This work presents a detailed numerical study of the dynamical behavior of convection in a spherical shell, as applied to mantle convection. From both 2-dimensional (120 radial and 360 tangential points) and 3-dimensional (60 radial levels and spherical harmonics up to order and degree l=33, m=33), we show that for a spherical shell (with inner to outer radii ratio &eegr;=62) convection becomes time-dependent, with l=2 dominating, at a Rayleigh number of about 31 times supercritical for a constant viscosity, base-heated configuration. This secondary instability is characterized by oscillatory time-dependence, with higher frequencies involved, at slightly higher Rayleigh numbers. In the process of illustrating the onset of time-dependence, we extend our analysis to show that the onset of weak turbulence in spherical-shell convection takes place at about 60 times the critical Rayleigh number via a quasi-periodic mode.