Low-resolution global ocean models display hysteresis behaviour when forced with an anomalous freshwater input into the northern North Atlantic. Of central importance in this hysteresis behaviour is the existence of so-called saddle-node bifurcations. In this paper, focus is on the physical characterization of the multiple equilibrium regime using a fully implicit global ocean model for which bifurcation diagrams can be explicitly computed. The physics of the position of the relevant saddle-node bifurcation in parameter space is clarified and a modification of an earlier suggested diagnostic of the presence of the multiple equilibrium regime is proposed. The relevance of this indicator for coupled climate models is shown by studying the sensitivity of the multiple equilibria regime to changes in the horizontal and vertical diffusivities of the model.