With thermal diffusivity $\kappa_T$, temperature obeys
$$
T_t+uT_x+wT_z=\kappa_T(T_{xx}+T_{zz}).
$$
Scale $z$ with $h_0$, $x$ with $b_0=\epsilon h_0$, and $w$ with $w_0$. Continuity gives the cross-plume velocity scale $u_0\sim(b_0/h_0)w_0=\epsilon w_0$, 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
$$
\boxed{\frac{\text{vertical diffusion}}{\text{horizontal diffusion}}=\frac{b_0^2}{h_0^2}=\epsilon^2\ll1}.
$$


Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-345/4/b/solution