Figure 3.3 (also see Plate 6) shows that the steady states of the two-box model can be calculated analytically and the dependence of the flux q (a measure of the strength of the THC) on the freshwater flux H can be examined. In the standard case (weak diffusion, lower curve), the THC is strongest (value arbitrarily set to 1) when the freshwater flux H vanishes. With higher H, the THC weakens. If H = 0.3, no steady-state solution with q > 0 is possible. However, for H > 0.1, there is a second stable equilibrium, a reverse mode of the THC, with q < 0. This flow pattern strengthens as H increases. For a certain parameter range, here 0.1 < H < 0.3, three equilibria are possible; it is readily shown that the middle one (on the dotted part of the curve) is not a stable solution. In that range of H, the model exhibits hysteresis.
The presence of hysteresis is strongly dependent on model parameters. This is shown for a case in which the effect of the horizontal mixing due to gyre transports is increased (strong diffusion). Hysteresis disappears (upper curve); and for progressively increasing freshwater forcing, the THC smoothly approaches zero and—again smoothly—turns into the reverse mode for H = 0.56. For any given freshwater forcing, there is a unique and stable THC. The presence of multiple equilibria of the THC therefore depends strongly on the model formulation, parameterization of processes, and choice of parameter values.