Signatures Consistent with Multifrequency Tipping in the Atlantic Meridional Overturning Circulation

The early detection of tipping points, which describe a rapid departure from a stable state, is an important theoretical and practical challenge. Tipping points are most commonly associated with the disappearance of steady-state or periodic solutions at fold bifurcations. We discuss here multifrequency tipping ($M$ tipping), which is tipping due to the disappearance of an attracting torus. $M$ tipping is a generic phenomenon in systems with at least two intrinsic or external frequencies that can interact and, hence, is relevant to a wide variety of systems of interest. We show that the more complicated sequence of bifurcations involved in $M$ tipping provides a possible consistent explanation for as yet unexplained behavior observed near tipping in climate models for the Atlantic meridional overturning circulation. More generally, this Letter provides a path toward identifying possible early warning signs of tipping in multiple-frequency systems.

Figure 1

Dynamics near an $M$ tipping event of Eq. (1) for $\epsilon =0$. (a) Hysteresis loop in parameter $c$ for ${\tau }_n=0.953$, represented by their amplitude $\mathrm{amp}[h(t)]$. Stable solutions are shown in blue, while those with one and two unstable Floquet multipliers are shown in green and red, respectively. (b) The $2:7$ resonance tongue in the $(c,{\tau }_n)$ plane. The color scheme shows amplitudes of stable solutions taken from simulations while slowly increasing $c$. The inset is an enlargement of the resonance tongue where it folds; see text for a list of bifurcation types. (c) Time series with linearly increasing $c$ corresponding to the gray-shaded region in (a).

Figure 2

Hysteresis loops of (a) the GENIE-2 model as reproduced under Creative Common license (CC BY 4.0) from Ref. [23], and (b) the conceptual DDE model (1) for ${\tau }_n=0.953$, $\mathrm{\Delta }c=1\times {10}^{-6}\text{per time unit}$, and $\epsilon =0.0005$.

Figure 3

Additional multistability near the tipping event observed in (a) a global ocean GCM as reproduced with permission from Springer Nature from Ref. [25], and (b) the conceptual DDE model (1) for ${\tau }_n=0.953$, $\mathrm{\Delta }c=1\times {10}^{-6}\text{per time unit}$, and $\epsilon =0.0005$.

Figure 4

Intermittent times series in (a) ECBilt-CLIO model with a freshwater perturbation of 5, 7.5, and 10 mSv at 5000 years as reproduced under Creative Common license (CC BY-NC-SA 2.5) from Ref. [28], and (b) the conceptual DDE model (1) for $c=2.966,{\tau }_n=0.953$, and $\epsilon =0.001$, with a perturbation $\mathrm{\Delta }c$ in parameter $c$ of 0.003, 0.0045, and 0.006.