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

A new approach is proposed to use the covariant scalar equations of the σ-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 σ-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.