Bifurcation analysis of a two-delay model for the Atlantic Meridional Overturning Circulation

The Atlantic Meridional Overturning Circulation (AMOC) is a climate-relevant ocean current system responsible for the meridional heat transport in the Atlantic. The AMOC strength is affected by a meridional density difference, where the density in the northern North Atlantic is controlled by an advective and a convective feedback. Here, we introduce and study a conceptual mathematical model of the variability of the AMOC strength by representing these feedbacks as delayed terms in a scalar delay differential equation (DDE) for the salinity in the northern North Atlantic. After scaling and without external input, this DDE has the associated delay times (Formula presented) (Formula presented) and (Formula presented) (Formula presented) as its only parameters. We perform a numerical bifurcation analysis of this deceptively simple-looking DDE AMOC model with the continuation software package DDE-BifTool. We find and characterize intricate dynamical regimes, including those exhibiting complicated oscillations associated with homoclinic connections. These results are presented as bifurcation diagrams in the (Formula presented) (Formula presented) -plane, where we identify a codimension-two Belyakov transition as an organizing center for nearby complicated dynamics. Moreover, we present a detailed analysis of different attractor regions in the (Formula presented) (Formula presented) -plane, which we identify by computing the (strong) unstable manifold of a physically relevant equilibrium. As a general picture, we find that attractor regions repeat as the values of either (Formula presented) (Formula presented) or (Formula presented) (Formula presented) increase, including in physically relevant regions of these two (scaled) delay times. In this way, we clarify where different types of dynamics — such as periodic orbits of different periods, invariant tori, and chaotic dynamics — can be observed, and how they emerge or disappear.