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Use all coefficients exactly as printed and use Sv consistently for transports. The present salt contrast and density-driven friction coefficient areThusThe large numerical value of $k$ reflects division by a small dimensionless density contrast, about $2.07\times 10^{-6}$ 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 rootsThe critical freshwater forcing in a two-box overturning model occurs where the two roots coalesce:There is also a negative-flow branch for every $F>0$: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 writtenwhere $V_{\mathrm{eff}}>0$ summarizes the reservoir volumes. The linear slope at a positive root is $(A-2q)/V_{\mathrm{eff}}$: the upper branch is stable and the lower branch unstable. At a negative root it is $(2q-A)/V_{\mathrm{eff}}<0$, 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.