The thermal convection in a rotating spherical shell or a rotating sphere with a spherically symmetric gravity force has been studied as one of the fundamental problems in geophysics and astrophysics. A typical example of the convective flows under such situations is the flow in the Earth outer core. It is believed that the flow produces several geomagnetic phenomena such as generation and reversals of a dipole magnetic field. However, the relationship between the dynamical behavior of the fluid motion and the geomagnetic phenomena has not been well understood because the flow subject to high Taylor and high Rayleigh numbers is very complicated in both space and time.

The purpose of the present study is to understand the dynamical behavior of convection when the convective state is far from the onset, under relatively high Taylor number. We solve a full set of Boussinesq equations numerically using the spectral method in a rotating spherical shell. The numerical simulations are carried out for the Rayleigh numbers up to 90 times the critical value, above which convective motion starts. As long as the Rayleigh number is near the critical, a convective flow is driven by several vortices along the rotation axis outside of the cylindrical surface tangent to the inner sphere at the equator, inside of which the flow is stagnant. At the Rayleigh numbers exceeding 5 to 7 times the critical, the convection is active also inside the cylindrical surface (i.e. polar regions) where high temperature blobs are formed, that contributes high heat transport. Convection in the cylindrical surface is more active than that in the exterior. At the Rayleigh numbers over 60 times the critical, the activity of convection inside the cylinder approaches that outside, implying that the convection tends to be uniform along spheres.