A Simple Model of Millennial Oscillations of the Thermohaline Circulation

Abstract

Stommel's two-box model of thermohaline circulation is modified to include the possibility of convection. When reduced to a two-degrees-of-freedom dynamical system, the model exhibits the well-known multiple (thermal and haline) steady states as well as new convective thermal steady states. However, for some values of the control parameters (such as the freshwater flux) oscillations occur. Millennial period oscillatory regimes correspond to switches between the Stommel's haline fixed point and the convective thermal state, both of which are unstable in a window of precipitation values. The transitions between steady and oscillatory regimes at the boundaries of the window are global bifurcations, which in some cases have an infinite period character. This character is due either to the proximity of the Stommel saddle node bifurcation or to the infinite time it takes for the convection to resume when the system is in the haline regime. The oscillations bear a close relationship to those of Welander's flip-flop model. The physics of this class of millennial oscillations may be relevant to those observed in more complex OGCMs and may help to rationalize certain features of the millennial band oscillations that punctuate the last glacial period.

On the Linkage between Antarctic Surface Water Stratification and Global Deep-Water Temperature

“…Recent studies using simple box models, but allowing for dynamic control of overturning based on density differences, have shown the possibility of deep decoupling oscillations (Colin de Verdière et al, 2006;Colin de Verdière, 2007). This possibility is suppressed in our box model because the overturning rates are fixed.…”

“…In simple box models, the overturning circulation is typically parameterized as a function of density differences between boxes (Stommel, 1961;Colin de Verdière, 2007). This traditional approach is inadequate, however, for depicting mechanically-driven flows, such as Antarctic upwelling.…”

Stability of the Atlantic Meridional Overturning Circulation: A Review and Synthesis

“…However, the magnitude (a few Sverdrups) and typical periodicity (decadal to centennial) of such variations mismatch the observed millennial variability during the past glacial‐interglacial interval. In contrast, a possibility of pronounced self‐sustained oscillations on a millennial timescale has been shown by many models of lower complexity, from box models (e.g., Colin de Verdière, ; Colin de Verdière et al, ) and loop models (mapping the ocean overturning circulation onto a rotating wheel; e.g., Sévellec & Fedorov, , ; Winton & Sarachik, ) to simplified coupled models (e.g., Sakai & Peltier, ; , ; Weaver & Hughes, ; Weijer & Dijkstra, ), suggesting that one or more of AMOC steady states may be unstable. This appears to be especially true for the models configured for glacial conditions.…”

Millennial Variability in an Idealized Ocean Model: Predicting the AMOC Regime Shifts

“…Studies using zonally averaged and box models (some exposed to gradually varying surface freshwater fluxes) have demonstrated the existence of millennial oscillations for intermediate values of the freshwater flux and proposed different low-order conceptual dynamic systems describing millennial oscillation (Cessi 1996;Sakai and Peltier 1999;Colin de Verdi ere 2007;Colin de Verdi ere et al 2006;S evellec et al 2010). Different authors emphasize the role of salinity (Sakai and Peltier 1999;S evellec et al 2010), convection (Cessi 1996;Colin de Verdi ere et al 2006;Colin de Verdi ere 2007), or the interhemispheric bipolar seesaw (Stocker and Johnsen 2003) for these oscillations.…”

“…Different authors emphasize the role of salinity (Sakai and Peltier 1999;S evellec et al 2010), convection (Cessi 1996;Colin de Verdi ere et al 2006;Colin de Verdi ere 2007), or the interhemispheric bipolar seesaw (Stocker and Johnsen 2003) for these oscillations. Apparently, the period of the oscillations in these models depends on how far the system is from the bifurcation (Colin de Verdi ere et al 2006;S evellec et al 2010) and can go from about 1000 yr to infinity even for a relative, weak change in the oscillation amplitude, as expected for an infinite-period bifurcation (Strogatz 1994).…”