarimetric observations.
The results of the computations indicate that the polarization is more sensitive than the intensity to cloud microstructure, such as the particle size and shape. Multiple scattering does not wash out features in the polarization as effectively as it does in the intensity, because the polarization arises primarily from photons scattered once or a small number of times. Hence polarization measurements, particularly in the near infrared, are potentially a valuable tool for cloud identification and for studies of the microphysics of clouds.
The computations are made primarily at four wavelengths in the near infrared, from 1.2 to 3.4μ. The results for λ=1.2μ are also applicable to scattering at visual and ultraviolet wavelengths. The other wavelengths are selected to illustrate the basic scattering characteristics in the near infrared for reflection of sunlight from water clouds.
It is shown that the intensity computed with the exact theory including polarization differs by ≲1.0%1.0 from the intensity computed in the common scalar approximation in which the polarization is neglected. Therefore, when only the intensity is required, and not the polarization, it is possible in most cases to neglect polarization entirely.
An approximation obtained by setting the phase matrix elements P34(α) and P43 (α) equal to zero is proposed and tested. It is found that this introduces errors less than one part in 106 for the intensity and errors ≲0.0002 in the degree of polarization. This means that in computing the polarization properties for multiple scattering by spherical particles it is usually adequate to work with 3 by 3 matrices.
An examination is made of the accuracy of the polarization in the approximation in which it is assumed that multiply scattered photons are unpolarized. A modified version of this, which, in addition, takes advantage of the fact that diffracted fight is nearly unpolarized, is also tested. The modified approximation is found to yield an improved accuracy in most cases.
Another approximation,which can be termed a renormalization method, is described and tested. The method consists of modifying the phase matrix for single scattering so that the integrations over zenith angle can he performed with a small number of points. The, order of the approximation (the number of zenith angles in the integrations) can easily be varied and accuracies sufficient for practical applications can be obtained at low orders of approximation. The method is therefore useful for small computers.
Abstract
The intensity and polarization of sunlight reflected by terrestrial water clouds are computed with the doubling method. The calculations illustrate that this method can be effectively used in problem involving strongly anisotropic phase matrices. The method can therefore be used to derive information about planetary clouds, including those of the earth, from polarimetric observations.
The results of the computations indicate that the polarization is more sensitive than the intensity to cloud microstructure, such as the particle size and shape. Multiple scattering does not wash out features in the polarization as effectively as it does in the intensity, because the polarization arises primarily from photons scattered once or a small number of times. Hence polarization measurements, particularly in the near infrared, are potentially a valuable tool for cloud identification and for studies of the microphysics of clouds.
The computations are made primarily at four wavelengths in the near infrared, from 1.2 to 3.4μ. The results for λ=1.2μ are also applicable to scattering at visual and ultraviolet wavelengths. The other wavelengths are selected to illustrate the basic scattering characteristics in the near infrared for reflection of sunlight from water clouds.
It is shown that the intensity computed with the exact theory including polarization differs by ≲1.0%1.0 from the intensity computed in the common scalar approximation in which the polarization is neglected. Therefore, when only the intensity is required, and not the polarization, it is possible in most cases to neglect polarization entirely.
An approximation obtained by setting the phase matrix elements P34(α) and P43 (α) equal to zero is proposed and tested. It is found that this introduces errors less than one part in 106 for the intensity and errors ≲0.0002 in the degree of polarization. This means that in computing the polarization properties for multiple scattering by spherical particles it is usually adequate to work with 3 by 3 matrices.
An examination is made of the accuracy of the polarization in the approximation in which it is assumed that multiply scattered photons are unpolarized. A modified version of this, which, in addition, takes advantage of the fact that diffracted fight is nearly unpolarized, is also tested. The modified approximation is found to yield an improved accuracy in most cases.
Another approximation,which can be termed a renormalization method, is described and tested. The method consists of modifying the phase matrix for single scattering so that the integrations over zenith angle can he performed with a small number of points. The, order of the approximation (the number of zenith angles in the integrations) can easily be varied and accuracies sufficient for practical applications can be obtained at low orders of approximation. The method is therefore useful for small computers.
Abstract
Measurements of circular polarization of visible light from planets have recently been reported. It is pointed out here that the values measured for the circular polarization for Jupiter and Venus are of the magnitude expected for sunlight reflected by a cloudy planetary atmosphere. The variations of the sense of the polarization with phase angle and with location on the planetary disk are also consistent with expectations for reflection by clouds.
