# What is the minimum temperature difference to drive mantle convection?

The condition for convection to cease is that the Rayleigh number $$\mathrm{Ra}=\frac{g\rho\alpha L^3\Delta T}{\kappa\nu}\lessapprox 10^3.$$ For the Earth's mantle I have seen estimates of $$\mathrm{Ra}\approx 10^6$$ to $$10^8$$, which naively would mean we need to reduce $$\Delta T$$ by a factor of $$10^3$$ to $$10^5$$ for convection to cease. In reality this is an overestimate since the thermal expansion $$\alpha$$, the thermal diffusivity $$\kappa$$ and especially the viscosity $$\nu$$ are temperature dependent.

However, being a non-geologist I find it hard to find easy functional forms of these parameters in the literature since they also depend on a number of other things like pressure and are subject to big empirical uncertainties. Are there any canonical estimates for the necessary temperature difference, or qualitative models?