Critical freshwater forcing in a two-box overturning model ($F_c=A^2/(4B),q_c=A/2$)

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.