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Knut Stamnes and Henri Dale

Abstract

The recently developed matrix method to solve the discrete ordinate approximation to the radiative transfer equation in plane parallel geometry (Stamnes and Swanson, 1981) is extended to compute the full azimuthal dependence of the intensity, Comparing computed intensifies with those obtained by other established methods, we find that for phase functions typical of atmospheric haze 32 streams are sufficient for better than 1% agreement, while 16 streams yield an accuracy of about 1–5% except for angles close to the forward and backward directions for which the error is about 10–15%. The results of the intensity computations are summarized by presenting three-dimensional “stack plots” of the intensity as a function of polar and azimuthal angles. We also show that for flux calculations four streams suffice to obtain 1% accuracy, while eight streams yield an accuracy better than 0.1%.

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Knut Stamnes and Roy A. Swanson

Abstract

The difficulties inherent in the conventional numerical implementation of the discrete ordinate method (following Chandrasekhar's prescription) for solving the radiative transfer equation are discussed. A matrix formulation is developed to overcome these difficulties, and it is specifically shown that the order of the algebraic eigenvalue problem can be reduced by a factor of 2. An expression for the source function is derived and used to obtain angular distributions. By appealing to the reciprocity principle, it is shown that substantial computational shortcuts are possible if only integrated quantities such as albedo and transmissivity are required. Comparison of fluxes calculated by the present approach with those obtained by other methods shows that low-order discrete ordinate approximations yield very accurate results. Thus, the present approach offers an efficient and reliable computational scheme that lends itself readily to the solution of a variety of radiative transfer problems in realistic planetary atmospheres.

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Si-Chee Tsay, Knut Stamnes, and Kolf Jayaweera

Abstract

A radiation model is constructed that includes radiative interactions with atmospheric gases as well as parameterized treatments of scattering and absorption/emission by cloud droplets and haze particles. A unified treatment of solar and terrestrial radiation is obtained by using identical cloud and haze parameterization procedure for the shortwave and longwave region. The influence of the relative humidity of the haze particles is also considered. Snow conditions of the arctic region are simulated in terms of snow grain sizes and soot contamination in the surface layers. Data from the Arctic Stratus Cloud Experiment collected in 1980 are used for model comparisons and sensitivity studies under cloudy and hazy sky conditions.

During the arctic summer, stratus clouds are a persistent feature and decrease the downward flux at the surface by about 130–200 W m−2. Arctic haze in the summertime is important if it is above the cloud layer or in air with low relative humidity, and it decreases the downward flux at the surface by about 10–12 W m−2. By comparison the greenhouse effect of doubling the carbon dioxide amount increases the downward flux at the surface by about 4–7 W m−2 and can be offset by the background haze or by an increase in cloudiness of about 4%.

Assuming steady microstructures of stratus clouds, we find that in late June a clear sky condition results in more available downward flux for snow melt (yielding a melting rate of 9.3 em day−1) than does a cloudy sky condition (6 cm day−1). This is because the increase of infrared radiation diffused back to the surface by the cloud can not compensate for the reduction of solar radiation. When the snow starts to melt, the decreasing snow albedo further accelerates the melting process.

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Zhenyi Lin, Nan Chen, Yongzhen Fan, Wei Li, Knut Stamnes, and Snorre Stamnes

Abstract

The treatment of strongly anisotropic scattering phase functions is still a challenge for accurate radiance computations. The new delta-M+ method resolves this problem by introducing a reliable, fast, accurate, and easy-to-use Legendre expansion of the scattering phase function with modified moments. Delta-M+ is an upgrade of the widely used delta-M method that truncates the forward scattering peak with a Dirac delta function, where the “+” symbol indicates that it essentially matches moments beyond the first M terms. Compared with the original delta-M method, delta-M+ has the same computational efficiency, but for radiance computations, the accuracy and stability have been increased dramatically.

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