Thresholds and Nonlinearities

Considerable effort has gone into analyzing both model experiments and past records of climate change for evidence of abrupt changes. In nonlinear models, rapid changes resulting from crossing a threshold in some forcing function are common. Data on a number of aspects of the climate system suggest that it also has thresholds, multiple equilibria, and other features which can result in episodes of rapid change [ Broecker et al., 1985; Mayewski et al., 1993; McElroy, 1994]. The behavior of the thermohaline circulation of the oceans (THC) is one of the most frequently cited examples of nonlinear dynamics in the earth climate system and a potential source of rapid future change.

Discussion of the potential instability of the thermohaline circulation goes back several decades [e.g., Stommel, 1961] and has been a focal topic of numerical modeling and empirical studies [ Weaver et al., 1991; Manabe and Stouffer, 1994]. The Younger Dryas, a cold period 13-11,000 years before the present, was associated with a major change in ocean circulation that began over a period of 10-300 years and ended very rapidly (in 20 years) [ Boyle and Keigwin, 1987; Dansgaard et al., 1989; Mayewski et al., 1993]. Because of the strong suspicion that phenomena such as the Younger Dryas event may have resulted from a major and abrupt change in the THC, a number of modeling studies have addressed the stability of the THC. Models used range in complexity from simple analytical models to ocean general circulation models [ Marotzke, 1989; Weaver and Sarachik, 1991; Quon and Ghil, 1992]. These studies indicate that the THC has multiple equilibria. For example, Stocker et al. [1992] showed the system to have two stable states, resembling the circulation of the modern and last glacial maximum oceans. Transitions between the two states occurred smoothly as the high-latitude freshwater flux was varied, suggesting that changes in high-latitude precipitation minus evaporation and river runoff play an important role in controlling glacial-interglacial changes in ocean circulation. This result is consistent, although not identical, with the behavior of a number of other models of the THC [e.g., Birchfield, 1989; Weaver et al., 1991; Manabe and Stouffer, 1994].

Analyses reported in the 1990 Intergovernmental Panel on Climate Change (IPCC) Assessment [ Bretherton et al., 1990] focused attention on the role of the oceans in delaying an atmospheric warming in response to increasing carbon dioxide. Because a substantial fraction of the heat stored by the oceans is rapidly transported into the deep ocean in high-latitude regions (in the sinking phase of the thermohaline circulation), changes in the THC could influence the rate of atmospheric warming. Indeed, early GCM simulations showed that increasing CO led to a weakening of the THC and atmospheric warming, while decreasing CO had the opposite effects [ Stouffer et al., 1989; Bretherton et al., 1990]. Complex sensitivity of the THC to CO forcing is indicated in the three 500-year simulations carried out by Manabe and Stouffer [1994] with a coupled atmosphere-ocean model (Figure 3). These simulations were forced by increasing CO to either two or four times current levels, followed by a period of stable CO concentration. When atmospheric CO was doubled over 70 years and then held constant, the thermohaline circulation weakened during the first 150 simulated years and then gradually recovered to nearly its original strength by year 500. Global surface air temperature increased as CO increased (at 3.5C per century), followed by a drift towards a new equilibrium at a rate of 0.2-0.25C per century. With respect to temperature, the response of the coupled system to an increase to four times current CO (over 140 years) closely paralleled the doubling experiment, with temperature increases of 3.5C per century as CO increased (Figure 3). In the 4xCO experiment, however, the thermohaline circulation disappeared. As a result, the effective thermal inertia of the ocean decreased with simulated time because heat was no longer transferred rapidly to the deep ocean. Thus, the 4xCO simulation had nearly twice the rate of warming after stabilization of CO levels (0.4-0.5C per century) as compared to the CO doubling experiment. Changes in freshwater inputs (precipitation minus evaporation) arising from increased poleward transport of atmospheric water vapor in a warmer world caused the gradual capping of high-latitude waters by low-salinity water and led to the weakening (in the 2xCO simulation) and disappearance (in the 4xCO simulation) of the THC.

