Mantle plumes supply hot material to the asthenospheric channel which spreads beneath initially normal oceanic lithosphere at midplate hotspots. The fate of this material and the thermal structure of the lithosphere depend on the vigor of secondary convection through the steep themal gradient above the ponded plume material. It has been proposed either that (1) secondary convection is so vigorous that material hotter than the normal mantle adiabat is cooled a short distance downstream of the hotspot such that the geotherm resembles that of younger lithosphere or (2) that the plume material retains its excess temperature for a significant time and the lithosphere thins slowly. A parameterized method of modeling transient secondary convection is developed to examine this question. Calculations indicate that secondary convection is likely to be sluggish unless partial melting greatly lowers viscosity. In addition, vigorous secondary convection would greatly retard upstream asthenospheric flow of plume material beneath the nose of the Hawaiian swell and asthenospheric flow of hot material away from hotspots on slowly moving plates like Cape Verde. This conclusion is compatible with limited seismic evidence that indicates that the lithosphere is not detectably thinned beneath the Hawaiian swell. ¿ American Geophysical Union 1994