The thermohaline hysteresis response to varying North Atlantic freshwater forcing is studied by means of a three-dimensional global ocean general circulation model (OGCM). The influence of vertical diffusivity is examined using a wide range of mixing coefficients. For sufficiently large vertical diffusivity the model shows a pronounced hysteresis behaviour, so that two equilibrium states of the thermohaline circulation are found: one mode with intense deep water formation in the North Atlantic ("on" mode), and one mode with strongly reduced convective activity ("off" mode). By decreasing the diffusivity, the two branches of the hysteresis merge. In addition, the effect of numerical diffusion is analysed by applying different advection schemes. A positive feedback between overturning strength and the numerical diffusivity of upstream advection is found, resulting in a reduced stability of the "on" mode. Comparing the results with those from zonally averaged models exhibits substantial differences regarding the stability properties of the thermohaline circulation. Freshwater transports by horizontal gyres have an important effect on the overturning circulation in the OGCM.