Nonlinear normal mode initialization equations, which provide required balance relations for atmospheric data, are considered in the generalized case of shallow water equations in arbitrary orthogonal coordinates. Using the concept of ellipticity in the sense of Douglis–Nirenberg, the conditions of well posedness of boundary value problems for balance equations are derived in the cases of constrained streamfunction, constrained potential vorticity, and constrained pressure fields.

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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.

In this study the problem of finding the conformal mapping from a sphere onto a plane with a given scale function independent of longitude is solved for an arbitrary spherical domain. The obtained results are compared with the well-known projections used in cartography and geophysical fluid dynamics. The problem of minimization of the distortion under conformal mappings is solved for domains in the form of the spherical disk. The distortions of some extensively used conformal mappings are compared with the distortions of orthogonal mappings.