This study quantifies the main characteristics of a terrain-following, G-coordinate through mathematical analyses of its covariant and contravariant basis vectors as well as the vertical coordinate of σ. A 3-D schema...This study quantifies the main characteristics of a terrain-following, G-coordinate through mathematical analyses of its covariant and contravariant basis vectors as well as the vertical coordinate of σ. A 3-D schematic of the σ-coordinate in a curvilinear coordinate system is provided in this study. The characteristics of the basis vectors were broken down into their ＂local vector charac- teristics＂ and ＂spatial distribution characteristics＂, and the exact expressions of the covariant; in addition, the con- travariant basis vectors of the G-coordinate used to eluci- date their detailed characteristics were properly solved. Through rewriting the expression of the vertical coordi- nate of G, a mathematical expression of all the cr-coor- dinate surfaces was found, thereby quantifying the so- called terrain-following characteristics and lack of flexi- bility to adjust the slope variation of G-coordinate sur- faces for the classic definition of G. Finally, an analysis on the range value of the vertical coordinate demonstrated that the general value range of G could be obtained by eliminating the G-coordinate surfaces below the Earth＇s surface. All these quantitative descriptions of the charac- teristics of G-coordinate were the foundation for improv- ing the G-coordinate or creating a new one.展开更多
基金supported by the National Natural Science Foundation of China under Grant Nos. 40821092,40633016,and 40875022
文摘This study quantifies the main characteristics of a terrain-following, G-coordinate through mathematical analyses of its covariant and contravariant basis vectors as well as the vertical coordinate of σ. A 3-D schematic of the σ-coordinate in a curvilinear coordinate system is provided in this study. The characteristics of the basis vectors were broken down into their ＂local vector charac- teristics＂ and ＂spatial distribution characteristics＂, and the exact expressions of the covariant; in addition, the con- travariant basis vectors of the G-coordinate used to eluci- date their detailed characteristics were properly solved. Through rewriting the expression of the vertical coordi- nate of G, a mathematical expression of all the cr-coor- dinate surfaces was found, thereby quantifying the so- called terrain-following characteristics and lack of flexi- bility to adjust the slope variation of G-coordinate sur- faces for the classic definition of G. Finally, an analysis on the range value of the vertical coordinate demonstrated that the general value range of G could be obtained by eliminating the G-coordinate surfaces below the Earth＇s surface. All these quantitative descriptions of the charac- teristics of G-coordinate were the foundation for improv- ing the G-coordinate or creating a new one.
