= Solution
\b[Illustration TODO.] The source's two-volume circulation sketch on page 4 is unavailable. Its exact arrows and layout cannot be recovered. The surviving flow law, <salinity>/<temperature> definitions and printed steady polynomial allow the calculation below under an explicit freshwater sign convention; no unavailable diagram values are invented.
Use $\Delta T=T_2-T_1>0$, $\Delta S=S_2-S_1$, and let positive $q$ be the thermally driven state with cold box 1 denser than warm box 2. Take positive $F$ to freshen box 1, represented in a fixed-volume <salt budget> by an equivalent salt removal $FS_1$ from that box and addition to the other reservoir. This is the freshwater convention compatible with the printed $+BF$ and positive $B$. It can represent a compensating saline <volume> transfer or the usual virtual-salt-flux approximation. The exact pipe assignment is not claimed to have been read from the missing illustration.
Exchange in either circulation direction reduces a <salinity> difference at rate proportional to $|q|\Delta S$. Steady salt balance therefore gives
$$
\boxed{|q|\Delta S=FS_1,\qquad \Delta S=\frac{FS_1}{|q|}}\qquad(q\ne0).
$$
The supplied linear <equation of state> gives
$$
\frac{\rho_1-\rho_2}{\rho_0}=\alpha\Delta T-\beta\Delta S,
\qquad q=k(\alpha\Delta T-\beta\Delta S).
$$
Eliminating $\Delta S$ yields the <two-box overturning model> equation
$$
\boxed{q|q|-A|q|+BF=0,\qquad A=k\alpha\Delta T,\qquad B=k\beta S_1}.
$$
Here $A$ and $B$ both have transport units when $q,F,k$ do, while $\alpha\Delta T$ and $\beta S_1$ are dimensionless. Although the source calls $q$ a mass flow, its stated Sverdrup units are <volume> per time; the common reference <density> cancels in the Boussinesq salt/<volume> approximation. At $F>0$, $q=0$ cannot be a finite-salinity steady state in this model. At $F=0$, zero flow needs separate treatment instead of division by $|q|$.
The circulation weakens when freshwater freshens the cold reservoir, and the weaker circulation removes the resulting <salinity> contrast less effectively. This <salt-advection feedback> is the mechanism behind multiple steady states, not an additional externally imposed switch.
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