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Abstract
The Gaussian integration of moments is systematically discussed. It is shown that the well-known diffusivity-factor approximation is equivalent to a one-node Gaussian quadrature. The limit as the moment power approaches infinity in a one-node Gaussian quadrature produces a diffusivity factor of e 1/2 = 1.648 721 3, which is very close to the value of 1.66 suggested by Elsasser.
The errors due to the diffusivity-factor approximation are analyzed in a one-dimensional radiative transfer model. Generally, the results cannot be improved by using other one-node Gaussian quadrature schemes with different moments. More accurate results can be obtained by using higher-node Gaussian quadratures. It is found that the limit as the moment power approaches infinity always produces the best results. The computational advantage of the diffusivity-factor approximation is kept in the higher-node Gaussian quadratures. It is, therefore, feasible to implement the higher-node Gaussian quadratures in climate models.
Abstract
The Gaussian integration of moments is systematically discussed. It is shown that the well-known diffusivity-factor approximation is equivalent to a one-node Gaussian quadrature. The limit as the moment power approaches infinity in a one-node Gaussian quadrature produces a diffusivity factor of e 1/2 = 1.648 721 3, which is very close to the value of 1.66 suggested by Elsasser.
The errors due to the diffusivity-factor approximation are analyzed in a one-dimensional radiative transfer model. Generally, the results cannot be improved by using other one-node Gaussian quadrature schemes with different moments. More accurate results can be obtained by using higher-node Gaussian quadratures. It is found that the limit as the moment power approaches infinity always produces the best results. The computational advantage of the diffusivity-factor approximation is kept in the higher-node Gaussian quadratures. It is, therefore, feasible to implement the higher-node Gaussian quadratures in climate models.
Abstract
A hypothesis that the fractional scattering into the forward peak is related to solar zenith angle and single scattering albedo is proposed. Calculations show that this assumption can increase the accuracy of the δ-Eddington approximation. For the scattering conservative case this method can improve the results in the region of thin optical depth. For the scattering nonconservative case this method can reduce the errors for reflection and absorption in the region of small solar zenith angle, where the incoming solar energy is most significant.
Abstract
A hypothesis that the fractional scattering into the forward peak is related to solar zenith angle and single scattering albedo is proposed. Calculations show that this assumption can increase the accuracy of the δ-Eddington approximation. For the scattering conservative case this method can improve the results in the region of thin optical depth. For the scattering nonconservative case this method can reduce the errors for reflection and absorption in the region of small solar zenith angle, where the incoming solar energy is most significant.
Abstract
Various aspects of infrared radiative transfer through clouds are investigated. First, three solutions to the IR radiative transfer equation are presented and assessed, each corresponding to a different approximation for the Planck function. It is shown that the differences in results between solutions with linear and exponential dependence of the Planck source function are small for typical vertical resolutions in climate models. Second, a new perturbation-based approach to solving the IR radiative transfer equation with the inclusion of cloud scattering is presented. This scheme follows the standard perturbation method, and allows one to identify the zeroth-order equation with the absorption approximation and the first-order equation as including IR scattering effects. This enables one solution to accurately treat cloudy layers in which cloud scattering is included, and allows for an improved and consistent treatment of absorbing aerosol layers in the absence of cloud by using the zeroth-order equation. This new scheme is more simple and efficient compared to previous perturbation method work for treating infrared absorption and scattering. Last, a general method is devised for calculating the random, maximum, and slantwise overlap of cloud layers, which conveniently integrates into the two-stream radiative transfer solution in this work. For several random and maximum (or slantwise) overlap cloud cases with a wide variation of cloud fractions, the error in the cooling rate is generally less than 1 K day−1 and the error in the radiative flux is generally less than 3 W m−2.
Abstract
Various aspects of infrared radiative transfer through clouds are investigated. First, three solutions to the IR radiative transfer equation are presented and assessed, each corresponding to a different approximation for the Planck function. It is shown that the differences in results between solutions with linear and exponential dependence of the Planck source function are small for typical vertical resolutions in climate models. Second, a new perturbation-based approach to solving the IR radiative transfer equation with the inclusion of cloud scattering is presented. This scheme follows the standard perturbation method, and allows one to identify the zeroth-order equation with the absorption approximation and the first-order equation as including IR scattering effects. This enables one solution to accurately treat cloudy layers in which cloud scattering is included, and allows for an improved and consistent treatment of absorbing aerosol layers in the absence of cloud by using the zeroth-order equation. This new scheme is more simple and efficient compared to previous perturbation method work for treating infrared absorption and scattering. Last, a general method is devised for calculating the random, maximum, and slantwise overlap of cloud layers, which conveniently integrates into the two-stream radiative transfer solution in this work. For several random and maximum (or slantwise) overlap cloud cases with a wide variation of cloud fractions, the error in the cooling rate is generally less than 1 K day−1 and the error in the radiative flux is generally less than 3 W m−2.
