Finite Differences on t he Sphere

Even before any numerical considerations are confronted, we run into difficulties simply in specifying a coordinate system. It is important to distinguish between true scalarvalued and vector-valued functions. The value of a scalar function, say temperature, is independent of coordinate system. That is, the temperature at some point in space is the same regardless of how we define the position of the point. On the other hand, the individual components of a vector-valued function, such as the wind velocity, obviously differ depending on the coordinate system. In a spherical coordinate system, the lines of constant longitude converge at the poles, so longitude is multivalued at the poles. This means that the components of the wind vector are discontinuous at the poles, although the wind vector itself is perfectly well behaved at the pole.