Abstract
Measurements of circular polarization of visible light from planets have recently been reported. It is pointed out here that the values measured for the circular polarization for Jupiter and Venus are of the magnitude expected for sunlight reflected by a cloudy planetary atmosphere. The variations of the sense of the polarization with phase angle and with location on the planetary disk are also consistent with expectations for reflection by clouds.
Abstract
Calculations of the reflectivity of water clouds (liquid and ice particles) are compared to observations of terrestrial clouds in the near infrared. The presentation is divided into four parts which may be consulted individually. Section 3 presents new Mic scattering calculations of general interest, Sections 4–7 compare multiple-scattering results to cloud observations, Section 8 suggests a revision in the optical constants of ice for λ ≈ 3 μ, and the Appendix details several methods which substantially reduce the work load in multiple-scattering computations.
Our results indicate that it is possible to use the spectral variation of the reflectivity to derive the size of the cloud particles and their phase (liquid or solid) as well as the total optical depth of the clouds. Typical results show dense cirrus clouds to have an optical depth ≥10 and to be composed of ice particles of mean radius 15–20 μ the cumulus clouds which were analyzed showed a more variable, but usually smaller, particle size.
In spectral regions where the single-scattering albedo is high it is found that most of the gas absorption takes place within the clouds rather than above them.
Abstract
Calculations of the reflectivity of water clouds (liquid and ice particles) are compared to observations of terrestrial clouds in the near infrared. The presentation is divided into four parts which may be consulted individually. Section 3 presents new Mic scattering calculations of general interest, Sections 4–7 compare multiple-scattering results to cloud observations, Section 8 suggests a revision in the optical constants of ice for λ ≈ 3 μ, and the Appendix details several methods which substantially reduce the work load in multiple-scattering computations.
Our results indicate that it is possible to use the spectral variation of the reflectivity to derive the size of the cloud particles and their phase (liquid or solid) as well as the total optical depth of the clouds. Typical results show dense cirrus clouds to have an optical depth ≥10 and to be composed of ice particles of mean radius 15–20 μ the cumulus clouds which were analyzed showed a more variable, but usually smaller, particle size.
In spectral regions where the single-scattering albedo is high it is found that most of the gas absorption takes place within the clouds rather than above them.
Abstract
The intensity and polarization for single scattering by large spherical particles are computed using both the exact Mie theory and the approximation of ray optics. It is found that the ray-tracing method can yield accurate results for particle size parameters in the range of interest for some meteorological applications, where the size parameter is the ratio of the particle circumference to the wavelength of the incident light. Since this method is practical for application to nonspherical particles, it should be of use in studies of cloud microstructure. The ray-optics method is also useful in the case of spherical particles because it provides a physical explanation for features which occur in the exact theory.
The ray-optics calculations include Fraunhofer diffraction as well as geometrical reflection and refraction; rays undergoing one or two internal reflections, which give rise to the observable rainbows, are also included. Calculations are made for non-absorbing and absorbing spheres for several refractive indices in the range 1.1 ≤ nr ≤ 2.0. Comparisons between the ray-optics approximation and the exact Mie theory are made for nr = 1.33 and 1.50. It is found that the two methods are in close agreement, if the particle size parameter is ≳400. It is also shown that, to a good approximation, the ray-optics solution may often be used to obtain the entire phase matrix for single scattering.
Abstract
The intensity and polarization for single scattering by large spherical particles are computed using both the exact Mie theory and the approximation of ray optics. It is found that the ray-tracing method can yield accurate results for particle size parameters in the range of interest for some meteorological applications, where the size parameter is the ratio of the particle circumference to the wavelength of the incident light. Since this method is practical for application to nonspherical particles, it should be of use in studies of cloud microstructure. The ray-optics method is also useful in the case of spherical particles because it provides a physical explanation for features which occur in the exact theory.
The ray-optics calculations include Fraunhofer diffraction as well as geometrical reflection and refraction; rays undergoing one or two internal reflections, which give rise to the observable rainbows, are also included. Calculations are made for non-absorbing and absorbing spheres for several refractive indices in the range 1.1 ≤ nr ≤ 2.0. Comparisons between the ray-optics approximation and the exact Mie theory are made for nr = 1.33 and 1.50. It is found that the two methods are in close agreement, if the particle size parameter is ≳400. It is also shown that, to a good approximation, the ray-optics solution may often be used to obtain the entire phase matrix for single scattering.
