Two-box overturning model ($q|q|-A|q|+BE=0$)

A reduced model with warm and cold mixed reservoirs whose signed exchange volume flux is proportional to their density difference. For fixed temperature contrast and reference-salinity freshwater forcing, the stationary salt budget is $|q|\Delta S=E{S}_0$. A linear equation of state then gives $q|q|-k\alpha \Delta T|q|+k\beta S_{0}E=0$. Positive circulation has two branches below a saddle-node bifurcation; the stronger positive branch and the reversed branch are stable under the usual salt budget dynamics.

For $q|q|-A|q|+BF=0$ with positive $A,B$, the two positive roots $q_{\pm }=(A\pm \sqrt{A^2-4BF})/2$ merge at the displayed saddle-node bifurcation. The negative root $q_{\mathrm{rev}}=(A-\sqrt{A^2+4BF})/2$ exists for every positive $F$. Under standard fixed-temperature salt-contrast dynamics, the stronger positive and reversed branches are stable, while the weaker positive branch is unstable. A linear ramp $F=F_0(1+st)$ reaches the fold after $(F_c/F_0-1)/s$, but the equilibrium flow changes before that time; a specified closeness tolerance generally gives an earlier limit.

A circulation change alters the salinity contrast and hence the density difference that supports the circulation. In a thermally dominated positive branch, weaker exchange removes the imposed salinity contrast less effectively, making the high-latitude reservoir fresher and weakening the circulation further. This positive feedback permits bistability and a saddle-node bifurcation.