Coordinate Transformation on a Sphere Using Conformal Mapping

Abstract

When setting up global ocean circulation models one faces the problem of including the Arctic Ocean where the traditional spherical coordinate system has a singularity at the pole. In addition, in regional model applications one has to deal with open boundaries where assumptions are made about the normally poorly known boundary conditions. Here an analytical reversible coordinate transformation on a sphere that preserves the orthogonality and the shape of infinitesimal figures is presented. Starting from a standard spherical coordinate system, the transformation is able to map the North and South Poles to two arbitrary locations of the earth and this is readily done with the aid of a conformal mapping in the extended complex plane. The resulting coordinate system will have enhanced resolution along the geodesic curve between the new poles. Examples are given where the transformation is used to strongly increase the resolution in a particular region of interest in the model domain.