Abstract
The near infrared reflectivity of ice clouds is computed and compared to observations of Venus. The difficulty in making an exact correction for CO2 absorption precludes the possibility of either establishing or absolutely ruling out ice particles as the primary cloud constituent; however, it is possible to conclude that the clouds are not optically thick and composed of large ice crystals. There is still disagreement as to the quantitative significance of the infrared spectra, but, if it is assumed that a 20% depression may exist in the continuum near 2.0 μ then optically thin clouds (τ≲5–10) of ice particles with radii ≲4 μ are compatible with the observations. It is shown that there is a small amount of positive evidence for such clouds.
Abstract
The near infrared reflectivity of ice clouds is computed and compared to observations of Venus. The difficulty in making an exact correction for CO2 absorption precludes the possibility of either establishing or absolutely ruling out ice particles as the primary cloud constituent; however, it is possible to conclude that the clouds are not optically thick and composed of large ice crystals. There is still disagreement as to the quantitative significance of the infrared spectra, but, if it is assumed that a 20% depression may exist in the continuum near 2.0 μ then optically thin clouds (τ≲5–10) of ice particles with radii ≲4 μ are compatible with the observations. It is shown that there is a small amount of positive evidence for such clouds.
Abstract
The polarization of reflected sunlight is computed for model atmospheres of Venus as a function of location on the apparent planetary disk. The calculations are for both homogeneous (uniformly mixed) and layered models, as required to investigate the vertical distribution of particles. The results are compared with available observations, which are few in number and of poor spatial resolution. The results for the homogeneous and layered models are also integrated over the planet and compared with whole-disk observations.
It is shown that the Rayleigh scattering observed in the polarization of Venus originates primarily from within the visible clouds, rather than from above the clouds. The photon mean free path is ∼5 km at the 50 mb pressure level, which is well within the visible clouds. Thus the visible “clouds” are actually a very diffuse hazy region. This visible cloud layer extends at least up to the level where the pressure is ∼10 mb.
The results indicate that the atmosphere behaves, for this type of observation, more nearly as the so-called “homogeneous model” than as the “reflecting layer model.” However, there is some indication in the data that the turbidity (ratio of cloud particle opacity to Rayleigh opacity) increases with depth into the atmosphere. This conclusion receives stronger support from a comparison of particle number densities obtained from the polarization data (∼30 particles cm−3 at the 50 mb level) with the number densities obtained from other observations which refer on the average to higher and lower levels in the atmosphere.
Abstract
The polarization of reflected sunlight is computed for model atmospheres of Venus as a function of location on the apparent planetary disk. The calculations are for both homogeneous (uniformly mixed) and layered models, as required to investigate the vertical distribution of particles. The results are compared with available observations, which are few in number and of poor spatial resolution. The results for the homogeneous and layered models are also integrated over the planet and compared with whole-disk observations.
It is shown that the Rayleigh scattering observed in the polarization of Venus originates primarily from within the visible clouds, rather than from above the clouds. The photon mean free path is ∼5 km at the 50 mb pressure level, which is well within the visible clouds. Thus the visible “clouds” are actually a very diffuse hazy region. This visible cloud layer extends at least up to the level where the pressure is ∼10 mb.
The results indicate that the atmosphere behaves, for this type of observation, more nearly as the so-called “homogeneous model” than as the “reflecting layer model.” However, there is some indication in the data that the turbidity (ratio of cloud particle opacity to Rayleigh opacity) increases with depth into the atmosphere. This conclusion receives stronger support from a comparison of particle number densities obtained from the polarization data (∼30 particles cm−3 at the 50 mb level) with the number densities obtained from other observations which refer on the average to higher and lower levels in the atmosphere.
Abstract
The linear polarization of sunlight reflected by Venus is analyzed by comparing observations with extensive multiple scattering computations. The analysis establishes that Venus is veiled by a cloud or haze layer of spherical particles. The refractive index of the particles is 1.44±0.015 at λ=0.55 μm with a normal dispersion, the refractive index decreasing from 1.46±0.015 at λ=0.365 μm to 1.43±0.015 at λ=0.99 μm. The cloud particles have a narrow size distribution with a mean radius of ∼1 μm; specifically, the effective radius of the size distribution is 1.05±0.10 μm and the effective variance is 0.07±0.02. The particles exist at a high level in the atmosphere, with the optical thickness unity occurring where the pressure is about 50 mb.
The particle properties deduced from the polarization eliminate all but one of the cloud compositions which have been proposed for Venue. A concentrated solution of sulfuric acid (H2SO4-H2O) provides good agreement with the polarization data.
