The Approximation of Vector Functions and Their Derivatives on the Sphere

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Abstract
A vector basis is defined for vector functions that are given on the sphere. It is shown that the basis is orthogonal, that it is complete and that the convergence properties of the spectral representation are determined by the smoothness of the vector function in cartesian coordinates. The method for analyzing a vector function in terms of this basis uses the methods for analyzing scalar functions in terms of surface harmonics. A vector function that is smooth in cartesian coordinates and nonzero at the pole will be discontinuous in spherical coordinates. This is a result of the discontinuous spherical coordinate system rather than the vector function itself. This, in turn, induces singulari- ties in certain surface derivatives on the sphere, even though the function is bounded and differentiable in cartesian coordinates. Many of the individual terms in a partial differential equation on the sphere are unbounded at the poles; however, the cancela- tion between these terms is sufficient to yield a bounded result. This does not create a problem from a theoretical point of view; however, it becomes a significant problem from a computational point of view. Indeed, the finer the grid the more pronounced the difficulty becomes near the pole where the computation fails completely. In this paper we show how to identify the unbounded terms and group them into bounded expres- sions. We also show how to compute these bounded expressions in a stable way that avoids determining their individual unbounded terms when the function is given in terms of the vector basis.

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