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| Characteristics of a Terrain-Following Sigma Coordinate |
| LI Yi-Yuan,WANG Bin,WANG Dong-Hai |
| 1. State Key Laboratory of Numerical Modeling for Atmospheric Sciences and Geophysical Fluid Dynamics, Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029, China,1. State Key Laboratory of Numerical Modeling for Atmospheric Sciences and Geophysical Fluid Dynamics, Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029, China; 2. Center for Earth System Science, Tsinghua University, Beijing 100084, China,3. State Key Laboratory of Severe Weather, Chinese Academy of Meteorological Sciences, Beijing 100081, China |
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Abstract This study quantifies the main characteristics of a terrain-following, sigma-coordinate through mathematical analyses of its covariant and contravariant basis vectors as well as the vertical coordinate of sigma. A 3-D schematic of the sigma-coordinate in a curvilinear coordinate system is provided in this study. The characteristics of the basis vectors were broken down into their “local vector characteristics” and “spatial distribution characteristics”, and the exact expressions of the covariant; in addition, the contravariant basis vectors of the sigma-coordinate used to elucidate their detailed characteristics were properly solved. Through rewriting the expression of the vertical coordinate of sigma, a mathematical expression of all the sigma-coordinate surfaces was found, thereby quantifying the so-called terrain-following characteristics and lack of flexibility to adjust the slope variation of sigma-coordinate surfaces for the classic definition of sigma. Finally, an analysis on the range value of the vertical coordinate demonstrated that the general value range of sigma could be obtained by eliminating the sigma-coordinate surfaces below the Earth’s surface. All these quantitative descriptions of the characteristics of sigma-coordinate were the foundation for improving the sigma-coordinate or creating a new one.
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Received: 12 January 2011
Revised: 07 March 2011
Accepted: 07 March 2011
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