The onset of time-dependence and some subsequent transitions in two-dimensional Rayleigh--B¿nard cells of aspect ratios a=2.5 and a=4 are studied numerically for a Boussinesq fluid of infinite Prandtl number. Oscillatory convection starts at a Rayleigh number of only 21,000 (a=2.5) or 12,000 (a=4.0) for stress-free boundaries. Above Ra=100,000 the cells with a=4.0 break down into several short cells, whereas single cells with a=2.5 remain stable to at least Ra=500,000. With rigid boundaries the tendency towards break-up into short cells is found to be more pronounced. Long-wavelength cells, modulated by travelling boundary-layer instabilities, appear to be a likely form of convection in the earth's mantle. ¿ American Geophysical Union 1987