While there is consensus among models on the existence of multiple steady states in the THC [ Marotzke and Willebrand, 1991; Manabe and Stouffer, 1994], the sensitivity of the ocean (the real one) is more problematic. Empirical evidence indicates that the thermohaline circulation has been in its current state since the Younger Dryas episode 11 kyr BP. This implies stability over global temperature fluctuations of <2C [ Folland et al., 1990]. Results such as Manabe and Stouffer's [1994] may suggest a greater sensitivity of the state of the THC than is implicit in this record.

Current debate over the sensitivity of models versus the real world hinges on an important technicality of ocean modeling. Ocean models are integrated to a quasi-steady state (usually) using what are known as ``restoring'' boundary conditions. With restoring boundary conditions, fluxes of heat and fresh water are calculated from the difference between the simulated state of the ocean (temperature, salinity) and an observed climatology (usually that of Levitus [1982]). The difference in state is converted into a flux via multiplication by a restoring coefficient in units of 1/time [ Weaver and Sarachik, 1991; Wang and Birchfield, 1992; Huang, 1993; Tziperman et al., 1994]. The choice of timescale in this restoring coefficient is important to the behavior of the model [ Tziperman et al., 1994]. Model results show that even when air-sea fluxes are calculated, movement towards an unstable regime can be induced by varying freshwater fluxes (i.e., ``mixed'' model boundary conditions: restoring for temperature and prescribed for salinity) [ Weaver et al., 1991; Stocker et al., 1992; Huang, 1993; Tziperman et al., 1994].

A number of studies have shown that the THC obtained under restoring boundary conditions (model spin-up) may be unstable upon transition to mixed boundary conditions [ Wang and Birchfield, 1992; Tziperman et al., 1994]. This could have serious implications for the use of ocean models initialized under restoring conditions, because it is unknown whether the instability of the models (both mixed boundary conditions and coupled atmosphere-ocean models) after transition from initializing conditions is a real feature of the system. Tziperman et al. [1994] showed that there are both stable and unstable regimes with respect to the transition to mixed boundary conditions (from restoring) and concluded that the real ocean may be stable, but near the stability transition point with respect to freshwater forcing. In contrast, based on estimates of current poleward water vapor transport in the atmosphere (from current levels of freshwater inputs to the high latitude ocean regions), Wang and Birchfield [1992] concluded that the real ocean should be far from the critical region of transition between stable and unstable states.

In addition to the thermohaline circulation, other potential thresholds in the climate system exist. Several workers have speculated about the existence of thresholds in the terrestrial carbon cycle. Most studies of the carbon cycle have concluded that a sink of 1-2 Gt (1 Gt = 10 metric tons) carbon per year must exist in the terrestrial biosphere [e.g., Tans et al., 1990]. This sink is frequently ascribed to the fertilization of land plants by increasing carbon dioxide, a nonlinear phenomenon which saturates at some concentration of atmospheric CO, possibly dependent upon interactions with the nitrogen cycle [ Comins and McMurtrie, 1993; Melillo et al., 1993]. If CO fertilization is the mechanism for a terrestrial sink and if it were to saturate, then the fraction of excess CO remaining in the atmosphere would increase. Such a change in terrestrial carbon fluxes might then abruptly alter CO forcing of the climate system at some point in the future [ Houghton and Woodwell, 1989; Woodwell, 1992]. Thus, a threshold-like phenomenon in the climate system could be driven by a threshold in forcing, rather than by a dynamic instability of the physical system. McElroy [1994] has suggested, based on the correlation between high atmospheric CO and warm climate during the Cretaceous, that climate sensitivity inferred from models might be too low and that as-yet-unknown processes might alter climate greatly to a more equable state (warm, pole-to-pole) if CO passes some threshold. Because of the great vulnerability of socioeconomic and biological systems to discontinuities and high rates of climate change, studies of thresholds and nonlinearities in the climate and associated biogeochemical cycles are crucial areas of active research in the coming years.