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

Abstract 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 τ and σ 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 ( τ , σ ) -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 ( τ , σ ) -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 σ or τ 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.