Comparative Analysis of Conformal Mappings Used in Limited-Area Models of Numerical Weather Prediction

Abstract

Conformal separable projections from a sphere onto a plane are introduced to generalize the concept of conformal stereographic, conic, and cylindrical projections. The concept of equivalence of projections is used for partition of all considered projections into equivalence classes. The variation coefficient is defined as the ratio between maximum and minimum mesh sizes of numerical grids. The problem of minimization of this coefficient inside each equivalence class is studied. The obtained variation coefficients from all classes are compared and the principal relation among stereographic, cylindrical, and conic projections is established. The stereographic conformal projection is indicated as that which generates the “best” numerical grids for numerical weather prediction limited-area models.

On grid generation for numerical models of geophysical fluid dynamics

“…1, the variation coefficient of the classical conformal mappings is plotted as a function of the spherical radius a . The "best" conic and cylindrical mappings were found using the results in [5]. It can be seen that the "best" stereographic mappings assure visibly better uniformity than two others for spherical radius greater than 4000 km.…”

“…In [5,6] it was shown that the variation coefficients of the "best" stereographic, cylindrical and conic mappings satisfy the following inequality: str < cyl < con . Therefore, it is sufficient to minimize the variation coefficient in the class of stereographic mappings in order to find the "best" projection among the classical ones.…”

Some problems of conformal mappings of spherical domains

“…It was shown in [2], that such cylindrical mapping is tangent to the sphere at the centerpoint P 0 ofΩ γ (that is, P 0 lies on the "new" equator of rotated spherical coordinates) and its distortion coefficient is…”

“…The most used approach is an application of conformal mappings from a sphere onto a plane because these transformations maintain a simple form of the governing equations and also assure local isotropy and smoothness of the variation of physical mesh sizes on computational grid. This way, the problem of the generation of the most uniform mappings appears again (see [2], [3], [8], [15]). …”

Comparison of different spatial grids for numerical schemes of geophysical fluid dynamics

“…Our aim is to verify through numerical experiments up to what extent the theoretical results presented in [3][4][5] for domains of a specific geometry are applicable in the practical case of the rectangular computational domains. The paper is structured as follows.…”