Four-thirds convective heat-transfer law ($q\propto (\Delta T{)}^{4/3}$)

A high-Rayleigh number closure $\mathrm{Nu}=C{\mathrm{Ra}}^{1/3}$ gives $q=C{k}_T[g\beta /(\nu \kappa ){]}^{1/3}(\Delta T{)}^{4/3}$, independent of layer depth. It assumes boundary-layer-controlled heat transfer, a well-mixed interior, fixed material properties and a fixed Prandtl number regime. It is a model, not a universal asymptotic law for every turbulent convection regime or for an impulsive initial conductive transient.

A well-mixed layer of depth $h$ losing heat through an unstable cold upper boundary has bulk excess temperature $\theta =T-T_0$. With four-thirds convective heat-transfer law flux $q=Ck[g\beta /(\nu \kappa ){]}^{1/3}{\theta }^{4/3}$, its heat capacity per area is $kh/\kappa$. Thus $\dot{\theta }=-K{\theta }^{4/3}$ with $K=C\kappa [g\beta /(\nu \kappa ){]}^{1/3}/h$, giving the displayed cooling law. It excludes the initial conductive transient and the late low-Rayleigh conduction regime.