Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 332 (Fluid Dynamics of the Solid Earth)

Write the given pressure-dependent ratio of permeability of a porous medium to sponge thickness aswhere $K$ is constant. The fluid pressure drop across the sponge is the hydrostatic pressure of the current,because the air pressures above and below are equal. Therefore Darcy's law gives the downward volume flux per unit horizontal area asThus the sponge drainage obeys a power law in the current thickness, as described by pressure-dependent Darcy drainage.

Let $z$ measure height above the sponge. Under the long-wave approximation, hydrostatic pressure gives $p_x=\rho g{H}_x$, and the horizontal balance for viscous fluid flow isThe no-slip boundary condition $u(0)=0$ and stress-free boundary condition $u_z(H)=0$ giveIntegrating this lubrication theory profile gives the horizontal volume flux per unit span

Local mass conservation includes the downward loss found in part i:Consequently the required nonlinear partial differential equation isIf the imposed inlet flux is $Q_0{t}^{\alpha }$ and the moving front is $x=x_N(t)$, sufficient boundary conditions areFor a current released onto a dry substrate one also takes the initial condition $H(x,0)=0$ away from the source. The front position is part of this moving-boundary problem.

Seek a similarity solution in the dimensionally completed formwhere the fixed scales absorb $A,D$, and $Q_0$. Its inlet volume flux scales asso the prescribed source requiresThe time derivative, horizontal lubrication theory term, and pressure-dependent Darcy drainage term scale respectively asEquating the three exponents givesSolving these linear equations together with the inlet condition yields the similarity exponents of a draining gravity currentEquivalently,and, up to constant dimensional scales,

For $\alpha =2$, part iii gives $a=1$, $b=2$, and $\beta =1$. PutSubstitution into the nonlinear partial differential equation gives the ordinary differential equationwith boundary conditions

To find the leading edge, set $y={\eta }_N-\eta$ and suppose $F\sim C{y}^p$. The two singular terms in the ordinary differential equation have orders $y^{4p-2}$ and $y^{p-1}$. Their exponents agree only when $p=1/3$. Their leading coefficients then satisfywhereas $F$ and the constant drainage $D$ are lower-order terms. HenceThe draining gravity current therefore has a one-third-power leading edge and its flux vanishes there.

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.

Let $d(z,t)$ be the local water-film thickness. To first order in the interface amplitudes,The lubrication theory flux down the vertical surface, with a no-slip boundary condition at the ice and a stress-free boundary condition at the water-air interface, isFor the unperturbed film $p_z=0$, so

The linearized Young–Laplace equation gives the capillary-pressure perturbationBecause the prescribed volume flux is uniform, its first-order perturbation must vanish. Linearization of the flux law givesThus, with $\Gamma =\gamma /(3\rho g)$,and henceThe complex amplitude ratio records the phase shift caused by surface tension in the long-wave approximation.

The unperturbed temperature is linear in each material. Continuity of the conductive heat flux at $x=h$ givesOn definingthis becomes $T_m-T_h=ϵ(T_h-T_a)$. Therefore

Let ${\rho }_i$ be the ice mass density and $L_f$ its latent heat of fusion per unit mass. The ice is isothermal in this model, so the Stefan condition equates latent-heat production to the heat flux conducted through the water:Consequently the unperturbed lateral solidification speed isThe material parameters are the thermal conductivities $k_w,k_a$, ice density ${\rho }_i$, and specific latent heat $L_f$; the geometric thermal length is $\delta$.

Write the base-state gradients asThe base heat flux and Stefan condition relations are

Under the quasi-stationary approximation, and with advective heat transport neglected, each temperature perturbation satisfies the Laplace equation. The decaying normal-mode solutions areKeeping the displaced ice interface at $T_m$ omits the curvature correction described by the Gibbs--Thomson relation.
The fixed melting temperature at the displaced ice interface givesExpanding temperature and conductive heat flux continuity at the displaced outer interface gives

Put $r=k_a/k_w$. Since $G_a=G_w/r$, the temperature condition in the limits $r\ll 1$ and $\alpha h\ll 1$ givesAlthough $r$ is small, the product $rC$ is therefore order one and cannot be discarded. The flux condition then givesThis is why the stated asymptotic warning matters.

The perturbed Stefan condition at $x=0$ isUsing $B\sim {\eta }_2{G}_w$ and $-k_w{G}_w={\rho }_i{L}_{f}V$ yieldsSubstitution of the interface-amplitude ratio from part i produces the thin-film icicle-ripple instability dispersion relation

Set $b=\Gamma h{\alpha }^3$. Rationalizing this complex number givesso its growth rate and imaginary part areDifferentiating the growth rate with respect to the wavenumber shows that its positive stationary point satisfiesIt is the unique global maximum, and henceAt this wavenumber $b=1/\sqrt{5}$. With the convention $e^{i\alpha z+\sigma t}$, a constant phase travels with phase velocityThereforeBecause $z$ increases downward, the negative sign means that the icicle ripples migrate upward with speed $\sqrt{5}V/6$.