Abstract
The global horizontal structure of atmospheric entropy has been investigated. In energy balance models, the horizontal distribution of the atmospheric internal entropy production rate has been obtained. Based on the entropy balance relation, this work is of rigorous thermodynamics foundation. In the models, the radiation entropy has been evaluated through the effective temperature method. It is found that with the increase of latitude, the internal entropy production decreases and the entropy production corresponding to the thermal conduction increases. In addition, the atmospheric entropy structure problem under ice age conditions is discussed.
Abstract
The global horizontal structure of atmospheric entropy has been investigated. In energy balance models, the horizontal distribution of the atmospheric internal entropy production rate has been obtained. Based on the entropy balance relation, this work is of rigorous thermodynamics foundation. In the models, the radiation entropy has been evaluated through the effective temperature method. It is found that with the increase of latitude, the internal entropy production decreases and the entropy production corresponding to the thermal conduction increases. In addition, the atmospheric entropy structure problem under ice age conditions is discussed.
Abstract
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Abstract
No abstract available.
Abstract
A scheme that can handle cloud infrared scattering based on the absorption approximation is developed. In a two-stream mode, the new scheme produces more accurate results than those from the modified two-stream discrete ordinate method. For low and middle clouds, the two-stream version of the scheme produces a flux error less than 1 W m−2 and a heating rate error less than 0.5 K day−1. With high clouds, the errors in calculated fluxes and heating rates are less than 1.4 W m−2 and 1.5 K day−1, respectively. The four-stream version of the proposed scheme is slightly inferior to the four-stream discrete ordinate method. However, as opposed to the discrete ordinate technique, this scheme treats cloud-free layers the same as the absorption approximation. Therefore, numerically, it is much more efficient. Considering the radiative transfer module only, in a two-stream mode, the new scheme, which considers multiple scattering, uses only about 50% more CPU time than the absorption approximation method for a 100-layer column atmosphere with 20 cloudy layers.
Abstract
A scheme that can handle cloud infrared scattering based on the absorption approximation is developed. In a two-stream mode, the new scheme produces more accurate results than those from the modified two-stream discrete ordinate method. For low and middle clouds, the two-stream version of the scheme produces a flux error less than 1 W m−2 and a heating rate error less than 0.5 K day−1. With high clouds, the errors in calculated fluxes and heating rates are less than 1.4 W m−2 and 1.5 K day−1, respectively. The four-stream version of the proposed scheme is slightly inferior to the four-stream discrete ordinate method. However, as opposed to the discrete ordinate technique, this scheme treats cloud-free layers the same as the absorption approximation. Therefore, numerically, it is much more efficient. Considering the radiative transfer module only, in a two-stream mode, the new scheme, which considers multiple scattering, uses only about 50% more CPU time than the absorption approximation method for a 100-layer column atmosphere with 20 cloudy layers.
Abstract
This paper presents a four-stream extension of the δ-Eddington approximation by considering the higher-order spherical harmonic expansion in radiative intensity. By using the orthogonality relation of the spherical harmonic functions, the derivation of the solution is fairly straightforward. Calculations show that the δ-four-stream spherical harmonic expansion approximation can reduce the errors in reflection, transmission, and absorption substantially in comparison with the δ-Eddington approximation. For the conservative scattering case, the error of the new model is generally less than 1% for optical thickness greater than unity except for gracing incident solar beam. For nonconservative scattering cases (single scattering albedo ω=0.9), the error is less than 5% for optical thickness greater than unity, in contrast to errors of up to 20% or more under the δ-Eddington approximation. This model can also predict the azimuthally averaged intensity to a good degree of accuracy. The computational time for this model is not as intensive as for the rigorous numerical methods, owing to the analytical form of the derived solution.
Abstract
This paper presents a four-stream extension of the δ-Eddington approximation by considering the higher-order spherical harmonic expansion in radiative intensity. By using the orthogonality relation of the spherical harmonic functions, the derivation of the solution is fairly straightforward. Calculations show that the δ-four-stream spherical harmonic expansion approximation can reduce the errors in reflection, transmission, and absorption substantially in comparison with the δ-Eddington approximation. For the conservative scattering case, the error of the new model is generally less than 1% for optical thickness greater than unity except for gracing incident solar beam. For nonconservative scattering cases (single scattering albedo ω=0.9), the error is less than 5% for optical thickness greater than unity, in contrast to errors of up to 20% or more under the δ-Eddington approximation. This model can also predict the azimuthally averaged intensity to a good degree of accuracy. The computational time for this model is not as intensive as for the rigorous numerical methods, owing to the analytical form of the derived solution.
