Past exam of the year 2 of the earth sciences (geology) course of the University of Oxford / 2022 / Paper 1 / 10 / a

There are four well-mixed reservoirs: Mediterranean surface/deep $(M_s,M_d)$ and Black Sea surface/deep $(B_s,B_d)$. Use the actual drawing’s arrow endpoints: $F_2$ leaves $M_s$, $F_5$ enters $M_d$ from $B_s$, and $F_6$ starts in $M_s$ and enters $B_d$. With $F_i$ as positive water-mass transport rates, the four steady conservation of mass equations areAdding them cancels every internal transport and gives the whole-system requirement $F_1+F_9=F_2$. There is no stated evaporation, precipitation or other external transport to add.

The schematic endpoints are unusual for the conventional two-layer exchange picture: its Gibraltar outflow is drawn in the surface box, and its Black Sea surface inflow descends into the Mediterranean deep box. Moving $F_2$ to $M_d$, $F_5$ to $M_s$, and the source of $F_6$ to $M_d$ would define a different, more conventional idealization. The displayed equations consistently follow the supplied drawing; the same endpoint convention is used in the following two parts.

Let $S_{Ms},S_{Md},S_{Bs},S_{Bd}$ be salt mass fractions, with reservoir water masses denoted by the corresponding $M$ symbols. Let $S_A$ be Atlantic inflow salinity and $S_D$ the Danube inflow salinity, approximately zero for fresh water. With the diagram endpoint convention from part i, instantaneous mixing givesEvery outgoing salt transport uses the salinity of its donor box. The water masses are constant because the water balances are at steady state, even while the salinities evolve. If reservoir water masses varied, the left side would instead be $d(M_j{S}_j)/dt$.

Summing yields $d({\sum }_j{M}_j{S}_j)/dt=F_1{S}_A+F_9{S}_D-F_2{S}_{Ms}$, a direct whole-system salt check. Fresh water carries essentially no salt, but its dilution effect is still present through the water balance and the resulting transport terms.

For a constant-mass, well-mixed box the mean residence time is stored water mass divided by its total outgoing water-mass transport. Using the source figure’s endpoints,At water steady state, each denominator can equivalently be replaced by that box’s summed inflow. If transports are specified as volume rates, use volume rather than mass in the numerator, or convert with density consistently. The drawn maximum basin depths alone do not supply layer volumes, plan areas or transport magnitudes, so no numerical residence times follow. Internal recycling counts as exit from a particular box; these times are not the time to escape the combined sea system.