The stagnant layer transports <heat> by <thermal conduction>, so its upward <heat flux> is
$$
F_s=\frac{k(T_h-T_s)}h.
$$
At the maximum-flux interface from part iii,
$$
T_h-T_s=(T_m-T_s)-\frac35(T_w-T_m).
$$
Equating $F_s$ to $F_{\max}$ gives
$$
\boxed{
h=\frac1{C_0}
\left(\frac{\operatorname{Ra}_c\nu\kappa}{g\alpha}\right)^{1/3}
\frac{(T_m-T_s)-\frac35(T_w-T_m)}
{(T_w-T_m)^{5/3}}}.
$$
The stagnant layer has positive thickness only if
$$
T_w<T_m+\frac53(T_m-T_s).
$$
For an ice-covered freshwater lake, take $T_s\simeq0\,{}^\circ\mathrm C$ and $T_m\simeq4\,{}^\circ\mathrm C$. The largest interior temperature compatible with this <penetrative convection in an ice-covered lake> model is therefore
$$
\boxed{T_{w,\max}\simeq4+\frac53(4)
=10.7\,{}^\circ\mathrm C}.
$$

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-332/2/iv/solution