We choose to look, first, at the thermohaline circulation (THC) because it has been thoroughly examined over many years in computer simulations, and its bifurca-tional structure is quite well understood.

The remarkable global extent of the THC is well known. In the Atlantic it is closely related to, and helps to drive, the North Atlantic Current (including the Drift), and the Gulf Stream: so its variation could significantly affect the climate of the British Isles and Europe. It exhibits multi-stability and can switch abruptly in response to gradual changes in forcing which might arise from global warming. Its underlying dynamics are summarised schematically in Figure 3.3 (below) adapted from the paper by Rahmstorf et al. (2005), which itself drew on the classic paper of Stommel (1961). This shows the response, represented by the overturning strength of the circulation (q), versus the forcing control, represented by the freshwater flux (from rivers, glaciers, etc.) into the North Atlantic (¡). The suggestion is that anthropogenic (man-induced) global warming may shift this control parameter,

Overturning, q (Sv)

Overturning, q (Sv)

Subcritical |
or |
S , |
Advective | |||

pitchfork |
Fold |
✓ |
spin-down | |||

\ |
y |
- ^ « |
THC 'off' |

Freshwater forcing (Sv)

Freshwater forcing (Sv)

Figure 3.3 A schematic diagram of the thermohaline response showing the two bifurcations and the associated hysteresis cycle (Rahmstorf 2000). The subcritical pitchfork bifurcation will be observed in very simple models, but will be replaced by a fold in more elaborate ones: see, for example, Figure 3.5(b). Note that 1 Sv is 106 cubic metres per second, which is roughly the combined flow rate of all rivers on Earth.

past the fold bifurcation at a critical value of ^ = ^crit (= 0.2 in this highly schematic diagram). The hope is that by tuning a climate model to available climatological data we could determine ^crit from that model, thereby throwing some light on the possible tipping of the real climate element.

The question of where the tipping shows in models has been addressed in a series of papers by Dijkstra & Weijer (2003, 2005), Dijkstra et al. (2004), and Huisman et al. (2009) using a hierarchy of models of increasing complexity. The simplest model is a box model consisting of two connected boxes of different temperatures and salinity representing the North Atlantic at low and high latitudes. For this box model it is known that two stable equilibria coexist for a large range of freshwater forcing. The upper end of the model hierarchy is a full global ocean circulation model.

Using this high-end model, Dijkstra & Weijer (2005) applied techniques of numerical bifurcation theory to delineate two branches of stable steady-state solutions. One of these had a strong northern overturning in the Atlantic while the other had hardly any northern overturning, confirming qualitatively the sketch shown in Figure 3.3. Finally, Huisman et al. (2009) have discovered four different flow regimes of their computer model. These they call the Conveyor (C), the Southern

Sinking (SS), the Northern Sinking (NS) and the Inverse Conveyor (IC), which appear as two disconnected branches of solutions, where the C is connected with the SS and the NS with the IC. The authors argue that these findings show, significantly, that the parameter volume for which multiple steady states exist is greatly increased.

An intuitive physical mechanism for bi-stability is the presence of two potential wells (at the bottom of each is a stable equilibrium) separated by a saddle, which corresponds to the unstable equilibrium. Applying a perturbation then corresponds to a temporary alteration of this potential energy landscape. Dijkstra et al. (2004) observed that this picture is approximately true for ocean circulation if one takes the average deviation of water density (as determined by salinity and temperature) from the original equilibrium as the potential energy. They showed, first for a box model and then for a global ocean circulation model, that the potential energy landscape of the unperturbed system defines the basins of attraction fairly accurately. This helps engineers and forecasters to determine whether a perturbation (for example, increased freshwater influx) enables the bi-stable system to cross from one basin of attraction to the other.

Concerning the simple box models of the THC, we might note their similarity to the atmospheric convection model in which Lorenz (1963) discovered the chaotic attractor: this points to the fact that we must expect chaotic features in the THC and other climate models. See Dijsktra (2008) for a summary of the current state of ocean modelling from a dynamical systems point of view, and, for example, Tziperman et al. (1994) and Tziperman (1997) for how predictions of ocean models connect to full global circulation models.

Building on these modelling efforts, ongoing research is actively trying to predict an imminent collapse at the fold seen in the models (for example, Figure 3.3) from bifurcational precursors in time series. Held & Kleinen (2004) use the LDR (described earlier in Section 3.4 and in Table 3.5) as the diagnostic variable that they think is most directly linked to the distance from a bifurcation threshold. They demonstrate its use to predict the shutdown of the North Atlantic thermohaline circulation using the oceanic output of CLIMBER2, a predictive coupled model of intermediate complexity (Petoukhov et al., 2000). They make a 50000-years transient run with a linear increase in atmospheric CO2 from 280 to 800 parts per million (ppm), which generates within the model an increase in the fresh water forcing which is perturbed stochastically. This run results in the eventual collapse of the THC as shown in Figure 3.4.

In Figure 3.4(a) the graph (corresponding approximately to the schematic diagram of Figure 3.3) is fairly linear over much of the timescale: there is no adequate early prediction of the fold bifurcation in terms of path curvature. The graph of Figure 3.4(b) shows the variation of the first-order autoregressive coefficient

q(Sv) ■— |
' 1 t | |

0.2 0.4 |
0.6 |
0.8 1.0 Target |

Moving window |
i i ■ | |

95% error zone |
t |

Figure 3.4 Results of Held & Kleinen (2004) which give a good prediction of the collapse of the thermohaline circulation induced by a four-fold linear increase of CO2 over 50 000 years in a model simulation. Collapse present at t & 0.8 in (a) is predicted to occur when the propagator, c = ARC(1), shown in (b), or its linear fit, reaches +1.

Figure 3.4 Results of Held & Kleinen (2004) which give a good prediction of the collapse of the thermohaline circulation induced by a four-fold linear increase of CO2 over 50 000 years in a model simulation. Collapse present at t & 0.8 in (a) is predicted to occur when the propagator, c = ARC(1), shown in (b), or its linear fit, reaches +1.

or propagator, ARC(1), which is described in Section 3.4. Unlike the response diagram of q(t), the time-series of ARC(1), although noisy, allows a fairly good prediction of the imminent collapse using the linear fit drawn: the fairly steady rise of ARC(1) towards its critical value of + 1 is indeed seen over a very considerable timescale. Notice that the linear fit is surrounded by a 95 per cent zone, giving probability bounds to the collapse time. These bounds emphasise that much more precise predictions will be needed before they can be used to guide policy on whether to implement geo-engineering proposals.

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