The response of the THC to a perturbation of the surface freshwater balance can be illustrated in the two-box model of Stommel (1961), which is explained in Box 3.2. A new addition to the original system is horizontal diffusion. It reflects the transport of salinity by the ocean gyres, the midlatitude systems of ocean circulation characterized by swift currents near the western boundary (for example, the Gulf Stream) and slower return flows farther eastward, occurring at virtually the same depth. Diffusion has a dramatic influence on the structure of the solutions (Box 3.3). For weak diffusion, the model exhibits hysteresis, which enables the existence of abrupt change in response to slow changes in the forcing; strong diffusion eliminates hysteresis (Plate 6). As shown in Plate 6, from their arbitrary starting point, both models initially migrate toward the steady state for the freshwater forcing H = 0.2, where large H indicates large transfer of freshwater through the atmosphere from the low-latitude to the high-latitude ocean (or its computational equivalent, large transfer of salt through the atmosphere from the high-latitude to the low-latitude ocean). When H is increased slowly, both models follow their equilibrium curves, as indicated by the arrows. The standard model has a threshold at H = 0.3, and an abrupt transition towards the other, “reverse” equilibrium occurs (orange curve). From then on, the model system follows the lower equilibrium curve; the hysteresis is shown by the orange curve remaining on the lower, red branch, even after the freshwater forcing has returned to its original value of 0.2. It would require a reduction of H to below 0.1 to force a return to the upper branch; if H stays above 0.1, changes remain permanent even after the perturbation of H is removed. The response of the diffusive model to changes in H is completely different (green curve). At each instant, the change in the THC scales with the forcing and no abrupt transition is observed. This model version has only one equilibrium solution for any given freshwater flux H, and, in contrast with the version with multiple equilibria, it exhibits only reversible changes (Plate 6).
A different way of illustrating this behavior is depicted in Figure 3.4, which shows the time evolution of the THC in response to a slow increase in freshwater forcing followed by an equally slow decrease. The diffusive model approaches zero THC strength essentially on the time scale of the change in forcing. In contrast, the standard case starts out with a slow