Abstract
Parameterizations of absorptance depth for ammonium sulfate [(NH4)2SO4], ammonium bisulfate (NH4HSO4), and sulfuric acid (H2SO4) in the infrared are provided for an eight-band model (covering 340–2500 cm−1) and for 32 individual wavenumbers in order to generate other band schemes. The parameterization is simple in form and in its dependence on relative humidity.
It is found that the aerosol surface infrared forcing can cancel about 12%–24% aerosol surface solar forcing in a clear sky condition. Also the existence of clouds could enhance the ratio of aerosol surface infrared forcing to the aerosol surface solar forcing. In contrast to the solar case, a small mode size distribution does not always produce a larger aerosol surface forcing. Also it is found that the aerosol surface forcing is dependent on the aerosol location. Very simple analysis is presented to help understand the related physics on sulfate aerosol infrared radiative forcing.
Abstract
Parameterizations of absorptance depth for ammonium sulfate [(NH4)2SO4], ammonium bisulfate (NH4HSO4), and sulfuric acid (H2SO4) in the infrared are provided for an eight-band model (covering 340–2500 cm−1) and for 32 individual wavenumbers in order to generate other band schemes. The parameterization is simple in form and in its dependence on relative humidity.
It is found that the aerosol surface infrared forcing can cancel about 12%–24% aerosol surface solar forcing in a clear sky condition. Also the existence of clouds could enhance the ratio of aerosol surface infrared forcing to the aerosol surface solar forcing. In contrast to the solar case, a small mode size distribution does not always produce a larger aerosol surface forcing. Also it is found that the aerosol surface forcing is dependent on the aerosol location. Very simple analysis is presented to help understand the related physics on sulfate aerosol infrared radiative forcing.
Abstract
The effects of atmospheric spherical curvature and refraction and their impact on radiative transfer have been studied. It is shown that formulas employed in GCMs for atmospheric curvature and refraction underestimate the effect of effective solar pathlength. A new parameterization is therefore proposed. It is emphasized that the atmospheric curvature effect on radiative transfer is a localized problem with height dependence. A method corresponding to the local effective pathlength factor is proposed. This rigorous scheme enables variations in both the pathlength and the gaseous amount along a solar direct beam to be accurately evaluated in the radiative transfer process. The results of the rigorous scheme can be used as the benchmark to the proposed parameterizations for the effective pathlength factor. It is found that the new parameterization proposed in this note has better results in flux and heating rates when compared to other parameterizations.
Abstract
The effects of atmospheric spherical curvature and refraction and their impact on radiative transfer have been studied. It is shown that formulas employed in GCMs for atmospheric curvature and refraction underestimate the effect of effective solar pathlength. A new parameterization is therefore proposed. It is emphasized that the atmospheric curvature effect on radiative transfer is a localized problem with height dependence. A method corresponding to the local effective pathlength factor is proposed. This rigorous scheme enables variations in both the pathlength and the gaseous amount along a solar direct beam to be accurately evaluated in the radiative transfer process. The results of the rigorous scheme can be used as the benchmark to the proposed parameterizations for the effective pathlength factor. It is found that the new parameterization proposed in this note has better results in flux and heating rates when compared to other parameterizations.
Abstract
For radiative transfer in a thin atmosphere, an analytical four-stream isosector approximation for solar radiative transfer is presented. This approximation method is based on the assumption of four spherical sectors of isotropic intensities. Calculations show that the four-stream isosector approximation model substantially improves the accuracy in reflection, transmission, and absorption with respect to the Coakley–Chýlek model. For an optical thickness less than unity, the four-stream isosector approximation has errors mostly under 5%, in contrast to errors up to 20% or higher for the Coakley–Chýlek model. This four-stream isosector approximation can be applied to atmospheric aerosol layers or thin cirrus clouds.
Abstract
For radiative transfer in a thin atmosphere, an analytical four-stream isosector approximation for solar radiative transfer is presented. This approximation method is based on the assumption of four spherical sectors of isotropic intensities. Calculations show that the four-stream isosector approximation model substantially improves the accuracy in reflection, transmission, and absorption with respect to the Coakley–Chýlek model. For an optical thickness less than unity, the four-stream isosector approximation has errors mostly under 5%, in contrast to errors up to 20% or higher for the Coakley–Chýlek model. This four-stream isosector approximation can be applied to atmospheric aerosol layers or thin cirrus clouds.
