Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 345 / 4 / b / Solution

With thermal diffusivity ${\kappa }_T$, temperature obeysScale $z$ with $h_0$, $x$ with $b_0=ϵh_0$, and $w$ with $w_0$. Continuity gives the cross-plume velocity scale $u_0\sim (b_0/h_0)w_0=ϵw_0$, so both advective terms scale as $w_0\Delta T/h_0$. Horizontal and vertical diffusion scale as ${\kappa }_T\Delta T/b_0^2$ and ${\kappa }_T\Delta T/h_0^2$, respectively. Their ratio is