Viscous overturn of a dense crust (${\tau }_s\sim [{\mu }_s/(\Delta \rho gA\sqrt{\kappa }){]}^{2/3}$)

A dense viscous crust of thickness $a$ above lighter liquid can overturn on a Rayleigh-Taylor instability time ${\mu }_s/(\Delta \rho ga)$. If diffusive growth gives $a=A\sqrt{\kappa t}$, matching its age to that time gives ${\tau }_s\sim [{\mu }_s/(\Delta \rho gA\sqrt{\kappa }){]}^{2/3}$. A renewed crust then supplies average solid-volume flux $J_s\sim (\Delta \rho g/{\mu }_s{)}^{1/3}(A^2\kappa {)}^{2/3}$. This scale assumes that heat supplied by the interior does not arrest the crust before instability.