3D Radiative Transfer in Weakly Inhomogeneous Medium. Part I: Diffusive Approximation

V. L. Galinsky Center for Atmospheric Sciences, Center for Clouds, Chemistry and Climate, Scripps Institution of Oceanography, University of California, San Diego, La Jolla, California

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V. Ramanathan Center for Atmospheric Sciences, Center for Clouds, Chemistry and Climate, Scripps Institution of Oceanography, University of California, San Diego, La Jolla, California

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Abstract

The solution of the three-dimensional radiative transfer equation in weakly horizontally inhomogeneous medium has been obtained in the diffusion approximation using the expansion of the three-dimensional delta-Eddington approximation. The solution approach, referred as the gradient correction (GC) method, expands the horizontal fluxes and the source function in terms of the horizontal gradient of the extinction coefficient and/or the cloud-top boundary. In the transfer equation, only the zeroth- and first-order gradient terms are retained and hence the following limitations apply. First, the length of the horizontal variations of optical properties of the medium should be large in comparison to the mean radiative transport length. Second, the ratio of the vertical to horizontal scales should be small enough so that fluxes from boundaries may be neglected.

Since there are no restrictions to the amplitude of the optical properties variations, this method may even be applicable to a medium with strong horizontal variations of optical properties, as long as scales of the variations are large enough in comparison to the radiative transport length. The analytical solutions are in excellent agreement with the more accurate numerical solutions. The solution also shows the solar zenith angle dependence of the albedo, similar to that observed in analyses of satellite imagery.

The GC approach may be useful as a fast and computationally inexpensive method both for the correction of the independent pixel approximation used for extraction of cloud fields from satellite imagery and possibly for the calculation of the radiation fluxes in climate models.

Corresponding author address: Dr. Vitaly Galinsky, Center for Atmospheric Sciences, Center for Clouds, Chemistry and Climate, Scripps Institution of Oceanography, University of California, San Diego, 9500 Gilman Drive, MC # 0221, La Jolla, CA 92093-0221.

Email: vitaly@fiji.ucsd.edu

Abstract

The solution of the three-dimensional radiative transfer equation in weakly horizontally inhomogeneous medium has been obtained in the diffusion approximation using the expansion of the three-dimensional delta-Eddington approximation. The solution approach, referred as the gradient correction (GC) method, expands the horizontal fluxes and the source function in terms of the horizontal gradient of the extinction coefficient and/or the cloud-top boundary. In the transfer equation, only the zeroth- and first-order gradient terms are retained and hence the following limitations apply. First, the length of the horizontal variations of optical properties of the medium should be large in comparison to the mean radiative transport length. Second, the ratio of the vertical to horizontal scales should be small enough so that fluxes from boundaries may be neglected.

Since there are no restrictions to the amplitude of the optical properties variations, this method may even be applicable to a medium with strong horizontal variations of optical properties, as long as scales of the variations are large enough in comparison to the radiative transport length. The analytical solutions are in excellent agreement with the more accurate numerical solutions. The solution also shows the solar zenith angle dependence of the albedo, similar to that observed in analyses of satellite imagery.

The GC approach may be useful as a fast and computationally inexpensive method both for the correction of the independent pixel approximation used for extraction of cloud fields from satellite imagery and possibly for the calculation of the radiation fluxes in climate models.

Corresponding author address: Dr. Vitaly Galinsky, Center for Atmospheric Sciences, Center for Clouds, Chemistry and Climate, Scripps Institution of Oceanography, University of California, San Diego, 9500 Gilman Drive, MC # 0221, La Jolla, CA 92093-0221.

Email: vitaly@fiji.ucsd.edu

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