Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 76 / 2

The Rayleigh number compares destabilizing thermal buoyancy with viscous and thermal diffusion:The conductive heat flux is $k\Delta T/h$, and the Nusselt number is the actual upward heat flux divided by this value. The four-thirds convective heat-transfer law is the empirical large-Rayleigh closure $\mathrm{Nu}=C{\mathrm{Ra}}^{1/3}$. It givesThe layer depth cancels: thin thermal boundary layers control the resistance while convection mixes the interior. The dimensionless coefficient $C$ depends on the boundary and material-property regime; the two boundary-layer temperature drops can be absorbed into its convention. This is not a universal law for all Rayleigh-Bénard convection. Below convective onset the stable conductive solution instead has $\mathrm{Nu}=1$, and an impulsive initial conductive transient is not described by the steady large-Rayleigh closure.

Cooling from above makes dense cold fluid overlie lighter warm fluid, so the upper boundary layer is unstable and can mix the interior. Assume high Rayleigh number, rapid interior mixing, constant properties, negligible wall heat capacity and a quasi-steady cooling flux obeying the four-thirds convective heat-transfer law. Write $\theta (t)=T_{\mathrm{bulk}}(t)-T_0$ and use a coefficient $C$ appropriate to this single cooling boundary. The volumetric heat capacity is $\rho c_p=k/\kappa$, so the area-wise heat balance isWith $K=C\kappa [g\beta /(\nu \kappa ){]}^{1/3}/h$, integration gives convective cooling under the four-thirds heat-transfer law:The power-law decay is a bulk approximation between the initial boundary transient and the later stage when the decreasing temperature contrast no longer maintains strong convection. It is not an exact spatially uniform solution satisfying the cold boundary itself.

Cooling from below is different: dense cold fluid now lies below warm fluid and stabilizes the layer. With no other imposed motion, cooling is by thermal conduction, and temperature becomes depth dependent. Let $z=0$ be the cooled bottom and $z=h$ the insulating top. Solve ${\theta }_t=\kappa {\theta }_{zz}$ with $\theta (0,t)=0$, ${\theta }_z(h,t)=0$ and $\theta (z,0)=\Delta T$. The mixed-boundary eigenfunctions have ${\lambda }_n=(n+1/2)\pi /h$, givingIts volume-averaged temperature isThe late conductive decay is exponential on the timescale $4h^2/({\pi }^2\kappa )$, rather than the convective $t^{-3}$ bulk law. This assumes the usual positive thermal expansion coefficient and no destabilizing compositional stratification.