In recent years, numerical weather prediction models have progressed from finite difference methods to spectral methods in order to obtain more accurate forecasts in less computational time. A switch from "pure" finite difference to "pure" spectral techniques, however, may not be the best approach in all cases. Indeed, for some problems a mixed approach with finite difference approximations in one coordinate and spectral approximations in the remaining coordinates may provide comparably accurate results in less computational time than either pure method alone. This is particularly evident for the case of a hurricane where a state of near-circular symmetry typically prevails. For such a problem, a semi-spectral (SS) method using finite difference approximations in the radial direction and spectral methods in the azimuthal direction is a natural choice. While the effectiveness of pure methods for certain problems in fluid dynamics and atmospheric dynamics is well documented, comparatively little work has been undertaken to examine the effectiveness of SS schemes. This is particularly evident in problems concerning the vorticity dynamics of the hurricane near-core region. Because of the nearly circular flow of a strong hurricane, we hypothesize that SS formulations may prove useful in helping to elucidate the asymmetric vorticity dynamics in the hurricane's near-core region. This study examines the effectiveness of the semi-spectral approach for two classes of problems. The first problem concerns the redistribution of vorticity anomalies within a "master" vortex possessing a vorticity profile which decreases monotonically with radius. The second problem concerns barotropic instability, vortex breakdown, and vorticity mixing that is observed in mature hurricanes.