New types of convective instability and associated nonlinear phenomena in rapidly rotating spherical systems have been discovered through numerical simulations. The Prandtl number, defined as the ratio of the viscous and thermal diffusivities of a fluid, Pr=&ngr;/&kgr;, plays a crucial role in determining the fundamental features of both the instabilities and the corresponding nonlinear convection. The results shed new light on regimes of convection in the earth's core and the atmospheres of the major planets. ¿American Geophysical Union 1991