We examine the Galerkin (including single-mode and Lorenz-type) equations for convection in a sphere to determine which physical processes are neglected when the equations of motion are truncated too severely. We test our conclusions by calculating solutions to the equations of motion for different values of the Rayleigh number and for different values of the limit of the horizontal spatial resolution. We show how the gross features of the flow such as the mean temperature gradient, central temperature, boundary-layer thickness, kinetic energy and temperature variance spectra, and energy production rates are affected by truncation in the horizontal direction. We find that the transitions from steady-state to periodic, and then to aperiodic convection depend not only on Rayleigh number but also very strongly on the horizontal resolution of the calculation. All of our models are well resolved in the vertical direction, so the transitions do not appear to be due to poorly resolved boundary layers. One of the effects of truncation is to enhance the high-wavenumber end of the kinetic energy and thermal variance spectra. Our numerical examples indicate that, as long as the kinetic energy spectrum decreases with wavenumber, a truncation gives a qualitatively correct solution.