A terrain-following coordinate （a-coordinate） in which the computational form of pressure gradient force （PGF） is two-term （the so-called classic method） has significant PGF errors near steep terrain. Using the ...A terrain-following coordinate （a-coordinate） in which the computational form of pressure gradient force （PGF） is two-term （the so-called classic method） has significant PGF errors near steep terrain. Using the covariant equations of the a-coordinate to create a one-term PGF （the covariant method） can reduce the PGF errors. This study investigates the factors inducing the PGF errors of these two methods, through geometric analysis and idealized experiments. The geometric analysis first demonstrates that the terrain slope and the vertical pressure gradient can induce the PGF errors of the classic method, and then generalize the effect of the terrain slope to the effect of the slope of each vertical layer （φ）. More importantly, a new factor, the direction of PGF （a）, is proposed by the geometric analysis, and the effects of φ and a are quantified by tan φ.tan a. When tan φ.tan a is greater than 1/9 or smaller than -10/9, the two terms of PGF of the classic method are of the same order but opposite in sign, and then the PGF errors of the classic method are large. Finally, the effects of three factors on inducing the PGF errors of the classic method are validated by a series of idealized experiments using various terrain types and pressure fields. The experimental results also demonstrate that the PGF errors of the covariant method are affected little by the three factors.展开更多
Although G-coordinate is one of the most popular methods used in marine and estuarine modeling, it has long suffered from the so-called ＂steep boundary problem＂, namely, the PGF problem. In this paper, a new method ...Although G-coordinate is one of the most popular methods used in marine and estuarine modeling, it has long suffered from the so-called ＂steep boundary problem＂, namely, the PGF problem. In this paper, a new method called the ＂σ-sharpen immersed boundary method＂ （σ-SIBM） is put forward. In this method, the virtual flat bottom boundary is creatively introduced in regions with the steep boundary and is taken as the boundary of numerical domain. By this, OH/Ox of numerical domain changes to be a controllable value and the steep bottom problem is then transformed to the non-conforming boundary problem, which is, in turn, solved by the SIBM. The accuracy and efficiency of the σ-sharpen immersed boundary method （σ-SIBM） has been showed by both comparative theoretical analysis and classical numerical tests. First, it is shown that the σ-SIBM is more effective than the z-level method, in that σ-SIBM needs special treatment only in the steep section, but the z-level method needs the special treatment in each grid note. Second, it is superior to the p-method in that it is not restricted by the density distribution. This paper revisits the classical seamount numerical test used in numerous studies to prove the sigma errors of the pressure gradient force （PGFE） and their long-term effects on circulation. It can be seen that, as for the maximum erroneous velocity and kinetic energy, the value of σ-SIBM is much less than that of the z-level method and the traditional σ-method.展开更多
基金jointly supported by the National Basic Research Program of China[973 Program,grant number 2015CB954102]National Natural Science Foundation of China[grant numbers41305095 and 41175064]
文摘A terrain-following coordinate （a-coordinate） in which the computational form of pressure gradient force （PGF） is two-term （the so-called classic method） has significant PGF errors near steep terrain. Using the covariant equations of the a-coordinate to create a one-term PGF （the covariant method） can reduce the PGF errors. This study investigates the factors inducing the PGF errors of these two methods, through geometric analysis and idealized experiments. The geometric analysis first demonstrates that the terrain slope and the vertical pressure gradient can induce the PGF errors of the classic method, and then generalize the effect of the terrain slope to the effect of the slope of each vertical layer （φ）. More importantly, a new factor, the direction of PGF （a）, is proposed by the geometric analysis, and the effects of φ and a are quantified by tan φ.tan a. When tan φ.tan a is greater than 1/9 or smaller than -10/9, the two terms of PGF of the classic method are of the same order but opposite in sign, and then the PGF errors of the classic method are large. Finally, the effects of three factors on inducing the PGF errors of the classic method are validated by a series of idealized experiments using various terrain types and pressure fields. The experimental results also demonstrate that the PGF errors of the covariant method are affected little by the three factors.
基金supported by the National Natural Science Foundation of China(Grant Nos.51209239,51109194)"985 Project"of Minzu Univer-sity of China(Grant No.MUC98507-08)
文摘Although G-coordinate is one of the most popular methods used in marine and estuarine modeling, it has long suffered from the so-called ＂steep boundary problem＂, namely, the PGF problem. In this paper, a new method called the ＂σ-sharpen immersed boundary method＂ （σ-SIBM） is put forward. In this method, the virtual flat bottom boundary is creatively introduced in regions with the steep boundary and is taken as the boundary of numerical domain. By this, OH/Ox of numerical domain changes to be a controllable value and the steep bottom problem is then transformed to the non-conforming boundary problem, which is, in turn, solved by the SIBM. The accuracy and efficiency of the σ-sharpen immersed boundary method （σ-SIBM） has been showed by both comparative theoretical analysis and classical numerical tests. First, it is shown that the σ-SIBM is more effective than the z-level method, in that σ-SIBM needs special treatment only in the steep section, but the z-level method needs the special treatment in each grid note. Second, it is superior to the p-method in that it is not restricted by the density distribution. This paper revisits the classical seamount numerical test used in numerous studies to prove the sigma errors of the pressure gradient force （PGFE） and their long-term effects on circulation. It can be seen that, as for the maximum erroneous velocity and kinetic energy, the value of σ-SIBM is much less than that of the z-level method and the traditional σ-method.
