The observed quasi-rigid behaviour of surface plates in the course of their relative motion is a consequence of the high viscosity which obtains in the cold near surface region. By assigning a particular constant velocity as the upper boundary condition in a numerical model of mantle convection, we have investigated the effect of lithospheric rigidity on the variation of oceanic bathymetry and heat flow as a function of ocean floor age. Predicted variations of both bathymetry and heat flow at the surface of mantle wide convection cells which are partially heated from within, exhibit the same qualitative behaviour as data compiled for the major ocean basins. The bathymetry varies with distance x from the spreading centre initially as x1/2 but subsequently flattens with respect to an x1/2 reference curve whereas the heat flow closely follows an x-1/2 decay over most of the convection cell. Consequently the viability of the mechanism for sea floor flattening proposed by Jarvis and Peltier (1980) is increased when a constant surface velocity, characteristic of rigid plates, is incorporated in the model. This model successfully predicts both that the bathymetry should flatten and that the heat flow should not.