Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 43 / 4

For stationary horizontally homogeneous means, incompressibility gives ${\bar{w}}_z=0$ and the impermeable boundary makes $\bar{w}=0$ throughout. With no imposed horizontal mean pressure gradient, the balances reduce toThe last equation includes the vertical Reynolds stress; omitting it requires an additional small-normal-stress approximation. The total vertical buoyancy flux is $B_0$ and the horizontal momentum flux is a signed constant $J_M$ with $|J_M|=U_{\ast }^2$. These are fluxes per unit reference density.

The local down-gradient closure defines eddy viscosity and eddy diffusivity byA parcel moving upwards typically retains the smaller momentum or scalar value from its previous height in an increasing mean profile; its fluctuation then correlates negatively with upward motion. This motivates positive coefficients, but is a closure assumption, not a universal theorem about turbulent or convective fluxes. The mean gradients obeyFor positive applied traction on a fluid above a bottom boundary, $J_M=+U_{\ast }^2$ and the mean velocity decreases away from the forcing boundary. The usual atmospheric convention specifies the stress absorbed by the boundary and has the opposite sign of $J_M$; scalar and Richardson-number results are unchanged. At large Reynolds number and Péclet number, outside the molecular wall sublayers, $K_M\gg \nu$ and $K_B\gg \kappa$, so molecular transport can be neglected in these flux-gradient relations. Large bulk numbers do not justify doing this exactly at a smooth solid wall.

Choose the length convention ${\ell }_m=-U_{\ast }^3/B_0$; a dimensionless multiplier can be absorbed into the similarity functions. Dimensional analysis under the stated local dependence givesTo obtain a log-linear expansion one also needs a regular neutral limit: take $f_M=a_M+d_M\zeta +O({\zeta }^2)$ and $f_B=a_B+d_B\zeta +O({\zeta }^2)$, with positive $a_M,a_B$. Dimensions alone do not specify these constants or guarantee differentiability. Let $s_u=-J_M/U_{\ast }^2$ denote the shear sign. In the high-number regime of part (a), integrate the reciprocal expansions from a reference height $z_r>0$ to obtain the near-neutral log-linear surface-layer profilesTheir leading terms are logarithmic; the first buoyancy-dependent corrections are linear in height. The additive constants require boundary or reference data and cannot be obtained from the prescribed fluxes alone.

The gradient Richardson number isThus $|z/{\ell }_m|\ll 1$ is nearly neutral, shear-dominated turbulence: the buoyancy contribution is small relative to shear production. Stable flux has $B_0<0$, ${\ell }_m>0$ and positive Richardson number; heating has $B_0>0$, ${\ell }_m<0$ and negative Richardson number. The smallness condition is on the magnitude, not an automatically satisfied signed inequality for every negative $z/{\ell }_m$.

Use positive jumps $\Delta T=T-T_{\infty }$ and $\Delta S=S-S_{\infty }$ for the hot salty lower reservoir. The Boussinesq density excess below the interface is proportional to $\beta \Delta S-\alpha \Delta T$. Static stability therefore requireswith one the neutral limiting value. For $R_{\rho }<1$ the lower layer is lighter overall and ordinary overturning convection replaces the statically stable interface.

In the stable hot-below, salty-below configuration, thermal diffusivity exceeds salt diffusivity. Heat transfer can destabilize thin adjacent fluid regions before the stabilizing salt difference is erased, causing the diffusive regime of double-diffusive convection. As $R_{\rho }\to \infty$, stabilization by salt dominates, interfacial convective exchange is strongly suppressed, and the specified heat-flux model tends to zero at fixed $\Delta T$. This limit is not a claim that molecular heat conduction itself ceases.

Let $T_0=T_i-T_{\infty }>0$, $S_0=S_i-S_{\infty }>0$ and $R_0=\beta S_0/(\alpha T_0)$. At steady volume, the weir outflow equals the inflow $Q$. The well-mixed heat and salt balances giveWith $\theta =(T_i-T)/T_0$ and $\eta =(S_i-S)/S_0$, the flux ratio implies ${\eta }_s=q{\theta }_s$, where $q=R_F/R_0$. The actual interface ratios are $\Delta T=T_0(1-\theta )$, $\Delta S=S_0(1-\eta )$ and $R_{\rho }=R_0(1-\eta )/(1-\theta )$.

The original PDF uses $R_{\rho }^{-2}$ in the heat-flux law; the converted TeX's exponent $-3$ is a transcription error. Using the PDF yieldsSubstitute this into the heat balance to obtain the double-diffusive overflow reservoir relationsFor the usual heat-dominated positive upward buoyancy flux, $0\le R_F<1$ and $R_0>1$ give $0\le q<1$. Then the right-hand side after substituting ${\eta }_s=q{\theta }_s$ decreases from $C$ to zero as ${\theta }_s$ runs from zero to one; there is a unique physical steady root. The net upward buoyancy flux is $B_0=g\alpha F_T(1-R_F)$.

With inflow and overflow both zero and fixed reservoir volume $V$, the transient balances are $V\dot{T}=-A{F}_T$ and $V\dot{S}=-A{F}_S$. ThereforeThe last equality uses the initial steady state. Define $k_c=Ab(\alpha T_0{)}^{1/3}/(V{R}_0^2)=CQ/V$, where the latter $Q$ is the pre-shutdown flow. The post-shutdown double-diffusive reservoir cooling equation isSeparate variables. An explicit primitive of $(1-q\theta {)}^2/(1-\theta {)}^{10/3}$ isDifferentiate it to verify the three powers. HenceFor $q<1$, $\theta$ increases monotonically to one only at infinite time, with $1-\theta \sim [3(1-q{)}^2/(7k_{c}t){]}^{3/7}$ up to the choice of time origin. The reservoir approaches $T_{\infty }$, but its salinity tends to $S_i-q{S}_0$, leaving the positive salt contrast $(1-q)S_0$. Thus $R_{\rho }\to \infty$ and the double-diffusive flux shuts down asymptotically. The assumption of a stationary interface and continued applicability of this flux parameterization is part of the reservoir model, not a claim about every actual hydrothermal pool.