= Solution
Use all coefficients exactly as printed and use Sv consistently for transports. The present salt contrast and density-driven <friction> coefficient are
$$
\Delta S=\frac{0.5\times35}{15}=1.1667\ \mathrm{g\,kg^{-1}},\qquad
k=\frac{15}{(2\times10^{-7})(15)-(8\times10^{-7})(1.1667)}.
$$
Thus
$$
\boxed{k=7.26\times10^6\ \mathrm{Sv}=7.26\times10^{12}\ \mathrm{m^3\,s^{-1}},\quad
A=21.774\ \mathrm{Sv},\quad B=203.226\ \mathrm{Sv}}.
$$
The large numerical value of $k$ reflects division by a small dimensionless <density> contrast, about $2.07\times10^{-6}$ with the supplied effective expansion coefficients. It is not the actual overturning transport, which remains $15$ Sv.
For $q>0$, the steady equation is a quadratic with roots
$$
\boxed{q_\pm=\frac{A\pm\sqrt{A^2-4BF}}2},\qquad 0\le F\le F_c=\frac{A^2}{4B}.
$$
The <critical freshwater forcing in a two-box overturning model> occurs where the two roots coalesce:
$$
\boxed{F_c=0.58324\ \mathrm{Sv},\qquad q_c=A/2=10.887\ \mathrm{Sv}}.
$$
There is also a negative-flow branch for every $F>0$:
$$
\boxed{q_{\rm rev}=\frac{A-\sqrt{A^2+4BF}}2<0}.
$$
At the present forcing, the three roots are $15.000$, $6.774$ and $-3.950$ Sv. The prescribed present state is therefore the upper positive branch, not an arbitrary choice among the roots.
Stability needs a <salinity> evolution law, not just the algebraic steady equation. For fixed <temperature> contrast and reference <salinity>, the standard salt-contrast dynamics can be written
$$
\dot{\Delta S}=\frac{FS_1-|q|\Delta S}{V_{\rm eff}},\qquad
\dot q=-\frac{q|q|-A|q|+BF}{V_{\rm eff}},
$$
where $V_{\rm eff}>0$ summarizes the reservoir volumes. The linear slope at a positive root is $(A-2q)/V_{\rm eff}$: the upper branch is stable and the lower branch unstable. At a negative root it is $(2q-A)/V_{\rm eff}<0$, so the reversed branch is stable. The two stable states below the fold have different attraction basins, demonstrating <bistability>. Their stability is conditional on this fixed-temperature, fixed-reference-salinity closure.
As forcing increases quasistatically from $0.5$ Sv, the normal branch weakens and reaches a <saddle-node bifurcation> at $F_c$. Beyond it no positive steady circulation remains, so the reduced dynamics move toward the stable reversed branch. The reversal is <salinity> dominated: the fresh cold reservoir is no longer the denser one. Recovery under a later forcing change can depend on the system's state and feedback history.
\Image[/past-exam-of-the-physics-of-atmospheres-and-oceans-course-of-the-university-of-oxford/2015-two-box-circulation-branches.png]
{title=Steady transports and their stability in the explicitly stated freshwater convention; present state and critical freshwater forcing are marked}
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