Marginal-stability thermal boundary-layer scaling ($\delta \propto {\theta }^{-1/3},E\propto {\theta }^{4/3}$)

For constant viscosity and thermal properties, marginal local Rayleigh number gives $\rho g\alpha \theta {\delta }^3/(\eta \kappa )\sim R_c$. Conduction through the resulting thermal boundary layer gives $E\sim k\theta /\delta \propto {\theta }^{4/3}$. This determines exponents, not an exact prefactor. The closure applies when the layer is thin compared with the convecting interior and its temperature contrast is positive.