  The scaling of plate-tectonic convection is investigated by simulating
thermal convection with pseudoplastic rheology and strongly
temperature-dependent viscosity. The effect of mantle melting is also explored
with additional depth-dependent viscosity. Heat-flow scaling can be constructed
with only two parameters, the internal Rayleigh number and the lithospheric
viscosity contrast, the latter of which is determined entirely by rheological
properties. The critical viscosity contrast for the transition between
plate-tectonic and stagnant-lid convection is found to be proportional to the
square root of the internal Rayleigh number. The relation between mantle
temperature and surface heat flux on Earth is discussed on the basis of these
scaling laws, and the inverse relationship between them, as previously
suggested from the consideration of global energy balance, is confirmed by this
fully dynamic approach. In the presence of surface water to reduce the
effective friction coefficient, the operation of plate tectonics is suggested
to be plausible throughout the Earth history.