Abstract
The linear polarization of sunlight reflected by Venus is analyzed by comparing observations with extensive multiple scattering computations. The analysis establishes that Venus is veiled by a cloud or haze layer of spherical particles. The refractive index of the particles is 1.44±0.015 at λ=0.55 μm with a normal dispersion, the refractive index decreasing from 1.46±0.015 at λ=0.365 μm to 1.43±0.015 at λ=0.99 μm. The cloud particles have a narrow size distribution with a mean radius of ∼1 μm; specifically, the effective radius of the size distribution is 1.05±0.10 μm and the effective variance is 0.07±0.02. The particles exist at a high level in the atmosphere, with the optical thickness unity occurring where the pressure is about 50 mb.
The particle properties deduced from the polarization eliminate all but one of the cloud compositions which have been proposed for Venue. A concentrated solution of sulfuric acid (H2SO4-H2O) provides good agreement with the polarization data.
Abstract
The spectrum of sunlight reflected by Jupiter is analyzed by comparing observations of Woodman et al. (1979) with multiple-scattering computations. The analysis yields information on the vertical cloud structure at several latitudes and on the abundance of CH4 and NH3 in the atmosphere of Jupiter.
The abundance of CH4, is (1.8±0.4) × 10−3 for [CH4]/[H2], which corresponds to a carbon abundance 2±0.4 times that in the atmosphere of the sun for currently accepted values of the solar composition. The quoted limits for the abundance include the effects of uncertainties in the cloud and haze structure. The abundance of NH3 is (2.8±1.0) × 10−4 for [NH3]/[H2] in the region between 1 bar and 3–5 bars, corresponding to a nitrogen abundance 1.5±0.5 times that in the atmosphere of the sun. Thus nitrogen is at least as abundant on Jupiter as on the sun, and it may exceed the abundance in the solar atmosphere by a factor as great as that for carbon. These abundances suggest that all ices (and rocks) are overabundant on Jupiter by a factor approximately 2 or more, providing an important constraint on models for the formation of Jupiter from the primitive solar nebula.
Clouds of mean visible optical depth approximately 10 exist in both belts and zones at a pressure level of several hundred millibars. The pressure level of the clouds, the gaseous NH3 abundance, the mean temperature profile and the Clausius-Clapeyron relation together suggest that these clouds are predominantly ammonia crystals and place the cloud bottom at 600–700 mb. Beneath this “ammonia” cloud region is an optically thick cloud layer at 3–5 bars; this cloud may be composed of H20. The region between these two cloud layers is relatively transparent. Thus NH4SH clouds, assumed to be optically thick in all previous multi-layered cloud models for Jupiter, are optically thin or broken, if they exist.
A diffuse distribution of aerosols (“haze”) exists between approximately 150 and 400–500 mb, i.e., above the main ammonia cloud region. These aerosols are at least 1 μm in diameter. The ultraviolet absorption occurs in both the haze region and the ammonia cloud region. The decreasing absorption with increasing wavelength is due to an increasing single scattering albedo rather than a decreasing aerosol optical depth as in the “Axel dust” model. Thus the spectral variation of albedo reflects a changing bulk absorption coefficient of the material composing the aerosols and is diagnostic of the aerosol composition.
Ratio spectra of the North Tropical Zone (NTrZ) and North Equatorial Belt (NEB) imply that the scatterers in the 150–500 mb haze region (which may include ammonia “cirrus") reach to higher altitudes over the NTrZ than over the NEB. But the tops of the more optically dense main “cloud” layer appear to reach to higher altitudes over the NEB, implying that the usual picture of the zones as regions of rising motions and enhanced ammonia cloudiness is too simple. The total optical thickness of aerosols in the haze and cloud regions is greater in the zone than in the belt, but there is more ultraviolet-absorbing aerosol in the belt. Ten parameters are needed to describe the vertical distribution of aerosol properties to satisfy only the spectra of Woodman et al., suggesting that the atmospheric dynamics and cloud physics an Jupiter are extremely complex.
Abstract
The spectrum of sunlight reflected by Jupiter is analyzed by comparing observations of Woodman et al. (1979) with multiple-scattering computations. The analysis yields information on the vertical cloud structure at several latitudes and on the abundance of CH4 and NH3 in the atmosphere of Jupiter.
The abundance of CH4, is (1.8±0.4) × 10−3 for [CH4]/[H2], which corresponds to a carbon abundance 2±0.4 times that in the atmosphere of the sun for currently accepted values of the solar composition. The quoted limits for the abundance include the effects of uncertainties in the cloud and haze structure. The abundance of NH3 is (2.8±1.0) × 10−4 for [NH3]/[H2] in the region between 1 bar and 3–5 bars, corresponding to a nitrogen abundance 1.5±0.5 times that in the atmosphere of the sun. Thus nitrogen is at least as abundant on Jupiter as on the sun, and it may exceed the abundance in the solar atmosphere by a factor as great as that for carbon. These abundances suggest that all ices (and rocks) are overabundant on Jupiter by a factor approximately 2 or more, providing an important constraint on models for the formation of Jupiter from the primitive solar nebula.
