Editorial Type:
Article
Article Type:
Research Article

Analytical Infrared Delta-Four-Stream Adding Method from Invariance Principle

Feng Zhang Key Laboratory of Meteorological Disaster, Ministry of Education, Nanjing University of Information Science and Technology, Nanjing, China
Collaborative Innovation Center on Forecast and Evaluation of Meteorological Disasters, Nanjing University of Information Science and Technology, Nanjing, China
State Key Laboratory of Numerical Modeling for Atmospheric Sciences and Geophysical Fluid Dynamics, Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing, China

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Kun Wu Key Laboratory of Meteorological Disaster, Ministry of Education, Nanjing University of Information Science and Technology, Nanjing, China
State Key Laboratory of Severe Weather, Chinese Academy of Meteorological Sciences, Beijing, China

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Jiangnan Li Canadian Center for Climate Modeling and Analysis, Environment and Climate Change Canada, University of Victoria, Victoria, British Columbia, Canada

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Quan Yang Key Laboratory of Meteorological Disaster, Ministry of Education, Nanjing University of Information Science and Technology, Nanjing, China
State Key Laboratory of Severe Weather, Chinese Academy of Meteorological Sciences, Beijing, China

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Jian-Qi Zhao State Key Laboratory of Numerical Modeling for Atmospheric Sciences and Geophysical Fluid Dynamics, Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing, China

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Jian Li State Key Laboratory of Severe Weather, Chinese Academy of Meteorological Sciences, Beijing, China

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Print Publication:
01 Oct 2016
DOI:
https://doi.org/10.1175/JAS-D-15-0317.1
Page(s):
4171–4188
Restricted access

Abstract

The single-layer solutions using a four-stream discrete ordinates method (DOM) in infrared radiative transfer (IRT) have been obtained. Two types of thermal source assumptions—Planck function exponential and linear dependence on optical depth—are considered. To calculate the IRT in multiple layers with a vertically inhomogeneous atmosphere, an analytical adding algorithm has been developed by applying the infrared invariance principle. The derived adding algorithm of the delta-four-stream DOM (δ-4DDA) can be simplified to work for the delta-two-stream DOM (δ-2DDA).

The accuracy for monochromatic emissivity is investigated for both δ-2DDA and δ-4DDA. The relative error for the downward emissivity can be as high as 15% for δ-2DDA, while the error is bounded by 2% for δ-4DDA. By incorporating δ-4DDA into a radiation model with gaseous transmission, δ-4DDA is much more accurate than δ-2DDA. Also, δ-4DDA is much more efficient, since it is an analytical method. The computing time of δ-4DDA is about one-third of the corresponding inverse matrix method.

Earth System Modeling Center Contribution Number 124.

Denotes Open Access content.

Corresponding author address: Feng Zhang, College of Atmospheric Science, Nanjing University of Information Science and Technology, 219 Ning Liu Road, Nanjing 21004, China. E-mail: feng_zhang126@126.com

Abstract

The single-layer solutions using a four-stream discrete ordinates method (DOM) in infrared radiative transfer (IRT) have been obtained. Two types of thermal source assumptions—Planck function exponential and linear dependence on optical depth—are considered. To calculate the IRT in multiple layers with a vertically inhomogeneous atmosphere, an analytical adding algorithm has been developed by applying the infrared invariance principle. The derived adding algorithm of the delta-four-stream DOM (δ-4DDA) can be simplified to work for the delta-two-stream DOM (δ-2DDA).

The accuracy for monochromatic emissivity is investigated for both δ-2DDA and δ-4DDA. The relative error for the downward emissivity can be as high as 15% for δ-2DDA, while the error is bounded by 2% for δ-4DDA. By incorporating δ-4DDA into a radiation model with gaseous transmission, δ-4DDA is much more accurate than δ-2DDA. Also, δ-4DDA is much more efficient, since it is an analytical method. The computing time of δ-4DDA is about one-third of the corresponding inverse matrix method.

Earth System Modeling Center Contribution Number 124.

Denotes Open Access content.

Corresponding author address: Feng Zhang, College of Atmospheric Science, Nanjing University of Information Science and Technology, 219 Ning Liu Road, Nanjing 21004, China. E-mail: feng_zhang126@126.com
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