Commun. Comput. Phys., 12 (2012), pp. 1033-1050.


A New Approach to Implement Sigma Coordinate in a Numerical Model

Yiyuan Li 1, Donghai Wang 2, Bin Wang 3*

1 State Key Laboratory of Numerical Modeling for Atmospheric Sciences and Geophysical Fluid Dynamics, Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing, China.
2 State Key Laboratory of Severe Weather, Chinese Academy of Meteorological Sciences, Beijing, China.
3 State Key Laboratory of Numerical Modeling for Atmospheric Sciences and Geophysical Fluid Dynamics, Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing, China; and Ministry of Education Key Laboratory for Earth System Modeling, and Center for Earth System Science, Tsinghua University, Beijing, China.

Received 3 March 2011; Accepted (in revised version) 23 September 2011
Available online 28 March 2012
doi:10.4208/cicp.030311.230911a

Abstract

This study shows a new way to implement terrain-following sigma-coordinate in a numerical model, which does not lead to the well-known "pressure gradient force (PGF)" problem. First, the causes of the PGF problem are analyzed with existing methods that are categorized into two different types based on the causes. Then, the new method that bypasses the PGF problem all together is proposed. By comparing these three methods and analyzing the expression of the scalar gradient in a curvilinear coordinate system, this study finds out that only when using the covariant scalar equations of sigma-coordinate will the PGF computational form have one term in each momentum component equation, thereby avoiding the PGF problem completely. A convenient way of implementing the covariant scalar equations of sigma-coordinate in a numerical atmospheric model is illustrated, which is to set corresponding parameters in the scalar equations of the Cartesian coordinate. Finally, two idealized experiments manifest that the PGF calculated with the new method is more accurate than using the classic one. This method can be used for oceanic models as well, and needs to be tested in both the atmospheric and oceanic models.

AMS subject classifications: 86-08, 86A05, 86A10

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Key words: Pressure gradient force (PGF), terrain-following sigma coordinate, non-orthogonal, basis vectors, numerical modeling, computational errors.

*Corresponding author.
Email: liyiyuan@mail.iap.ac.cn (Y. Li), d.wang@hotmail.com (D. Wang), wab@lasg.iap.ac.cn (B. Wang)
 

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