Pressure gradient errors in a covariant method of implementing theσ-coordinate:idealized experiments and geometric analysis

A new approach is proposed to use the covariant scalar equations of the a-coordinate （the covariant method）, in which the pressure gradient force （PGF） has only one term in each horizontal momentum equation, and the PGF errors are much reduced in the computational space. In addition, the validity of reducing the PGF errors by this covariant method in the computational and physical space over steep terrain is investigated. First, the authors implement a set of idealized experiments of increasing terrain slope to compare the PGF errors of the covariant method and those of the classic method in the computational space. The results demonstrate that the PGF errors of the covariant method are consistently much-reduced, compared to those of the classic method. More importantly, the steeper the terrain, the greater the reduction in the ratio of the PGF errors via the covariant method. Next, the authors use geometric analysis to further investigate the PGF errors in the physical space, and the results illustrates that the PGF of the covariant method equals that of the classic method in the physical space; namely, the covariant method based on the non-orthogonal a-coordinate cannot reduce the PGF errors in the physical space. However, an orthogonal method can reduce the PGF errors in the physical space. Finally, a set of idealized experiments are carried out to validate the results obtained by the geometric analysis. These results indicate that the covariant method may improve the simulation of variables relevant to pressure, in addition to pressure itself, near steep terrain.

本文针对经典σ坐标的气压梯度误差(PGF误差),采用多种地形展开理想试验,对比经典σ坐标的经典方案和协变方案的PGF误差。结果表明:计算空间中,协变方案始终能减小经典方案的误差,地形越陡,效果越明显。然而,几何分析和理想试验均表明:协变方案仅能减小计算空间的误差,不能减小物理空间的误差;相比经典方案,正交地形追随坐标能同时减小计算空间和物理空间的误差。