A Solution to the Pole Problem for the Shallow Water Equations on a Sphere

Journal article, 2014

We consider a reduced gridding technique for the shallow water equations on a sphere, based on spherical coordinates. In a small vicinity of the poles, a longitudinal derivative is discretized at a grid-point on a parallel, by using points on the great circle through the grid-point and tangent to the parallel. Centered one-dimensional interpolation formulas are used in this process and also in connecting adjacent segments in the reduced grid. The remaining spatial discretization is obtained by simply replacing derivatives by centered equidistant finite difference approximations. Numerical experiments for scalar advection equations and for the well-known Rossby-Haurwitz test example indicate that the methods developed work surprisingly well. Some advantages are that (i) a fairly uniform grid, with many reductions or segments, can be used, (ii) order of approximation $2p$ in the spatial discretizations requires only $4p+1$ points and (iii) the local and simple structure of the schemes will make efficient implementation on massively parallel computer systems possible. The paper is an attempt towards global numerical weather prediction models, by first analyzing the pole problem for reduced latitude-longitude grids.