mechanical analogue of such behavior; Box 3.1 presents a sketch of hysteresis of the THC resulting from perturbations in the freshwater forcing.

The simplest variant of a hysteresis loop is a useful tool to discuss the different possibilities for how the climate system can respond to changes in some controlling variable. More freshwater increases the buoyancy of the surface waters and tends to reduce the strength of the THC; less freshwater makes the THC more stable. This is illustrated by the change in the location of the system on the upper branch of the hysteresis. As long as perturbations do not exceed thresholds, the system response is weak (Figure 3.1a). An abrupt change is triggered on the crossing of a threshold. A small change in forcing can then cause large additional perturbations (Figure 3.1b). If the initial state of the ocean-atmosphere system is a unique equilibrium, the system jumps back to the original state once the perturbation has ceased; the abrupt change is reversible. However, if other equilibria exist, the perturbation can cause an irreversible change (Figure 3.1c), unless a perturbation is applied that has the opposite sense of the original one and is large enough; in Figure 3.1c, state 3 would have to be pushed to the left, beyond the upward-pointing branch.

Experiments with simplified ocean-circulation and climate models have helped to discover the possible hysteresis behavior of the atmosphere-ocean system. As shown in Box 3.1, hysteresis is one manifestation of multiple equilibria in a nonlinear system. The existence of hysteresis for the THC was first shown by Stocker and Wright (1991), who used a simplified model. For some values of the freshwater balance of the North Atlantic, the THC can be either in a strong or in a collapsed state. Numerous studies with a variety of ocean models coupled to simple representations of the atmosphere have demonstrated the existence of hysteresis (e.g., Mikolajewicz and Maier-Reimer, 1994; Rahmstorf and Willebrand, 1995); this is a robust property of such models. Obviously, in more-complex models, the hysteresis can consist of a number of sub-branches nested in a complicated way. However, it is unclear whether the hysteresis behavior would persist in more-realistic coupled models, particularly if the ocean component has spatial resolution believed to be necessary to be quantitatively consistent with observations. Likewise, it is unclear where the climate system is now on the hysteresis curve of the Atlantic THC: What is its structure? Does it have thresholds? If so, how close is the threshold? The following discussion demonstrates how model- and parameter-dependent the answer can be.

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