Results of a similarity theory for spherical mantle convection are presented. The single-mode mean field equations are analyzed for convection which is so vigorous that temperature disturbances become localized in thin thermal boundary layers. Our purpose is to study effects of spherical geometry, density interfaces, heat source distribution, and cell size. Steady state solutions are found for isoviscous spherical shells in which the field of motion is spatially periodic in a single spherical harmonic degree l. Calculations are carried out over the range 2?l?40 and for various fractions of internal versus base heating. Three configurations are examined: (1) convection in a single layer of cells extending through the whole mantle, (2) convection in two layers, separated by a density interface at 670-km depth, and (3) convection in a single layer terminating at 670 km. Results of these calculations are used to give estimates of surface horizontal velocities in terms of the heat loss, viscosity stratification, amount of internal heating, and depth of circulation. The surface velocity is most strongly affected by the thickness of the convecting shell. Deep mantle convection can achieve surface velocities which agree with observed plate speeds, while convection restricted to the upper mantle does not, at least on the scale of the major plates. The temperature distribution is strongly affected by the spherical geometry and by the presence of density interfaces. The principal difference between convection in one and two layers is that the latter produces a 'hot' lower mantle, while the former produces a 'warm' one.