THE VECTOR HARMONIC TRANSFORM METHOD FOR SOLVING PARTIAL-DIFFERENTIAL EQUATIONS IN SPHERICAL GEOMETRY

The development of computational methods for solving partial differential equations in spherical geometry is complicated by problems induced by the spherical coordinate system itself. Even though the solution is smooth in Cartesian coordinates, in spherical coordinates the components of vector fields such as the wind are multivalued at the Poles and the differential equations have unbounded terms. For example, the total derivative of the velocity is unbounded at the poles. Here, the vector harmonic transform method for the effective treatment of these problems is presented. Vector fields such as the wind are expanded in terms of vector harmonics, and scalar fields such as pressure and temperature are expanded in terms of scalar harmonics. Unbounded terms in the differential equation are grouped into bounded expressions that are evaluated by their formal application to the spectral expansions. The method can be applied to any differential equation without introducing scalar-dependent variables, such as divergence or vorticity, or without raising the order of the differential equations. The method can be implemented on either a Gauss or equally spaced latitudinal grid with points located at the poles because the method does not contain any divisions by the cosine of the latitude. The computational requirements are comparable to traditional spectral methods.