Clouds of mean visible optical depth approximately 10 exist in both belts and zones at a pressure level of several hundred millibars. The pressure level of the clouds, the gaseous NH3 abundance, the mean temperature profile and the Clausius-Clapeyron relation together suggest that these clouds are predominantly ammonia crystals and place the cloud bottom at 600–700 mb. Beneath this “ammonia” cloud region is an optically thick cloud layer at 3–5 bars; this cloud may be composed of H20. The region between these two cloud layers is relatively transparent. Thus NH4SH clouds, assumed to be optically thick in all previous multi-layered cloud models for Jupiter, are optically thin or broken, if they exist.
A diffuse distribution of aerosols (“haze”) exists between approximately 150 and 400–500 mb, i.e., above the main ammonia cloud region. These aerosols are at least 1 μm in diameter. The ultraviolet absorption occurs in both the haze region and the ammonia cloud region. The decreasing absorption with increasing wavelength is due to an increasing single scattering albedo rather than a decreasing aerosol optical depth as in the “Axel dust” model. Thus the spectral variation of albedo reflects a changing bulk absorption coefficient of the material composing the aerosols and is diagnostic of the aerosol composition.
Ratio spectra of the North Tropical Zone (NTrZ) and North Equatorial Belt (NEB) imply that the scatterers in the 150–500 mb haze region (which may include ammonia “cirrus") reach to higher altitudes over the NTrZ than over the NEB. But the tops of the more optically dense main “cloud” layer appear to reach to higher altitudes over the NEB, implying that the usual picture of the zones as regions of rising motions and enhanced ammonia cloudiness is too simple. The total optical thickness of aerosols in the haze and cloud regions is greater in the zone than in the belt, but there is more ultraviolet-absorbing aerosol in the belt. Ten parameters are needed to describe the vertical distribution of aerosol properties to satisfy only the spectra of Woodman et al., suggesting that the atmospheric dynamics and cloud physics an Jupiter are extremely complex.
Abstract
A stochastic model for hourly temperatures for Big Spring, Tex., has been developed. The governing parameters were deduced from an 11-year developmental sample, and give hourly temperatures as a function of harmonics representing annual and diurnal variations, and a first-order Markov chain process. The latter incorporates adjustments for the seasonal variation of the serial (hour-to-hour) correlation coefficient, and for the seasonal and diurnal variations of the variability and non-normality of frequency distributions of hourly temperatures. Each of the characteristics is given explicitly as a function of hour of the year.
Two 10-year samples were generated and compared to the developmental sample. Criteria were established to determine how well the model duplicates nature. The variability of mean monthly temperature and the frequency of occurrence of low diurnal ranges are underestimated. However, the model gives good estimates of the duration of temperatures below 32°F, and above 65° and 90°F, and of the frequency distribution of monthly 3, 6, 12, 24, 72 and 144 h maximum and minimum temperatures.
The general applicability of the model and its utility are discussed. The model could be used to determine the effects of climatic trends, e.g., a gradual cooling, on the average length of the growing season, the mean number of heating/cooling degree days, and other temperature-related parameters.
Abstract
A stochastic model for hourly temperatures for Big Spring, Tex., has been developed. The governing parameters were deduced from an 11-year developmental sample, and give hourly temperatures as a function of harmonics representing annual and diurnal variations, and a first-order Markov chain process. The latter incorporates adjustments for the seasonal variation of the serial (hour-to-hour) correlation coefficient, and for the seasonal and diurnal variations of the variability and non-normality of frequency distributions of hourly temperatures. Each of the characteristics is given explicitly as a function of hour of the year.
Two 10-year samples were generated and compared to the developmental sample. Criteria were established to determine how well the model duplicates nature. The variability of mean monthly temperature and the frequency of occurrence of low diurnal ranges are underestimated. However, the model gives good estimates of the duration of temperatures below 32°F, and above 65° and 90°F, and of the frequency distribution of monthly 3, 6, 12, 24, 72 and 144 h maximum and minimum temperatures.
The general applicability of the model and its utility are discussed. The model could be used to determine the effects of climatic trends, e.g., a gradual cooling, on the average length of the growing season, the mean number of heating/cooling degree days, and other temperature-related parameters.
