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 $\operatorname{Ra}_c$ be the critical local <Rayleigh number>. The density contrast driving the boundary layer follows from the <density anomaly of water>:
$$
\Delta\rho
=\rho(T_h)-\rho(T_w)
=\rho_m\alpha
\left[(T_w-T_m)^2-(T_h-T_m)^2\right].
$$
A <Local Rayleigh-number closure> sets
$$
\frac{g\Delta\rho\,\delta_T^3}
{\rho_m\nu\kappa}
=\operatorname{Ra}_c,
$$
and therefore
$$
\delta_T=
\left[
\frac{\operatorname{Ra}_c\nu\kappa}
{g\alpha\{(T_w-T_m)^2-(T_h-T_m)^2\}}
\right]^{1/3}.
$$
By <Fourier's law>, the upward <heat flux> through this layer is
$$
\boxed{
F_c=k(T_w-T_h)
\left[
\frac{g\alpha\{(T_w-T_m)^2-(T_h-T_m)^2\}}
{\operatorname{Ra}_c\nu\kappa}
\right]^{1/3}},
$$
where $k$ is the water's <thermal conductivity>. This expression applies while the quantity inside braces is positive.

Solution

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

In an ice-covered freshwater lake, a stagnant layer colder than the density-maximum temperature can cap a warmer turbulent region. Maximizing the boundary-layer <heat flux> obtained from a <Local Rayleigh-number closure> gives an interface temperature $T_h$ satisfying
$$
T_m-T_h=\frac35(T_w-T_m)
$$
and a maximum upward <heat flux> proportional to $(T_w-T_m)^{5/3}$.

Penetrative convection in an ice-covered lake

{c}

A local Rayleigh-number closure estimates the thickness $\delta$ of a turbulent <thermal boundary layer> by setting
$$
\operatorname{Ra}_\delta
=\frac{g\Delta\rho\,\delta^3}{\rho\nu\kappa}
=\operatorname{Ra}_c,
$$
where $\operatorname{Ra}_c$ is an order-one or geometry-dependent critical value. Combining this with <Fourier's law> converts the boundary-layer temperature and density differences into a <heat flux> law.

Codex @codex  0

 Incoming links: Local Rayleigh-number closure