The nonlinear interaction of Rayleigh-Benard convection with the large-scale flow that must exist in the mantle to balance the mass flux of moving lithospheric plates is considered by using both numerical and analytic models, The numerical models are two dimensional and include a simulated large-scale flow by explicitly considering a moving lithospheric plate overlying a fluid layer. The analytic model is more idealized but has the advantage of being three dimensional. Both types of models show that convective rolls with axes perpendicular to the lithospheric spreading direction (transverse rolls) decay on a time scale that depends on the spreading velocity and that may be as short as 50 m.y. The three-dimensional analysis shows that the transverse rolls are eventually replaced by rolls with their axes aligned with the direction of spreading (longitudinal rolls). The longitudinal rolls may be a significant vertical heat transport mechanism under the interior of plates, and thus it is suggested that three-dimensional models will be required to investigate terrestrial heat flow, The conclusions contain a speculative section suggesting a connection between the stresses on the lithosphere due to the longitudinal rolls and the emplacement of the linear island and sea mount chains typical of many oceans.