Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 332 / 2 / ii / Solution

Let ${\delta }_T$ be the thickness of the thermal boundary layer immediately below $z=h$, let $\nu$ be the kinematic viscosity, let $\kappa$ be the thermal diffusivity, and let ${\mathrm{Ra}}_c$ be the critical local Rayleigh number. The density contrast driving the boundary layer follows from the density anomaly of water:A Local Rayleigh-number closure setsand thereforeBy Fourier's law, the upward heat flux through this layer iswhere $k$ is the water's thermal conductivity. This expression applies while the quantity inside braces is positive.