On Computing Viscous Forces in Map Coordinates with a Variable Scale

Abstract

Equations are first derived in shape-preserving coordinates for the spatial derivatives of the unit vectors, the gradient and Laplacian of a scalar, the divergence and vorticity of a vector, the advective acceleration in the equations of motion, the strain-rate tensor, and the viscous forces per unit mass. A shape-preserving projection is defined here as one in which the map scale, though spatially variable, is independent of the orientation of an infinitesimal line segment. Shape-preserving projections are also conformal. Examples are stereographic and Mercator projections. The results are then extended to the case where the map scale factors in the two horizontal coordinate directions are different.