A distinctive property of plate tectonics as a natural form of convection is the variety of large horizontal scales exhibited by the plates in contrast to the marked regularity of scale seen in other natural convecting systems and laboratory experiments. Two idealized model problems are studied to gain some insight into the physical properties of a convecting layer that can lead to a multiplicity of large horizontal scales. The first model determines the most unstable wavelength for convection in a fluid having anisotropic viscosity. When the resistance to horizontal motions and the associated dissipation is made small by means of the anisotropy, large horizontal scales become most unstable. The general conclusion is that convection cells of large horizontal size compared to the layer depth are possible whenever the dissipation associated with horizontal motions is small, regardless of what material property of the systme is responsible for this reduction in dissipation. The models suggest that the low-viscosity zone believed to exist at the base of the lithospheric plates may play an important role in allowing plate tectonics to exhibit a variety of horizontal scales.