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

The density anomaly of water makes mass density increase with temperature up to $T_m$ and decrease above $T_m$. Since temperature increases with depth in the stated layer, the density profile first increases to its maximum ${\rho }_m$ at the $T=T_m$ isotherm and then decreases toward the warm bottom.

The upper region has denser water below lighter water and hence stable density stratification. Below the density maximum, density decreases downward, so lighter water lies beneath denser water and gives unstable density stratification. The lower region therefore overturns by thermal convection, while the cold upper region remains a stagnant cap. This coexistence is penetrative convection.

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.

Setso $0<x<\Delta$. Apart from material constants, the heat flux in part ii isThe derivative of its logarithm vanishes whenwhich gives the maximizing interface temperatureSubstitution into the flux law giveswhereThus the proportionality coefficient depends on the thermal conductivity, nonlinear thermal-expansion coefficient $\alpha$, kinematic viscosity, thermal diffusivity, gravity, and critical Rayleigh number. This is the maximum-flux law for penetrative convection in an ice-covered lake.

The stagnant layer transports heat by thermal conduction, so its upward heat flux isAt the maximum-flux interface from part iii,Equating $F_s$ to $F_{\max }$ givesThe stagnant layer has positive thickness only ifFor an ice-covered freshwater lake, take $T_s\simeq 0{}^{\circ }\mathrm{C}$ and $T_m\simeq 4{}^{\circ }\mathrm{C}$. The largest interior temperature compatible with this penetrative convection in an ice-covered lake model is therefore

Define the dimensionless stagnant-layer depth and time byso $H^2/\kappa$ is the thermal diffusion time. The maximizing interface temperature givesEquating conductive and convective heat fluxes therefore gives the algebraic relation

Since $k=\rho c_p\kappa$, where $c_p$ is the specific heat capacity, the bulk heat balance in the convecting depth $H-h$ becomesRetaining the heat capacity of the thin stagnant layer changes this only by relative order $\eta$.

For $\mathcal{F}\gg 1$ and $\theta =O(1)$, the flux relation gives $\eta =O({\mathcal{F}}^{-1})$. The leading equation is consequentlySeparating variables and using $\theta (0)=1$ giveshence

The approximation in part v requires the dimensionless stagnant depthto remain small. Since $0<\theta \le 1$ makes the numerator order one, this condition isUsing the solution from part v, the equivalent time range isor, at the level of asymptotic equivalence,The cooling becomes appreciable on the shorter scale $\tau =O({\mathcal{F}}^{-1})$, so these ranges overlap widely when $\mathcal{F}\gg 1$.

The Rayleigh number based on the convecting depth isThe definition of $\mathcal{F}$ implicit in part iii givesIt follows thatIf $1-\eta =O(1)$ and $\theta \gg {\mathcal{F}}^{-3/5}$, then $\mathrm{Ra}\gg O({\mathcal{F}}^{9/5})$. The thermal convection therefore remains strongly supercritical throughout the range in which the thin stagnant-layer approximation is valid.