The development of computational methods for solving partial differential equations on the surface of the sphere is complicated by problems induced by the spherical coordinate system itself. The horizontal velocity components are discontinuous at the poles in spherical coordinates even though they are smooth in Cartesian coordinates. This is simply because latitude increases as the pole is approached and immediately begins to decrease after the pole is passed. Therefore, velocity, as measured relative to the spherical coordinate system, changes sign on opposite sides of the pole. These discontinuities create a fundamental problem associated with solving differential equations on the surface of the sphere; namely, they induce unbounded terms in the differential equations in the neighborhood of the poles (Swarztrauber, 1993). Indeed, the total derivative of the velocity with respect to time is unbounded. Its latitudinal component, in a very small neighborhood of the pole, is shown on the Home Page. Although these terms are unbounded, they combine with other unbounded terms to form bounded differential expressions. For example, the total derivative of the velocity combines with the "metric" terms to obtain the fluid acceleration. Discontinuous velocities and unbounded terms create a host of computational problems that are collectively refered to as the "pole problem."

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