On Conformal Conic Mappings of Spherical Domains

Andrei Bourchtein,Ludmila Bourchtein

doi:10.1155/2014/840280

Abstract

The problem of the generation of homogeneous grids for spherical domains is considered in the class of conformal conic mappings. The equivalence between secant and tangent projections is shown and splitting the set of conformal conic mappings into equivalence classes is presented. The problem of minimization of the mapping factor variation is solved in the class of conformal conic mappings. Obtained results can be used in applied sciences, such as geophysical fluid dynamics and cartography, where the flattening of the Earth surface is required.

Highlights

The problem of the generation of homogeneous grids for spherical domains is one of the oldest problems of cartography and geodesy, and it is the important part of developing efficient numerical schemes for geophysical simulations, in particular, for atmosphere-ocean dynamics
Computational grids based on the spherical coordinates are highly nonhomogeneous that cause the problems for both dynamical and physical parts of numerical schemes [1,2,3]
The most efficient way to circumvent this problem is an application of conformal mappings from a sphere onto a plane, because these transformations usually maintain a simpler form of the governing equations and assure local isotropy and smoothness of variation of the physical mesh sizes on computational grids [1,2,3,4,5,6]

Summary

Introduction

The problem of the generation of homogeneous grids for spherical domains is one of the oldest problems of cartography and geodesy, and it is the important part of developing efficient numerical schemes for geophysical simulations, in particular, for atmosphere-ocean dynamics. The most efficient way to circumvent this problem is an application of conformal mappings from a sphere onto a plane, because these transformations usually maintain a simpler form of the governing equations and assure local isotropy and smoothness of variation of the physical mesh sizes on computational grids [1,2,3,4,5,6]. As a measure of the homogeneity of the computational grid one can use the ratio between the maximum and minimum values of the mapping factor over the considered domain:. This criterion is suitable for generation of computational grids for explicit and semi-implicit schemes [1, 7, 8]. We consider the problem of minimization of α in the class of conic mappings, which are standard official cartographic projections for intermediate and large-scale regions of the Earth surface [9,10,11,12] and which are frequently used in the modeling of atmosphere and ocean dynamics in the middle and low latitudes [13,14,15,16,17,18,19,20]

Equivalence Classes of the Conic Mappings

Optimal Choice of Conic Mappings