Three-dimensional numerical models are used to study the planform of thermal convection in rectangular boxes, imposing uniform velocity at the surface. This problem is relevant to the structure of mantle convection under moving tectonic plates. Different planforms can be obtained depending on the Rayleigh number, viscosity structure and length of the plate. If there is a thin low viscosity layer in the uppermost part of the model, the solution consists of narrow ascending plumes below the divergent edge of neighboring plates as well as descending sheets below lines of plate convergence. If the experimental box is long, well developed drifting columnar blobs arise from the unstable lower boundary layer, giving the solution an oscillatory character. ¿ American Geophysical Union 1990