Past exam of the geophysical fluid dynamics course of the University of Oxford / 2025 / 2 / a / Solution

Define the salinity contrast $\Delta S=S_1-S_2$, positive when the warm box is saltier. The linear equation of state gives the cold-minus-warm density contrast and signed volume fluxHere $q>0$ represents warm upper water travelling poleward and cold deep water returning equatorward. The $\beta$ in this question is the haline density coefficient, unrelated to the planetary vorticity gradient in Question 1. The imposed heating and cooling are taken to maintain $\Delta T$.

For the positive circulation shown in the source diagram, the exact stationary salt budget of the cold box isThe precipitation adds no salt. Replacing the salinity multiplying the small compensating volume flux by a reference $S_0$ gives $q\Delta S\simeq E{S}_0$. When circulation reverses, the upstream box for the exchange reverses too: the homogenizing exchange is $|q|$, not $q$. In the same reference-salinity approximation the two stationary salt budgets reduce toFor example, when $q=-r<0$ and $r>E$, the exact cold-box balance is $(r-E)S_1-r{S}_2=0$, giving $r\Delta S=E{S}_1$ and the same leading approximation.

Substitute $\Delta S=E{S}_0/|q|$ into the density-driven volume flux and multiply by $|q|$:Both $A$ and $B$ have the units of volume flux, so every term has units of squared volume flux. The absolute value expresses the fact that transport in either direction reduces the salinity contrast, while the density contrast sets its direction.

The requested equation is thus the reference-salinity, or virtual salt flux, version of the two-box overturning model. If the finite compensating flow is retained exactly with unequal $S_1,S_2$, one has the appropriate upstream salinity on the right-hand side instead of exactly $S_0$. The approximation assumes $|\Delta S|/S_0\ll 1$ and $E/|q|\ll 1$. Also, for $E>0$, $q=0$ cannot be a steady solution: the imposed freshwater transport would build an uncompensated salinity contrast.