The "Pole Problem"
A description of the Home Page graphic
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
Last updated July 14, 1998.
Mail comments to Paul Swarztrauber.